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3e1f1a54a5ab5882	Korrelationstage 2021 korrel21 Poster Prizes poster prize of the 1st poster session - Alla Bezvershenko, Universität zu Köln poster prize of the 2nd poster session - Alessio Lerose, University of Geneva poster prize of the 3rd poster session - Tiago Mendes Santos, MPIPKS Dresden poster prize of the 4th poster session - Anne-Maria Visuri, University of Bonn Floquet vortex states induced by light carrying the orbital angular momentum Ahmadabadi, Iman We propose a scheme to create an electronic Floquet vortex state by irradiating the circularly-polarized laser light carrying non-zero orbital angular momentum on the two-dimensional semiconductor. We analytically and numerically study the properties of the Floquet vortex states, using methods analogous to the techniques used for the analysis of superconducting vortex states, and we show that such Floquet vortex states have a wide tunability range. To illustrate the impact of such tunability, we propose how such states could be used for quantum information processing. Interaction of a Neel-type skyrmion and a superconducting vortex Andriyakhina, Elizaveta Superconductor-ferromagnet heterostructures hosting vortices and skyrmions are new area of an interplay between superconductivity and magnetism. We study an interaction of a Neel-type skyrmion and a Pearl vortex in thin heterostructures due to stray fields. Surprisingly, we find that it can be energetically favorable for the Pearl vortex to be situated at some nonzero distance from the center of the Neel-type skyrmion. The presence of a vortex-antivortex pair is found to result in increase of the skyrmion radius. Our theory predicts that a spontaneous generation of a vortex-anti-vortex pair is possible under some conditions in the presence of a Neel-type skyrmion. Exotic phases of cluster-forming systems Angelone, Adriano I will present my recent results on bosonic systems featuring extended-range interactions, of interest for experiments with cold Rydberg-dressed atoms. In my previous work, I proved these Hamiltonians to host a wide variety of interesting physical phenomena, including (super)solid phases of clusters of particles, as well as out-of-equilibrium glass and superglass states (the latter displaying the coexistence of glassy physics and superfluidity). In this talk, I will discuss my demonstration, in the ground-state regime of this class of models, of a novel type of phase transition between two supersolid states characterized by different crystalline and superfluid exchange structures. I will then discuss my results on the out-of-equilibrium counterparts of the states mentioned above, which I prove to be glasses and (super)solids (the latter featuring crystalline structures in general remarkably different from their ground-state counterparts) in an energy range which would allow their observation in experimental realizations. Out of equilibrium dynamics of polarons in a Bose-Einstein Condensate Ardila, Luis Broken-Symmetry Ground States of Quantum Magnetism on the Pyrochlore Lattice Astrakhantsev, Nikita The spin-1/2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram. Besides hosting several ordered phases, the case with only nearest-neighbor antiferromagnetic interactions is debated for potentially realizing a spin-liquid ground state. Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and several complementary variational techniques. Specifically, we employ a Pfaffian-type many-variable Monte Carlo ansatz and convolutional neural network quantum states for calculations with up to $3\times 4^3$ and $3 \times 3^3$ spins, respectively. We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit. Our main results are (1) a determination of the phase transition between the putative spin liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry breaking tendencies. We find clear indications for spontaneously broken inversion and rotational symmetry, calling the quantum spin-liquid scenario into question. Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets. Effect of disorder on topological charge pumping in the Rice-Mele model Bertok, Eric Recent experiments with ultracold quantum gases have successfully realized integer-quantized topological charge pumping in optical lattices. Motivated by this progress, we study the effects of static disorder on topological Thouless charge pumping. We focus on the half-filled Rice-Mele model of free spinless fermions and consider random diagonal disorder. In the instantaneous basis, we compute the polarization, the entanglement spectrum, and the local Chern marker. As a first main result, we conclude that the space-integrated local Chern marker is best suited for a quantitative determination of topological transitions in a disordered system. In the time-dependent simulations, we use the time-integrated current to obtain the pumped charge in slowly periodically driven systems. As a second main result, we observe and characterize a disorder-driven breakdown of the quantized charge pump. There is an excellent agreement between the static and the time-dependent ways of computing the pumped charge. The topological transition occurs well in the regime where all states are localized on the given system sizes and is therefore not tied to a delocalization-localization transition of Hamiltonian eigenstates. For individual disorder realizations, the breakdown of the quantized pumping occurs for parameters where the spectral bulk gap inherited from the band gap of the clean system closes, leading to a globally gapless spectrum. As a third main result and with respect to the analysis of finite-size systems, we show that the disorder average of the bulk gap severely overestimates the stability of quantized pumping. A much better estimate is the typical value of the distribution of energy gaps, also called mode of the distribution. Dicke transition in open many-body systems determined by fluctuation effects Bezvershenko, Alla In recent years, one important experimental achievement was the strong coupling of quantum matter and quantum light. Realizations reach from ultracold atomic gases in high-finesse optical resonators to electronic systems coupled to THz cavities. The dissipative nature of the quantum light field and the global coupling to the quantum matter leads to many exciting phenomena such as the occurrence of dissipative quantum phase transition to self-organized exotic phases. Here we develop a new approach which combines a mean-field approach with a perturbative treatment of fluctuations beyond mean-field, which becomes exact in the thermodynamic limit. We argue that these fluctuations are crucial in order to determine the mixed state (finite temperature) character of the transition and to unravel universal properties of the self-organized states. We validate our results by comparing to time-dependent matrix-product-state calculations. Spin and charge order in doped spin-orbit coupled Mott insulators Biderang, Mehdi We study a two-dimensional single band Hubbard Hamiltonian with antisymmetric spin-orbit coupling. We argue that this is the minimal model to understand the electronic properties of locally non-centrosymmetric transition-metal (TM) oxides such as Sr$_2$IrO$_4$. Based on exact diagonalizations of small clusters and the random phase approximation, we investigate the correlation effects on charge and magnetic order as a function of doping and of the TM-oxygen-TM bond angle $\theta$. For small doping and $\theta\lesssim$ 15° we find dominant commensurate in-plane antiferromagnetic fluctuations while ferromagnetic fluctuations dominate for $\theta\gtrsim$ 25°. Moderately strong nearest-neighbor Hubbard interactions can also stabilize a charge density wave order. Furthermore, we compare the dispersion of magnetic excitations for the hole-doped case to resonant inelastic X-ray scattering data and find good qualitative agreement. Spectroscopy of doped quantum magnets -- new directions Bohrdt, Annabelle Single-particle spectral functions, which are usually measured using photoemission experiments in electron systems, contain direct information about fractionalization and the quasiparticle excitation spectrum. In this talk, I will present recent developments that enable angle-resolved photo-emission spectroscopy (ARPES) of the one- and two-dimensional Fermi-Hubbard model using ultra cold atoms in optical lattices. I will discuss numerical results for the one-dimensional t-J model, where a sharp asymmetry in the distribution of spectral weight appears, that can be explained by a slave-fermion mean-field theory of the spin excitations (spinons). By employing a time-dependent ARPES protocol, akin to pump probe experiments in solids, we directly reveal interaction effects between the spinons. While in one dimension the spin (spinon) and charge (chargon) excitations are deconfined, several theories suggest that in two dimensions, dopants can be understood as bound states of these partons. Recent progress in the microscopic description of mobile dopants allows us to conjecture a one-to-one relation of the one-dopant spectral function and the properties of the constituting spinons in the undoped parent antiferromagnet (AFM). Using time-dependent matrix product state calculations of the spectral function of a single hole doped into a two-dimensional Heisenberg AFM, we thoroughly test this hypothesis and obtain excellent agreement with our semi-analytical predictions. We directly probe the microscopic nature of the spinon-chargon bound states through a new extension of ARPES, which uncovers long-lived rotational resonances. Similar to Regge trajectories in high-energy physics, which reflect the quark structure of mesons, we establish a linear dependence of the rotational energy on the super-exchange coupling. Our findings suggest that the rich physics of lightly doped cuprates may originate from an emergent parton structure. Skyrmion and Tetarton Lattices in Twisted Bilayer Graphene Bömerich, Thomas Recent experiments on twisted bilayer graphene show an anomalous quantum Hall (AQH) effect at filling of three electrons per moiré unit cell. The AQH effect arises in an insulating state with both valley and ferromagnetic order. We argue that weak doping of such a system leads to the formation of a novel topological spin texture, a "double-tetarton lattice". The building block of this lattice, the "double-tetarton", is a spin configuration which covers 1/4 of the unit sphere twice. In contrast to skyrmion lattices, the net magnetization of this magnetic texture vanishes. Only at large magnetic fields are more conventional skyrmion lattices recovered. But even for large fields the addition of a single charge to the ferromagnetic AQH state flips hundreds of spins. Our analysis is based on the investigation of an effective nonlinear sigma model which includes the effects of long-ranged Coulomb interactions. Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model Bonetti, Pietro Maria We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping $p$ and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with Néel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the $(p,T)$ phase diagram centered around $15\%$ hole doping. Bosonic continuum theory of one-dimensional lattice anyons Bonkhoff, Martin Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations. Quantum phases of two-dimensional Z2 gauge theory coupled to single-component fermion matter Borla, Umberto We investigate the rich quantum phase diagram of Wegner's theory of discrete Ising gauge fields interacting with U(1) symmetric single-component fermion matter hopping on a two-dimensional square lattice. In particular limits the model reduces to (i) pure $Z_2$ even and odd gauge theories, (ii) free fermions in a static background of deconfined $Z_2$ gauge fields, (iii) the kinetic Rokhsar-Kivelson quantum dimer model at a generic dimer filling. We develop a local transformation that maps the lattice gauge theory onto a model of $Z_2$ gauge-invariant spin 1/2 degrees of freedom. Using the mapping, we perform numerical density matrix renormalization group calculations that corroborate our understanding of the limits identified above. Moreover, in the absence of the magnetic plaquette term, we reveal signatures of topologically ordered Dirac semimetal and staggered Mott insulator phases at half-filling. At strong coupling, the lattice gauge theory displays fracton phenomenology with isolated fermions being completely frozen and dimers exhibiting restricted mobility. In that limit, we predict that in the ground state dimers form compact clusters, whose hopping is suppressed exponentially in their size. We determine the band structure of the smallest clusters numerically using exact diagonalization. Realizing a symmetry-protected topological phase in the antiferromagnetic spin-1/2 Hubbard ladder Bourgund, Dominik The spin-1 Haldane chain is the paradigmatic example of symmetry protected topological (SPT) phases, which are characterized by non-local order parameters and edge states. Here we report on the experimental realization of such a phase using ultracold fermions in optical lattices. Site-resolved potential shaping allows us to create a tailored spin-1/2 ladder geometry needed to explore the topologically nontrivial Haldane phase. Harnessing the full spin and density resolution of our Fermi-gas microscope, we detect a finite non-local string correlator in the bulk and localized spin-1/2 states at the edges. We confirm the robustness of the state by tuning the ratio of the leg to rung coupling of the ladder. We finally go beyond the spin model and explore the effect of charge fluctuations on the SPT phase in the general Hubbard regime. Electronuclear Quantum Criticality Brando, Manuel We present here a rare example of electronuclear quantum criticality in a metal. The compound YbCu4.6Au0.4 is located at an unconventional quantum critical point (QCP). In this material the relevant Kondo and RKKY exchange interactions are very low, of the order of 1K. Furthermore, there is a strong competition between antiferromagnetic and ferromagnetic correlations, possibly due to geometrical frustration within the fcc Yb sublattice. This causes strong spin fluctuations which prevent the system to order magnetically. Because of the very low Kondo temperature the Yb3+ 4f-electrons couple weakly with the conduction electrons allowing the coupling to the nuclear moments of the 171Yb and 173Yb isotopes to become important. Thus, the quantum critical fluctuations observed at the QCP derive not from purely electronic states but from entangled electronuclear states. This is evidenced in the anomalous temperature and field dependence of the specific heat at low temperatures. Finite-momentum energy dynamics in a Kitaev magnet Brenig, Wolfram We study the energy-density dynamics at finite momentum of the two-dimensional Kitaev spin-model on the honeycomb lattice. Due to fractionalization of magnetic moments, the energy relaxation occurs through mobile Majorana matter, coupled to a static $\mathbb{Z}_{2}$ gauge field. At finite temperatures, the $\mathbb{Z}_{2}$ flux excitations act as an emergent disorder, which strongly affects the energy dynamics. We show that sufficiently far above the flux proliferation temperature, but not yet in the classical regime, gauge disorder modifies the coherent low-temperature energy-density dynamics into a form which is almost diffusive, with hydrodynamic momentum scaling of a diffusion-kernel, which however remains retarded, primarily due to the presence of two distinct relaxation channels of particle-hole and particle-particle nature. Relations to thermal conductivity are clarified. Our analysis is based on complementary calculations in the low-temperature homogeneous gauge and a mean-field treatment of thermal gauge fluctuations, valid at intermediate and high temperatures. Chiral transitions in chains of Rydberg atoms Chepiga, Natalia Investigation of the nature of commensurate-incommensurate transition out of period-p phase has a long history that goes back to the study of absorbed monolayers on surfaces. The problem has been revived by recent experiments on Rydberg atoms in 1D trap with the phase diagram dominated by lobes of integer periodicities p=2,3,4,5… Recent development of constrained DMRG algorithm that takes a full advantage of Rydberg blockade brings the study of chiral melting to a completely new level of accuracy. In my talk I will shown that transitions out of period-3 and period-4 phases change their nature along the critical lines: conformal points in the three-state Potts and Ashkin-Teller universality classes are surrounded by direct chiral transitions followed by the opening of floating phases. Numerical detection of chiral transitions brings the first consistent explanation of dynamical critical exponent z>1 deduced from recent Harvard experiments. I will explain that an appearance of the chiral transition is a generic feature of phases where the number of particles is not conserved because the Luttinger liquid parameter of the floating phase changes along the Pokrovsky-Talapov transition and can thus reach the value at which the floating phase becomes unstable. Colmenárez Gómez, Luis Andres Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems. We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r,t) = [Ai(t), Bi+r] of local operators. Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/rα with distance r. Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1. The asymptotic form of the tails of C(r,t) ∝ t/rα is described by short-time perturbation theory, which is valid at short times and long distances. Effect of nonmagnetic dilution on magnetic properties of frustrated oxyborates Contreras Medrano, Cynthia Magnetic oxyborates of the 3d transition metals are good examples of strongly correlated systems in which magnetic frustration and structural disorder have often been observed [1,2]. Low-dimensional substructures are characteristic of these compounds, and they are present in the form of ladders in ludwigites, ribbons in warwickites, and planes in hulsites. These compounds have shown complex and intriguing physical properties that depend on their elementary composition. Nonmagnetic ions are usually used to reduce magnetic interactions in an effort to understand these complex magnetic compounds. However, heterometallic ludwigites with different nonmagnetic ions displayed distinct properties as metamagnetic transition, partial magnetic order, spin-glass, fluctuations, making these compounds fundamentally more interesting. A summary of these effects is listed in this presentation. Kondo Breakdown in a Spin-1/2 Chain of Adatoms on a Dirac Semimetal Danu, Bimla We consider a spin-1/2 Heisenberg chain coupled via a Kondo interaction to two-dimensional Dirac fermions. The Kondo interaction is irrelevant at the decoupled fixed point, leading to the existence of a Kondo-breakdown phase and a Kondo-breakdown critical point separating such a phase from a heavy Fermi liquid. We reach this conclusion on the basis of a renormalization group analysis, large-N calculations as well as extensive auxiliary-field quantum Monte Carlo simulations. We extract quantities such as the zero-bias tunneling conductance which will be relevant to future experiments involving adatoms on semimetals such as graphene. Resonant inelastic x-ray scattering study of vector chiral ordered kagome antiferromagnet Datta, Trinanjan We study the resonant inelastic x-ray scattering (RIXS) features of vector chiral ordered kagome antiferromagnets. Utilizing a group theoretical formalism that respects lattice site symmetry, we calculated the L-edge magnon contribution for the vesignieite compound BaCu$_3$V$_2$O$_8$(OH)$_2$. We show that polarization dependence of the L-edge RIXS spectrum can be used to track magnon branches. We predict a non-zero L-edge signal in the non-cross $\pi-\pi$ polarization channel. At the K-edge, we derived the two-site effective RIXS and Raman scattering operator for two-magnon excitation in vesignieite using the Shastry–Shraiman formalism. Our derivation considers spin-orbit coupling effects in virtual hopping processes. We find vector chiral correlation (four-spin) contribution that is proportional to the RIXS spectrum. Our scattering operator formalism can be applied to a host of non-collinear non-coplanar magnetic materials at both the L and K-edge. We demonstrate that vector chiral correlations can be accessed by RIXS experiments. Construction of low-energy symmetric Hamiltonians and Hubbard parameters for twisted multilayer systems using ab-initio input Davydov, Arkadiy A computationally efficient workflow for obtaining the low-energy symmetric tight-binding Hamiltonians for twisted multilayer systems is presented in this work. We apply this scheme to the twisted bilayer graphene at the first magic angle. As the initial step, the full-energy tight-banding Hamiltonian is generated by the Slater-Koster model with parameters fitted to ab-initio data at larger angles. Then, the low-energy symmetric four-band and twelve-band Hamiltonians are constructed using the maximum-localization procedure subjected to the crystal and the developed time-reversal symmetry constraint. Finally, we compute extended Hubbard parameters for both models within the constrained random phase approximation (cRPA) for screening, which again respect the symmetries. The relevant data and results of this work are freely available via an online repository. The workflow is straightforwardly transferable to other twisted multi-layer materials. Multifractality meets entanglement: relation for non-ergodic extended states De Tomasi, Giuseppe It is now well established that entanglement plays a central role on the thermalization process of quantum many-body systems. On the other hand, ergodicity is deeply connected to the notion of chaos, which implies also an equipartition of the wave-function over the available many-body Fock states, which is usually quantified by multi-fractal analysis. In this talk, I will discuss a link between ergodic properties extracted from entanglement entropy and the ones from multi-fractal analysis [1]. I will show a generalization of the work of Don. N. Page [2] for the entanglement entropy, to the case of non-ergodic but extended (NEE) states. By implementing the NEE states with a new and simple class of random states, which live in a fractal of the Fock space, I will compute, both analytically and numerically, its von Neumann/Renyi entropy. Remarkably, I will show that the entanglement entropies can still present a fully ergodic behavior, even though the wave-function lives in a vanishing ratio of the full Hilbert space in the thermodynamic limit. In the final part of the talk, I will apply the aforementioned results to analyze the breakdown of thermalization in kinematically constrained models having Fock/Hilbert space fragmentation [3]. References: [1] Phys. Rev. Lett. 124, 200602 (2020) [2] Phys. Rev. Lett. 71, 1291 (1993) [3] Phys. Rev. B 100, 214313 (2019) Extraction of many-body Chern number from a single wave function Dehghani, Hossein The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature. In this paper, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian. For FQH states, our method requires one additional integer invariant as input: the number of 2 Non-Abelian Bloch oscillations in higher-order topological insulators Di Liberto, Marco Bloch oscillations (BOs) are a fundamental phenomenon by which a wave packet undergoes a periodic motion in a lattice when subjected to a force. Observed in a wide range of synthetic systems, BOs are intrinsically related to geometric and topological properties of the underlying band structure. This has established BOs as a prominent tool for the detection of Berry-phase effects, including those described by non-Abelian gauge fields. In this work, we unveil a unique topological effect that manifests in the BOs of higher-order topological insulators through the interplay of non-Abelian Berry curvature and quantized Wilson loops. It is characterized by an oscillating Hall drift synchronized with a topologically-protected inter-band beating and a multiplied Bloch period. We elucidate that the origin of this synchronization mechanism relies on the periodic quantum dynamics of Wannier centers. Our work paves the way to the experimental detection of non-Abelian topological properties through the measurement of Berry phases and center-of-mass displacements. The quasi-1D Lieb-Liniger gas in a trap with time-periodically modulated interactions Eggert, Sebastian We consider the exactly solvable Lieb-Liniger model of quasi-1D interacting bosonic atoms with time-periodically modulated interactions. Using recent advances for a Floquet-Bogoliubov rotation, we are able to obtain the Floquet eigenstates exactly in the long-wavelength limit, i.e. for wave-numbers below a cut-off $q_c$. We observe a dramatic change of behavior in the appearance of resonant density waves as the frequency $\omega$ is lowered through the corresponding energy scale, which corresponds to around $2 \pi \times$kHz in typical experimental systems. The wavelength of the resonant waves is proportional to the square-root of the average density, becoming shorter near the edges of the confining trap, while in the center the maximum is in the $\mu$m range. Breakdown of the Doniach picture: From single impurity physics to correlated lattice models Eickhoff, Fabian In order to understand the competition between the RKKY and Kondo driven screening of local moments and to shed some new light on Nozières exhaustion argument, we present a low energy mapping of multi-impurity Anderson models with $N_f$ correlated sites to a cluster model coupled to $N_c$ independent conduction bands. This mapping becomes exact in the wide band limit, is applicable for arbitrary strengths of interaction and doesn't rely on any kind of symmetry. The rigorous mathematical criterion for determining $N_c$ replaces the phenomenological exhaustion argument and reveals that there are always insufficient Kondo screening channels available in the periodic Kondo or Anderson model: The singlet ground state formation must be driven by the inter-cluster spin correlations. As an application of the mapping we study the effect of local vacancies in multi impurity models using the NRG. These vacancies in graphene or in Heavy Fermions induce decoupled bound single particle states that lead to the formation of local moments. We address the puzzling question how these local moments can be screened and what determines the additionally emerging low temperature scale. Universal aspects of constrained quantum systems out of equilibrium Feldmeier, Johannes As advances in the technology of quantum simulation provide increasing control over complex many-body systems, the study of their out-of-equilibrium properties is gaining attention. While a full theoretical description of the unitary evolution is in general challenging to obtain, we can rely on a notion of universality to describe the quantum thermalization process: after a local equilibration time, emergent hydrodynamics describes the diffusive transport of conserved quantities through the system, accompanied by a linear-in-time growth of entanglement. Here we discuss how such universal aspects of the late time dynamics in quantum many-body systems can be drastically altered in the presence of constraints. In particular, we first show how systems that are constrained to conserve the dipole moment of an associated global charge generally display anomalously slow subdiffusive transport, consistent with recent cold atom experiments. We investigate these systems numerically using classically simulable circuits and construct an analytical hydrodynamic theory that yields the correct scaling of local correlations at late times. We then go on to demonstrate how the presence of constraints in such fractonic quantum matter - characterized by excitations with restricted mobility - can further lead to an anomalously slow growth of entanglement. Variational wave functions for spin-phonon models Ferrari, Francesco The existence and stability of spin-liquid phases represent central topics in the field of frustrated magnetism. In the last few years, a large theoretical effort has been devoted to proposing and studying frustrated spin models which could host spin-liquid ground states. Although several examples of well-established spin-liquid phases are now available, the question of the stability of these states to the coupling between spins and lattice distortions (i.e. phonons) has been scarcely investigated. As suggested by the well-known one-dimensional case, the effect of lattice deformations could cause a Peierls instability of spin-liquid phases towards the formation of valence-bond order. We present a variational framework for the study of spin-phonon models in which lattice distortions affect the exchange interaction between the spins. Our method, based on Jastrow-Slater wave functions and Monte Carlo sampling, provides a full quantum treatment of both spin and phonon degrees of freedom. We first assess the accuracy of our variational scheme by comparing our results for the spin-Peierls chain with other numerical methods [1]. Then, we discuss the effects of the spin-phonon coupling on the spin liquid phase of the square lattice J1-J2 model. [1] F. Ferrari, R. Valenti, F. Becca, Phys. Rev. B 102, 125149 (2020). Quantized Electrochemical Transport in Weyl Semimetals Flores Calderón, Rafael Álvaro We show that under the effect of an external electric field and a gradient of chemical potential, a topological electric current can be induced in Weyl semimetals without inversion and mirror symmetries. We derive analytic expressions for the nonlinear conductivity tensor and show that it is nearly quantized for small tilting when the Fermi levels are close to the Weyl nodes. When the Van Hove point is much larger than the largest Fermi level, the band structure is described by two linearly dispersing Weyl fermions with opposite chiralities. In this case, the electrochemical response is fully quantized in terms of fundamental constants and the scattering time, and it can be used to measure directly the topological charge of Weyl points. We show that the electrochemical chiral current may be derived from an electromagnetic action similar to axion electrodynamics, where the position-dependent chiral Fermi level plays the role of the axion field. This posits our results as a direct consequence of the chiral anomaly. Semi-classical Dynamics of Magnetic Field-Induced Phases in Kitaev Magnets Franke, Oliver The Kitaev honeycomb model with a finite $\Gamma$ interaction shows a multitude of magnetic field induced phases between the low field zigzag and the high field polarised regime. These intermediate phases often posses large unit cells at finite magnetic fields and their relation with the experimentally measured phase diagram of RuCl$_3$ is unclear. Here, we study these competing phases with advanced stochastic LLG simulations and present their static and dynamic structure factors. There are sharp low-frequency spin-waves which broaden quickly at elevated frequencies. While the different intermediate orders with large unit cells are clearly distinguishable in their structure factors for low-temperatures their broad response for experimentally relevant temperatures looks remarkably similar. $^{93}$Nb NMR study of 2D superconductivity in van der Waals heterostructures Frassineti, Jonathan The purpose of this work is the study of physical symmetries and charge-related properties in a bulk superlattice, consisting of the transition metal dichalcogenide (TMD) superconductor 2H-niobium disulfide (2H-NbS$_{2}$) and a commensurate block layer. This study will be conducted by performing Nuclear Magnetic Resonance (NMR) experiments on $^{93}$Nb nuclei, which are suitable to this technique due to their high sensitivity to NMR and high abundancy. Considerable efforts have been carried out to reach 2D superconductivity in transition metal dichalcogenides (TMDs). These materials present intrinsically strong spin-orbit coupling and inversion symmetry breaking, which may yield to exotic forms of superconductivity in the clean limit, when the Pippard coherence length, $\xi_{0}$, is smaller than the electronic mean-free path, $l$, i.e $\frac{\xi_{0}}{l} \ll 1$. High-quality H-NbS$_{2}$ monolayers with electronic mobilities more than three orders of magnitude larger than in bulk 2H-NbS$_{2}$ can be realized in a bulk single crystal superlattice formed with an appropriate block layer. Correspondingly, we show that this material is a clean limit 2D superconductor exhibiting a BKT transition at $T_{BKT}$ = 0.82 K. The fundamental structural unit in hexagonal TMDs is the H-MX$_{2}$ layer where M and X are a transition metal and chalcogen, respectively. This structure breaks inversion symmetry in the layer plane owing to the trigonal prismatic coordination of X around M, and as a result yields an out-of-plane (Ising) spin-texture. For thin flakes deposited on substrates, the substrate-flake interface breaks mirror symmetry and yields an in-plane (Rashba) spin-texture. The simultaneous breaking of mirror and inversion symmetry leads to a mixed spin texture on the Fermi surface composed of both Ising and Rashba components. These spin-textures, and the resulting physics, are suppressed in the bulk limit where the overall unit cell preserves inversion symmetry. NMR experiments can be performed in order to elucidate the change in physical properties due to the symmetry breaking in the heterostructure. Varying the orientation of the applied magnetic field could allow us to clarify the effect of the symmetry breaking along the stacking direction. Furthermore, we could expose the main differences between bulk 2H-NbS$_{2}$ and monolayers of 1H-NbS$_{2}$ stacked onto the block layer. In conclusion, the combination of the transition metal dichalcogenide (TMD) superconductor 2H-niobium disulfide (2H-NbS$_{2}$) and the block layer provides a powerful example of high spin-orbit coupling effects which influence superconductivity and electronic transport properties. Robustness and invariance of entanglement in symmetry-protected topological phases at and away from the phase transition. Fromholz, Pierre Gapped topological phases of matter display exclusive entanglement properties that could prove useful in topological quantum computers. Much of these properties are unknown, in particular for symmetry-protected topological phases (SPTP). In my presentation, I will summarize a series a work (some of which I contributed to) that shows that the ground state of SPTP at low-dimension displays one long-range entanglement between the edge and that this entanglement can be extracted using the « disconnected entanglement entropy » SD. I show that this quantity is measurable (although with difficulties), that it is quantized, robust to disorder, and robust to quenches in the topological regime. I finally show that the quantity can be used at phase transition to obtain seemingly universal critical exponent, making SD a non-local analogous to an order parameter. Localization effects in the disordered two-dimensional Bose-Hubbard-model Geißler, Andreas In recent years experiments have shown localization effects consistent with the notion of many-body localization (MBL) for high-energy many-body states of the disordered Bose-Hubbard model in one and two-dimensional ultra-cold atomic lattice gases [1] as well as the related superfluid to Bose-glass ground state transition in three dimensions [2]. A proper theoretical understanding of MBL phenomena depends on knowledge about the full eigenstate spectrum. Therefore, exact numerical studies have been limited to small system sizes. In contrast, the Bose-glass phase can already be understood via the ground state [3]. So, by applying the fluctuation operator expansion method [4] to obtain beyond mean-field insight into the full fluctuation spectrum [3,5], I present the scaling analysis of both phenomena within a single framework. With the collection of obtained critical points, we are able to map out a phase diagram for the ground state and to characterize the mobility edge of the many-body quasiparticle excitations as extended and thermal. In the thermodynamic limit the shape of the mobility edge suggests the absence of a complete spectral localization transition. For a confined finite-size system, on the other hand, the method predicts a transition as observed in experiment as well a pattern dependence of the critical disorder strength at fixed energy density of the initial states. [1] C. D’Errico et al., PRL 113, 095301 (2014); J.-y. Choi et al., Science 352, 1547 (2016) [2] C. Meldgin et al., Nature Physics 12, 646 (2016) [3] A. Geissler, arXiv: 2011.10104 [4] A. Geissler et al., PRA 98, 063635 (2018) [5] A. Geissler, G. Pupillo, PRR 2, 042037 (2020) Higher-Order Weyl Semimetals Ghorashi, Sayed Ali Akbar Criticality and phase diagram study of the long-range quantum Ising chain Gonzalez Lazo, Eduardo Zero-temperature and finite-temperature phase transitions of the quantum Ising chain are studied. Long-range (falling off as $1/r^\alpha$ , where $r$ is the distance between two spins in units of lattice spacing) ferromagnetic interaction among the spins are present in the model. This work characterize the critical behavior and phases for $\alpha=0.05$ and $\alpha=1.5$ using Path Integral Monte Carlo calculations. The thermodynamic limit behavior is studied applying Finite-Size-Scaling techniques to the obtained finite-sizes results. The results support the existence of a finite temperature transition for the studied values of $\alpha$, where the correlation length critical exponent has the same value through all the critical line. Composite Topological Structures in Superconductor-Ferromagnet Heterostructures Görzen, Lucas Magnet-superconductor hybrids are promising platforms that can allow the manipulation of topological defects of the superconducting order parameter by controlling the motion of structures in the magnetization field. We simulate the adiabatic motion of domain walls and magnetic vortices in a ferromagnetic thin film proximity-coupled to a superconducting layer hosting a superconducting vortex. The model assumption for the ferromagnetic layer is that the energy density contributions only consist of uniaxial anisotropy and exchange energy. Remarkably, we find that whether a superconducting vortex can be moved crucially depends on the chirality of the domain walls. Specifically Néel domain walls can \enquote{push} superconducting vortices. Magnetic vortex structures can generate spiral-like structures in the pairing potential of the superconductor if they exhibit a (counter) clockwise rotating magnetization. Furthermore, we apply and benchmark an efficient numerical method for the self-consistent calculation of the superconducting order parameter. Our method, based on Green’s functions in which the spectral density of non-interacting systems is approximated by Chebyshev polynomials, can drastically speed up the calculation for large system sizes and naturally lends itself to parallelization. Higher Order Auxiliary Field Quantum Monte Carlo Methods Goth, Florian The auxiliary field quantum Monte Carlo (AFQMC) method has been a workhorse in the field of strongly correlated electrons for a long time and has found its most recent implementation in the ALF package (alf.physik.uni-wuerzburg.de). The utilization of the Trotter decomposition to decouple the interaction from the non-interacting Hamiltonian makes this method inherently second order in terms of the imaginary time slice. We show that due to the use of the Hubbard-Stratonovich transformation (HST) a semigroup structure on the time evolution is imposed that necessitates the introduction of a new family of complex-hermitian splitting methods for the purpose of reaching higher order. We will give examples of these new methods and study their efficiency, as well as perform comparisons with other established second and higher order methods in the realm of the AFQMC method. Fluctuation control of non-thermal orbital orders Grandi, Francesco Multi-minima free energy surfaces represent many physical situations [1], such as different orbital orders in transition metal compounds. In this class of systems, fluctuations of the order parameters are essential in determining the shape of the free energy. Already at equilibrium, restoring forces are of entropic origin through the order-by-disorder mechanisms [2], and fluctuations can therefore be expected to be important for the nonequilibrium dynamics. This might open non-equilibrium pathways to control the dynamics of the order parameter and even stabilize states otherwise unstable at low temperatures [3]. Here, we describe the dynamics induced by suitable time-varying protocols in the 120° compass model using time-dependent Ginzburg-Landau theory [4], and we propose to use the momentum-resolved spectrum of the fluctuations to map out the instantaneous form of the potential, what should be soon achievable in time-resolved inelastic X-ray scattering experiments. One of the protocols we analyze is a time modulation of the exchange couplings that mimics the action of oscillating electric fields that have been suggested to modify the intensity of the exchange interactions for both orbital and spin degrees of freedom. In orbital models, this can lead to a force that acts directly on the order parameter and that can be used to switch the state of the system between equivalent configurations. We particularly study the interplay between this external force and the non-thermal entropic one during an orbital switching event. In the spirit of the control of non-thermal orders by light-manipulation of the fluctuations, we analyze a similar model that, in equilibrium, has a free energy that hosts several stable solutions and, above a critical temperature Tc, several metastable states induced by the order-by-disorder mechanism. After a sudden excitation of the fluctuations, we find it is possible to transiently stabilize the metastable state even if the temperature of the order parameter is below Tc. [1] Sun et al., Phys. Rev. X 10, 021028 (2020) [2] Nussinov et al., Rev. Mod. Phys. 87, 1 (2015) [3] Grandi and Eckstein, arXiv (2021) [4] Dolgirev et al., Phys. Rev. B 101, 174306 (2020) Momentum-resolved conductivity of strongly interacting bosons in optical lattice Grygiel, Barbara In recent years a significant progress in experimental techniques of trapping, cooling, and manipulating atomic gases has allowed for study of transport properties of these systems. In this talk I would like to present momentum-dependent conductivity of strongly interacting bosons in optical lattice. We use the Bose-Hubbard model in quantum rotor approach, which allows us to describe the superfluid-Mott insulator phase transition. Moreover, this approach takes into account the spatial fluctuations, thus it captures the influence of lattice geometry and spatially dependent gauge potentials. The conductivity is derived as a response function to a small, spatially non-uniform synthetic electric field. We present the momentum-resolved conductivity for square and cubic lattices both in the superfluid and Mott insulator phases. We also show that additional conductivity channels appear at non-zero temperature. The analysis of the conductivity in the case of uniformly filled lattice allows us to determine the group velocity of the excitation, which could provide a close link to experimental results. Possible inversion symmetry breaking in the $S=1/2$ pyrochlore Heisenberg magnet Hagymasi, Imre We address the ground state properties of the long-standing and much-studied three dimensional quantum spin liquid candidate, the $S=\frac 1 2$ pyrochlore Heisenberg antiferromagnet. By using $SU(2)$ DMRG, we are able to access cluster sizes of up to 128 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground state energy, $E_0/N_\text{sites} = -0.490(6) J$, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures. Information Dynamics in a Model with Hilbert Space Fragmentation Hahn, Dominik Fluctuations and symmetry effects in many body self-organization in a dissipative cavity Halati, Catalin-Mihai We investigate the full quantum evolution of ultracold interacting bosonic atoms on a chain and coupled to an optical cavity. Extending the time-dependent matrix product state techniques and the many-body adiabatic elimination techniques to capture the global coupling to the cavity mode and the open nature of the cavity, we examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We show that the fluctuations beyond the mean-field state give a mixed state character to the dissipative phase transition and self-organized steady states. In the case of ideal bosons coupled to the cavity, the open system exhibits a strong symmetry which leads to the existence of conservation laws and multiple steady states. We find that the introduction of a weak breaking of the strong symmetry by a small interaction term leads to a direct transition from multiple steady states to a unique steady state. High magnetic field studies on atacamite, Cu$_2$Cl(OH)$_3$, a model compound for the $S = 1/2$ sawtooth chain Heinze, Leonie The mineral atacamite, Cu$_2$Cl(OH)$_3$, represents a model compound of the $S = 1/2$ sawtooth chain with both AFM couplings $J \sim 340$ K along the chain and $J' \sim 100$ K within the sawteeth, deduced from density functional theory [1]. We have extensively characterized the magnetic phase diagram of atacamite for ${\bf H} \parallel b$ axis: In low fields and below $T_{\rm N} = 8.4$ K a long-range ordered AFM state (propagation vector ${\bf q} = (1/2, 0, 1/2)$) is present [2]. Further, we have probed the high magnetic field region of the phase diagram by means of pulsed magnetic field measurements and have observed a flattening of the magnetization at $M = M_{\rm sat}/2$, which is entered for magnetic fields $> 31.5$ T [1]. We find that the flattening of the magnetization is unrelated to the known $1/2$-magnetization plateau of a quantum sawtooth chain, but might instead be understood as field-driven canting of a 3D network of weakly coupled sawtooth chains. [1] L. Heinze et al., arxiv:1904.07820 [cond mat.str el], [2] L. Heinze et al., Physica B 536, 377 (2018). Charge order from structured coupling in VSe2 Henke, Jans Charge order - ubiquitous among correlated materials - is customarily described purely as an instability of the electronic structure. However, the resulting theoretical predictions often do not match high-resolution experimental data. A pertinent case is 1T-VSe2, whose single-band Fermi surface and weak-coupling nature make it qualitatively similar to the Peierls model underlying the traditional approach. Despite this, its Fermi surface is poorly nested, the thermal evolution of its charge density wave (CDW) ordering vectors displays an unexpected jump, and the CDW gap itself evades detection in direct probes of the electronic structure. We demonstrate that the thermal variation of the CDW vectors is naturally reproduced by the electronic susceptibility when incorporating a structured, momentum-dependent electron-phonon coupling, while the evasive CDW gap presents itself as a localized suppression of spectral weight centered above the Fermi level. Our results showcase the general utility of incorporating a structured coupling in the description of charge ordered materials, including those that appear unconventional. Ref: SciPost Phys. 9, 056 (2020). doi: 10.21468/SciPostPhys.9.4.056 Real Space Classification of 2D Many-body Topological Phases Herzog-Arbeitman, Jonah The topological phases of non-interacting electrons have been exhaustively classified by their symmetries and spatial dimensions, culminating in a modern electronic band theory. More recently, a formalism using Real Space Invariants (RSIs) has enumerated the topological invariants in all 2D crystalline phases in terms of the Wannier functions which comprise the many-body groundstate. In this work, we generalize this classification to 2D interacting systems, and we find that interaction-stable topological phases are finite in number. We provide the topological invariants for each phase in terms of many-body RSIs for the 17 wallpaper groups with and without time-reversal symmetry and spin-orbit coupling, and connect the many-body RSIs to the band representation in the non- interacting limit. Our results show that all single-particle strong topology is stable to interactions. However, we find that some single-particle fragile phases may be trivialized, and we construct an analytically solvable interacting Hamiltonian which demonstrates this. Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet Hirschmann, Moritz Topological semimetals and metals may contain nodal points or lines, i.e., zero- or one-dimensional crossings in the energy bands. In the present work we discuss an extension to two-dimensional nodal features. These nodal planes are enforced in systems described by certain nonsymmorphic space groups. We give criteria to predict nodal planes and consider in the process paramagnetic as well as magnetic space groups. Based on an analysis of symmetry eigenvalues we identify space groups with a necessarily non-zero Chern number associated to the nodal planes. The arguments are supported by minimal models and explicit calculation of the topological invariants. We have identified a number of materials with topological nodal planes, among them MnSi in its ferromagnetic phase. Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons Hisano Natori, Willian Massashi The coupling between spin and orbital degrees of freedom in Kugel-Khomskii models can enhance quantum fluctuations that prevent any spin-orbital order. The proposal of new systems that implement such Hamiltonians turns the computation of quantum spin-orbital liquids' signatures into a timely problem. In this talk, we discuss the exact dynamical correlations for the quantum spin-orbital liquid phases of an SU(2)-symmetric Kitaev honeycomb lattice model. This model is treated as a Hamiltonian of strongly interacting multiples of a j=3/2 total angular momentum. We show that the spin fractionalizes into S=1 fermionic magnons, whose dynamic correlation function can be analytically studied. We also show that the dynamical correlations of the total angular momentum can be exactly calculated using the same techniques developed to compute the S=1/2 Kitaev model's dynamics. We discuss how resonant inelastic x-ray scattering (RIXS) can uncover the fermionic magnons, while neutron scattering provides a mixed contribution of these particles and $Z_2$ gauge excitations. This work exemplifies how the phenomenology of quantum spin-orbital liquids differs from their spin-only counterparts, as well as the complementary roles of RIXS and INS in studying these systems. Magneto-thermodynamics of the $J_1$-$J_2$ Heisenberg antiferromagnet on the square lattice Honecker, Andreas We investigate the finite-temperature properties of the $J_1$-$J_2$ Heisenberg antiferromagnet on the square lattice in the presence of an external magnetic field. We focus on the highly frustrated regime around $J_2 \approx J_1/2$. The $H$-$T$ phase diagram is investigated with particular emphasis on the finite-temperature transition into the ``up-up-up-down'' state that is stabilized by thermal and quantum fluctuations and manifests itself as a plateau at one half of the saturation magnetization in the quantum case. Furthermore, we discuss the enhanced magnetocaloric effect associated to the ground-state degeneracy that arises at the saturation field for $J_2=J_1/2$. Computations for the spin-1/2 system are carried out using finite-temperature Lanczos and quantum typicality approaches. Fisher zeros and persistent coherence in M-qubit non-unitary quantum circuits Hooley, Chris We present a model of many-body quantum dynamics with measurements and post-selection that exhibits a panoply of space- and/or time-ordered phases, from ferromagnetic order to spin-density waves to time crystals. We demonstrate that these phases, including the inherently non-equilibrium dynamical ones, correspond to the complex-temperature equilibrium phases of the exactly solvable square-lattice anisotropic Ising model. Our results include: an explicit construction of the quantum circuit with local one- and two-spin gates; exact solutions onM-leg ladders that already exhibit decoherence-free subspaces that seed the 2D behavior; analytic continuation of the (partial) Onsager solution in the thermodynamic limit; numerical tensor network computations in the presence of an external magnetic field; and insights obtained using an exact fermionized solution. Characterization of strongly disordered many-body systems from one-particle measures Hopjan, Miroslav Typical experiments, designed to detect the many-body delocalization-localization transition, measure the dynamical properties of such systems [1]. However, much work has been done to provide evidence of the transition in the structure of eigenstates. In our recent works [2, 3], we introduce a new quantitative measure for the Fock-space localization [4], computed in the eigenstates. It has a distinct behaviour in the delocalized and localized phase, observed both for bosons [2] and fermions [3], and is potentially useful for the analysis of future experiments. Its scaling properties in the interacting systems are distinct from those in non-interacting systems [3] which points at a different mechanism for the transitions. Moreover, in fermionic systems, we extract a spatial subsystem entropy from the one-particle density matrix (OPDM) and observe that such entropy provides an upper bound on the entanglement entropy [5]. Interestingly, in the MBL regime, the OPDM entropy exhibits the main features of localization, i.e., the area law of eigenstates and the logarithmic growth with time after a quantum quench [5], and it thus provides an additional diagnostic tool for experiments. [1] See, e.g., Lukin et al. Science 364, 256 (2019), Choi et al, Science 352, 1547 (2016) [2] M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020) [3] M. Hopjan, G. Orso and F. Heidrich-Meisner, in preparation [4] S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Phys. Rev. Lett. 115, 046603 (2015) [5] M. Hopjan, F. Heidrich-Meisner and V. Alba, arxiv:2011.02200 (2020) Dynamic Structure Factor of Disordered Coupled-Dimer Heisenberg models Hörmann, Max We investigate the impact of quenched disorder on the zero-temperature dynamic structure factor of coupled-dimer Heisenberg models on two-dimensional bilayers on the square, triangular and Kagome lattice. Using perturbative continuous unitary transformations, the effects on quasiparticles are investigated [1,2]. The disorder leads to intriguing quantum structures in dynamical correlation functions well observable in spectroscopic experiments. [1] M. Hörmann, P. Wunderlich, K. P. Schmidt, Phys. Rev. Lett. 121, 167201 (2018) [2] M. Hörmann and K. P. Schmidt, Physical Review B 102.9 (2020): 094427. Gutzwiller-projected trial states for quantum magnets at finite temperature Horn, Friederike Quantum gas microscopy can achieve single-site and spin-resolved detection of ultracold atoms in optical lattices. Being able to create such snapshots from a Hamiltonian provides an important link between theory and experiment. Here we propose a variational Monte Carlo method to sample the ground state of the 1D and 2D antiferromagnetic Heisenberg Hamiltonian at finite temperature. We construct a Gutzwiller projected density matrix from the eigenstates of a fermionic mean field approximation of the Heisenberg Hamiltonian. This enables us to compute the expectation value of the energy and approximate the entropy as a function of the mean fields and effective coupling constant. Minimizing the free energy we can thus obtain a variational ground state. We will present first results in a one dimensional system. Floquet driving enforced chiral hinge modes without quasi-energy gaps Huang, Biao We demonstrate in a 3D periodically driven model that an intricate type of chiral hinge mode shows up whose quasi-energy spectrum, surprisingly, fully mixes with the bulk spectrums. Such chiral modes exhibit robustness in a considerable range of Hamiltonian parameters and defect strengths. The existence and robustness of these chiral hinge modes can be traced back to an interplay between the boundary geometry and the peculiar bulk dispersions characteristic of a periodically driven system. A tentative topological theory is also formulated to describe such an unusual boundary mode living without quasi-energy gaps. As a by-product, the model we propose also coexists with a Floquet Weyl semimetal phase that can be straightforwardly realized in optical lattices. Ref: Biao Huang, Viktor Novičenko, André Eckardt, Gediminas Juzeliūnas, "Accumulation of chiral hinge modes and its interplay with Weyl physics in a three-dimensional periodically driven lattice system", arXiv:2101.08281 Thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet Hutak, Taras The spin-half pyrochlore Heisenberg antiferromagnet (PHAF) is one of the most challenging problems in the field of highly frustrated quantum magnetism. We calculate thermodynamic properties of this model by interpolating between the low- and high-temperature behavior. For that, we follow ideas developed in detail by B.Bernu and G.Misguich [1] and use for the interpolation the entropy exploiting sum rules [the so-called entropy method (EM)]. We complement the EM results for the specific heat $c(T)$, the entropy $s(T)$, and the susceptibility $\chi(T)$ by the high-temperature expansion data up to order 13 [2]. The EM provides reliable data for the whole temperature region for the PHAF [3]. We do not find hints neither for an extra low-temperature peak nor an extra shoulder below the main maximum for $c(T)$. However, the absence of an extra low-temperature feature goes hand in hand with a significant shift of the single maximum towards $T\approx0.25$. A gapless spectrum is more favorable than a gapped one, i.e., most likely there is power-law low-temperature behavior of $c(T)$. Although best results are for an exponent $\alpha=2$, other exponents ($\alpha=1, 3/2, 5/2, 3$) cannot be excluded. We predict a ground-state energy $e_{0}\approx-0.52$. Our EM data for the susceptibility $\chi(T)$ in comparison with data obtained by diagrammatic Monte Carlo [4] provide further evidence for a gapless spectrum with a ground-state energy $e_{0}\approx-0.52$. We compare our findings with the ones obtained recently by other groups [5-7]. [1] B. Bernu and G. Misguich, Phys. Rev. B 63, 134409 (2001); G. Misguich and B. Bernu, Phys. Rev. B 71, 014417 (2005). [2] A. Lohmann, H.-J. Schmidt, and J. Richter, Phys. Rev. B 89, 014415 (2014). [3] O. Derzhko, T. Hutak, T. Krokhmalskii, J. Schnack, and J. Richter, Phys. Rev. B 101, 174426 (2020). [4] Y. Huang, K. Chen, Y. Deng, N. Prokof'ev, and B. Svistunov, Phys. Rev. Lett. 116, 177203 (2016). [5] R. Schäffer, I. Hagymási, R. Moessner, and D.J. Luitz, Phys. Rev. B 102, 054408 (2020). [6] I. Hagymási, R. Schäffer, R. Moessner, and D. J. Luitz, arXiv:2010.03563. [7] N. Astrakhantsev, T. Westerhout, A. Tiwari, K. Choo, A. Chen, M. H. Fischer, G. Carleo, and T. Neupert, arXiv:2101.08787. Towards a Topological Quantum Chemistry description of correlated systems: the case of the Hubbard diamond chain Iraola, Mikel The recently introduced topological quantum chemistry (TQC) framework has provided a description of universal topological properties of all possible band insulators in all space groups based on crystalline unitary symmetries and time reversal. While this formalism filled the gap between the mathematical classification and the practical diagnosis of topological materials, an obvious limitation is that it only applies to weakly interacting systems-which can be described within band theory. It is an open question to which extent this formalism can be generalized to correlated systems that can exhibit symmetry protected topological phases which are not adiabatically connected to any band insulator. In this work we address the many facettes of this question by considering the specific example of an extended version of a Hubbard diamond chain. This model features a Mott insulator, a trivial insulating phase and an obstructed atomic limit phase. Here we discuss the nature of the Mott insulator and determine the phase diagram and topology of the interacting model with infinite density matrix renormalization group calculations, variational Monte Carlo simulations and with many-body topological invariants. We then proceed by considering a generalization of the TQC formalism to Green's functions combined with the concept of topological Hamiltonian to identify the topological nature of the phases, using cluster perturbation theory to calculate the Green's functions. The results are benchmarked with the above determined phase diagram and we discuss the applicability and limitations of the approach and its possible extensions. Quantum critical points between spin liquids and long-range-ordered phases Janssen, Lukas Quantum spin liquids are exotic states of matter occurring in frustrated magnets. They feature fractionalized excitations interacting via an emergent gauge field and, in many cases, the absence of any long-range ordering. This makes their experimental or numerical identification a formidable task. On this poster, I argue that insight into the nature of putative quantum spin liquids can be gained by studying quantum phase transitions out of such states. In particular, I present an example of a quantum critical point between a U(1) Dirac spin liquid and a long-range-ordered phase, which we study using sign-problem-free quantum Monte Carlo simulations and a concomitant field-theoretical analysis. It will be shown that the presence of fractionalized excitations in the spin-liquid phase has significant consequences for the system's behavior near criticality. Quantum critical points adjacent to spin-liquid phases fall into novel fractionalized universality classes, the universal aspects of which will be discussed as well. Reference: [1] L. Janssen, W. Wang, M. M. Scherer, Z. Y. Meng, and X. Y. Xu, Confinement transition in the QED3-Gross-Neveu-XY universality class, Phys. Rev. B 101, 235118 (2020) Geometric Response of Chiral Superconductors Jiang, Qingdong Despite intense theoretical study and experimental effort, largely driven by possible applications in topological quantum computing, the most basic question about these materials - do they exist? - remains unsettled. This unsatisfactory situation arises because the experimental signatures which have been considered to date have proved either difficult to implement with sufficient precision, ambiguous to interpret, or both. Here we propose quantitatively clear and qualitatively striking signatures, based on geometric and dynamic generalizations of standard Josephson junctions, which could be decisive. Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation Jiménez-Galán, Alvaro Modern light generation technology offers extraordinary capabilities for sculpting light pulses, with full control over individual electric field oscillations within each laser cycle [1]. These capabilities are at the core of lightwave electronics - the dream of ultrafast lightwave control over electron dynamics in solids, on a few-cycle to sub-cycle timescale, aiming at information processing at tera-Hertz to peta-Hertz rates. At the same time, quantum materials encompass fascinating properties such as the possibility to harness extra electronic degrees of freedom, e.g., the valley pseudospin [2]. Previous works have established optical initialization of the valley pseudospin via resonant circular pulses [3,4,5], taking advantage of the optical valley selection rules. Still, manipulating and reading the valley degree of freedom on timescales shorter than valley depolarization and in a non-material-specific (non-resonant) way, remains a crucial challenge. Bringing the frequency-domain concept of topological Floquet systems to the few-femtosecond time domain, I will present an all-optical, non-resonant approach to control the injection of carriers into the valleys on a few-femtosecond timescale by controlling the sub-cycle structure of non-resonant driving fields, and read the valley pseudospin in graphene-like monolayers by using the imprint of the Berry curvature on the high harmonic generation spectrum [6]. Such valley control does not rely on the optical valley selection rule. Instead, the tailored field modifies the laser-driven band structure on a sub-cycle timescale, allowing ultrafast optical control of the topological properties of 2D graphene-like quantum materials [7]. References [1] F. Krausz et al., Rev. Mod. Phys., 81 163 (2009). [2] S.A. Vitale et al., Small, 14 1801483 (2018). [3] D. Xiao et al., Phys. Rev. Lett., 99 236809 (2007). [4] F. Langer et al., Nature 557 76 (2018). [5] S.A. Oliaei Motlagh et al., Phys. Rev. B 100 115431 (2019). [6] R.E.F. Silva et al., Nat. Phot. 13 849 (2019). [7] Á. Jiménez-Galán et al., Nat. Phot. 14 728 (2020). Unsupervised machine learning of topological phase transitions from experimental data Käming, Niklas Recently, machine learning methods have been shown to be an alternative way of localizing phase boundaries also from noisy and imperfect data and without the knowledge of the order parameter. Using unsupervised machine learning techniques including anomaly detection and influence functions we obtain the topological phase diagram of the Haldane model in a completely unbiased fashion from experimental data. We show that the methods can successfully be applied to experimental data at finite temperature and to data of Floquet systems, when postprocessing the data to a single micromotion phase. Our work provides a benchmark for unsupervised detection of new exotic phases in complex many-body systems. Quantum droplet phases in extended Bose-Hubbard models with cavity-mediated interactions Karpov, Peter The Bose-Hubbard model and its various extensions have been studied for more than 30 years. We show that, surprisingly, there is still a room for new physics there, demonstrating a variety of quantum droplet phases. Such quantum droplets are self-bound objects which have recently drawn significant attention in continuum models featuring competing repulsive and attractive interactions and describing, for example, dipolar gases and bosonic mixtures. Multimode optical cavities offer an alternative, rapidly developing experimental platform, for studying competing short- and long-range interactions. Here, differently e.g. from the case of dipolar gases, the long-range cavity-mediated interaction can be widely tuned in range and strength. We study a system of bosonic atoms trapped in a lattice inside a multimode optical cavity, which can be modeled by an extended Bose-Hubbard model with competing on-site repulsive and finite-range (cavity-mediated) attractive interactions. We use the canonical worm Quantum Monte Carlo algorithm to explore the phase diagram of the model. Our approach is numerically exact and applicable in arbitrary dimensions. Moreover, since we explicitly work in the canonical ensemble, we don't have to fine-tune the chemical potential and can deal with arbitrary occupation numbers (up to the total number of particles in the system). Thus we can successfully study the droplet phases, overcoming the difficulties of other more conventional methods like grand-canonical worm algorithms, stochastic series expansion, and DMRG. The canonical worm algorithm can be straightforwardly applied to a more broad class of other experimentally relevant models featuring competing repulsive and attractive interactions, for example, dipolar gases and bosonic mixtures. In addition to the previously studied density-wave and supersolid self-organized superradiant phases, the finite-range cavity-mediated attraction can lead to the formation of quantum self-bound droplets. The droplet phases dominate the phase diagram and can include both compressible superfluid/supersolid as well as incompressible Mott and density-wave droplets. Confinement and Mott transitions of dynamical charges in 1D lattice gauge theories Kebric, Matjaz Lattice gauge theories (LGTs) have become a valuable tool to study strongly correlated condensed matter systems. This becomes in particular interesting when gauge degrees of freedom are coupled to matter since they allow us to study the complex problem of confinement. However, when the lattice is doped and matter becomes dynamical the clear notion of confinement becomes complicated. Here we study a one-dimensional (1D) \Zt LGT model where the gauge fields are coupled to dynamical charges with confining \Zt electric field and repulsive nearest-neighbour interactions. We map our model to a local string-length Hamiltonian where we link the confinement in the \Zt LGT model to a broken translational symmetry in the string-length basis. In addition we study the Mott transition of the charges at a specific filling of $n=2/3$. We find that the metallic phase of the confined Luttinger liquid is characterized by a hidden off-diagonal quasi-long-range order. Furthermore we map the 1D \Zt LGT model to a $t-J_{z}$ model which can be implemented in cold atom experiments by using the Rydberg dressing schemes; thus, we propose a way to directly test our theoretical predictions. https://arxiv.org/abs/2102.08375 Statistical physics through the lens of real-space mutual information Koch-Janusz, Maciej Identifying the relevant coarse-grained degrees of freedom in a complex physical system is a key stage in developing powerful effective theories in and out of equilibrium. The celebrated renormalization group provides a framework for this task, but its practical execution in unfamiliar systems is fraught with ad hoc choices, whereas machine learning approaches, though promising, often lack formal interpretability. Recently, the optimal coarse-graining in a statistical system was shown to exist, based on a universal, but computationally difficult information-theoretic variational principle. This limited its applicability to but the simplest systems; moreover, the relation to standard formalism of field theory was unclear. Here we present an algorithm employing state-of-art results in machine-learning-based estimation of information-theoretic quantities, overcoming these challenges. We use this advance to develop a new paradigm in identifying the most relevant field theory operators describing properties of the system, going beyond the existing approaches to real-space renormalization. We evidence its power on an interacting model, where the emergent degrees of freedom are qualitatively different from the microscopic building blocks of the theory. Our results push the boundary of formally interpretable applications of machine learning, conceptually paving the way towards automated theory building. Discontinuous quantum and classical magnetic response of the pentakis dodecahedron Konstantinidis, Nikolaos The pentakis dodecahedron, the dual of the truncated icosahedron, consists of 60 edge-sharing triangles. It has 20 six- and 12 five-fold coordinated vertices, with the former forming a dodecahedron, and each of the latter connected to the vertices of one of the 12 pentagons of the dodecahedron. When spins mounted on the vertices of the pentakis dodecahedron interact according to the nearest-neighbor antiferromagnetic Heisenberg model, the two different vertex types necessitate the introduction of two exchange constants. As the relative strength of the two constants is varied the molecule interpolates between the dodecahedron and a molecule consisting only of quadrangles. The competition between the two exchange constants, frustration, and an external magnetic fi eld results in a multitude of ground-state magnetization and susceptibility discontinuities. At the classical level the maximum is ten magnetization and one susceptibility discontinuities when the 12 fi ve-fold vertices interact with the dodecahedron spins with approximately one-half the strength of their interaction. When the two interactions are approximately equal in strength the number of discontinuities is also maximized, with three of the magnetization and eight of the susceptibility. At the full quantum limit, where the magnitude of the spins equals 1/2, there can be up to three ground-state magnetization jumps that have the total $z$ spin component changing by $\Delta S^z = 2$, even though quantum fluctuations rarely allow discontinuities of the magnetization. The full quantum case also supports a $\Delta S^z = 3$ discontinuity. Frustration also results in nonmagnetic states inside the singlet-triplet gap. These results make the pentakis dodecahedron the molecule with the most discontinuous magnetic response from the quantum to the classical level. Tunable topological states hosted by unconventional superconductors with adatoms Kreisel, Andreas Chains of magnetic atoms, placed on the surface of s-wave superconductors, have been established as a laboratory for the study of Majorana bound states. In such systems, the breaking of time reversal due to magnetic moments gives rise to the formation of in-gap states, which hybridize to form one-dimensional topological superconductors. However, in unconventional superconductors even non-magnetic impurities induce in-gap states since scattering of Cooper pairs changes their momentum but not their phase. Here, we propose a realistic path for creating topological superconductivity, which is based on an unconventional superconductor with a chain of non-magnetic adatoms on its surface. The topological phase can be reached by tuning the magnitude and direction of a Zeeman field,such that Majorana zero modes at its boundary can be generated, moved and fused. To demonstrate the feasibility of this platform, we develop a general mapping of films with adatom chains to one-dimensional lattice Hamiltonians. This allows us to study unconventional superconductors such as Sr$_2$RuO$_4$ exhibiting multiple bands and an anisotropic order parameter. Kitaev quasiparticles in a proximate spin liquid: A many-body localization perspective. Kumar, Aman We study the stability of Kitaev quasiparticles in the presence of a perturbing Heisenberg interaction as a Fock space localization phenomenon. We identify parameter regimes where Kitaev states are localized, fractal, or delocalized in the Fock space of exact eigenstates, with the first two implying quasiparticle stability. Finite-temperature calculations show that a vison gap, and a nonzero plaquette Wilson loop at low temperatures, both characteristic of the deconfined Kitaev spin-liquid phase, persist far into the neighboring phase that has a concomitant stripy spin-density wave (SDW) order. The key experimental implication for Kitaev materials is that below a characteristic energy scale, unrelated to the SDW ordering, Kitaev quasiparticles are stable. Orbital Density Waves in Elemental Chalcogens Kłosiński, Adam Stimulated by recent works highlighting the indispensable role of Coulomb interactions in the formation of helical chains and chiral electronic order in the elemental chalcogens, we explore the p-orbital Hubbard model on a one-dimensional helical chain. By solving it in the Hartree approximation we find a stable ground state with a period-three orbital density wave [1]. We establish that the precise form of the emerging order strongly depends on the Hubbard interaction strength. In the strong coupling limit, the Coulomb interactions support an orbital density wave that is qualitatively different from that in the weak-coupling regime. We identify the phase transition separating these two orbital ordered phases, and show that realistic values for the inter-orbital Coulomb repulsion in elemental chalcogens place them in the weak coupling phase, in agreement with observations of the order in the elemental chalcogens. [1] A. Klosinski, A. M. Oles, J. van Wezel and K. Wohlfeld, arxiv:2103.05925 Anomalous Quantum Oscillations in a Heterostructure of Graphene on a Proximate Quantum Spin Liquid Leeb, Valentin The quasi two-dimensional Mott insulator $\alpha\text{-}{\text{RuCl}}_{3}$ is proximate to the sought-after Kitaev quantum spin liquid (QSL). In a layer of $\alpha\text{-}{\text{RuCl}}_{3}$ on graphene the dominant Kitaev exchange is further enhanced by strain. Recently, quantum oscillation (QO) measurements of such $\alpha\text{-}{\text{RuCl}}_{3}$ / graphene heterostructures showed an anomalous temperature dependence beyond the standard Lifshitz-Kosevich description. Here, we develop a theory of {\it anomalous QO} in an effective Kitaev-Kondo lattice model in which the itinerant electrons of the graphene layer interact with the correlated magnetic layer via spin interactions. At low temperatures a heavy Fermi liquid emerges such that the neutral Majorana fermion excitations of the Kitaev QSL acquire charge by hybridising with the graphene Dirac band. Using ab-initio calculations to determine the parameters of our low energy model we provide a microscopic theory of {\it anomalous QOs} with a non-LK temperature dependence consistent with our measurements. We show how remnants of fractionalized spin excitations can give rise to characteristic signatures in QO experiments. Influence matrix approach to quantum many-body dynamics Lerose, Alessio I will introduce an approach to study quantum many-body dynamics, inspired by the Feynman-Vernon influence functional. Its central object is the influence matrix (IM), which describes the effect of a Floquet many-body system on the dynamics of local subsystems. For translationally invariant systems, the IM obeys a self-consistency equation. For certain fine-tuned models, remarkably simple exact solutions appear, which represent perfect dephasers (PD), i.e., many-body systems acting as perfectly Markovian baths on their parts. Such PDs include dual-unitary quantum circuits investigated in recent works. In the vicinity of PD points, the system is not perfectly Markovian, but rather acts as a quantum bath with a short memory time. In this case, we demonstrate that the self-consistency equation can be solved using matrix-product states (MPS) methods, as the IM temporal entanglement is low. Using a combination of analytical insights and MPS computations, we characterize the structure of the IM in terms of an effective "statistical-mechanics" description for interfering intervals of local quantum trajectories and illustrate its predictive power by analytically deriving the relaxation rate of an impurity embedded in the system. In the last part of the talk, I will describe how to use these ideas to study the many-body localized (MBL) phase of strongly disordered interacting spin systems subject to periodic kicks. This approach allows to study exact disorder-averaged time evolution in the thermodynamic limit. MBL systems fail to act as efficient baths, and this property is encoded in their IM. I will discuss the structure of an MBL IM and link it to the onset of temporal long-range order. Large surface magnetization in noncentrosymmetric antiferromagnets Lund, Mike Alexander Thin-film antiferromagnets (AFs) with Rashba spin-orbit coupling are theoretically investigated. We demonstrate that the relativistic Dzyaloshinskii-Moriya interaction (DMI) produces a large surface magnetization and a boundary-driven twist state in the antiferromagnetic Néel vector. We predict a magnetization on the order of $2.3\times10^{4}$A/m, which is comparable to the magnetization of ferromagnetic semiconductors. Importantly, the magnetization is characterized by ultrafast terahertz dynamics and provides different approaches for efficiently probing and controlling the spin dynamics of AFs as well as detecting the antiferromagnetic DMI. Notably, the magnetization does not lead to any stray magnetic fields except at the corners where weak magnetic monopole fields appear. Shadow-band formation and recombination of optical excitations in a correlated band-insulator Manmana, Salvatore R. We study the time-evolution of single-particle spectral functions following an electron-hole excitation of a one-dimensional correlated band-insulator realized by a Hubbard model with a magnetic superlattice using matrix product states (MPS). For an excitation with a specified spin, we find the electron-electron interaction to induce recombination of the excitation by a spin-dependent redistribution of the weights. In the spin direction unaffected by the excitation, a shadow-band forms in the gap region. We compare this finding to the formation of excitons in extended Hubbard models without a superlattice. Bosonization of the Q= 0 continuum of Dirac fermions Mantilla Serrano, Sebastián Felipe We develop a bosonization formalism that captures non-perturbatively the interaction effects on the Q = 0 continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the Q = 0 neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single particle Green’s function. We apply this machinery to compute the interaction corrections to the optical conductivity of 2D Dirac Fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong coupling regime. Bond Dependent Spin-Orbital Exchange and Quantum Order-by-Disorder in CoTiO3 McClarty, Paul There has been a great deal of interest in bond-dependent anisotropic couplings in strong spin-orbit coupled magnets - especially iridates and ruthenates - that has brought new physics into focus. Recent theoretical work has proposed that such couplings can be significant in certain cobalt magnets where the spin-orbit coupling is sub-dominant [1]. Here we report on CoTiO3, an insulating ABC stacked honeycomb easy plane magnet that orders into a structure with ferromagnetic layers stacked antiferromagnetically with a spin wave spectrum that is known to host Dirac magnons [2]. Our high resolution inelastic neutron scattering data clearly shows the presence of a magnon gap of about 1meV that must arise through the presence of bond-dependent exchange couplings [3]. The spectral gap also provides strong evidence for the existence of a quantum order-by-disorder mechanism a very rare phenomenon that selects the long-ranged ordered magnetic structure through the effect of quantum fluctuations - that, in this material, crucially involves virtual crystal field excitations. The same couplings that lead to the spectral gap also cause the Dirac magnons to wind around one another in a double helix structure and we show that the experimental data is consistent with this scenario. We also show the presence of dispersive exciton modes with Dirac nodes. All the key features of the experiment are explicable through a multi-boson theory with spin-orbital exchange couplings. [1] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018); R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408 (2018). [2] B. Yuan, I. Khait, G.-J. Shu, F. C. Chou, M. B. Stone, J. P. Clancy, A. Paramekanti, and Y.-J. Kim, Phys. Rev. X 10, 011062 (2020). [3] M. Elliot, P. A. McClarty, D. Prabhakaran, R. D. Johnson, H. C. Walker, P. Manuel, and R. Coldea, arXiv:2007.04199. Time evolution of terahertz-pumped heavy-fermion systems Meirinhos, Francisco Francisco Meirinhos and Johann Kroha Physikalisches Institut & Bethe Center for Theoretical Physics, Universität Bonn, Germany The search and characterisation of new quantum phases of matter has recently been intensified by the application of terahertz (THz) spectroscopy in the time domain to heavy-fermion systems [1-3]. It was experimentally shown that a single-cycle terahertz laser pulse disrupts the strongly correlated (Kondo) ground state in heavy-fermion compounds such as $CeCu_{6-x}Au_x$ which recovers after a characteristic delay time $\tau_K^$, accompanied by the emission of a temporally confined terahertz echo pulse. In this way, time-domain terahertz spectroscopy provides direct access to both, the quasiparticle spectral weight and the characteristic time or energy scales, across a heavy-fermion quantum phase transition [1,2]. The transient nature of such non-equilibrium dynamics leads new and interesting many-body physics, raising questions about the established properties of quasi-particles. In the present work we develop the theoretical description of this heavy-fermion non-equilibrium dynamics. The electronic part of the system is represented by an Anderson model described by a time-dependent version of the non-equilibrium Non-Crossing Approximaton (NCA). The THz photons are treated as a quantum field with its own dynamics and coupled to the heavy fermion-system by a dipole interaction. In this way, incident THz pulses with arbitrary pulse shape can be implemented as an initial condition. At the same time, the photon quantum dynamics allows for re-emission of radiation and, thereby, the necessary release of energy during the relaxation dynamics to the heavy-fermion ground state. These coupled dynamics are solved by an efficient time-stepping algorithm. We also discuss the thermalisation to ambient temperature in terms of a Lindblad-like coupling to the electromagnetic environment as a bath. [1] C. Wetli, S. Pal, J. Kroha, K. Kliemt, C. Krellner, O. Stockert, H. v. Löhneysen, and M. Fiebig, Time-resolved collapse and revival of the Kondo state near a quantum phase transition, Nature Phys. {\bf 14}, 1103 (2018) [2] S. Pal, C. Wetli, F. Zamani, O. Stockert, H. v. Löhneysen, M. Fiebig, and J. Kroha, Phys. Rev. Lett. {\bf 122}, 096401 (2019) [3] C.-J. Yang, S. Pal, F. Zamani, K. Kliemt, C. Krellner, O. Stockert, H. v. Löhneysen, J. Kroha, and Manfred Fiebig, Phys. Rev. Research {\bf 2}, 033296 (2020) Unsupervised Learning Universal Critical Behavior via the Intrinsic Dimension Mendes Santos, Tiago The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics. Here, we study how the minimum number of variables needed to accurately describe the important features of a data set - the intrinsic dimension (ID) - behaves in the vicinity of phase transitions. We employ state-of-the-art nearest-neighbors-based ID estimators to compute the ID of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless. For all the considered cases, we find that the ID uniquely characterizes the transition regime. The finite-size analysis of the I d allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent $\nu$ in the case of continuous transitions. For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods. Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space. From black holes to Weyl semimetals Meng, Tobias Weyl semimetals are a recent example of solid state materials featuring a relativistic band structure. This analogy has lead to the intriguing proposal that Weyl semimetals can mimic various effects known from high energy physics, such as Klein tunnelling. In this talk, I will discuss how Weyl semimetals with (over-)tilted nodes connect to black hole matrices, and what experimentally observable consequences thereof are. Improved quantum transport calculations for interacting nanostructures Minarelli, Emma Nanoelectronics devices such as semiconductor quantum dots and single molecule transistors exhibit a rich range of physical behavior due to the interplay between orbital complexity, strong electronic correlations and device geometry. Understanding and simulating the quantum transport through such nanostructures is essential for rational design and technological applications. In this talk, I will discuss both electric and heat quantum transport for interacting mesoscopic quantum transport. For the electric conductance, calculations are developed under linear response and I demonstrate the improvement over standard methods with applications such as triple quantum dots and two-channel charge Kondo models using the numerical renormalization group technique (NRG). I will treat reformulations of the Meir-Wingreen formula in the context of non-proportionate coupling set-ups and by means of perturbative verification of the Ng ansatz; of the Oguri formula in non-Fermi Liquid states and of the Kubo formula for conductance. For the heat conductance, both non-equilibrium versus linear response formulation are discussed, in particular with respect to their viability using NRG. Electrical conductivity formulas for a general two-band model and their application Mitscherling, Johannes In recent years, there is an increasing interest in transport properties of multiband systems due to advances in experimental techniques. We focus on the longitudinal, the anomalous and the ordinary Hall conductivity for a general two-band model. This model captures a broad spectrum of systems with very different and rich physics like Chern insulators, ferromagnets, and spiral spin density waves. We will see in a simple and fundamental derivation how two criteria for a unique and physically motivated decomposition of the conductivity formulas naturally arise from the multiband structure of the model. Those criteria allow us to relate interband contributions to concepts of quantum geometry, namely the quantum metric and the Berry curvature. They lead to a decomposition whose individual scaling behaviors with respect to the scattering rate can be analyzed systematically. We exemplify the general analysis by several applications ranging from spiral magnetic order in cuprates to the quantum anomalous Hall effect in Chern insulators. Interaction-stabilized topological magnon insulator in ferromagnets Mook, Alexander Condensed matter systems admit topological collective excitations above a trivial ground state, an example being Chern insulators formed by Dirac bosons with a gap at finite energies. However, in contrast to electrons, there is no particle-number conservation law for collective excitations. This gives rise to particle number-nonconserving many-body interactions whose influence on single-particle topology is an open issue of fundamental interest in the field of topological quantum materials. Taking magnons in ferromagnets as an example, we uncover topological magnon insulators that are stabilized by interactions through opening Chern-insulating gaps in the magnon spectrum. This can be traced back to the fact that the particle-number nonconserving interactions break the effective time-reversal symmetry of the harmonic theory. Hence, magnon-magnon interactions are a source of topology that can introduce chiral edge states, whose chirality depends on the magnetization direction. Importantly, interactions do not necessarily cause detrimental damping but can give rise to topological magnons with exceptionally long lifetimes. We identify two mechanisms of interaction-induced topological phase transitions and show that they cause unconventional sign reversals of transverse transport signals, in particular of the thermal Hall conductivity. Our results demonstrate that interactions can play an important role in generating nontrivial topology. Reference: Alexander Mook, Kirill Plekhanov, Jelena Klinovaja, Daniel Loss, arXiv:2011.06543 (2020) Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms Morpurgo, Giacomo A supersolid is a counterintuitive phase of matter that combines the global phase coherence of a superfluid with a crystal-like self-modulation in space. Recently, such states have been experimentally realized using dipolar quantum gases. Here we investigate the response of a dipolar supersolid to an interaction quench that shatters the global phase coherence. We identify a parameter regime in which this out-of-equilibrium state rephases, indicating superfluid flow across the sample as well as an efficient dissipation mechanism. We find a crossover to a regime where the tendency to rephase gradually decreases until the system relaxes into an incoherent droplet array. Although a dipolar supersolid is, by its nature, ‘soft’, we capture the essential behaviour of the de- and rephasing process within a rigid Josephson junction array model. Yet, both experiment and simulation indicate that the interaction quench causes substantial collective mode excitations that connect to phonons in solids and affect the phase dynamics. Competition between X-Cube and Toric Code in three dimensions Mühlhauser, Matthias We investigate the competition of the X-Cube model with the 3D Toric code using variational mean-field calculations and high-order series expansions. We determine the complete phase diagram, which interestingly consists of four regions, i.e. apart from the topologically ordered toric code phase and the X-Cube fracton phase we find two regions which are adiabatically connected to classical spin-liquid phases. In the end we also investigate the effect of an additional magnetic field in x- or z-direction using variational mean-field calculations. Anyonic Molecules in Atomic Fractional Quantum Hall Liquids: A Quantitative Probe of Fractional Charge and Anyonic Statistics Muñoz de las Heras, Alberto We study the quantum dynamics of massive impurities embedded in a strongly interacting, two-dimensional atomic gas driven into the fractional quantum Hall (FQH) regime under the effect of a synthetic magnetic field. For suitable values of the atom-impurity interaction strength, each impurity can capture one or more quasihole excitations of the FQH liquid, forming a bound molecular state with novel physical properties. An effective Hamiltonian for such anyonic molecules is derived within the Born-Oppenheimer approximation, which provides renormalized values for their effective mass, charge, and statistics by combining the finite mass of the impurity with the fractional charge and statistics of the quasiholes. The renormalized mass and charge of a single molecule can be extracted from the cyclotron orbit that it describes as a free particle in a magnetic field. The anyonic statistics introduces a statistical phase between the direct and exchange scattering channels of a pair of indistinguishable colliding molecules and can be measured from the angular position of the interference fringes in the differential scattering cross section. Implementations of such schemes beyond cold atomic gases are highlighted—in particular, in photonic systems. Conductivity and thermoelectric coefficients of doped SrTiO$_3$ at high temperatures Nazaryan, Khachatur We developed a theory of electric and thermoelectric conductivity of lightly doped SrTiO$_3$ in the non-degenerate region $k_B T \geq E_F$, assuming that the major source of electron scattering is their interaction with soft transverse optical phonons present due to proximity to ferroelectric transition. We have used kinetic equation approach within relation-time approximation and we have determined energy-dependent transport relaxation time $\tau(E)$ by the iterative procedure. Using electron effective mass $m$ and electron-transverse phonon coupling constant $\lambda$ as two fitting parameters, we are able to describe quantitatively a large set of the measured temperature dependences of resistivity $R(T)$ and Seebeck coefficient $\mathcal{S}(T)$ for a broad range of electron densities studied experimentally in recent paper. In addition, we calculated Nernst ratio $\nu=N/B$ in the linear approximation over weak magnetic field in the same temperature range. Orbital magnetic moment of magnons Neumann, Robin It is commonly accepted that magnons---collective excitations in a magnetically ordered system---carry a spin of $1\hbar$ or, phrased differently, a magnetic moment of $g \mu_\text{B}$. In this talk, I demonstrate that magnons carry magnetic moment beyond their spin magnetic moment. Our rigorous quantum theory uncovers a magnonic orbital magnetic moment brought about by spin-orbit coupling. We apply our theory to two paradigmatic systems where the notion of orbital moments manifests itself in novel fundamental physics rather than just quantitative differences. In a coplanar antiferromagnet on the two-dimensional kagome lattice the orbital magnetic moment gives rise to an orbital magnetization. While the spin magnetization is oriented in the kagome plane, the orbital magnetization also has a finite out-of-plane component leading to ``orbital weak ferromagnetism.'' The insulating collinear pyrochlore ferromagnet Lu$_2$V$_2$O$_7$ exhibits a ``magnonic orbital Nernst effects,'' i.\,e. transversal currents of orbital magnetic moment induced by a temperature gradient. The orbital magnetization and the orbital Nernst effect in magnetic insulators are two signatures of the orbital magnetic moment of magnons. Special states in quantum many-body spectra Nielsen, Anne Exceptions to thermalization in quantum many-body systems provide interesting opportunities. Many-body localization, in which all states in the spectrum have area law entanglement, constitutes a strong violation of the eigenstate thermalization hypothesis, and quantum many-body scars instead give rise to a weak violation with a few nonthermal states embedded in a spectrum of thermal states. Here, we propose and demonstrate that one can similarly have a weak violation of many-body localization, in which one or a few states in a spectrum have above area law entanglement, while the rest of the states are many-body localized. We show that this can be achieved through a mechanism, in which emergent symmetry of the special states prevents many-body localization in these states. Reference: Phys. Rev. Lett. 125, 240401 (2020) Bosonic Pfaffian State in the Hofstadter-Bose-Hubbard Model Palm, Felix A. Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor $\nu=1$ and find strong indications that at $\alpha=1/6$ magnetic flux quanta per plaquette the ground state is a lattice analog of the continuum non-Abelian Pfaffian. We study the on-site correlations of the ground state, which indicate its paired nature at $\nu=1$, and find an incompressible state characterized by a charge gap in the bulk. We argue that the emergence of a charge density wave on thin cylinders and the behavior of the two- and three-particle correlation functions at short distances provide evidence for the state being closely related to the continuum Pfaffian. The signatures discussed here are accessible in current cold atom experiments and we show that the Pfaffian-like state is readily realizable in few-body systems using adiabatic preparation schemes. Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class Parisen Toldin, Francesco We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision Monte Carlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naive arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models. Ref: F. Parisen Toldin, arXiv:2012.00039, Phys. Rev. Lett. (2021), to be published Finite frequency backscattering current noise at a helical edge Pashinsky, Boris Magnetic impurities with sufficient anisotropy could account for the observed strong deviation of the edge conductance of 2D topological insulators from the anticipated quantized value. In this work we consider such a helical edge coupled to dilute impurities with an arbitrary spin S and a general form of the exchange matrix. We calculate the backscattering current noise at finite frequencies as a function of the temperature and applied voltage bias. We find that, in addition to the Lorentzian resonance at zero frequency, the backscattering current noise features Fano-type resonances at nonzero frequencies. The widths of the resonances are controlled by the spectrum of corresponding Korringa rates. At a fixed frequency the backscattering current noise has nonmonotonic behavior as a function of the bias voltage. A quantum Boltzmann equation for strongly correlated electrons Picano, Antonio Collective orders and photo-induced phase transitions in quantum matter can evolve on timescales which are orders of magnitude slower than the femtosecond processes related to electronic motion in the solid. Quantum Boltzmann equations can potentially resolve this separation of timescales, but are often constructed within a perturbative framework. Here we derive a quantum Boltzmann equation which only assumes a separation of timescales (taken into account through the gradient approximation for convolutions in time), but is based on a non-perturbative scattering integral, and makes no assumption on the spectral function such as the quasiparticle approximation. In particular, a scattering integral corresponding to non-equilibrium dynamical mean-field theory is evaluated in terms of an Anderson impurity model in a non-equilibrium steady state with prescribed distribution functions. This opens the possibility to investigate dynamical processes in correlated solids with quantum impurity solvers designed for the study of non-equilibrium steady states. Prethermal phases of matter in dimension 1, 2, and 3 Pizzi, Andrea Many-body systems under a high-frequency drive spend an exponentially long time in a prethermal regime. The recent investigation of this regime for the realization of novel prethermal phases of matter has been severely limited by the computational challenges associated with the exponentially large Hilbert space of quantum many-body systems. We explore prethermal phases of matter in classical many-body systems undergoing driven Hamiltonian dynamics in one, two and three dimensions. First, we show that the phenomenology of known 1D quantum prethermal phases of matter is virtually the same when going classical, which suggests that these phenomena should in essence be thought of as robust to quantum fluctuations, rather than dependent on them. Second, we make use of efficient classical simulations to study the interplay between dimensionality and interaction range. For instance, we show that, in contrast to 1D, nontrivial prethermal phases can emerge in 3D even in short-range interacting systems. As concrete examples, we focus on higher-order and fractional discrete time crystals, prethermal phases of matter breaking the time translational symmetry of a drive with unexpectedly large and possibly fractional periods. Our work paves the way towards the study of novel prethermal phases of matter beyond the (few) known one-dimensional examples. Yu-Shiba-Rusinov states of single magnetic molecule in an s−wave superconductor Pradhan, Saurabh We use the numerical renormalization group theory to investigate the Yu-Shiba-Rusinov (YSR) bound state properties of single magnetic molecules placed in an s-wave superconducting substrate. The molecule consists of a large core spin and a single orbital, coupled via exchange interaction. The critical Coulomb interaction for the singlet/doublet transition decreases in the presence of this exchange interaction for both Ferro and anti-ferromagnetic couplings. The number of YSR states also increases to two pairs, however, in the singlet phase, one of the pairs has zero spectral weight. We explore the evolution of the in-gap states using the Anderson model. Away from the particle-hole symmetry point, the results suggest a doublet-singlet-doublet transition as the on-site energy is lowered while keeping the Coulomb interaction fixed. To understand these results, we write down an effective model for the molecule in the limit of a large superconducting order parameter. Qualitatively, it explains the various phase transitions and spectral nature of the in-gap states. Doping a Topological Insulator Rachel, Stephan The search for topological superconductors is one of the most pressing and challenging questions in condensed matter and material research. Despite some early suggestions that "doping a topological insulator" might be a successful recipe to find topological superconductors, until today there is no general understanding of the relationship of the topology of the superconductor and the topology of its underlying normal state system. One of the major obstacles is the strong effect of the Fermi surface and the subsequent pairing tendencies, usually preventing a detailed analysis and comparison between different topological superconducting systems. Here we present an analysis of doped topological insulators where the dominant Fermi surface effects have been removed. Our approach allows us to study and compare superconducting instabilities of different insulating normal state systems and reveal the potential of doping a topological insulator. Non-equilibrium evolution of Bose-Einstein condensate deformation in temporally controlled weak disorder Radonjic, Milan We consider a time-dependent extension of a perturbative mean-field approach to the homogeneous dirty boson problem by considering how switching on and off a weak disorder potential affects the stationary state of an initially equilibrated Bose-Einstein condensate by the emergence of a disorder-induced condensate deformation. We find that in the switch on scenario the stationary condensate deformation turns out to be a sum of an equilibrium part [1], that actually corresponds to adiabatic switching on the disorder, and a dynamically-induced part, where the latter depends on the particular driving protocol [2]. If the disorder is switched off afterwards, the resulting condensate deformation acquires an additional dynamically-induced part in the long-time limit, while the equilibrium part vanishes. We also present an appropriate generalization to inhomogeneous trapped condensates. Our results demonstrate that the condensate deformation represents an indicator of the generically non-equilibrium nature of steady states of a Bose gas in a temporally controlled weak disorder. [1] K. Huang and H.-F. Meng, Phys. Rev. Lett. 69, 644 (1992) [2] Milan Radonjić and Axel Pelster, SciPost Phys. 10, 008 (2021) Finite temperature phase diagram of ultracold bosons in a two-dimensional optical Hubbard lattice Ray, Sayak The Bose-Hubbard model (BHM) is well celebrated for its success to describe the phases and dynamics of ultracold interacting bosons in an optical lattice. In this work we compute the equilibrium phase diagram of the BHM at finite temperature using the cluster mean field (CMF) theory and show that this method is capable of describing the phase boundary accurately in good agreement with previous quantum Monte Carlo (QMC) studies as well with experiments. We calculate the condensate density in the superfluid (SF) phase, vanishing of which indicates the SFto- normal-fluid (NF) phase boundary. The compressibility, involving particle-hole excitations, is calculated in the SF, NF and Mott insulator phases. It provides an estimate of the crossover temperature from the MI to the NF state. Our method offers an advantage to analyze the effect of correlations systematically with increasing cluster size followed by a finite-cluster-size scaling, and at the same time overcomes the low temperature difficulties of QMC. Ulli Pohl, Sayak Ray, and Johann Kroha Physikalisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Nußallee 12, 53115, Bonn, Germany Fracton excitations in classical frustrated kagome spin models Reuther, Johannes Fractons are topological quasiparticles with limited mobility. While there exists a variety of models hosting these excitations, typical fracton systems require rather complicated many-particle interactions. Here, we discuss fracton behavior in the more common physical setting of classical kagome spin models with frustrated two-body interactions only. We investigate systems with different types of elementary spin degrees of freedom (three-state Potts, XY, and Heisenberg spins) which all exhibit characteristic subsystem symmetries and fracton-like excitations. The mobility constraints of isolated fractons and bound fracton pairs in the three-state Potts model are, however, strikingly different compared to the known type-I or type-II fracton models. One may still explain these properties in terms of type-I fracton behavior and construct an effective low-energy tensor gauge theory when considering the system as a 2D cut of a 3D cubic lattice model. Our extensive classical Monte-Carlo simulations further indicate a crossover into a low temperature glassy phase where the system gets trapped in metastable fracton states. Moving on to XY spins, we find that in addition to fractons the system hosts fractional vortex excitations. As a result of the restricted mobility of both types of defects, our classical Monte-Carlo simulations do not indicate a Kosterlitz-Thouless transition but again show a crossover into a glassy low-temperature regime. Finally, the energy barriers associated with fractons vanish in the case of Heisenberg spins, such that defect states may continuously decay into a ground state. These decays, however, exhibit a power-law relaxation behavior which leads to slow equilibration dynamics at low temperatures. Generative Model Learning For Molecular Electronics Rigo, Jonas The use of single-molecule transistors in nanoelectronics devices requires a deep understanding of the generalized'quantum impurity'models describing them. Microscopic models comprise molecular orbital complexity and strong electron interactions while also treating explicitly conduction electrons in the external circuit. No single theoretical method can treat the low-temperature physics of such systems exactly. To overcome this problem, we use a generative machine learning approach to formulate effective models that are simple enough to be treated exactly by methods such as the numerical renormalization group, but still capture all observables of interest of the physical system. We illustrate the power of the new methodology by application to the single benzene molecule transistor. Analysis of electronic properties of twisted bilayer graphene using exact diagonalization Rodrigues, Alina Twisted bilayer graphene (TBG), a structure created through the misalignment of two graphene sheets stacked one on top of the other, became an object of a great interest due to strong electronic correlations present in these systems [1-2]. The source of the correlations has not yet been explained, however its consequences were noticed in the experiments, where insulating and superconducting states were observed [3-4]. A significant effort is now being put into understanding the nature of these correlations [5-6]. Presented results are based on the exact diagonalization (ED) method which allows for precise analysis of the electronic correletions. It is however limited to small system studies. In our research an extension of the ED method was applied that truncates the dimension of the Hilbert space through dismissing electron configurations that have relatively small diagonal element in the Hamiltonian. We have analysed different system sizes and electron configurations. [1] J. M. B. Lopes dos Santos, N. M. R. Peres, A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)\\ [2] E. Bistritzer and Allan H. MacDonald, PNAS 108, 12233 (2011)\\ [3] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori \& P. Jarillo-Herrero, Nature 556, 80 (2018)\\ [4] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras \& P. Jarillo-Herrero, Nature 556, 43 (2018)\\ [5] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018)\\ [6] J. F. Dodaro, S. A. Kivelson, Y. Schattner, X. Q. Sun, and C. Wang, Phys. Rev. B 98, 075154 (2018)\\ Metastability in open quantum systems Rose, Dominic We expand on a recently developed theory for metastability in open quantum systems to allow for the understanding and analysis of emergent classical metastability. We first develop the theory required to measure the accuracy of a classical approximation to a generic quantum dynamics at long-times. We then use this theory to construct a general algorithmic approach to finding a good classical approximation to this long-time dynamics. We apply this theory to two systems; an open quantum Ising model and an open quantum generalization of the glassy East model. These exhibit strongly intermittent emission dynamics characteristic of systems with competing dynamical phases. We show that for appropriate parameters these systems dynamics display pronounced metastability, i.e., the system relaxes first to long-lived metastable states, before eventual relaxation to the true stationary state. From the spectral properties of the quantum master operator we characterise the low-dimensional manifold of metastable states for these models. We show that the long time dynamics can be accurately approximated by a classical stochastic dynamics between the metastable phases, directly related to the intermittent dynamics observed in quantum trajectories. In particular we demonstrate that the effective dynamics between the metastable states of the quantum East model is simply an adjustment of the long-time dynamics of the classical East model. Magnon driven bidirectional domain wall motion in kagome antiferromagnets Salimath, Akshaykumar Antiferromagnet based insulator spintronics is emerging as a revolution in quest for next generation ultra-low power technologies. In metallic spintronics, the physical moment of electrons contributes towards joule heating. In contrast, in antiferromagnet insulators, spin information is carried by magnetic excitations called magnons resulting in ultra-low power information transfer and processing. Antiferromagnets are abundant in nature and are associated with much richer physics. Recently, a class of non-collinear antiferromagnets with kagome lattice structure have garnered significant interest due to their inherent magnetic frustration. The large exchange interaction in these materials result in resonance frequencies in several THz range. For their ultimate applications in spintronics as an active layer, it is important to understand efficient manipulation of magnetic textures in these materials. In this work, we gain numerical and theoretical insights into the magnon dispersion bands in 1D kagome antiferromagnets, followed by magnon driven dynamics of domain wall. We observe that the domain wall can be efficiently manipulated by y-polarized spin waves in these materials. Remarkably, in our simulations, we observe bidirectional domain wall motion by tuning the frequency of the spin waves around the non-dispersive magnon band. The simulations are performed in our homegrown atomistic LLG solver. We substantiate the results through analytics derived from the Lagrangian formalism. The scattering of the magnons near the flat band which result in domain wall motion away from the source can be explained through WKB approximation, while the reflectionless behavior away from the flat band which result in domain wall motion towards the source can be explained with modified Po ̈schl-Teller potential problem. Our results are particularly interesting for racetrack memory applcations owing to the fact that we can control the direction of the information bits just by varying the frequency of the source rather than the polarity. Quantum Monte Carlo simulation of generalized Kitaev models Sato, Toshihiro We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm for the generalized Kitaev model. This phase pinning strategy greatly reduces the severity of the negative sign problem and opens a window of temperatures relevant to experiments where exact quantum Monte Carlo simulations can be carried out. We demonstrate this by carrying out extensive simulations of thermodynamical and dynamical properties of the Kitaev-Heisenberg model. Our numerical data reveals finite temperature properties of ordered and Kitaev spin-liquid phases inherent to the Kitaev-Heisenberg model. Pyrochlore $S=\frac{1}{2}$ Heisenberg antiferromagnet at finite temperature Schäfer, Robin We use a combination of three computational methods to investigate the notoriously difficult frustrated three dimensional pyrochlore $S=1/2$ quantum antiferromagnet, at finite temperature, $T$: canonical typicality for a finite cluster of $2\times 2 \times 2$ unit cells (i.e. $32$ sites), a finite-$T$ matrix product state method on a larger cluster with $48$ sites, and the numerical linked cluster expansion (NLCE) using clusters up to $25$ lattice sites, including non-trivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T = 0.57 J$, which is stable across finite size clusters and converged in the series expansion. At $T\approx 0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47 k_B \ln2$, which is relatively large compared to other frustrated models at this temperature. We also observe a non-monotonic dependence on $T$ of the magnetization at low magnetic fields, reflecting the dominantly non-magnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material NaCaNi$_2$F$_7$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material Dy$_2$Ti$_2$O$_7$. From observations to complexity of quantum states via unsupervised learning Schmitt, Markus The vast complexity is a daunting property of generic quantum states that poses a significant challenge for theoretical treatment, especially for non-equilibrium setups. Therefore, it is vital to recognize states which are locally less complex and thus describable with (classical) effective theories. We use unsupervised learning with autoencoder neural networks to detect the local complexity of time-evolved states by determining the minimal number of parameters needed to reproduce local observations. The latter can be used as a probe of thermalization, to assign the local complexity of density matrices in open setups and for the reconstruction of underlying Hamiltonian operators. Our approach is an ideal diagnostics tool for data obtained from (noisy) quantum simulators because it requires only practically accessible local observations. Fine-Grained Tensor Network Methods Schmoll, Philipp We develop a strategy for tensor network algorithms that allows to deal very efficiently with lattices of high connectivity. The basic idea is to fine-grain the physical degrees of freedom, i.e., decompose them into more fundamental units which, after a suitable coarse-graining, provide the original ones. Thanks to this procedure, the original lattice with high connectivity is transformed by an isometry into a simpler structure, which is easier to simulate via usual tensor network methods. In particular this enables the use of standard schemes to contract infinite 2d tensor networks - such as Corner Transfer Matrix Renormalization schemes - which are more involved on complex lattice structures. We prove the validity of our approach by numerically computing the ground-state properties of the ferromagnetic spin-1 transverse-field Ising model on the 2d triangular and 3d stacked triangular lattice, as well as of the hard-core and soft-core Bose-Hubbard models on the triangular lattice. Our results are benchmarked against those obtained with other techniques, such as perturbative continuous unitary transformations and graph projected entangled pair states, showing excellent agreement and also improved performance in several regimes. Thermodynamics of the N=42 kagome lattice antiferromagnet and magnon crystallization in the kagome lattice antiferromagnet Schnack, Jürgen For the paradigmatic frustrated spin-half Heisenberg antiferromagnet on the kagome lattice we performed large-scale numerical investigations of thermodynamic functions by means of the finite-temperature Lanczos method for system sizes of up to N = 42. We present the dependence of magnetization as well as specific heat on temperature and external field and show in particular that a finite-size scaling of specific heat supports the appearance of a low-temperature shoulder below the major maximum. We also present numerical evidence for the crystallization of magnons below the saturation field at non-zero temperatures for the highly frustrated spin-half kagome Heisenberg antiferromagnet. This phenomenon can be traced back to the existence of independent localized magnons or equivalently flat-band multi-magnon states. We present a loop-gas description of these localized magnons and a phase diagram of this transition, thus providing information for which magnetic fields and temperatures magnon crystallization can be observed experimentally. The emergence of a finite-temperature continuous transition to a magnon-crystal is expected to be generic for spin models in dimension D>1 where flat-band multi-magnon ground states break translational symmetry. Non-equilibrium Floquet steady states of time-periodic driven Luttinger liquids Schneider, Imke Imke Schneider$^{1,3}$, Serena Fazzini$^1$, Piotr Chudzinski$^2$, Christoph Dauer$^1$, and Sebastian Eggert$^1$ 1) Physics Department and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany 2) School of Mathematics and Physics, Queens University Belfast, Belfast, UK 3) Institute of Physics, University of Augsburg, 86135 Augsburg, Germany Time-periodic driving facilitates a wealth of novel quantum states and quantum engineering. The interplay of Floquet states and strong interactions is particularly intriguing, which we study using time-periodic fields in a Luttinger liquid with periodically changing interactions. By developing a time-periodic operator algebra, we are able to obtain the complete set of explicit steady state solutions of the time-dependent Schrödinger equation in terms of a Floquet-Bogoliubov ansatz and known analytic functions. Complex valued Floquet eigenenergies occur when integer multiples of the driving frequency approximately match twice the dispersion energy, which correspond to resonant states. Including damping effects we show that this resonant behavior leads to a large number of density excitations. This setup is one or the rare cases, where a complete Floquet solution can be obtained exactly for a time-periodically driven many-body system. Identifying non-thermal excitations in experiment Schuckert, Alexander Quantum many-body systems are expected to reach local thermal equilibrium during their non-equilibrium dynamics. Recently, however, several systems have been discovered which do not thermalize due to the presence of non-thermal excitations, for example in Rydberg atom chains and confining gauge theories. We show how to identify such excitations directly in experiment without theory input by measuring two-time correlation functions. We present several protocols to measure two-time correlation functions in quantum gas microscopes, trapped ions, superconducting qubits and Rydberg atoms. As examples, we show that in trapped ion chains, confined excitations can be unambiguously identified. Moreover, in the constraint Rydberg atom chain, quantum many-body scars can be identified directly in experiment. Our protocols show how to probe non-thermal excitations in the "quantum advantage" regime inaccessible to numerical methods. Nematic quantum criticality in a Dirac semimetal Schwab, Jonas We consider Dirac fermions, as realized by a pi-flux tight binding model on a square lattice coupled to an Ising model in a transverse field. The coupling is chosen such that spontaneous ordering of the Ising spins triggers a meandering of the Dirac fermions and thereby a nematic deformation of the Fermi surface. We consider two models where the nematic transition reduces the initial $C_{4V}$ ($C_{2V}$) point-group symmetry to $C_{2V}$ ($2V$). Auxiliary field quantum Monte Carlo simulations reveal continuous transitions in both cases. In contrast to mass generation transitions, nematic terms explicitly break the Lorentz symmetry and do not open a single particle gap. They define new and unexplored quantum phase transitions in semi-metals. Microscopic theory of infinite layer nickelates Schwemmer, Tilman The observation of superconductivity in strontium-doped NdNiO$_2$ and the following discovery of a superconducting dome in this material and its sister compound: Sr-doped PrNiO$_2$ provide a new avenue to the study of superconductivity in layered oxide materials. We analyze superconducting instabilities of Sr doped NdNiO$_2$ from various vantage points. Starting with first-principles calculations, we adopt a minimal three-orbital model tight binding model, which captures the key low-energy degrees of freedom. We study superconductivity in both models using the random phase approximation. We then approach the problem from a strong coupling perspective, and study the dominant pairing instability in the associated $t$−$J$ model limit. In all instances, the dominant pairing tendency is found in the d$_{x^2−y^2}$ channel, analogous to the cuprate superconductors. Magnetostriction in the $J$-$K$-$\Gamma$-$J_3$ model on the honeycomb lattice Schwenke, Alexander Using the numerical linked cluster expansion (NLCE) [1], we investigate thermodynamic and magnetoelastic properties of the $J$-$K$-$\Gamma$-$J_3$ spin-$\frac{1}{2}$ model on the honeycomb lattice in the presence of a magnetic field $B$. Apart from the specific heat and the magnetic susceptibility, we focus in particular on the linear magnetostriction coefficient $\lambda(B,T)$. As a prime result and based on expansions up to order $\sim O(9)$, we find clear indications for a field-induced transition in $\lambda(B,T)$. Employing exchange parameters as proposed for $\alpha\text{-RuCl}_3$, our results are very similar to recently observed experimental data [2] on this proximate quantum spin-liquid candidate material. [1]: M. Rigol et al., Phys. Rev. Lett. 97, 187202 (2006). [2]: S. Gass et al., Phys. Rev. B 101, 245158 (2020). Electric, thermal and thermoelectric transport at intermediate temperatures Schwiete, Georg Electric, thermal, and thermoelectric transport in correlated electron systems probe different aspects of the many-body dynamics, and thus provide complementary information. These are well studied in the low- and high-temperature limits, while the experimentally important intermediate regime, in which elastic and inelastic scattering are both important, is less understood. To fill this gap, we provide comprehensive solutions of kinetic equations for single band metallic systems and compensated metals in the presence of an electric field and a temperature gradient. We explore the role of the momentum dependence of the impurity scattering rate in the presence of inelastic scattering processes. Specifically, we show that inelastic processes only mildly affect the electric conductivity, but can generate a non-monotonic dependence of the Seebeck coefficient on temperature and even a change of sign. We explore the magnetic field dependence of the Seebeck coefficient and also discuss the Nernst effect. Finally, we address the Lorenz ratio for an impure compensated metal motivated by recent experiments on WP2, which observed a pronounced minimum of the Lorenz ratio at intermediate temperatures. Long-Range Hybrid Random Unitary Circuit using Clifford gates Sharma, Shraddha We explore the out-of-equilibrium phases of a special class of system consisting of alternate layers of measurements and unitary matrices, referred to as 'Hybrid Random Quantum Circuits'. In these systems the competition between entangling nature of unitary evolution and disentangling nature of measurements leads to a volume-law to area-law transition in the scaling of entanglement entropy at a fixed non-zero value of rate of measurement. It has been shown that these phase transitions in special analytical limit belongs to the Universality class of specific statistical model's phase transitions. We use 2-site and 2-qubit Clifford unitary gates and projection in Z-direction of the spin as measurements. Starting from a spin chain initially in product state, we explore the effect of range of the unitary circuit in space and in time on the values of the critical point and critical exponents to probe the Universality class of phase transition in the scaling of von-Neumann entnaglement entropy. Pentanuclear Spirocyclic Ni4Ln Derivatives: Field Induced Slow Magnetic Relaxation in the Dysprosium and Erbium Analogues Shukla, Pooja Five pentanuclear heterometallic isostructural complexes, [Ni4Ln(L)2(LH)2(CH3CN)3Cl]·xH2O·yCH3OH {Ln = YIII (1), GdIII (2), TbIII (3), DyIII (4) and ErIII (5)} [for 1 and 2, x = 2, y = 1; for 3, x = 6, y = 2; for 4, x = 5, y = 1; for 5, x = 2, y = 2] were prepared by the reaction of (E)‐2‐(hydroxymethyl)‐6‐{[(2‐hydroxyphenyl)imino]methyl}‐4‐methylphenol (LH3) with LnCl3·6H2O and Ni(OAc)2·4H2O in the presence of tetrabutylammonium hydroxide (TBA‐OH) base. The structural characterization reveals that compounds 1–5 contain a spirocyclic pentanuclear core [Ni4Ln(µ3‐O)4(µ2‐O)4]3+ where two triangular motifs [Ni2Ln(µ3‐O)2(µ2‐O)2]3+ are fused together through a common vertex of the LnIII ion. The central LnIII ion forms an eight‐coordinated distorted triangular dodecahedron geometry, while the nickel(II) ions form a distorted octahedron geometry. Comprehensive dc magnetic studies reveal that antiferromagnetic exchange interaction exists between the NiII centres. The ac susceptibility measurement revels that dysprosium and erbium analogue shows field induced slow magnetic relaxation with an anisotropic barrier (Ueff) of 25.12 cm–1 and 22.13 cm–1 respectively. Fermi polaron dynamics in Open quantum systems Sighinolfi, Matteo In ultracold atomic gases a widely studied topic is the impurity problem, where some impurities (e.g. atoms of different species or with different spin) are immersed in a medium of so called majority atoms. When the medium is made of fermionic atoms an impurity forms a quesiparticle called Fermi polaron that results from the dressing of the bare impurities with the excitation of the medium. Static properties of the Fermi polaron are well known and widely studied [1] while the dynamics is less understood. We investigate two different systems of Fermi polaron: the first is a collection of impurities in a fermionic medium, while the second system is composed by a single polaron coupled to a different atomic level via a Rabi frequency. The first system is described with a Keldysh formalism in analogy to what has been done for a quark-gluon plasma in Ref. [2] and the formation of bound state of impurities due to an induced polaron-polaron interaction is discussed. The second system is modelled on the experiment described in Ref [3], where the Rabi coupling is used to investigate the dynamics of the polaron population. Again we use Keldysh formalism, while recently a theoretically study with variational method has been made [4]. With our method we derive quantum Boltzmann equations for the populations in the relevant atomic levels in a similar way to what has been done for quantum Zeno effect in Fermi polarons in dissipative systems [5]. Main properties of the Boltzmann equations and their relation with polaron static properties are shown. Finally, theoretical predictions are compared directly to experimental results where they are also discussed. Bibliography: [1] Massignan, P., Zaccanti, M., & Bruun, G. M. (2014). Polarons, dressed molecules and itinerant ferromagnetism in ultracold Fermi gases. Reports on Progress in Physics, 77(3), 034401. [2] Blaizot, J. P., De Boni, D., Faccioli, P., & Garberoglio, G. (2016). Heavy quark bound states in a quark–gluon plasma: Dissociation and recombination. Nuclear Physics A, 946, 49-88. [3] Scazza, F., Valtolina, G., Massignan, P., Recati, A., Amico, A., Burchianti, A., ... & Roati, G. (2017). Repulsive Fermi polarons in a resonant mixture of ultracold Li 6 atoms. Physical review letters, 118(8), 083602. [4] Adlong, H. S., Liu, W. E., Scazza, F., Zaccanti, M., Oppong, N. D., Fölling, S., ... & Levinsen, J. (2020). Quasiparticle lifetime of the repulsive Fermi polaron. Physical Review Letters, 125(13), 133401. [5] Wasak, T., Schmidt, R., & Piazza, F. (2019). Quantum-Zeno Fermi-Polaron. arXiv preprint arXiv:1912.06618. Fulde-Ferrel-Larkin-Ovchinnikov phase and exponential ground state degeneracy in one-dimensional Fermi gas with attractive interactions Singh Roy, Monalisa We examine the properties of a one-dimensional (1D) Fermi gas with attractive intrinsic (Hubbard) interactions in the presence of spin-orbit coupling and Zeeman fields. Such a system can be realized in the setting of ultracold atoms confined in a 1D optical lattice, and has been proposed to host exotic topological phases and edge modes. In absence of any external fields, this system shows a trivial Bardeen–Cooper–Schrieffer (BCS) phase. Introduction of Zeeman field takes the system to a Fulde-Ferrel-Larkin-Ovchinnikov phase (FFLO), where the quasi-long range superconducting order co-exists with magnetic order in the system, as indicated by its pair momentum distribution. Next, we explore the effect of spin-orbit coupling in this system. We find that the addition of a smooth parabolic potential yields a phase with exponentially decaying pair binding and excitation energy gaps, which is expected to be associated with topological edge modes in the system. However, we show that this ground state degeneracy is susceptible to local impurities, and argue that the exponential splitting in the clean system is similar to a phase with only conventional order. References: [1] A. E. Feiguin, F. Heidrich-Meisner, G. Orso, and W. Zwerger, Lect. Not. Phys. 836, 503 (2011). [2] Y.-a. Liao, A. Rittner, T. Paprotta, et al., Nature 467, 567 (2010). [3] J. Ruhman, E. Berg, and E. Altman, Phys. Rev. Lett. 114, 100401 (2015). [4] M. Singh Roy, M. Kumar, Jay D. Sau, and S. Tewari, Phys. Rev. B 102, 125135 (2020). Incommensurate time crystalline dynamics in an atom-cavity system Skulte, Jim Periodically driven atoms in a high finesse optical cavity enjoy a very rich phase diagram. By off resonant driving the equilibrium properties of the system can be renormalised in a controlled fashion, while resonant driving allows for new non- equilibrium phases such as time crystalline phases and dynamical density wave orders as recently reported. In this talk, I will discuss the emergence of an incommensurate time crystal by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long lived, and map out the dynamical phase diagram. I will further present preliminary experimental results that confirm our theoretical predictions. Fragile topological flat bands through adatom superlattice engineering on graphene Skurativska, Anastasiia Magic-angle twisted bilayer graphene has received a lot of attention due to its flat bands with potentially non-trivial topology that lead to intricate correlated phases. However, control over the fabrication and thus system parameters of such devices is limited. We propose a single graphene sheet with adatoms periodically placed on top as an alternative system that realizes flat bands. Performing first principle calculations, we obtain realistic spectra for feasible transition-metal adatoms. Further group-theoretical analysis reveals the fragile nature of topology in some of the flat bands. We study the bulk-boundary correspondence associated with the fragile topology of the flat bands by building a minimal tight-binding and numerically examine the corner-localized in-gap states, which are a consequence of the filling anomaly resulting from the nontrivial topology. The high control over system parameters makes this system particularly interesting for experimental investigations. Weak breaking of translational symmetry in Z2 Topological ordered states Sodemann Villadiego, Inti We study Z2 topologically ordered states enriched by translational symmetry by employing a recently developed 2D bosonization approach that implements an exact Z2 charge-flux attachement in the lattice. This allows us to develop a theory of ‘weak symmetry breaking' of translations, which is a remarkable phenomenon beyond traditional symmetry fractionalization. In a weakly symmetry broken state, the ground state remains fully translational invariant, but the symmetry is 'broken' by its anyon quasiparticles. This phenomenon is accompanied by a series of amusing properties such as ground state degeneracy that depends on system size and the emergence of edge gapless Majorana modes. We construct a plethora of exactly solvable models in periodic lattices and also in cylinders and open lattices, that provide an exact illustrations of these anomalies and of the dispersive Majorana gapless boundary modes. Quantum magnetism and topological superconductivity in Yu-Shiba-Rusinov chains Steiner, Jacob Recent experiments probe subgap excitations in dilute chains of magnetic adatoms on superconducting substrates. In these chains, direct overlap of the adatom d orbitals is negligible, while their Yu-Shiba-Rusinov (YSR) states are still hybridizing. Such YSR chains have also been proposed as a setting for topological superconductivity in the framework of models which assume a frozen texture of classical spins. Motivated by these experiments, we consider quantum spin chains on superconducting substrates and explore their ground state as well as their excitation spectra as relevant for STM experiments. We find that the physics is considerably richer than that of their classical relatives. Scattering of mesons in quantum simulators Surace, Federica Maria Simulating real-time evolution in theories of fundamental interactions represents one of the central challenges in contemporary theoretical physics. Cold-atom platforms stand as promising candidates to realize quantum simulations of non-perturbative phenomena in gauge theories, such as vacuum decay and hadron collisions. In this talk, I will demonstrate that present-day quantum simulators can imitate linear particle accelerators, giving access to S-matrix measurements of elastic and inelastic meson collisions in low dimensions. Considering for definiteness a $(1+1)$-dimensional $\mathbb{Z}_2$-lattice gauge theory realizable with Rydberg-atom arrays, I will discuss protocols to observe and measure selected meson-meson scattering processes. I will also present a benchmark theoretical study of scattering amplitudes and numerical simulations of realistic wavepacket collisions. Many-body localization and quantum many-body systems with artificial gauge fields Suthar, Kuldeep The phenomenon of many-body localization (MBL) is attracting significant theoretical and experimental interests over the past few years. The signatures of MBL have been observed in recent cold-atom experiments of optical lattices. The recent experimental advances of artificial gauge fields motivate us to explore the static and dynamical properties of MBL with magnetic flux. We show that the breaking of time-reversal invariance leads to delocalization of spin sector, and the use of spin-dependent uncorrelated disorder potential recovers the complete localization. In the later part, we shall discuss the quantum phase transitions of dipolar bosons and disordered Bose-Hubbard model with synthetic flux, the finite-temperature phase diagrams, the role of anisotropy, and the density-driven staggered superfluidity of dipolar interacting bosons. Extended Hubbard model in (un)doped monolayer and bilayer graphene: Selection rules and organizing principle among competing orders Szabó, András In this talk I present the effects of generic short-range electronic interactions in monolayer and Bernal bilayer graphene. Typically, at zero doping insulating phases (such as charge-density-wave, antiferromagnet, quantum anomalous and spin Hall insulators) prevail at the lowest temperature, while gapless nematic or smectic liquids stabilize at higher temperatures. On the other hand, at finite doping the lowest temperature ordered phase is occupied by a superconductor. Besides anchoring such an organizing principle among the candidate ordered phases, I also establish a selection rule between them and the interaction channel responsible for the breakdown of the original Fermi liquid. In addition, I demonstrate the role of the normal state band structure in selecting the pattern of symmetry breaking from a soup of preselected incipient competing orders. As a direct consequence of the selection rule, while an antiferromagnetic phase develops in undoped monolayer and bilayer graphene, the linear (biquadratic) band dispersion favors condensation of a spin-singlet nematic (translational symmetry breaking Kekul\'e) superconductor in doped monolayer (bilayer) graphene, when the on site Hubbard repulsion dominates in these systems. On the other hand, nearest-neighbor (next-nearest-neighbor) repulsion accommodates charge-density-wave (quantum spin Hall insulator) and $s+if$ ($s$-wave) pairing at zero and finite chemical doping in both systems, respectively. Theory of magnetocaloric effect in V12 molecular magnet: role of quantum level crossings Szalowski, Karol Molecular nanomagnets offer a plethora of interesting properties, emerging from the interplay of the geometry and magnetic interactions in small quantum spin clusters. Among them, polyoxovanadates constitute a highly interesting class of cluster nanomagnets. The paper reports a computational study of low-temperature thermodynamic properties of V12 polyoxovanadate molecular magnet, focused on the description of the magnetocaloric effect [1]. The low-temperature magnetic properties of V12 are modeled using an anisotropic quantum Heisenberg Hamiltonian for an ensemble of non-interacting square tetramers. The exchange integrals between the spins S = 1/2 are taken from the experiment [2]. The exact thermodynamic description of the utilized model is constructed using analytic and numerical approach within the canonical ensemble formalism. The quantities of interest are: magnetic entropy and specific heat, isothermal entropy change, refrigerant capacity, adiabatic temperature change as well as magnetic Grüneisen ratio. The energy spectrum of the system of interest exhibits two quantum level crossings between the non-degenerate ground states with different total spin as the external magnetic field is increased. The critical fields for both crossings belong to experimentally accessible range. The importance of quantum level crossing for the low-temperature thermodynamics is demonstrated and emphasized throughout the study. In particular, the residual entropy related to the quantum level crossings is crucial for the magnetocaloric response close to the ground state. For V12 molecular magnet, a robust range of inverse magnetocaloric effect is predicted for cryogenic temperature range and for a significant span of magnetic fields. In particular, the quadratic dependence of the entropy change on the magnetic field amplitude is found for the range of inverse magnetocaloric effect. For the remaining range of temperatures and magnetic fields, a direct magnetocaloric effect occurs. A divergent behaviour of magnetic Grüneisen ratio is predicted at the quantum level crossing points. [1] K. Szałowski, Materials 13, 4399 (2020); doi:10.3390/ma13194399. [2] R. Basler et al., Inorg. Chem. 41, 5675 (2002); doi:10.1021/ic0202099. Large scale QMC calculations of the Fermi Velocity renormalization in graphene: bridging the gap between numerics and experiment Ulybyshev, Maksim We report on the results of recent Quantum Monte Carlo (QMC) simulations of the effects of electron-electron interactions in graphene. With the help of the Hybrid Monte Carlo algorithm we could achieve unprecedentedly large sample size (up to $102 \times 102$ lattice cells) in fully non-perturbative QMC calculations with strong long-range Coulomb interaction. Thus we could get deeply enough in the infrared regime to directly observe the logarithmic behavior of the Fermi Velocity and to compare its values with both experiment and low-energy effective field theory, which is essentially 2+1D Quantum Electrodynamics. Additional comparison was done with the results of the lattice perturbation theory in one loop and Random Phase approximation. These comparisons allowed us to judge on the numerical accuracy of certain interacting tight-binding Hamiltonian and strongly correlated 2+1D Quantum Electrodynamics in the description of the experimental data for free standing graphene. Entanglement entropy scaling transition under competing monitoring protocols Van Regemortel, Mathias Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the non-equilibrium steady state, we investigate the build-up of entanglement in the quantum trajectories. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics. Quantum dynamics with variational classical networks Verdel Aranda, Roberto Dynamics in correlated quantum matter is a hard problem, as its exact solution generally involves a computational effort that grows exponentially with the number of constituents. While remarkable progress has been witnessed in recent years for one-dimensional systems, much less has been achieved for interacting quantum models in higher dimensions, since they incorporate an additional layer of complexity. In this work, we employ a variational method that allows for an efficient and controlled computation of the dynamics of quantum many-body systems in one and higher dimensions. The approach presented here introduces a variational class of wave functions based on complex networks of classical spins akin to artificial neural networks, which can be constructed in a controlled fashion. We illustrate the performance of our method by studying quantum quenches in one- and two-dimensional models. The present work not only supplies a framework to address purely theoretical questions but could also be used to provide a theoretical description of experiments in quantum simulators, which have recently seen an increased effort targeting two-dimensional geometries. Importantly, our method can be applied to any quantum many-body system with a well-defined classical limit. This work is based on the paper: R. Verdel, M. Schmitt, Y.P. Huang, P. Karpov, M. Heyl, arXiv:2007.16084 (2020). Beyond Topological Quantum Chemistry Vergniory, Maia Topological quantum chemistry (TQC) framework has provided a complete description of the universal properties of all possible atomic band insulators in all space groups considering the crystalline unitary symmetries. It links the chemical and symmetry structure of a given material with its topological properties. While this formalism filled the gap between the mathematical classification and the practical diagnosis of topological materials, an obvious limitation is that it only applies to weakly interacting systems. It is an open question to which extent this formalism can be generalized to correlated systems that can exhibit symmetry protected topological Mott insulators. In this talk I will first introduce TQC and its application and then I will address this question by combining cluster perturbation theory and topological Hamiltonians within TQC. This simple formalism will be applied to calculate to the phase diagram of a representative model. The results are compared to numerically exact calculations from density matrix renormalization group and variational Monte Carlo simulations together with many-body topological invariants. Transport through a dissipative superconducting contact Visuri, Anne-Maria In superconducting contacts, the coherent tunneling of a single quasiparticle together with Cooper pairs leads to a sub-gap current structure [1]. This phenomenon, also called multiple Andreev reflections, is well known in condensed-matter superconducting junctions. Current-voltage characteristics consistent with multiple Andreev reflections were also measured in a cold-atom setup where two superfluids are coupled by a quantum point contact [2]. Futher cold-atom experiments have probed transport in the presence of local particle losses [3]. Motivated by such experiments, we investigate theoretically, using the Keldysh formalism, whether a local particle loss at the contact interferes with the multiple Andreev reflection process, and what kind of current-voltage characteristics it leads to. [1] Blonder, Klapwijk, Tinkham, Phys. Rev. B 25, 4515 (1982) [2] Husmann et al., SCience 350, 1498 (2015) [3] Corman et al., Phys. Rev. A 100, 053605 (2019) Two-triplon excitations of the Kitaev-Heisenberg-Bilayer Wagner, Erik We study the spectrum of a bilayer of Kitaev magnets on the honeycomb lattice coupled by Heisenberg exchange in its quantum-dimer phase for strong interlayer coupling. Using the perturbative Continuous Unitary Transformation (pCUT) we perform series expansion starting from the fully dimerized limit, to evaluate the elementary excitations, reaching up to and focusing on the two-triplon sector. In stark contrast to conventional bilayer quantum magnets, and because of the broken $SU(2)$-invariance, as well as the intralayer directional compass-exchange, the bilayer Kitaev magnet is shown to exhibit a rich structure of two-triplon scattering-state continua, as well as several collective two-triplon (anti)bound states. Direct physical pictures for the occurrence of the latter are provided and the (anti)bound states are studied versus the stacking type, the spin components, and the exchange parameters. In addition to the two-triplon spectrum, we investigate a corresponding experimental probe and evaluate the magnetic Raman-scattering intensity. We find a very strong sensitivity to the two-triplon interactions and the scattering geometry, however a signal from the (anti)bound states only in very close proximity to the continuum. Perturbative instability of non-ergodic phases in non-Abelian quantum chains Ware, Brayden An important challenge in the field of many-body quantum dynamics is to identify non-ergodic states of matter beyond many-body localization (MBL). Strongly disordered spin chains with non-Abelian symmetry and chains of non-Abelian anyons are natural candidates, as they are incompatible with standard MBL. In such chains, real space renormalization group methods predict a partially localized, non-ergodic regime known as a quantum critical glass (a critical variant of MBL). We argue that such tentative non-ergodic states are perturbatively unstable using an analytic computation of the scaling of off-diagonal matrix elements and accessible level spacing of local perturbations. Our results indicate that strongly disordered chains with non-Abelian symmetry display either spontaneous symmetry breaking or ergodic thermal behavior at long times; we identify the relevant length and time scales for thermalization. Fermi-polaron lasing in monolayer charge-tunable semiconductors Wasak, Tomasz We study the relaxation dynamics of driven, charge-tuneable monolayer semiconductors in the weak coupling to cavity photons. The itinerant electrons dress the optically pumped excitons to form two Fermi-polaron branches, which are termed as attractive and repulsive polarons. After excitation, a repulsive polaron quickly decays to an attractive polaron at higher momentum via the formation of trions (electron-exciton molecules). The electrons subsequently mediate a slower momentum-relaxation of the attractive polaron, which accumulates population at the edge of the light-cone region around zero momentum where the radiative recombination is the dominant loss channel. Due to the bosonic nature of exciton polarons, around the point where the decay into the attractive polaron overcomes the radiative and non-radiative loss, stimulated processes lead to a transition toward a lasing regime. The latter is characterized by a superlinear increase of light emission as well as extended spatiotemporal coherence. Many-body polaronic effects reduce the emission linewidth below the bare exciton linewidth set by nonradiative loss, as the excitation is partially stored in the electron cloud via the virtual formation of trions. Improving topological superconductivity in two- and three-dimensional Josephson junctions Wastiaux, Aidan As opposed to the numerous theoretical developments in the field of topological heterostructures hosting robust quasiparticles, difficulties are piling up for experimentalists on their way to building realistic and tunable setups with usable topological states. We address this widespread issue in a specific platform involving a planar Josephson junction made of semiconductor with strong spin-orbit coupling [ref] by proposing easy-to-reach regimes of parameters with enhanced stability of the Majorana end states. Moreover, the extension of those findings to a three-dimensional model provides henceforth a new flexible platform for realizing chiral Majorana edge states. Possible setups using Van der Waals heterostructures are suggested. Valence-bond order in two-dimensional antiferromagnets coupled to quantum phonons Weber, Manuel The search for valence-bond-solid (VBS) phases in 2D antiferromagnets (AFMs) has attracted a lot of interest due to the proposal of a continuous deconfined quantum phase transition that is beyond the Landau-Ginzburg-Wilson paradigm. While VBS order often appears in frustrated spin systems, large-scale quantum Monte Carlo studies have mainly concentrated on a class of designer Hamiltonians called J-Q models that can be simulated without a sign problem. It is of current interest to find VBS order also in more realistic models. In 1D, a coupling to phonons naturally leads to a dimerization via the spin-Peierls instability, but it is still an open question whether this is also the case in 2D. Here, we use a recently developed quantum Monte Carlo method for retarded interactions to show that a VBS state with Kekule pattern can arise in a spin-Peierls model on the honeycomb lattice. While the AFM—VBS transition is clearly first order for low phonon frequencies, it is tuned towards weakly first-order with increasing quantum lattice fluctuations. Our study reveals that retardation effects have a significant impact on the AFM—VBS transition. Moreover, we discuss our results in relation to frustrated spin models, electron-phonon models, and Dirac systems. The Mott Transition as a Topological Phase Transition Wong, Patrick We show that the Mott metal-insulator transition in the standard one-band Hubbard model can be understood as a topological phase transition. Our approach is inspired by the observation that the mid-gap pole in the self-energy of a Mott insulator resembles the spectral pole of the localized surface state in a topological insulator. We use numerical renormalization group--dynamical mean-field theory to solve the infinite-dimensional Hubbard model and represent the resulting local self-energy in terms of the boundary Green's function of an auxiliary tight-binding chain without interactions. The auxiliary system is of generalized Su-Schrieffer-Heeger model type; the Mott transition corresponds to a dissociation of domain walls. Revisiting the problem of the single hole in an antiferromagnet Wrzosek, Piotr Propagation of the single hole introduced into the antiferromagnetic ground state is one of the most studied problems in "cuprate physics", for it can be solved in a relatively controlled manner. Recently a renewed interest into this topic has been triggered by the possibility of simulating hole-doped antiferromagnets in the cold atom experiments [1]. In this contribution I would like to discuss some of our most recent studies on the propagation of the single hole in the antiferromagnet using the magnon language with a special attention paid to the interaction between the magnons [2-3]. To this end, I will introduce an intuitive picture which explains why the electron's spin and charge degrees of freedom can separate in a one-dimensional lattice, though a similar situation cannot occur in two dimensions [2]. Next, I will show that the string potential, which is believed to be felt by the hole moving in a two-dimensional Ising antiferromagnet, is significantly destroyed by the magnon-magnon interactions [3]. [1] C. S. Chiu et al., Science 365, 251 (2019); J. Koepsell et al., Nature 572, 358 (2019). [2] K. Bieniasz et al., SciPost Phys. 7, 066 (2019). [3] P. Wrzosek et al., Phys. Rev. B 103, 035113 (2021). Exceptional Spin Liquids Yang, Kang We establish the appearance of a qualitatively new type of spin liquid with emergent exceptional band-touching behaviours when coupling to the environment. We consider an open system of the Kitaev model generically coupled to an external environment. In extended parameter regimes, the usual band crossings of the emergent Majorana fermions from the original model are split into exceptional band crossings. In glaring contrast to the original gapless phase of the honeycomb model which requires time-reversal symmetry, this new phase is stable against all perturbations. The system also displays a large sensitivity to boundary conditions resulting from the non-Hermitian skin effect with telltale experimental consequences. Our results point to the emergence of new classes of spin liquids in open systems which might be generically realized due to unavoidable couplings with the environment. Hall effect in Sr2RuO4 under <100> uniaxial pressure Yang, Po-Ya The Hall coefficient of Sr2RuO4 goes through two sign changes, at $\sim$120 K and 30 K. It has been proposed that this temperature dependence is due to strong orbital differentiation of the inelastic scattering rates, which is a predicted consequence of strong Hund’s coupling. Here, in order to probe this hypothesis, we report the Hall resistance of Sr2RuO4 under tunable uniaxial stress. The gamma Fermi surface sheet of Sr2RuO4 is driven through a Lifshitz transition by uniaxial pressure along a <100> direction. At a temperature where resistivity is dominated by electron-electron scattering, the Hall coefficient becomes less electron-like while approaching the van Hove singularity, which supports the Hund’s coupling scenario in the three-band system. At very low temperature, however, despite the change in topology of the Fermi surface structure both the Hall resistivity and longitudinal resistivity are essentially unchanged across the Lifshitz transition, which is not expected in any simple model of transport in Sr2RuO4. One-dimensional ultracold bosons in shallow quasiperiodic systems: Bose glass phase and fractal Mott lobes Yao, Hepeng The emergence of a compressible insulator phase, known as the Bose glass, is characteristic of the interplay of interactions and disorder in correlated Bose fluids. While widely studied in tight-binding models, its observation remains elusive owing to stringent temperature effects. In this talk, I will present our results about the study of ultracold bosons in shallow 1D quasiperiodic potentials. First, I will start with the non-interacting case. Thanks to the exact diagonalization techniques, we determine the critical properties and the Hausdorff fractal dimension of the system [1]. Then, we move to the study of the interacting case based on the results of the ideal bosons. With the quantum Monte Carlo calculations, we compute the phase diagram of Lieb-Liniger bosons in shallow quasiperiodic potentials [2]. A Bose glass, surrounded by superfluid and Mott phases, is found. At finite temperature, we show that the melting of the Mott lobes is characteristic of a fractal structure and find that the Bose glass is robust against thermal fluctuations up to temperatures accessible in experiments. [1] H. Yao, H. Khoudli, L. Bresque, L. Sanchez-Palencia. Phys. Rev. Lett. 123, 070405 (2019). [2] H. Yao, T. Giamarchi, L. Sanchez-Palencia,Phys. Rev. Lett. 125(6), 060401 (2020). Residual bulk viscosity of a disordered 2D electron gas Zakharov, Vladimir The nonzero bulk viscosity signals breaking of the scale invariance. We demonstrate that a disorder in two-dimensional noninteracting electron gas in a perpendicular magnetic field results in the nonzero disorder-averaged bulk viscosity. We derive analytic expression for the bulk viscosity within the self-consistent Born approximation. This residual bulk viscosity provides the lower bound for the bulk viscosity of 2D interacting electrons at low enough temperatures. https://arxiv.org/abs/2102.10533 Orthogonal quantum many-body scars Zhao, Hongzheng Quantum many-body scars have been put forward as counterexamples to the Eigenstate Thermalization Hypothesis. These atypical states are observed in a range of correlated models as long-lived oscillations of local observables in quench experiments starting from selected initial states. The long-time memory is a manifestation of quantum non-ergodicity generally linked to a sub-extensive generation of entanglement entropy, the latter of which is widely used as a diagnostic for identifying quantum many-body scars numerically as low entanglement outliers. Here we show that, by adding kinetic constraints to a fractionalized orthogonal metal, we can construct a minimal model with orthogonal quantum many-body scars leading to persistent oscillations with infinite lifetime coexisting with rapid volume-law entanglement generation. Our example provides new insights into the link between quantum ergodicity and many-body entanglement while opening new avenues for exotic non-equilibrium dynamics in strongly correlated multi-component quantum systems. Subdiffusive dynamics and critical quantum correlations in a disorder-free localized Kitaev honeycomb model out of equilibrium Zhu, Guo-Yi Disorder-free localization has recently emerged as a mechanism for ergodicity breaking in homogeneous lattice gauge theories. In this work we show that this mechanism can lead to unconventional states of quantum matter as the absence of thermalization lifts constraints imposed by equilibrium statistical physics. We study a Kitaev honeycomb model in a skew magnetic field subject to a quantum quench from a fully polarized initial product state and observe nonergodic dynamics as a consequence of disorder-free localization. We find that the system exhibits a subballistic power-law entanglement growth and quantum correlation spreading, which is otherwise typically associated with thermalizing systems. In the asymptotic steady state the Kitaev model develops volume-law entanglement and power-law decaying dimer quantum correlations even at a finite energy density. Our work sheds light onto the potential for disorder-free localized lattice gauge theories to realize quantum states in two dimensions with properties beyond what is possible in an equilibrium context.
f260cece3fb318f0	Why Potassium bromide is used in infrared spectroscopy? Similarly, you may ask, why do we use KBr in IR spectroscopy? This method exploits the property that alkali halides become plastic when subjected to pressure and form a sheet that is transparent in the infrared region. Potassium bromide (KBr) is the commonest alkali halide used in the pellets. Degassing is performed to eliminate air and moisture from the KBr powder. What does bromine do to the body? Bromine is corrosive to human tissue in a liquid state and its vapors irritate eyes and throat. Bromine vapors are very toxic with inhalation. Humans can absorb organic bromines through the skin, with food and during breathing. Organic bromines are widely used as sprays to kill insects and other unwanted pests. Is KBr an ionic compound? The bond between K and Br in KBr is considered ionic. An electron is essentially transferred from K to Br, resulting in the formation of the ions K+ and Br-, which are then held together by electrostatic attraction. Chemical bonds can be purely ionic, purely covalent or have characteristics of both. What does a nujol mull do? To obtain an IR spectrum of a solid, a sample is combined with Nujol in a mortar and pestle or some other device to make a mull (a very thick suspension), and is usually sandwiched between potassium- or sodium chloride plates before being placed in the spectrometer. What is in the fingerprint region? What is the functional group region? The functional group region runs from 4000 cm-1to 1450 cm-1 , and the fingerprint region from 1450 cm-1to 500 cm-1 . A typical IR spectrum looks something like the one below. The functional group region contains relatively few peaks. These are typically associated with the stretching vibrations of functional groups. What is the Fermi resonance? A Fermi resonance is the shifting of the energies and intensities of absorption bands in an infrared or Raman spectrum. It is a consequence of quantum mechanical mixing. The phenomenon was explained by the Italian physicist Enrico Fermi. What is an overtone in IR? In vibrational spectroscopy, an overtone band is the spectral band that occurs in a vibrational spectrum of a molecule when the molecule makes a transition from the ground state (v=0) to the second excited state (v=2), where v is the vibrational quantum number that one gets after solving the Schrödinger equation for What is a combination band? The two bands are usually a fundamental vibration and either an overtone or combination band. The wavefunctions for the two resonant vibrations mix according to the harmonic oscillator approximation, and the result is a shift in frequency and a change in intensity in the spectrum. What are the hot band? In molecular vibrational spectroscopy, a hot band is a band centred on a hot transition, which is a transition between two excited vibrational states, i.e. neither is the overall ground state. What are aromatic overtones? Aromatic overtones: In infrared spectroscopy, a series of small peaks (usually three or four) typically found in the ~2000 cm-1 to ~1700 cm-1 range. Caused by overtones (harmonics) of the benzene ring vibrational modes having stretching frequencies in the infrared spectrum’s fingerprint region. What is hot band steel? hot band (hot-rolled steel) A coil of steel rolled on a hot-strip mill (hot-rolled steel). It can be sold in this form to customers or further processed into other finished products. What does the acronym for hot stand for? HOTAcronymDefinitionHOTHolographic One-TubeHOTHot Oil TracedHOTHigher Order Term (mathematics)HOTHerald of Truth (ministry) What does Hott stand for? What does HOTT stand for?Rank Abbr.MeaningHOTTHordes of the Things (gaming rules)HOTTHealth Occupations for Today and Tomorrow (various states)HOTTHands-On Turret Trainer (US Army)HOTTHuman Ovarian Thecal-Like Tumor (endocrinology) What is HR steel? It is also used to produce sheet metal. Cold Rolled SteelA rolling process at temperatures that are close to normal room temperature are used to create cold rolled steel. This increases the strength of the finished product through the use of strain hardening by as much as 20 percent. What does Hot Rolling do to steel? Cold rolling occurs with the metal below its recrystallization temperature (usually at room temperature), which increases the strength via strain hardening up to 20%. It also improves the surface finish and holds tighter tolerances. Is cold or hot rolled steel cheaper? What is a 1018 steel? What is 4140 steel used for? 4140 cold finished annealed is a chromium-molybdenun alloy steel that can be oil hardened to relatively high hardenability. The chromium content provides good hardness penetration, and the molybdenum imparts uniformity of hardness and high strength. What is 1045 steel used for? 1045 is a medium tensile low hardenability carbon steel generally supplied in the black hot rolled or occasionally in the normalised condition, with a typical tensile strength range 570 – 700 Mpa and Brinell hardness range 170 – 210 in either condition.Characterised by fairly good strength and impact properties, plus What does bromine do to the body? Is bromate harmful to humans? But, the problem is that this additive is also linked to cancer. In 1999, the International Agency on Research for Cancer declared that potassium bromate was a possible human carcinogen, which means that it possibly causes cancer. To no one surprise, the food industry says that potassium bromate is perfectly safe. What does bromine do to you? Originally posted 2022-03-31 03:07:37. Leave a Comment
d0ee658cba1120df	Dirac sea From Example Problems Jump to navigation Jump to search The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. It was invented by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932. The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, a special case of the Schrödinger equation that is consistent with special relativity, that Dirac had formulated in 1928. Although the equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for each quantum state possessing a positive energy E, there is a corresponding state with energy -E. This is not a big difficulty when we are looking at an isolated electron, because its energy is conserved and we can simply choose not to introduce any negative-energy electrons. However, it becomes serious when we start to think about how to include the effects of the electromagnetic field, because a positive-energy electron would be able to shed energy by continuously emitting photons, a process that could continue without limit as the electron descends into lower and lower energy states. Real electrons clearly do not behave in this way. Dirac's solution to this was to turn to the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle, which means that no two electrons can share a single energy state. Dirac hypothesized that what we think of as the "vacuum" is actually the state in which all the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy. Dirac also pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively-charged particle. Initially, Dirac identified this hole as a proton. However, Robert Oppenheimer pointed out that an electron and its hole would be able to annihilate each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atoms would not exist. Hermann Weyl also noted that a hole should act as though it has the same mass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932 when the positron was discovered by Carl Anderson, with all the physical properties predicted for the Dirac hole. Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite negative electric charge filling all of space. In order to make any sense out of this, one must postulate that the electrons in the sea do not produce any net electric field, or contribute to the total energy or momentum of a system. They also must not interact with one another, or the interactions would be infinitely strong, which would throw into doubt the picture of individual, more or less independent particles that we started out with (the Dirac equation). The development of quantum field theory in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture is much more convincing, especially since it recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular, the problem of the vacuum possessing infinite energy is not so much resolved as swept under the carpet. It is interesting to note that Dirac's idea is absolutely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge – the charge of the ionic lattice – that cancels out the electric charge of the sea. In fiction The Dirac Sea provides a mechanism for time-travel in Geoffrey A. Landis' Nebula Award-winning short story "Ripples in the Dirac Sea". It also appears in the sci-fi anime Neon Genesis Evangelion - noted for its high-quality technobabble - as an expanding void that sucks up matter during the attack of the 12th Angel, Leliel, in episode 16. It is mentioned again in episode 17 as a possible reason for the mysterious disappearance of NERV's Nevada branch. See also de:Dirac-See he:הים של דיראק nl:Diraczee sl:Diracovo morje
dd952ea3b8917c78	Graphite to Graphene… in a Kitchen Blender A photograph showing a ball-and-stick model of graphene near a typical kitchen blender. Image: NaturPhilosophieThe Wonder Material Ten years ago, the discovery of the wonder material – Graphene – was announced. Graphene is thin, stronger than steel, flexible, non-metallic, yet electrically conductive. For all these reasons, graphene promises to transform electronics, as well as other technologies. Because of its potential in industry, researchers have been looking for ways to make defect-free graphene in large amounts. If graphene sounds exotic, the atomic element that makes it really isn’t. Nothing, but Carbon Carbon is an almighty element. Familiar, but intriguing. The 15th most abundant element in the Earth’s crust, and the 4th most abundant element in the Universe by mass after hydrogen, helium and oxygen, carbon is also present in all known life forms, including the human body where it is the 2nd most abundant element by mass (about 18.5%) after oxygen. Together with the unique diversity of organic compounds and their unusual polymer-forming ability at temperatures commonly encountered on Earth, the abundance of carbon makes it the chemical basis for all known life. The physical properties of carbon vary widely according to its allotropic form, which means it can look like radically different stuff, depending on its molecular structure. Carbon can be brittle, or it can be immensely strong… Graphite vs Diamonds Graphite is mixed with clay to produce the lead in pencils. Diamonds ARE a girl’s best friend… 😉 And their exceptional durability has come to symbolise eternity (or is it, “an” eternity?) in a relationship. Graphite and diamond are both allotropes of carbon. So, why do they look so different? Graphite is opaque and black, while diamond is highly transparent. Graphite is soft enough to leave a streak on paper. Diamond is the hardest naturally-occurring material. Two diagrams showing the chemical structure of diamond compared to the chemical structure of graphite. To understand why those allotropes behave so differently, we can take a look at the molecular arrangement of the carbon atoms in both materials. Diamond has a crystalline molecular structure, whereas graphite looks like a stack of seemingly unconnected smaller molecules. Effectively, graphite is a material made up of many layers of graphene stacked on top of one another. Actually, it’s not just the appearance that differs. While diamond has a very low electrical conductivity, graphite is an excellent electrical conductor. But graphene is special. The theory of graphene was first explored by P.R. Wallace in 1947, as a starting point for understanding the electronic properties of 3D graphite. Enters Graphene… A photograph showing a stick-and-ball model of graphene.Graphene research has gone a long way since the substance was first isolated in 2004. The initial findings were reported in the academic journal Science. Research was informed by theoretical descriptions of graphene’s composition, structure and properties, which had all been calculated decades earlier. High-quality graphene also proved to be surprisingly easy to isolate, making more research possible. In 2010, Manchester University researchers Andre Geim and Konstantin Novoselov shared the Nobel Prize in Physics for their discovery of graphene. Geim and Novoselov famously used sticky tape to peel off the layers of graphene from graphite. They pulled graphene layers from graphite and transferred them onto thin SiO 2 on a silicon wafer in a process called either micromechanical cleavage or the ‘Scotch tape’ technique. A photograph showing how to make graphene, using graphite and sticky tape.Graphene can be described a one-atom-thick sheet of graphite arranged in a honeycomb structure. It is the basic structural element of other carbon allotropes, including graphite, charcoal, carbon nanotubes, and fullerenes. It can also be considered as an indefinitely large aromatic molecule. Three diagrams showing the way carbon atoms are structured in fullerene, nanotubes and graphene materials. Currently, graphene can be grown atom-by-atom via chemical vapour depositionHowever, while this process can produce metre-scale sheets of graphene, they also contain defects which can inhibit their physical properties. Most recently, Graphene Nano-Ribbons (GNRs) have been prepared by the oxidative treatment of carbon nanotubes and by plasma etching of nanotubes embedded in polymer films. Properties of Graphene and Nano-ribbons The atomic structure of isolated, single-layer graphene was studied by transmission electron microscopy (TEM) on sheets of graphene suspended between bars of a metallic grid. Electron diffraction patterns showed the expected honeycomb lattice. Suspended graphene also showed rippling of the flat sheet, with amplitude of about one nanometre. Chemically, graphene is the most reactive form of carbon, owing to the lateral availability of carbon atoms. Graphene burns at very low temperature (e.g. 350 °C). The electronic properties of graphene also have some similarity with carbon nanotubes. Graphene is a semi-metal or zero-gap semiconductor. Electrons propagating through graphene’s honeycomb lattice effectively lose their mass, producing quasi-particles that are described by a 2D analogue of the Dirac equation, rather than the Schrödinger equation for spin-1/2 particles. Electron waves in graphene propagate within a single-atom layer, making them sensitive to the proximity of other materials such as high-κ dielectrics, superconductors and ferromagnetics. Electrical Conductance Graphene electrons can move over tens or hundreds of microns without scattering, even at room temperature. Electrical resistance in 40-nanometre-wide nanoribbons of graphene changes in discrete steps following quantum mechanical principles. Electrons travel ballistically, similar to those observed in cylindrical carbon nanotubes. The ribbons’ theoretical conductance exceeds predictions by a factor of 10. The ribbons can act more like optical waveguides or quantum dots, allowing electrons to flow smoothly along the ribbons’ edges. By contrast, in conductors such as copper, electrical resistance increases in proportion to the length as electrons encounter impurities while moving through the conductor. Anomalous Quantum Hall Effect and Berry Phase The Buckminsterfullerene (or bucky-ball) is a spherical fullerene molecule with the formula C60. It has a cage-like fused-ring structure which resembles a football. Even though graphene on nickel Ni and silicon carbide SiC have both existed in the laboratory for several decades, the cleavage technique led directly to the first observation of the anomalous quantum Hall effect in graphene, providing direct evidence of graphene’s theoretically predicted Berry’s phase of massless Dirac fermions. The effect was reported soon after by Philip Kim and Yuanbo Zhang in 2005. Mechanically exfoliated graphene on SiO 2 (silicon dioxide) provided the first proof of the Dirac fermion nature of electrons. Mechanical Strength As of 2009, graphene was reported to be one of the strongest materials known with a breaking strength over 100 times greater than a hypothetical steel film of the same (thin) thickness, with a Young’s modulus (stiffness index) of 1 TPa and intrinsic strength of 130 GP, similar to Single Walled carbon NanoTubes (SWNTs). The Nobel announcement illustrated this fact by saying that a 1 square-metre hammock made of graphene would support a 4 kg cat, but only weigh as much as one of the cat’s whiskers! Electron mobility in graphene is extraordinarily high (15,000 cm2V-1s-1 at room temperature) and ballistic electron transport is reported to be on length scales comparable to that of SWNTs. One of the most promising aspects of graphene involves the use of GNRs. Cutting an individual graphene layer into a long strip can yield semiconducting materials where the band gap is tuned by the width of the ribbon. Bulk Production While graphene’s novel electronic and physical properties guarantee this material will be studied for years to come, there are some fundamental obstacles yet to overcome before graphene based materials can be fully utilised. Although the aforementioned methods of graphene preparation are effective, they have proved impractical for large-scale manufacturing. The most plentiful and inexpensive source of graphene is bulk graphite. Chemical methods for exfoliation of graphene from graphite provide the most realistic and scalable approach to graphene materials. Graphene layers are held together in graphite by enormous van der Waals forces. Overcoming these van der Waals forces is the major obstacle to graphite exfoliation. Until now, chemical efforts at graphite exfoliation have been focused mainly on intercalation, chemical derivatization, thermal expansion, oxidation-reduction, the use of surfactants, or a combination of these. Culinary Physics: Ready, Steady, Mix An Irish-UK team from Trinity College in Dublin tested out a variety of laboratory mixers, as well as kitchen blenders, as potential tools for manufacturing the wonder material. A photograph showing the standard type of kitchen blender used to "exfoliate" graphene and transmission electron microscope images of graphene flakes. A standard kitchen blender, and transmission electron microscope images of graphene flakes (Photo:CRANN) Jonathan Coleman and colleagues poured graphite powder (the stuff of pencil leads) into a blender, then added water and dishwashing liquid, mixing at high speed. The precise amount of dishwashing fluid that’s required is dependent on a number of different factors. The black solution containing graphene needs to be separated afterwards. But the researchers said their work “provides a significant step” towards deploying graphene in a variety of commercial applications. The results are reported in the journal Nature Materials. We could soon be finding graphene everywhere. And I mean EVERYWHERE. In addition to its potential uses in electronics, graphene might have applications in water treatment, oil spill clean-up, and even in the production of anything from stronger hosiery to thinner condoms. Hey! I just thought… Do you reckon graphene is a girl’s new best friend? 😀
e571e28523785326	I've been listening to a lot of Sean Carroll's Mindscape podcast recently, and in a recent episode with Rob Reid he discussed the Everettian or "many-worlds" approach to explaining the measurement problem in quantum mechanics. Towards the end of the episode they discussed an iPhone app that uses a quantum device connected to an HTTP API to "split the universe" by triggering a quantum measurement. Whether you believe the many-worlds theory or not, there's something very cool about having a quantum device just an HTTP call away… So I threw together an HttpClient to call the HTTP API to generate truly random numbers. And voila: you have a quantum random generator! And as an added bonus, you've split the universe multiple times to get it. The focus of this post isn't the HttpClient itself - that's a toy to scratch an itch more than anything else. Instead, I'm trying to put down my (exceedingly-basic) understanding of the quantum measurement problem, and the many-worlds approach. This is obviously a departure from my usual posts. I've always been casually interested in physics but I'm absolutely not a physicist, so I strongly suggest taking everything I say with a pinch of salt and imprecision! It's also a very brief consideration of the subject matter - I've added some podcast references at the end of the post from where I gleaned most of my understanding in this area! Quantum mechanics and the wave equation So what is quantum mechanics? Quantum mechanics is the single most successful theory we have to explain the world. There's no evidence from experimental physics that suggests there's a flaw in its description of the way the world works. It seems to be the way nature works. At its heart is the wave equation that describes the "quantum state" of the system. In classic Newtonian mechanics, the "state" of a system (e.g. a particle) is its location and its velocity - if you know those two properties, then you can completely determine the behaviour of the system by following Newton's laws. The wave equation is the quantum state, and it evolves according to the Schrödinger equation. A wave function corresponding to a particle traveling freely through empty space You can think of the wave equation as a cloud of possibilities. As Sean puts it: The wave function you should think of as a sort of a machine. You ask it a question, “Where is the electron?” For example. And it will say, “With a certain probability you will find the following answers to your question.” If all you care about is the position of one electron, then the wave function at every point in space has a value, and that value tells you the probability of seeing the electron there. One of the big questions is: what actually is the wave function? Is it describing a behaviour we observe, or is it something more fundamental? One of the biggest shifts for me was understanding that electrons aren't little balls. You can't think of it as a little ball that has a 90% chance to be in one place, and a 10% chance in the other place. That implies that the ball is always somewhere we just don't know where. The answer is that it's not really a ball. It's a weird wave thing that's everywhere at once (with varying possibilities). 2d cross sections of the probability density of various states of the hydrogen atom But how can that be so? We have huge amounts of technology that rely on our ability to manipulate electrons and other particles just as though they are little balls. How can both descriptions be correct? That brings us to the measurement problem. The measurement problem in quantum mechanics The wave equation isn't just a fancy way of thinking about probabilities – the implication is that the electron actually is in both places at once. It's in a "superposition" of all the possible states. But it's never possible for us to "see" an electron in that superposition state. When you measure or observe the electron, you only ever see it in a single place. You only ever see it as a particle. It seems then, that the act of looking at an electron causes it to behave differently, it "collapses the wave function". It's like a game of musical statues - everything is different (people are moving / the electron is behaving like a wave) until you look at it, and suddenly everyone acts casual (people freeze / the electron behaves like a particle). Wayne acting casual One of the big problems with this is it makes an "observer" a first-class citizen in physics. But what are the requirements to be an observer? Does it have to be a person? Is a camera an observer? What about a cat, or an amoeba? It's weird… The question of how (of if) the wave function collapses when you measure a quantum system is termed the measurement problem. There are a number of different approaches that attempt to address this problem, for example: • The Copenhagen interpretation which appears to say "it doesn't matter, the math works, stop complaining and do some real work" 🤷‍♂️. • Bohmian mechanics suggests the wave function is only part of the solution - there's extra hidden variables we don't know about. • Dynamical collapse theories suggest that wave function collapse happens spontaneously, but that the collapse of a single particle initiates the collapse of the entire measurement apparatus. • Hugh Everett's many-worlds interpretation suggests the wave function never collapses, rather that the universe "splits in two" - one in which the electron is in position A, one in which the electron is in position B. This many-worlds approach is the one favoured by Sean Carroll, and is the one of interest here. The many-worlds approach by Everett The many-worlds approach is in many ways the most mind-bending option. Don't be surprised if your gut reaction is that it's prosperous mumbo-jumbo! I'll try and explain as I best I understand it. The many-worlds approach suggests that when you see the wave function appear to collapse (due to an observation) it is actually the universe "splitting" into two branches. One in which the electron was in position A and one in which it was in position B. So what is an observation in the many-worlds approach? Everett said that the universe splits when a system becomes entangled with its environment (it decoheres). Any interaction between two systems will cause decoherance, and hence will cause the universe to split. But seeing as we're part of this universe, we (and our measuring equipment) are inherently quantum too! So any interaction we have with fundamental particles will inevitably be a quantum interaction, and the universe splits in two. In this universe, you observe the electron in position A for example, but in the other branch of the universe you observe it in position B. So in summary an "observation" is any time you cause a quantum interaction Or to use the famous Schrödinger's cat thought experiment, in one universe the cat is alive, in the other it's asleep (no cats are harmed in my thought experiments). Schrödinger's Cat, many worlds interpretation, with universe branching I haven't succeeded in getting my head fully around this yet. Both of the universes are "here" in some sense. The theory is not suggesting the universes are spatially distant (as in some multiverse theories), or that they're "wrapped up" in some higher dimensions (as in string theory). They're in the same "place" but evolving separately, and can't contact each other. Technically, they're two different vectors in Hilbert space, but that doesn't mean a lot to me conceptually! There's a lot more to the theory that is way beyond my capabilities, but the interesting notion is that every time you have a quantum interaction, you branch the wave function and the universe "splits in two". This is the principle that the Universe Splitter iPhone app relies on. Splitting the universe with a quantum measurement The Universe Splitter app is essentially a front-end to a "Quantis" brand quantum device made by ID Quantique. This device sends a photon towards a semi-reflective mirror, so that 50% of the time the photon passes through the mirror, and 50% of the time it is reflected. QRNG based on a Polarising Beam Splitter (PBS) QRNG based on a Polarising Beam Splitter (PBS). Figure from arXiv:1311.4547 [quant-ph]. Since this is a quantum observation, then every time the device fires the universe splits in two – one branch in which the photon was reflected, and one in which it passed through! That's the premise of the Universe Splitter app. If you commit to doing action A if the photon is reflected, and action B if it transmits, then you have created two separate universes: one in which you did action A, and one in which you did action B! This is quite a cute concept, but quantum randomness has a "real life" use too. It is the best (and only) source of true randomness. Radioactive decay is another source of randomness that can be traced back to the same quantum origin. So I got thinking - why don't we have a quantum random generator in .NET? Sure, you can purchase a quantum random generator, and communicate with it over USB etc. But I want a web service! After a bit of hunting, I found (what I assume) is the back-end the universe splitter app is using - an API provided by ETH Zürich. Creating a quantum random generator for .NET The "quantum random numbers as a service, QRNGaaS" allows you to call a very simple rest API and obtain random numbers generated using a Quantis random number generator. The API is very simple - you send a GET request, and you get back random numbers. For example, you can request 10 numbers between 0 and 9 (inclusive) curl http://random.openqu.org/api/randint?size=10&min=0&max=9 And you'll get JSON similar to the following [ 1, 3, 9, 8, 0, 4, 3, 6, 6, 3, 1] There's also an API for returning floating point numbers, and base64 encoded bytes. Unfortunately, I couldn't get the bytes API to work (500 Internal Server Error). Creating a .NET API to call this endpoint is pretty trivial. The code below is very much thrown together - it doesn't have any error checking, it creates its own HttpClient instead of having one injected via the constructor, it assumes the content will always be the expected format etc. But it works! using System; using System.Linq; using System.Net.Http; using System.Text.Json; using System.Threading.Tasks; public class QuantumRandomNumberGenerator private readonly HttpClient _httpClient; public QuantumRandomNumberGenerator() _httpClient = new HttpClient BaseAddress = new Uri("http://random.openqu.org/api/") public async Task<int[]> GetIntegers(int min, int max, int size) var url = $"randint?size={size}&min={min}&max={max}"; var stream = await _httpClient.GetStreamAsync(url); using(var document = await JsonDocument.ParseAsync(stream)) return document .Select(x => x.GetInt32()) This example uses the new System.Text.Json library in .NET Core 3.0 to efficiently parse the JSON response and return the array of integers. If you want to learn more about System.Text.Json I suggest reading the intro blog post. I also liked Stuart Lang's introduction. With our new QuantumRandomNumberGenerator defined, we can now generate truly random numbers: var qrng = new QuantumRandomNumberGenerator(); var values = await qrng.GetIntegers(0, 255, 10); Console.WriteLine(string.Join(", ", values)); // 32, 133, 183, 249, 208, 112, 76, 178, 44, 184 So, should you replace all your calls to RandomNumberGenerator with your shiny new QuantumRandomNumberGenerator. No, please don't. You really don't need true randomness in most places + the cost of calling an HTTP API for every new random number is clearly an issue. Add in the fact that it's a free service, no doubt exposed as a curiosity rather than providing any sorts of guarantees. Plus the fact you can only call the API over HTTP, not HTTPS. So please, don't put this into production. 😉 But do have fun thinking about all those universes you're creating. In one of them, the random list of numbers I generated above is all 0s, and in another, they're all 255s! 🤯
bfc01ed536051d92	Stability theory of solitary waves in the presence of symmetry. II. (English) Zbl 0711.58013 Summary: Consider an abstract Hamiltonian system which is invariant under a group of operators. We continue to study the effect of the group invariance on the stability of solitary waves [see part I, the authors, ibid. 74, 160- 197 (1987; Zbl 0656.35122)]. Applications are given to bound states and traveling wave solutions of nonlinear wave equations. 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 35Q55 NLS equations (nonlinear Schrödinger equations) 37C75 Stability theory for smooth dynamical systems Zbl 0656.35122 Full Text: DOI [1] Grillakis, M; Shatah, J; Strauss, W, Stability theory of solitary waves in the presence of symmetry, I, J. funct. anal., 74, 160-197, (1987) · Zbl 0656.35122 [2] Blanchard, P; Stubbe, J; Vazquez, L, Ann. inst. H. Poincaré, 47, 309-336, (1987) [3] Grillakis, M, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. pure appl. math., 41, 747-774, (1988) · Zbl 0632.70015 [4] Grillakis, M, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. pure appl. math., 43, 299-333, (1990) · Zbl 0731.35010 [5] Jones, C, An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation, J. differential equations, 71, 34-62, (1988) · Zbl 0656.35108 [6] Jones, C; Moloney, J, Instability of nonlinear waveguide modes, Phys. lett. A, 117, 175-180, (1986) [7] Oh, Y.-G, A stability theory for Hamiltonian systems with symmetry, J. geom. phys., 4, 163-182, (1987) · Zbl 0663.58013 [8] Strauss, W, Stability theory of solitary waves with invariants, Abstracts amer. math. soc., 7, 72, (Jan. 1986)
90e16596cae7337f	Page semi-protected From Wikipedia, the free encyclopedia (Redirected from Physical energy) Jump to navigation Jump to search The Sun in white light.jpg The Sun is the ultimate source of energy for most of life on Earth.[1] It derives its energy mainly from nuclear fusion in its core, converting mass to energy as protons are combined to form helium. This energy is transported to the sun's surface and released into space (mainly in the form of radiant (light) energy). Common symbols SI unitjoule Other units kW⋅h, BTU, calorie, eV, erg, foot-pound In SI base unitsJ = kg m2 s−2 DimensionM L2 T−2 In physics, energy is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light. Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The unit of measurement in the International System of Units (SI) of energy is the joule, which is the energy transferred to an object by the work of moving it a distance of one metre against a force of one newton. Common forms of energy include the kinetic energy of a moving object, the potential energy stored by an object's position in a force field (gravitational, electric or magnetic), the elastic energy stored by stretching solid objects, the chemical energy released when a fuel burns, the radiant energy carried by light, and the thermal energy due to an object's temperature. Mass and energy are closely related. Due to mass–energy equivalence, any object that has mass when stationary (called rest mass) also has an equivalent amount of energy whose form is called rest energy, and any additional energy (of any form) acquired by the object above that rest energy will increase the object's total mass just as it increases its total energy. For example, after heating an object, its increase in energy could in principle be measured as a small increase in mass, with a sensitive enough scale. Living organisms require energy to stay alive, such as the energy humans get from food and oxygen. Human civilization requires energy to function, which it gets from energy resources such as fossil fuels, nuclear fuel, or renewable energy. The processes of Earth's climate and ecosystem are driven by the radiant energy Earth receives from the Sun and the geothermal energy contained within the earth. In a typical lightning strike, 500 megajoules of electric potential energy is converted into the same amount of energy in other forms, mostly light energy, sound energy and thermal energy. Thermal energy is energy of microscopic constituents of matter, which may include both kinetic and potential energy. The total energy of a system can be subdivided and classified into potential energy, kinetic energy, or combinations of the two in various ways. Kinetic energy is determined by the movement of an object – or the composite motion of the components of an object – and potential energy reflects the potential of an object to have motion, and generally is a function of the position of an object within a field or may be stored in the field itself. While these two categories are sufficient to describe all forms of energy, it is often convenient to refer to particular combinations of potential and kinetic energy as its own form. For example, the sum of translational and rotational kinetic and potential energy within a system is referred to as mechanical energy, whereas nuclear energy refers to the combined potentials within an atomic nucleus from either the nuclear force or the weak force, among other examples.[citation needed] Some forms of energy (that an object or system can have as a measurable property) Type of energy Description Mechanical the sum of macroscopic translational and rotational kinetic and potential energies Electric potential energy due to or stored in electric fields Magnetic potential energy due to or stored in magnetic fields Gravitational potential energy due to or stored in gravitational fields Chemical potential energy due to chemical bonds Ionization potential energy that binds an electron to its atom or molecule Nuclear potential energy that binds nucleons to form the atomic nucleus (and nuclear reactions) Chromodynamic potential energy that binds quarks to form hadrons Elastic potential energy due to the deformation of a material (or its container) exhibiting a restorative force as it returns to its original shape Mechanical wave kinetic and potential energy in an elastic material due to a propagating oscillation of matter Sound wave kinetic and potential energy in a material due to a sound propagated wave (a particular type of mechanical wave) Radiant potential energy stored in the fields of waves propagated by electromagnetic radiation, including light Rest potential energy due to an object's rest mass Thermal kinetic energy of the microscopic motion of particles, a kind of disordered equivalent of mechanical energy Thomas Young, the first person to use the term "energy" in the modern sense. The word energy derives from the Ancient Greek: ἐνέργεια, romanizedenergeia, lit.'activity, operation',[2] which possibly appears for the first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure. In the late 17th century, Gottfried Leibniz proposed the idea of the Latin: vis viva, or living force, which defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the motions of the constituent parts of matter, although it would be more than a century until this was generally accepted. The modern analog of this property, kinetic energy, differs from vis viva only by a factor of two. Writing in the early 18th century, Émilie du Châtelet proposed the concept of conservation of energy in the marginalia of her French language translation of Newton's Principia Mathematica, which represented the first formulation of a conserved measurable quantity that was distinct from momentum, and which would later be called "energy". In 1807, Thomas Young was possibly the first to use the term "energy" instead of vis viva, in its modern sense.[3] Gustave-Gaspard Coriolis described "kinetic energy" in 1829 in its modern sense, and in 1853, William Rankine coined the term "potential energy". The law of conservation of energy was also first postulated in the early 19th century, and applies to any isolated system. It was argued for some years whether heat was a physical substance, dubbed the caloric, or merely a physical quantity, such as momentum. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat. These developments led to the theory of conservation of energy, formalized largely by William Thomson (Lord Kelvin) as the field of thermodynamics. Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, and Walther Nernst. It also led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.[4] Thus, since 1918, theorists have understood that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time. Units of measure Joule's apparatus for measuring the mechanical equivalent of heat. A descending weight attached to a string causes a paddle immersed in water to rotate. In 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. The most famous of them used the "Joule apparatus": a descending weight, attached to a string, caused rotation of a paddle immersed in water, practically insulated from heat transfer. It showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. In the International System of Units (SI), the unit of energy is the joule, named after Joule. It is a derived unit. It is equal to the energy expended (or work done) in applying a force of one newton through a distance of one metre. However energy is also expressed in many other units not part of the SI, such as ergs, calories, British Thermal Units, kilowatt-hours and kilocalories, which require a conversion factor when expressed in SI units. The SI unit of energy rate (energy per unit time) is the watt, which is a joule per second. Thus, one joule is one watt-second, and 3600 joules equal one watt-hour. The CGS energy unit is the erg and the imperial and US customary unit is the foot pound. Other energy units such as the electronvolt, food calorie or thermodynamic kcal (based on the temperature change of water in a heating process), and BTU are used in specific areas of science and commerce. Scientific use Classical mechanics In classical mechanics, energy is a conceptually and mathematically useful property, as it is a conserved quantity. Several formulations of mechanics have been developed using energy as a core concept. Work, a function of energy, is force times distance. This says that the work () is equal to the line integral of the force F along a path C; for details see the mechanical work article. Work and thus energy is frame dependent. For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball. The total energy of a system is sometimes called the Hamiltonian, after William Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have remarkably direct analogs in nonrelativistic quantum mechanics.[5] Another energy-related concept is called the Lagrangian, after Joseph-Louis Lagrange. This formalism is as fundamental as the Hamiltonian, and both can be used to derive the equations of motion or be derived from them. It was invented in the context of classical mechanics, but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy minus the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction). Noether's theorem (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law. In the context of chemistry, energy is an attribute of a substance as a consequence of its atomic, molecular, or aggregate structure. Since a chemical transformation is accompanied by a change in one or more of these kinds of structure, it is usually accompanied by a decrease, and sometimes an increase, of the total energy of the substances involved. Some energy may be transferred between the surroundings and the reactants in the form of heat or light; thus the products of a reaction have sometimes more but usually less energy than the reactants. A reaction is said to be exothermic or exergonic if the final state is lower on the energy scale than the initial state; in the less common case of endothermic reactions the situation is the reverse. Chemical reactions are usually not possible unless the reactants surmount an energy barrier known as the activation energy. The speed of a chemical reaction (at a given temperature T) is related to the activation energy E by the Boltzmann's population factor eE/kT; that is, the probability of a molecule to have energy greater than or equal to E at a given temperature T. This exponential dependence of a reaction rate on temperature is known as the Arrhenius equation. The activation energy necessary for a chemical reaction can be provided in the form of thermal energy. Basic overview of energy and human life. In biology, energy is an attribute of all biological systems, from the biosphere to the smallest living organism. Within an organism it is responsible for growth and development of a biological cell or organelle of a biological organism. Energy used in respiration is mostly stored in molecular oxygen[6] and can be unlocked by reactions with molecules of substances such as carbohydrates (including sugars), lipids, and proteins stored by cells. In human terms, the human equivalent (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for human metabolism, using as a standard an average human energy expenditure of 12,500 kJ per day and a basal metabolic rate of 80 watts. For example, if our bodies run (on average) at 80 watts, then a light bulb running at 100 watts is running at 1.25 human equivalents (100 ÷ 80) i.e. 1.25 H-e. For a difficult task of only a few seconds' duration, a person can put out thousands of watts, many times the 746 watts in one official horsepower. For tasks lasting a few minutes, a fit human can generate perhaps 1,000 watts. For an activity that must be sustained for an hour, output drops to around 300; for an activity kept up all day, 150 watts is about the maximum.[7] The human equivalent assists understanding of energy flows in physical and biological systems by expressing energy units in human terms: it provides a "feel" for the use of a given amount of energy.[8] Sunlight's radiant energy is also captured by plants as chemical potential energy in photosynthesis, when carbon dioxide and water (two low-energy compounds) are converted into carbohydrates, lipids, proteins and high-energy compounds like oxygen[6] and ATP. Carbohydrates, lipids, and proteins can release the energy of oxygen, which is utilized by living organisms as an electron acceptor. Release of the energy stored during photosynthesis as heat or light may be triggered suddenly by a spark in a forest fire, or it may be made available more slowly for animal or human metabolism when organic molecules are ingested and catabolism is triggered by enzyme action. All living creatures rely on an external source of energy to be able to grow and reproduce – radiant energy from the Sun in the case of green plants and chemical energy (in some form) in the case of animals. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as a combination of oxygen and food molecules, the latter mostly carbohydrates and fats, of which glucose (C6H12O6) and stearin (C57H110O6) are convenient examples. The food molecules are oxidized to carbon dioxide and water in the mitochondria and some of the energy is used to convert ADP into ATP: ADP + HPO42− → ATP + H2O The rest of the chemical energy of O2[9] and the carbohydrate or fat are converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains is used for other metabolism when ATP reacts with OH groups and eventually splits into ADP and phosphate (at each stage of a metabolic pathway, some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for work:[note 1] gain in kinetic energy of a sprinter during a 100 m race: 4 kJ gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3 kJ Daily food intake of a normal adult: 6–8 MJ It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical or radiant energy); most machines manage higher efficiencies. In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to the molecules it is built from. The second law of thermodynamics states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings").[note 2] Simpler organisms can achieve higher energy efficiencies than more complex ones, but the complex organisms can occupy ecological niches that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in ecology. As an example, to take just the first step in the food chain: of the estimated 124.7 Pg/a of carbon that is fixed by photosynthesis, 64.3 Pg/a (52%) are used for the metabolism of green plants,[10] i.e. reconverted into carbon dioxide and heat. Earth sciences In geology, continental drift, mountain ranges, volcanoes, and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior,[11] while meteorological phenomena like wind, rain, hail, snow, lightning, tornadoes and hurricanes are all a result of energy transformations in our atmosphere brought about by solar energy. Sunlight is the main input to Earth's energy budget which accounts for its temperature and climate stability. Sunlight may be stored as gravitational potential energy after it strikes the Earth, as (for example when) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbines or generators to produce electricity). Sunlight also drives most weather phenomena, save a few exceptions, like those generated by volcanic events for example. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, suddenly give up some of their thermal energy to power a few days of violent air movement. In a slower process, radioactive decay of atoms in the core of the Earth releases heat. This thermal energy drives plate tectonics and may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential energy storage of the thermal energy, which may later be transformed into active kinetic energy during landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store that has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy that has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks. Prior to this, they represent release of energy that has been stored in heavy atoms since the collapse of long-destroyed supernova stars (which created these atoms). In cosmology and astronomy the phenomena of stars, nova, supernova, quasars and gamma-ray bursts are the universe's highest-output energy transformations of matter. All stellar phenomena (including solar activity) are driven by various kinds of energy transformations. Energy in such transformations is either from gravitational collapse of matter (usually molecular hydrogen) into various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen). The nuclear fusion of hydrogen in the Sun also releases another store of potential energy which was created at the time of the Big Bang. At that time, according to theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy that can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into sunlight. Quantum mechanics In quantum mechanics, energy is defined in terms of the energy operator (Hamiltonian) as a time derivative of the wave function. The Schrödinger equation equates the energy operator to the full energy of a particle or a system. Its results can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of a slowly changing (non-relativistic) wave function of quantum systems. The solution of this equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta. In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic waves in a vacuum, the resulting energy states are related to the frequency by Planck's relation: (where is Planck's constant and the frequency). In the case of an electromagnetic wave these energy states are called quanta of light or photons. When calculating kinetic energy (work to accelerate a massive body from zero speed to some finite speed) relativistically – using Lorentz transformations instead of Newtonian mechanics – Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest energy: energy which every massive body must possess even when being at rest. The amount of energy is directly proportional to the mass of the body: For example, consider electronpositron annihilation, in which the rest energy of these two individual particles (equivalent to their rest mass) is converted to the radiant energy of the photons produced in the process. In this system the matter and antimatter (electrons and positrons) are destroyed and changed to non-matter (the photons). However, the total mass and total energy do not change during this interaction. The photons each have no rest mass but nonetheless have radiant energy which exhibits the same inertia as did the two original particles. This is a reversible process – the inverse process is called pair creation – in which the rest mass of particles is created from the radiant energy of two (or more) annihilating photons. In general relativity, the stress–energy tensor serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian approximation.[12] Energy and mass are manifestations of one and the same underlying physical property of a system. This property is responsible for the inertia and strength of gravitational interaction of the system ("mass manifestations"), and is also responsible for the potential ability of the system to perform work or heating ("energy manifestations"), subject to the limitations of other physical laws. In classical physics, energy is a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the energy–momentum 4-vector).[12] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of spacetime (= boosts). Some forms of transfer of energy ("energy in transit") from one object or system to another Type of transfer process Description Heat equal amount of thermal energy in transit spontaneously towards a lower-temperature object Work equal amount of energy in transit due to a displacement in the direction of an applied force Transfer of material equal amount of energy carried by matter that is moving from one system to another A turbo generator transforms the energy of pressurized steam into electrical energy Energy may be transformed between different forms at various efficiencies. Items that transform between these forms are called transducers. Examples of transducers include a battery (from chemical energy to electric energy), a dam (from gravitational potential energy to kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an electric generator), and a heat engine (from heat to work). Examples of energy transformation include generating electric energy from heat energy via a steam turbine, or lifting an object against gravity using electrical energy driving a crane motor. Lifting against gravity performs mechanical work on the object and stores gravitational potential energy in the object. If the object falls to the ground, gravity does mechanical work on the object which transforms the potential energy in the gravitational field to the kinetic energy released as heat on impact with the ground. Our Sun transforms nuclear potential energy to other forms of energy; its total mass does not decrease due to that itself (since it still contains the same total energy even in different forms) but its mass does decrease when the energy escapes out to its surroundings, largely as radiant energy. There are strict limits to how efficiently heat can be converted into work in a cyclic process, e.g. in a heat engine, as described by Carnot's theorem and the second law of thermodynamics. However, some energy transformations can be quite efficient. The direction of transformations in energy (what kind of energy is transformed to what other kind) is often determined by entropy (equal energy spread among all available degrees of freedom) considerations. In practice all energy transformations are permitted on a small scale, but certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces. Energy transformations in the universe over time are characterized by various kinds of potential energy, that has been available since the Big Bang, being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available. Familiar examples of such processes include nucleosynthesis, a process ultimately using the gravitational potential energy released from the gravitational collapse of supernovae to "store" energy in the creation of heavy isotopes (such as uranium and thorium), and nuclear decay, a process in which energy is released that was originally stored in these heavy elements, before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs or in civil nuclear power generation. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to kinetic and thermal energy in a very short time. Yet another example is that of a pendulum. At its highest points the kinetic energy is zero and the gravitational potential energy is at its maximum. At its lowest point the kinetic energy is at its maximum and is equal to the decrease in potential energy. If one (unrealistically) assumes that there is no friction or other losses, the conversion of energy between these processes would be perfect, and the pendulum would continue swinging forever. Energy is also transferred from potential energy () to kinetic energy () and then back to potential energy constantly. This is referred to as conservation of energy. In this isolated system, energy cannot be created or destroyed; therefore, the initial energy and the final energy will be equal to each other. This can be demonstrated by the following: The equation can then be simplified further since (mass times acceleration due to gravity times the height) and (half mass times velocity squared). Then the total amount of energy can be found by adding . Conservation of energy and mass in transformation Energy gives rise to weight when it is trapped in a system with zero momentum, where it can be weighed. It is also equivalent to mass, and this mass is always associated with it. Mass is also equivalent to a certain amount of energy, and likewise always appears associated with it, as described in mass-energy equivalence. The formula E = mc², derived by Albert Einstein (1905) quantifies the relationship between relativistic mass and energy within the concept of special relativity. In different theoretical frameworks, similar formulas were derived by J.J. Thomson (1881), Henri Poincaré (1900), Friedrich Hasenöhrl (1904) and others (see Mass-energy equivalence#History for further information). Part of the rest energy (equivalent to rest mass) of matter may be converted to other forms of energy (still exhibiting mass), but neither energy nor mass can be destroyed; rather, both remain constant during any process. However, since is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass (for example, 1 kg) from rest energy to other forms of energy (such as kinetic energy, thermal energy, or the radiant energy carried by light and other radiation) can liberate tremendous amounts of energy (~ joules = 21 megatons of TNT), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of an everyday amount energy is minuscule, which is why a loss of energy (loss of mass) from most systems is difficult to measure on a weighing scale, unless the energy loss is very large. Examples of large transformations between rest energy (of matter) and other forms of energy (e.g., kinetic energy into particles with rest mass) are found in nuclear physics and particle physics. Often, however, the complete conversion of matter (such as atoms) to non-matter (such as photons) is forbidden by conservation laws. Reversible and non-reversible transformations Thermodynamics divides energy transformation into two kinds: reversible processes and irreversible processes. An irreversible process is one in which energy is dissipated (spread) into empty energy states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states), without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another is reversible, as in the pendulum system described above. In processes where heat is generated, quantum states of lower energy, present as possible excitations in fields between atoms, act as a reservoir for part of the energy, from which it cannot be recovered, in order to be converted with 100% efficiency into other forms of energy. In this case, the energy must partly stay as thermal energy and cannot be completely recovered as usable energy, except at the price of an increase in some other kind of heat-like increase in disorder in quantum states, in the universe (such as an expansion of matter, or a randomization in a crystal). As the universe evolves with time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or as other kinds of increases in disorder). This has led to the hypothesis of the inevitable thermodynamic heat death of the universe. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do work through a heat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), continues to decrease. Conservation of energy The fact that energy can be neither created nor destroyed is called the law of conservation of energy. In the form of the first law of thermodynamics, this states that a closed system's energy is constant unless energy is transferred in or out as work or heat, and that no energy is lost in transfer. The total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. Whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it is found that the total energy of the system always remains constant.[13] While heat can always be fully converted into work in a reversible isothermal expansion of an ideal gas, for cyclic processes of practical interest in heat engines the second law of thermodynamics states that the system doing work always loses some energy as waste heat. This creates a limit to the amount of heat energy that can do work in a cyclic process, a limit called the available energy. Mechanical and other forms of energy can be transformed in the other direction into thermal energy without such limitations.[14] The total energy of a system can be calculated by adding up all forms of energy in the system. Richard Feynman said during a 1961 lecture:[15] There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. Most kinds of energy (with gravitational energy being a notable exception)[16] are subject to strict local conservation laws as well. In this case, energy can only be exchanged between adjacent regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the universe cannot change; this is a corollary of the local law, but not vice versa.[14][15] This law is a fundamental principle of physics. As shown rigorously by Noether's theorem, the conservation of energy is a mathematical consequence of translational symmetry of time,[17] a property of most phenomena below the cosmic scale that makes them independent of their locations on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable. This is because energy is the quantity which is canonical conjugate to time. This mathematical entanglement of energy and time also results in the uncertainty principle – it is impossible to define the exact amount of energy during any definite time interval (though this is practically significant only for very short time intervals). The uncertainty principle should not be confused with energy conservation – rather it provides mathematical limits to which energy can in principle be defined and measured. Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appear as system mass, whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it. In quantum mechanics energy is expressed using the Hamiltonian operator. On any time scales, the uncertainty in the energy is by which is similar in form to the Heisenberg Uncertainty Principle (but not really mathematically equivalent thereto, since H and t are not dynamically conjugate variables, neither in classical nor in quantum mechanics). In particle physics, this inequality permits a qualitative understanding of virtual particles, which carry momentum. The exchange of virtual particles with real particles is responsible for the creation of all known fundamental forces (more accurately known as fundamental interactions). Virtual photons are also responsible for the electrostatic interaction between electric charges (which results in Coulomb's law), for spontaneous radiative decay of excited atomic and nuclear states, for the Casimir force, for the Van der Waals force and some other observable phenomena. Energy transfer Closed systems Energy transfer can be considered for the special case of systems which are closed to transfers of matter. The portion of the energy which is transferred by conservative forces over a distance is measured as the work the source system does on the receiving system. The portion of the energy which does not do work during the transfer is called heat.[note 3] Energy can be transferred between systems in a variety of ways. Examples include the transmission of electromagnetic energy via photons, physical collisions which transfer kinetic energy,[note 4] and the conductive transfer of thermal energy. Energy is strictly conserved and is also locally conserved wherever it can be defined. In thermodynamics, for closed systems, the process of energy transfer is described by the first law:[note 5] where is the amount of energy transferred, represents the work done on or by the system, and represents the heat flow into or out of the system. As a simplification, the heat term, , can sometimes be ignored, especially for fast processes involving gases, which are poor conductors of heat, or when the thermal efficiency of the transfer is high. For such adiabatic processes, This simplified equation is the one used to define the joule, for example. Open systems Beyond the constraints of closed systems, open systems can gain or lose energy in association with matter transfer (this process is illustrated by injection of an air-fuel mixture into a car engine, a system which gains in energy thereby, without addition of either work or heat). Denoting this energy by , one may write Internal energy Internal energy is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal energy and not its absolute value, which is impossible to determine with thermodynamics alone.[18] First law of thermodynamics The first law of thermodynamics asserts that the total energy of a system and its surroundings (but not necessarily thermodynamic free energy) is always conserved[19] and that heat flow is a form of energy transfer. For homogeneous systems, with a well-defined temperature and pressure, a commonly used corollary of the first law is that, for a system subject only to pressure forces and heat transfer (e.g., a cylinder-full of gas) without chemical changes, the differential change in the internal energy of the system (with a gain in energy signified by a positive quantity) is given as where the first term on the right is the heat transferred into the system, expressed in terms of temperature T and entropy S (in which entropy increases and its change dS is positive when heat is added to the system), and the last term on the right hand side is identified as work done on the system, where pressure is P and volume V (the negative sign results since compression of the system requires work to be done on it and so the volume change, dV, is negative when work is done on the system). This equation is highly specific, ignoring all chemical, electrical, nuclear, and gravitational forces, effects such as advection of any form of energy other than heat and PV-work. The general formulation of the first law (i.e., conservation of energy) is valid even in situations in which the system is not homogeneous. For these cases the change in internal energy of a closed system is expressed in a general form by where is the heat supplied to the system and is the work applied to the system. Equipartition of energy The energy of a mechanical harmonic oscillator (a mass on a spring) is alternately kinetic and potential energy. At two points in the oscillation cycle it is entirely kinetic, and at two points it is entirely potential. Over a whole cycle, or over many cycles, average energy is equally split between kinetic and potential. This is an example of the equipartition principle: the total energy of a system with many degrees of freedom is equally split among all available degrees of freedom, on average. This principle is vitally important to understanding the behavior of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a distribution of energy between parts of a system. When an isolated system is given more degrees of freedom (i.e., given new available energy states that are the same as existing states), then total energy spreads over all available degrees equally without distinction between "new" and "old" degrees. This mathematical result is part of the second law of thermodynamics. The second law of thermodynamics is simple only for systems which are near or in a physical equilibrium state. For non-equilibrium systems, the laws governing the systems' behavior are still debatable. One of the guiding principles for these systems is the principle of maximum entropy production.[20][21] It states that nonequilibrium systems behave in such a way as to maximize their entropy production.[22] See also 1. ^ These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but the power output (in case of a sprinter) and the force (in case of a weightlifter). 2. ^ Crystals are another example of highly ordered systems that exist in nature: in this case too, the order is associated with the transfer of a large amount of heat (known as the lattice energy) to the surroundings. 3. ^ Although heat is "wasted" energy for a specific energy transfer (see: waste heat), it can often be harnessed to do useful work in subsequent interactions. However, the maximum energy that can be "recycled" from such recovery processes is limited by the second law of thermodynamics. 4. ^ The mechanism for most macroscopic physical collisions is actually electromagnetic, but it is very common to simplify the interaction by ignoring the mechanism of collision and just calculate the beginning and end result. 5. ^ There are several sign conventions for this equation. Here, the signs in this equation follow the IUPAC convention. 1. ^ Shuster, Michele; Vigna, Janet; Sinha, Gunjan (2011). Scientific American Biology for a Changing World. MacMillan. p. 90. ISBN 9780716773245. 2. ^ Harper, Douglas. "Energy". Online Etymology Dictionary. Archived from the original on October 11, 2007. Retrieved May 1, 2007. 3. ^ Smith, Crosbie (1998). The Science of Energy – a Cultural History of Energy Physics in Victorian Britain. The University of Chicago Press. ISBN 978-0-226-76420-7. 4. ^ Lofts, G; O'Keeffe D; et al. (2004). "11 – Mechanical Interactions". Jacaranda Physics 1 (2 ed.). Milton, Queensland, Australia: John Willey & Sons Australia Ltd. p. 286. ISBN 978-0-7016-3777-4. 5. ^ The Hamiltonian MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007 6. ^ a b Schmidt-Rohr, K. (2020). "Oxygen Is the High-Energy Molecule Powering Complex Multicellular Life: Fundamental Corrections to Traditional Bioenergetics” ACS Omega 5: 2221–33. 7. ^ "Retrieved on May-29-09". Archived from the original on 2010-06-04. Retrieved 2010-12-12. 8. ^ Bicycle calculator – speed, weight, wattage etc. "Bike Calculator". Archived from the original on 2009-05-13. Retrieved 2009-05-29.. 9. ^ Schmidt-Rohr, K (2015). "Why Combustions Are Always Exothermic, Yielding About 418 kJ per Mole of O2". J. Chem. Educ. 92 (12): 2094–2099. Bibcode:2015JChEd..92.2094S. doi:10.1021/acs.jchemed.5b00333. 10. ^ Ito, Akihito; Oikawa, Takehisa (2004). "Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model. Archived 2006-10-02 at the Wayback Machine" in Shiyomi, M. et al. (Eds.) Global Environmental Change in the Ocean and on Land. pp. 343–58. 11. ^ "Earth's Energy Budget". Archived from the original on 2008-08-27. Retrieved 2010-12-12. 12. ^ a b Misner, Thorne, Wheeler (1973). Gravitation. San Francisco: W.H. Freeman. ISBN 978-0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link) 13. ^ Berkeley Physics Course Volume 1. Charles Kittel, Walter D Knight and Malvin A Ruderman 14. ^ a b The Laws of Thermodynamics Archived 2006-12-15 at the Wayback Machine including careful definitions of energy, free energy, et cetera. 15. ^ a b Feynman, Richard (1964). The Feynman Lectures on Physics; Volume 1. US: Addison Wesley. ISBN 978-0-201-02115-8. 16. ^ "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws". 1918-07-16. Archived from the original on 2011-05-14. Retrieved 2010-12-12. 17. ^ "Time Invariance". Archived from the original on 2011-07-17. Retrieved 2010-12-12. 18. ^ I. Klotz, R. Rosenberg, Chemical Thermodynamics – Basic Concepts and Methods, 7th ed., Wiley (2008), p. 39 19. ^ Kittel and Kroemer (1980). Thermal Physics. New York: W.H. Freeman. ISBN 978-0-7167-1088-2. 20. ^ Onsager, L. (1931). "Reciprocal relations in irreversible processes". Phys. Rev. 37 (4): 405–26. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405. 21. ^ Martyushev, L.M.; Seleznev, V.D. (2006). "Maximum entropy production principle in physics, chemistry and biology". Physics Reports. 426 (1): 1–45. Bibcode:2006PhR...426....1M. doi:10.1016/j.physrep.2005.12.001. Further reading External links
a4e2d5b2843673a9	Equation Solving More about Equation Solving For more about equation solving please refer to another notebook of mine: Intelligence. There are so many methods and techniques to solve an equation. Here we will review only some of them. Ordinary Differential Equations There are many important equations in physics. Fig. 10 Taken from Riley’s book. The are many methods to solve an ODE, 1. Green’s function. 2. Series solution 3. Laplace transform 4. Fourier transform Green’s Function Definition of Green’s Function The idea of Green/s function is very simple. To solve a general solution of equation (1)\[\frac{d^2}{d x^2} y(x) + y(x) = f(x),\] where $f(x)$ is the source and some given boundary conditions. To save ink we define \[\hat L_x = \frac{d^2}{dx^2} + 1,\] which takes a function $y(x)$ to $f(x)$, i.e., (2)\[\hat L_x y(x) = f(x).\] Now we define the Green’s function to be the solution of equation (2) but replacing the source with delta function $\delta (x-z)$ \[\hat L_x G(x,z) = \delta(z-x).\] Why do we define this function? The solution to equation (1) is given by \[y(x) = \int G(x,z) f(z) dz.\] To verify this conclusion we plug it into the LHS of equation (1) \[\begin{split}& \left(\frac{d^2}{dx^2} +1 \right) \int G(x,z) f(z) dz \\ =& \int \left[ \left(\frac{d^2}{dx^2} +1 \right) G(x,z) \right] f(z) dz \\ =& \int \delta(z-x) f(z) dz \\ =& f(x),\end{split}\] in which we used one of the properties of Dirac delta distribution \[\int f(z) \delta(z-x) dz = f(x).\] Also note that delta function is even, i.e., $\delta(-x) = \delta(x)$. So all we need to do to find the solution to a standard second differential equation \[\left( \frac{d^2}{dx^2} + p(x) \frac{d}{dx} + q(x) \right)y(x) = f(x)\] is do the following. 1. Find the general form of Green’s function (GF) for operator for operator $\hat L_x$. 2. Apply boundary condition (BC) to GF. This might be the most tricky part of this method. Any ways, for a BC of the form $y(a)=0=y(b)$, we can just choose it to vanish at a and b. Otherwise we can move this step to the end when no intuition is coming to our mind. 3. Continuity at $n-2$ order of derivatives at point $x=z$, that is \[G^{(n-2)}(x,z) \vert_{x<z} = G^{(n-2)} \vert _{x>z} ,\qquad \text{at } x= z.\] 4. Discontinuity of the first order derivative at $x=z$, i.e., \[G^{(n-1)}(x,z)\vert_{x>z} - G^{(n-1)}(x,z)\vert_{x<z} = 1, \qquad \text{at } x= z.\] This condition comes from the fact that the integral of Dirac delta distribution is Heaviside step function. 5. Solve the coefficients to get the GF. 6. The solution to an inhomogeneous ODE $y(x)=f(x)$ is given immediately by If we haven’t done step 2 we know would have some unkown coefficients which can be determined by the BC. How to Find Green’s Function So we are bound to find Green’s function. Solving a nonhonogeneous equation with delta as source is as easy as solving homogeneous equations. We do this by demonstrating an example differential equation. The problem we are going to solve is \[\left(\frac{d^2}{dx^2} + \frac{1}{4}\right) y(x) = f(x),\] with boundary condition (3)\[y(0) = y(\pi) = 0.\] For simplicity we define First of all we find the GF associated with We just follow the steps. • The general solution to \[\hat L_x G(x,z) = 0\] is given by \[\begin{split}G(x,z) = \begin{cases} A_1\cos (x/2) + B_1 \sin(x/2), & \qquad x \leq z, \\ A_2\cos (x/2) + B_2 \sin(x/2), & \qquad x \geq z. \end{cases}\end{split}\] • Continuity at $x=z$ for the 0th order derivatives, \[G(z_-,z) = G(z_+,z),\] which is exactly (4)\[A_1\cos(z/2) + B_1 \sin(z/2) = A_2 \cos(z/2) + B_2\sin(z/2).\] • Discontinuity condition at 1st order derivatives, \[\left.\frac{d}{dx} G(x,z) \right\vert_{x=z_+} - \left.\frac{d}{dx} G(x,z) \right\vert_{x=z_-} = 1,\] which is (5)\[-\frac{A_2}{2}\sin\frac{z}{2} + \frac{B_2}{2} \cos\frac{z}{2} - \left( -\frac{A_1}{2}\sin\frac{z}{2} + \frac{B_1}{2}\cos\frac{z}{2} \right) = 1\] Now we combine ((4)) and ((5)) to eliminate two degrees of freedom. For example, we can solve out $A_1$ and $B_1$ as a function of all other coefficients. Here we have \[\begin{split}B_1 &= \frac{ - 2/\sin(z/2) }{\tan(z/2) + \cot(z/2)} + B_2 , \\ A_1 &= A_2 + B_2(\tan(z/2)-1) + \frac{2}{\sin(z/2) + \cot(z/2)\cos(z/2)}.\end{split}\] • Write down the form solution using $y(x) = \int G(x,z) f(z) dz$. Then we still have two unknown free coefficients $A_2$ and $B_2$, which in fact is to be determined by the BC equation (3). Series Solution A second order ODE, \[y''(x)+p(x) y'(x) + q(x)y(x)=0\] Wronskian of this is \[\begin{split}W(x) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix},\end{split}\] where $y_1$ and $y_2$ are linearly independent solutions, i.e., $c_1 y_1 + c_2 y_2=0$ is only satisfied when $c_1=c_2=0$. Wronskian is NOT zero if they are linearly independent. Singularities of an ODE is are defined when $p(x)$ or $q(x)$ or both of them have singular points. For example, Legendre equation \[(1-z^2) y'' - 2 z y' + l(l+1) y = 0\] has three singular points which are $z=\pm 1, \infty$ while $z=0$ is an ordinary point. Solution at Ordinary Points Series expansion of the solution can be as simple as \[y(z) = \sum_{n=0}^{\infty} a_n z^n,\] which converges in a radius $R$ where $R$ is the distance from $z=0$ to the nearest singular point of our ODE. Solution at Regular Singular Points Frobenius series of the solution \[y(z) = z^\sigma \sum_{n=0}^{\infty} a_n z^n.\] The next task is to find the indicial equation. If the roots are not differing by an integer, we just plug the two solutions to $\sigma$ in and find two solutions independently. If the roots differ by an integer, on the other side, we can only plug in the larger root and find one solution. As for the second solution, we need some other techniques, such as Wronskian method and derivative method. Wronskian method requires two expression of Wronskian, which are \[W(z) = C e^{-\int^z p(u) \mathrm du}.\] From the first expression, we have \[y_2(z) = y_1(z) \int^z \frac{W(u)}{y_1(u)^2} \mathrm d u.\] However, we don’t know $W(z)$ at this point. We should apply the second expression of Wronskian, \[y_2(z) = y_1(z) \int^z \frac{C e^{-\int^z p(u) \mathrm du}}{y_1(u)^2} \mathrm d u,\] where the constant $C$ can be set to 1 as one wish. The derivative method is on my to do list. Comparing With A General Form For equation that take the following form, \[y'' + \frac{1 - 2a}{x} y' + \left( (b c x^{c-1})^2 + \frac{a^2 - p^2 c^2}{x^2} \right) y = 0,\] where $y\equiv y(x)$, we can write down the solutions immediately, \[y(x) = x^a \mathscr {Z}_p (b x^c),\] in which $\mathscr {Z}_p$ is the solution to Bessel equation, i.e., is one kind of Bessel function with index $p$. A Pendulum With A Uniformly Chaning String Length As an example, let’s consider the case of length changing pendulum, \[\frac{d}{dt} \left( m l^2 \dot{\theta}\right) = - m g l \sin\theta \approx = - m g l \theta.\] Notice that l is a function of time and \[l = l_0 + v t.\] Then the equation can be rewritten as \[\frac{d^2}{dl^2}\theta + \frac{2}{l} \frac{d}{dl} \theta + \frac{g/v^2}{l} \theta = 0.\] Comparing with the general form, we have one of the possible solutions \[\begin{split}a & = -1/2, \\ pc & = 1/2, \\ c & = 1/2, \\ p & = 1, \\ b & = 2\sqrt{g}/v.\end{split}\] This solution should be \[\begin{split}\theta &= l^a \mathscr{Z}_p(b l^c) \\ & = \frac{1}{\sqrt{l}} J_1(\frac{2\sqrt{g}}{v} \sqrt{l}).\end{split}\] Airy Equatioin Time-independent Schrödinger equation with a simple potential, \[\ddot{\Psi} + \alpha x \Psi = 0.\] Comparing it with general form, we should set \[\begin{split}a & = 1/2, \\ \lvert p c \rvert & = 1/2, \\ c & = 3/2, \\ b^2 c^2 & = \alpha^2.\end{split}\] So the two possible solutions are \[\begin{split}\Psi_1(x) & = \sqrt{x} \mathscr{Z}_{1/3}(2/3 \alpha x^{3/2}), \\ \Psi_2(x) & = \sqrt{x} \mathscr{Z}_{-1/3}(2/3 \alpha x^{3/2}).\end{split}\] The general solution is \[\Psi(x) = a \Psi_1(x) + b \Psi_2(x).\] Second Order Differential Equations and Gauss’ Equation Gauss’ equation has the form \[z(z-1)\frac{d^2}{dz^2} u(z) + \left[(a+b+1)z -c \right] \frac{d}{dz} u(z) + a b u(z) =0,\] which has a solution of the hypergeometric function form \[u(z) = {}_2 F_{1}(a,b;c;z).\] The interesting part about this equation is that its Papperitz symbol is \[\begin{split}\begin{Bmatrix} 0 & 1 & \infty & \\ 0 & 0 & a & z \\ 1-c & c-a-b & b & \end{Bmatrix} ,\end{split}\] in which the first three columns are the singularities at points $0,1,\infty$ while the last column just points out that the argument of this equation is $z$. This means, in some sense, the solution to any equation with three singularities can be directly written down by comparing the equation with Gauss’ equation. If you care, the actual steps are changing variables, rewriting the equation into Gauss’ equation form, writing down the solutions. Integral Equations Neumann Series AKA WKB For differential equation, whenever the highest derivative is multiplied by a small parameter, try this. But generally, the formalism is the following. First of all, we use Hilbert space $\mathscr L[a,b;w]$ which means the space is defined on $[a,b]$ with a weight $w$, i.e., \[\braket{f}{g} = \int_a^b dx w(x) f(x) g(x).\] Quantum Mechanics Books Notice that this is very different from the notation we used in most QM books. What is the catch? Try to write down $\braket{x}{u}$. It’s not that different because one can alway go back to the QM notation anyway. With the help of Hilbert space, one can alway write down the vector form of some operators. Suppose we have an equation \[\hat L u(x) = f(x),\] where $\hat L=\hat I + \hat M$. So the solution is simply \[\begin{split}u(x) &= {\hat L}^{-1} f(x)\\ &=(\hat I + \hat M)^{-1} f(x) .\end{split}\] However, it’s not a solution until we find the inverse. A most general approach is the Neumann series method. We require that \[\\| \hat M u \\| < \gamma \\| u \\|,\] where $\gamma\in (0,1)$ and should be independent of $x$. As long as this is satisfied, the equation can be solved using Neumann series, which is an iteration method with \[\begin{split}u(x)&=u_0(x)+ \delta u_1(x) + \delta^2 u_2(x) +\cdots \\ u_0(x) & = f(x).\end{split}\] As an example, we can solve this equation \[(\hat I + \ket{t}\bra{\lambda}) u(t) = f(t).\] We define $\hat M = \ket{t}\bra{\lambda}$ and check the convergence condition for $\lambda$. Step one is always checking condition of convergence. Step two is to write down the series and zeroth order. Then we reach the key point. The iteration relation is \[u_n(t) + \int_0^1 ds su_{n-1}(s) = 0.\] One can write down $u_1$ imediately \[u_1(t) = -\int_0^1 ds s u_0(s).\] Keep on going. Using Dyads in Vector Space For the same example, where $\hat L=\hat I + \hat M$, we can solve it using vector space if operator is linear. Suppose we have a $\hat M=\ket{a}\bra{b}$, the equation, in some Hilbert space, is \[\ket{u} + \ket{a}\braket{b}{u} = \ket{f}.\] Multiplying through by $\bra{b}$, we have \[\braket{b}{u} + \braket{b}{a}\braket{b}{u} = \braket{b}{f},\] which reduces to a linear equation. We only need to solve out $\braket{b}{u}$ then plug it back into the original equation. Back to top
7e02d191d1cdecd8	Michael Trott Even More Formulas… for Everything—From Filled Algebraic Curves to the Twitter Bird, the American Flag, Chocolate Easter Bunnies, and the Superman Solid July 18, 2013 — Comments Off Here are some of the non-mathematical laminae that Wolfram\|Alpha knows closed-form equations for: shape lamina Assume we want a filled curve instead of just the curve itself. For closed curves, say the James Bond logo, we could just take the curve parametrizations and fill the curves. As a graphics object, filling a curve is easy to realize by using the FilledCurve function. James Bond curve 007 curve For the original curves, we had constructed closed-form Fourier series-based parametrizations. While the FilledCurve function yields a visually filled curve, it does not give us a closed-form mathematical formula for the region enclosed by the curves. We could write down contour integrals along the segment boundaries in the spirit of Cauchy’s theorem to differentiate the inside from the outside, but this also does not result in “nice” closed forms. So, for filled curves, we will use another approach, which brings us to the construction of laminae for various shapes. The method we will use is based on constructive solid geometry. We will build the laminae from simple shaped regions that we first connect with operations such as AND or OR. In a second step, we will convert the logical operations by mathematical functions to obtain formulas of the form f(x, y) > 0 for the region that we want to describe. The method of conversion from the logical formula to an arithmetic function is based on Rvachev’s R-function theory. Let’s now construct a geometrically simple shape using the just-described method: a Mitsubishi logo-like lamina, here shown as a reminder of how it looks. As this sign is obviously made from three rhombi, we define a function polygonToInequality that describes the interior of a single convex polygon. A point is an interior point if it lies on the inner side of all the line segments that are the boundaries of the polygon. We test the property of being inside by forming the scalar product of the normals of the line segments with the vector from a line segment’s end point to the given point. It is simple to write down the vertices of the three rhombi, and so a logical formula for the whole logo. The last equation can be considerably simplified. The translation of the logical formula into a single inequality is quite simple: we first write all inequalities with a right-hand side of zero and then translate the Or function to the Max function and the And function to the Min function. This is the central point of the Rvachev R-function theory. By using more complicated translations, we could build right-hand sides of a higher degree of smoothness, but for our purposes Min and Max are sufficient. The points where the right-hand side of the resulting inequality is greater than zero we consider part of the lamina, otherwise the points are outside. In addition to just looking nicer and more compact, the single expression, as compared to the logical formula, evaluates to a real number everywhere. This means, in addition to just a yes/no membership of a point {x,y}, we have actual function values f(x, y) available. This is an advantage, as it allows for plotting f(x, y) over an extended region. It also allows for a more efficient plotting than the logical formula because function values around f(x, y) = 0 can be interpolated. So we obtain the following quite simple right-hand side for the inequality that characterizes the Mitsubishi logo. And the resulting image looks identical to the one from the logical formula. Plotting the right-hand side for the inequality as a bivariate function in 3D shows how the parts of the inequality that are positive emerge from the overall function values. Now, this type of construction of a region of the plane through logical formulas of elementary regions can be applied to more regions and to regions of different shapes, not necessarily polygonal ones. In general, if we have n elementary building block regions, we can construct as many compound regions as there are logical functions in n variables. The function BooleanFunction enumerates all these 2^2^n possibilities. The following interactive demonstration allows us to view all 65,536 configurations for the case of four ellipses. We display the logical formula (and some equivalent forms), the 2D regions described by the formulas, the corresponding Rvachev functions, and the 3D plot of the Rvachev R-function. The region selected is colored yellow. Cutting out a region from not just four circles, but seven, we can obtain the Twitter bird. Here is the Wolfram\|Alpha formula for the Twitter bird. (Worth a tweet?) By drawing the zero-curves of all of the bivariate quadratic polynomials that appear in the Twitter bird inequality as arguments of max and min, the disks of various radii that were used in the construction become obvious. The total bird consists of points from seven different disks. Some more disks are needed to restrict the parts used from these six disks. Here are two 3D versions of the Twitter bird as 3D plots. As the left-hand side of the Rvachev R-equation evaluates to a number, we use this number as the value (possibly modified) in the plots. We can also use the closed-form equation of the Twitter bird to mint a Twitter coin. The boundary of the laminae described by Rvachev-R functions has the form f(x, y) = 0. Generalizing this to f(x, y) = g(z) naturally extrudes the 2D shape into 3D, and by using a function that increases with \|z\|, we obtain closed 3D surfaces. Here this is done for g(z) ~ (z^2) for the Twitter bird (we also add some color and add a cage to confine the bird). The use g(z) ~ (z^2) means z ~ ± f(x y)^(1/2) at the boundaries, and the infinite slope of the square root functions gives a smooth bird surface near the z = 0 plane. Now we will apply the above-outlined construction idea to a slightly more complicated example: we will construct an equation for the United States’ flag. The most complicated-looking part of the construction is a single copy of the star. Using the above function polygonToInequality for the five triangle parts of the pentagram and the central pentagon, we obtain after some simplification the following form for a logical function describing a pentagram. Here is the pentagram shown, as well the five lines that occur implicitly in the defining expression of the pentagram. The detailed relative sizes of the stars and stripes are specified in Executive Order 10834 (“Proportions and Sizes of Flags and Position of Stars”) of the United States government. Using the data from this document and assuming a flag of height 1, it is straightforward to encode the non-white parts of the US flag in the following manner. For the parallel horizontal stripes we use a sin(a y) (with appropriately chosen a) construction. The grid of stars in the left upper corner of the flag is made from two square grids, one shifted against the other (a 2D version of an fcc lattice). The Mod function allows us to easily model the lattice arrays. This gives the following closed-form formula for the US flag. Taking the visual complexity of the flag into account, this is quite a compact description. Making a plot of this formula gives—by construction—the American flag. We can apply a nonlinear coordinate transformation to the inequality to let the flag flow in the wind. And using a more quickly varying map, we can construct a visual equivalent of Jimi Hendrix‘s “Star-Spangled Banner” from the Rainbow Bridge album. As laminae describe regions in 2D, we can identify the plane with the complex plane and carry out conformal maps on the complex plane, such as for the square root function or the square. Here are the four maps that we will apply to the flag. Column transformations And these are the conformally mapped flags. The next interactive demonstration applies a general power function z -> (shift + scale z)α to the plane containing the flag. (For some parameter values, the branch cut of the power function can lead to folded over polygons.) So far we have used circles and polygons as the elementary building blocks for our lamina. It is straightforward to use more complicated shapes. Let’s model a region of the plane that approximates the logo of the largest US company—the apple from Apple. As this is a more complicated shape, calculating an equation that describes it will need a bit more effort (and code). Here is an image of the shape to be approximated. So, how could we describe a shape like an apple? For instance, one could use osculating circles and splines (see this blog entry). Here we will go another route. Algebraic curves can take a large variety of shapes. Here are some examples: Similar to the Fourier series approximation properties that we used before for the curves, we now build on the Stone–Weierstrass theorem, which guarantees that any continuous function can be approximated by polynomials. We we look for an algebraic curve that will approximate the central apple shape. To do so, we first extract again the points that form the boundary of the apple. (To do this, we reuse the function pointListToLines from the first blog post of this series, mentioned previously.) We assume that the core apple shape is left-right symmetric and select points from the left side (meaning the side that does not contain the bite). The following Manipulate allows us to quickly to locate all points on the left side of the apple. To find a polynomial p (x, y)= 0 that describes the core apple, we first use polar coordinates (with the origin at the apple’s center) and find a Fourier series approximation of the apple’s boundary in the form equation. The use of only the cosine terms guarantees the left-right symmetry of the resulting apple. We rationalize the resulting approximation and find the corresponding bivariate polynomial in Cartesian coordinates using GroebnerBasis. After expressing the cos(kcos) terms in terms of just cos(cos) and sin(cos), we use the identity cos(cos)^2+sin(cos)^2 = 1 to eliminate cos(cos) and sin(cos) to obtain a single polynomial equation in x and y. As we rounded the coefficients, we can safely ignore the last digits in the integer coefficients of the resulting polynomial and so shorten the result. Here is a slightly simplified version. Here is the resulting apple as an algebraic curve. Now we need to take a bite on the right-hand side and add the leaf on top. For both of these shapes, we will just use circles as the geometric shapes. The following interactive Manipulate allows the positioning and sizing of the circles, so that they agree with the Apple logo. The initial values are chosen so that the circles match the original image boundaries. (We see that the imported image is not exactly left-right symmetric.) So, we finally arrive at the following inequality describing the Apple logo. Now that we have discussed how to make laminae of various shapes, let’s have some fun and use a 2D lamina from Wolfram\|Alpha to derive a variety of new 2D and 3D images from it. Like in the case of curves, there are nearly unlimited possibilities. First, as a small reward for all of the above implemented code, let’s reward ourselves with some chocolate. Starting with the Easter bunny lamina. We can easily extract the polygons from this 2D graphic and construct a twisted bunny in 3D. The Rvachev R-form allows us to immediately make a 3D Easter bunny made from milk chocolate and strawberry-flavored chocolate. By applying the logarithm function, the parts where the defining function is negative are not shown in the 3D plot, as they give rise to complex-valued function values. We can also make the Easter bunny age within seconds, meaning his skin gets more and more wrinkles as he ages. We carry out this aging process by taking the polygons that form the lamina and letting them undergo a Brownian motion in the plane. Let’s now play with some car logo-like laminae. We take a Yamaha-like shape; here are the corresponding region and 3D plot. We could, for instance, take the Yamaha lamina and place 3D cones in it. And in the next example, we use a Volkswagen-like shape to construct a half-sphere with a pattern inside. By forming a weighted mixture between the Yamaha equation and the Volkswagen equation, we can form the shapes of Yamawagen and Volksmaha. Next we want to construct another, more complicated, 3D object from a 2D lamina. We take the Superman insignia. superman lamina The function superman[{x, y}] returns True if a point in the x, y plane is inside the insignia. And here is the object we could call the Superman solid. (Or, if made from the conjectured new supersolid state of matter, a super(man)solid.) It is straightforwardly defined through the function superman. The Superman solid engraves the shape of the Superman logo in the x, y plane as well as in the x, y plane into the resulting solid. Viewed from the front as well as from the side, the projection is the Superman insignia. To stay with the topic of Superman, we could take the Bizarro curve and roll it around to form a Bizarro-Superman cake where Superman and Bizarro face each other as cake cross sections. Superman cake This cake we can then refine by adding some kryptonite crystals, here realized through elongated triangular dipyramid polyhedra. Next, let’s use a Batman insignia-shaped lamina and make a quantum Batman out of it. Batman lamina We will solve the time-dependent Schrödinger equation for a quantum particle in a 2D box with the Batman insignia as the initial condition. More concretely, assume the initial wave function is 1 within the Batman insignia and 0 outside. So, the first step is the calculation of the 2D Fourier coefficients. Numerically integrating a highly oscillating function over a domain with sharp boundaries using numerical integration can be challenging. The shape of the Batman insignia suggests that we first integrate with respect to y and then with respect to x. The lamina can be conveniently broken up into the following subdomains. All of the integrals over y can be calculated in closed form. Here is one of the integrals shown. To calculate the integrals over x, we need to multiply the integralsWRTy with sin(k Π x) and then integrate. Because k is the only parameter that is changing, we use the (new in Version 9) function ParametricNDSolveValue. We calculate 200^2 Fourier coefficients. This relatively large number is needed to obtain a good solution of the Schrödinger equation. (Due to the discontinuous nature of the initial conditions, for an accurate solution, even more modes would be needed.) Using again the function xyArray from above, here is how the Batman logo would look if it were to quantum-mechanically evolve. We will now slowly end our brief overview on how to equationalize shapes through laminae. As a final example, we unite the Fourier series approach for curves discussed in the first blog post of this series with the Rvachev R-function approach and build an apple where the bite has the form of the silhouette of Steve Jobs, the Apple founder who suggested the name Mathematica. The last terms of the following inequality result from the Fourier series of Jobs’ facial profile. Brilliant!!! Really impressive Posted by Fernando July 18, 2013 at 2:22 pm And this is what mathematicians do for recreation. I didn’t understand much of it, but I found it to be a delightful stream of consciousness and a fun illustration of a practical [?] application of math priinciples using Mathematica. Posted by Richard Johnstone July 18, 2013 at 3:33 pm This is not human!! Posted by Sebastian August 22, 2014 at 6:03 pm
d40667c2d627615e	torsdag 24 april 2014 Quantum Mechanics as Gift from God More Intelligent than Man • Quantum mechanics is, with relativity, the essence of the big conceptual revolution of the physics of the 20th century. • Now, do we really understand quantum mechanics? • It is probably safe to say that we understand its machinery pretty well; in other words, we know how to use its formalism to make predictions in an extremely large number of situations, even in cases that may be very intricate. • Heinrich Hertz, who played such a crucial role in the understanding of electromagnetic waves in the 19th century (Hertzian waves), remarked that, sometimes, the equations in physics are “more intelligent than the person who invented them” [182]. • The remark certainly applies to the equations of quantum mechanics, in particular to the Schrödinger equation, or to the superposition principle: they contain probably much more substance that any of their inventors thought, for instance in terms of unexpected types of correlations, entanglement, etc. • It is astonishing to see that, in all known cases, the equations have always predicted exactly the correct results, even when they looked completely counter-intuitive. • Conceptually, the situation is less clear. • Nevertheless, among all intellectual constructions of the human mind, quantum mechanics may be the most successful of all theories since, despite all efforts of physicists to find its limits of validity (as they do for all physical theories), and many sorts of speculation, no one for the moment has yet been able to obtain clear evidence that they even exist. Future will tell us if this is the case; surprises are always possible! Laloe illuminates the fact that modern physicists (and nobody else) do not understand the modern physics of quantum mechanics, and do not even pretend to do so, as a conceptual revolution away from classical physics based on understanding. The argument is that the linear Schrödinger equation must be more intelligent than Schrödinger, since Schrödinger admittedly could not understand it and nobody else has ever claimed to understand it either. If the difference between science and religion is that science is all about understanding, while religion leaves understanding to divinity, modern physics appears to be more religion than science. But it is hard to understand that an equation that cannot be solved, always predicts exactly the correct results! It is more easy to believe that any observation made can be claimed to fit exactly with the equation, since checking is impossible. It would be more convincing if observation was somewhat different from theory. No, We Don't Understand Quantum Mechanics, But There Is Hope. Yes, QM is a strange world. The Preface of book Do We Really Understand Quantum Mechanics by Franck Laloe supplemented by an article with the same title, tells the truth about quantum mechanics: • In many ways, quantum mechanics QM is a surprising theory... because it creates a big contrast between its triumphs and difficulties. • On the one hand, among all theories, quantum mechanics is probably one of the most successful achievements of science. The applications of quantum mechanics are everywhere in our twentyfirst century environment, with all sorts of devices that would have been unthinkable 50 years ago. • On the other hand, conceptually this theory remains relatively fragile because of its delicate interpretation – fortunately, this fragility has little consequence for its efficiency. • The reason why difficulties persist is certainly not that physicists have tried to ignore them or put them under the rug! • Actually, a large number of interpretations have been proposed over the decades, involving various methods and mathematical techniques. • We have a rare situation in the history of sciences: consensus exists concerning a systematic approach to physical phenomena, involving calculation methods having an extraordinary predictive power; nevertheless, almost a century after the introduction of these methods, the same consensus is far from being reached concerning the interpretation of the theory and its • This is reminiscent of the colossus with feet of clay. • The difficulties of quantum mechanics originate from the object it uses to describe physical systems, the state vector (wave function) $\Psi$. • Without any doubt, the state vector is a curious object to describe reality! The message is that QM a formidable achievement of the human intellect which is incredibly useful in practice, but like a colossus with feet of clay has a main character flaw, namely that it is a curious way to describe reality and as such not understood by physicists. There are two ways the handle if a physical theory is not understood because it is so curious, either the theory is dismisssed as being seriously flawed or the curiosity is chosen as a sign that the theory is correct and beyond questioning by human minds. The reason QM is so mysterious is that the wave function $\Psi =\Psi (x_1,x_2,…,x_N)$ for an atom or molecule with $N$ electrons depends on $N$ independent three-dimensional space variables $x_1$, $x_2$,…, $x_N$, together with time, thus is a function in $3N$ space dimensions plus time and as such has no direct real physical meaning since real physics takes place in $3$ space dimensions. The wave function $\Psi$ is introduced as a the solution to a linear multi-dimensional linear wave equation named Schrödinger's equation of the form • $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$, where $H$ is a Hamiltonian operator acting on wave functions. The mysticism of QM thus originates from Schrödinger's equation and is manifested by the fact that there is no real derivation of Schrödinger's equation from basic physical laws. Instead, Schrödinger's equation is motivated as a purely formal manipulation of classical Hamiltonian mechanics without physical meaning. The main trouble with QM based on a linear multi-d Schrödinger equation is thus the physical interpretation of the multi-d wave function and the accepted answer to this enigma is to view • $\vert\Psi (x_1,…,x_N)\vert^2$ as a probability distribution of a particle configuration described by the coordinates $(x_1,…,x_N)$ representing human knowledge about a physics and not physics itself. Epistemology of what we can know is thus allowed to replace ontology of what is. The linear multi-d Schrödinger equation thus lacks connection to physical reality. Moreover, because of its many dimensions the equation cannot be solved (analytically or computationally), and the beautiful net result is that QM is based on an equation without physical meaning which cannot be solved. No wonder that physicists still after 100 years of hard struggle do not really understand QM. But since Schrödinger's linear multi-d equation lacks physical meaning (and neither can be solved) there is no compelling reason to view it as the foundation of atomistic physics. It appears to be more constructive to consider instead systems of non-linear Schrödinger equations in $N$ three-dimensional wave functions $\psi_1(x),…,\psi_N(x)$ with $x$ a 3d space coordinate, in the spirit of of Hartree models, as physically meaningful computable models of potentially great practical usefulness. Sums of such wave functions then play a basic role and have physical meaning, to be compared the standard setting with $\Psi (x_1,…,x_N)$ in the form of Slater determinants as sums of muli-d products $\psi (x_1)\psi (x_2)…\psi (x_N)$ of complicated unphysical nature. tisdag 22 april 2014 Omodern Matematikundervisning Utan Ansvariga Matematiker Matematikinstitutionerna vid KTH och Chalmers skickar varje år en ny larmrapport om ytterligare försämrade matematikkunskaper hos nyantagna teknologer och nu var det dags igen: Med larmrapporten friskriver sig högskolematematikerna från sitt ansvar att se till att landets matematikutbildning är modern och funktionell, genom att skylla på skolmatematiken: • De högskolelärare som SvD pratar med är överens om att studenternas svaga grundkunskaper gjort att utbildningsnivån vid högskolorna sänkts undan för undan. • Visst har vi anpassat nivån, men det är inget folk vill tala högt om. • För svag matteundervisning i grundskola och gymnasium, i kombination med en för generös betygsättning, ligger bakom problemen. Men skolmatematiken är en (förenklad) variant av högskolematematiken och anledningen att skolmatematiken inte längre fungerar är att högskolematematiken är omodern och inte motsvarar datorsamhällets nya möjligheter och behov. När jag försöker få högskolematematikerna att bära sitt ansvar och modernisera utbildningen möts jag av oförstående och uppgivenhet och mitt öppna brev till Svenska Matematikersamfundet och Nationalkommitten för Matematik leder ingenstans. Se också mitt inlägg i kommande maj-nummer av SMS-Bulletinen. torsdag 17 april 2014 Extremism of Modern Physics as Bluff Poker Physics Modern physics has been driven into an increasingly extremist position with focus on extremely small or large spatial or temporal scales or extremely large energies. When problems were met on a certain (extreme) scale, the study was directed to yet more extreme scales and energies, as in a steadily increasing bet in a game of poker with little on hand to never get called. When LHC does not deliver, then the bet is raised to a new bigger more powerful LHC... When Einstein was pressed about the meaning of his special theory of relativity, he increased the bet to general relativity and when pressed about the meaning of general relativity he jumped the bet to cosmology... When physicists after the introduction of quantum mechanics faced questions about the electronic structure of atoms and molecules, they turned to the three orders of magnitude smaller proton and neutron forming atomic kernels, and then to the quarks forming the proton and neutron and then ultimately to string theory on scales 15 orders of magnitude smaller than the proton in an ulitmate attempt to find the origin of gravitation acting on cosmological scales. In each case the problems met on one scale were met by resort to smaller or larger scales, steadily increasing the bet and preventing a call. Today cosmology is directed to multiversa and inflation after Big Bang as the next step after Einstein's cosmology of general relativity supposedly all originating from string theory. But this may be the last possible bet and a call is approaching anticipated as a crisis in physics. onsdag 16 april 2014 Crisis in Physics vs Computational Physics The May14 issue of Scientific American asks the following questions: These questions naturally present themselves because modern theoretical physicists have driven themselves to search for the truth on scales which are either too small (string theory) or too big (cosmology) to be assessed experimentally. But theory without experiment may well be empty theory and that may be the meaning of the crisis. Of course, advocates of string theory like Lubos, forcefully denies that there is a crisis in physics. But there are other blog voicesand leading physicists show little hope.. But modern physicists have a new tool to use and that is computational physics, which offers an experimental laboratory without the scale limits of a physical laboratory. Computational physics needs computable models, but both quantum mechanics and general relativity are based on models which are not computable, and so there is a lot of work to be done. The question is if modern theoretical physicists have the right training to do this work. måndag 14 april 2014 Wanted: Constructive Physics Wanted: Constructive version of Schrödinger's equation! The book Constructive Physics by Y.I. Oshigov has an important message: • Only in the rebuilding of the gigantic construction of the modern physics in the constructive manner can open doors to the understanding of the complex processes in the sense of exact sciences. • The modern situation in physics looks like a crisis, and the genealogy of this crisis is the same as for the crisis in mathematics in the first third of the 20th century: this is the crisis in the axiomatic method. • Today we possess the more exact kit of instruments of the constructive mathematics: algorithms must replace formulas. • (The multidimensional wave function) harbors serious defects….it does not allow the computation of such functions already for a small number of particles, for example 10, let alone for the more complex systems. • This complexity barrier is principal. We should not think then that the quantum theory for many bodies gives such reliable answers to questions as it was the case in one particle case. In short, quantum mechanics based on Schrödinger's equation for a wave function in $3N$ space dimensions for $N$ particles (electrons or kernels) must be given a new constructive form. A real challenge! My answer is given as Many-Minds Quantum Mechanics. onsdag 9 april 2014 Popper: Realism vs Quantum Muddle vs Statistics Karl Popper starts out Quantum Theory and the Schism of Physics, as Vol III of Postscript to Logic of Scientific Discovery, with the following declaration: • Realism is the message of this book. • It is linked with objectivity…with rationalism, with the reality of the human mind, of human creativity, and human suffering. In Preface 1982: On a Realistic and Commonsense Interpretation of Quantum Mechanics, Popper gives his verdict: • Today, physics is in a crisis….This crisis is roughly as old as the Copenhagen interpretation of quantum mechanics. • In my view, the crisis is, essentially, due to two things: (a) the intrusion of subjectivism into physics; and (b) the victory of the idea that quantum theory has reached a complete and final truth. • Subjectivism in physics can be traced to several great mistakes. One is the positivism or idealism of Mach. Another is the subjectivist interpretation of the calculus of probability. • The central issue here is realism. That is to say, the reality of the physical world we live in: the fact that this world exists independently of ourselves; that it existed before life existed,…and that it will continue to exist long after we have all been swept away. • The subjectivist dogma was too deeply entrenched within the ruling interpretation of quantum mechanics, the so-called Copenhagen interpretation… this is how the great quantum muddle started….and the whole terminology, introduced in the early period of the theory, conspired to make the muddle worse and worse. • Another source of the crisis in physics is the persistence of the belief that quantum mechanics is final and complete. • Philosophers and physicists have been all too prone under the direct influence of Machian positivism, to take up idealist positions… • One of the things that this volume of the Postscript tries to to is to review many of the past arguments for idealism - which many current physicists still simply take for granted - and to show their error. But Popper, one of the greatest philosophers of science of the 20th century, talked to deaf ears and the crisis in physics is deepening every year… Another thing is that Popper deepened the crisis be dwelling deeper into the statistical interpretation of the wave function of Born as the basis of the Copenhagen interpretation. Popper thus identified the crisis but then himself got drowned by the muddle... Quantum Theory: Flight from Realism The book Quantum Theory and the Flight from Realism by Christopher Norris is introduced by: • Norris examines the premises of orthodox quantum theory as formulated most influentially by Bohr and Heisenberg….as requiring a drastic revision of principles which had hitherto defined the very nature of scientific method, casual explanation and rational enquiry. • Putting the case for a realist approach which adheres to well-tried scientific principles of casual reasoning and interference to the best explanation, Norris clarifies the debate… Norris continues: • In this book I examine various aspects of the near century-lonh debate concerning the conceptual foundation of quantum mechanics (QM) and the problems it has posed for physicists and philosophers from Einstein to the present. They include the issue of wave-particle dualism; the uncertainty attaching to measurements of particle location or momentum, the (supposedly) observer-induced "collapse of the wave-function"; and the evidence of remote superluminal interaction between widely separated particles. • It is important to grasp exactly how the problems arose and exactly why - on what scientific or philosophical grounds - any alternative (realist) contrual should have been so often and routinely ruled out as a matter of orthodox QM wisdom. This is an important book with the important mission of bringing realism back to physics after a century of anti-realist confusion ultimately corrupting all of science and with the adoption of climate alarmism by the American Physical Society as the tragic anti-realist irrational expression. tisdag 8 april 2014 Essence of Quantum Mechanics: Energy vs Frequency in Wave Models In Schrödinger's Equation: Smoothed Particle Dynamics we observed that Schrödinger's equation for Hydrogen atom with one electron (normalized to unit mass and charge) reads • $i\bar h\dot\psi + H\psi =0$, • $H\psi =\frac{\bar h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$, where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $\bar h$ Planck's (reduced) constant. In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form 1. $\bar h\dot\phi +H\chi =0$, 2. $\bar h\dot\chi - H\phi =0$. If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as • $\phi (x,t)=\cos(\omega t)\phi_E(x)$, $\chi (x,t)=\sin(\omega t)\phi_E(x)$, with $\bar h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$. The parallel to a harmonic oscillator (with $H$ the identity) is obvious. We see that the effect of the time derivative term is to connect energy $E$ to (angular) frequency $\omega$ by • $\bar h\omega = E$, • or $h\nu =E$, where $h=2\pi\bar h$ and $\nu =\frac{\omega}{2\pi}$ is frequency in Hertz, where $h$ acts as scale factor. Schrödinger's equation thus sets up a connection between frequency $\nu$, which can be observed as atomic emission lines, and a model of internal atomic energy $E$ as the sum of kinetic and potential energies of eigenfunctions of the Hamiltonian with the connection $\bar h\omega =h\nu = E$. Observations of atomic emission then show to fit with energy levels of the model, which gives support to the functionality of the model. The basic connection $\nu \sim E$ can also be seen in Planck's radiation law (with simplified high-frequency cut-off) • $R(\nu ,T)=\gamma T\nu^2$ for $\frac{h\nu}{kT} < 1$, where $R(\nu ,T)$ is normalized radiance as energy per unit time, with $\gamma =\frac{2k}{c^2}$, $T$ is temperature and $k$ is Boltzmann's constant, which gives an energy per cycle scaling with $\nu$ and a high frequency cut-off $h\nu$ scaling with atomic energy $kT$. The connection $h\nu =E$ also occurs in the law of photoelectricity • $h\nu = P + K$, where $P$ is the release energy and $K=eU$ is the kinetic energy of a released electron with $e$ the electron charge and $U$ the stopping potential. The atomic connection $h\nu =E$ between frequency and energy thus has both theoretical and experimental support, but it does not say that energy is "quantized" into discrete packets of energy $h\nu$ carried by particles named photons of frequency $\nu$. The relation $h\nu =E$ is compatible with wave models of both emission from atoms and radiation from clusters of atoms and if so by Ockham's razor particle models have no role to play. Atomic emission and radiation is a resonance phenomenon much like the resonance in a musical instrument, both connecting frequency to matter. Text books state that 1. Blackbody radiation and the photoelectric effect cannot be explained by wave models. 2. Hence discrete quanta and particles must exist. 3. Hence there is particle-wave duality. I give on Computational Blackbody Radiation evidence that 1 is incorrect, and therefore also 2 and 3. Without particles a lot of the mysticism of quantum mechanics can be eliminated and progress made. måndag 7 april 2014 The Strange Story of The Quantum: Physics as Mysticism The Strange Story of The Quantum by Banesh Hoffman bears witness to the general public about modern physics as mysticism: • This book is designed to serve as a guide to those who would explore the theories by which the scientist seeks to comprehend the mysterious world of the atom. • The story of the quantum is the story of a confused and groping search for knowledge…enlivened by coincidences such as one would expect to find only in fiction. • It is a story about turbulent revolution…and of the tempesteous emergence of a much chastened regime - Quantum Mechanics. • The magnificent rise of the quantum to a dominant position in modern science and philosophy is a story of drama and high adventure often well-nigh incredible. It is a chaotic tale…apparent chaos…nonsensical…intricate jagsaw…major discovery of the human mind. • Planck called his bundle or quota a QUANTUM of energy…This business of bundles of energy was unpardonable heresy, frightening to even the bravest physicist. Prandtl was by no means happy... But all was to no avail….to Max Planck had fallen the immortal honor of discovering them. • Einstein insisted...that each quantum of energy somehow must behave like a particle: a particle of light; what we call a photon…But how could a particle theory possibly hope to duplicate the indisputable triumphs of the wave theory? To go back to anything like the particle theory would be tantamount to admitting that the elaborately confirmed theory of electromagnetic phenomena was fundamentally false. Yet Einstein...was actually proposing such a step. • It is difficult to decide where science ends and mysticism begins….In talking of the meaning of quantum mechanics, physicists indulge in more or less mysticism according to their individual tastes. • Perhaps it is this which makes it seem so paradoxical. • Perhaps there is after all some innate logic in quantum theory. • The message of the quantum suddenly becomes clear: space and time are not fundamental. • Out of it someday will spring a new and far more potent theory…what will then survive of our present ideas no one can say… • Already we have seen waves an particles and causality and space and time all undermined. • Let us hasten to bring the curtain down in a rush lest something really serious should happen... Hoffman's book was first published in 1947. Since then the mysticism of modern physics has only become deeper... söndag 6 april 2014 Schrödinger's Equation: Smoothed Particle Dynamics Eigenfunctions of the Hamiltonian for the Hydrogen atom with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger's equation as a smoothed version of the particle dynamics of a harmonic oscillator. This is continuation of the previous post How to Make Schrödinger's Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge): • $ih\dot\psi + H\psi =0$, • $H\psi =\frac{h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$, 1. $h\dot\phi +H\chi =0$, 2. $h\dot\chi - H\phi =0$. We can see 1- 2 as an analog of the equation for a harmonic oscillator $\ddot u+\omega^2u=0$ written in system form (with $h=1$) • $\dot\phi + \omega\chi =0$ • $\dot\chi - \omega \phi = 0$, where $\phi =\dot u$ and $\chi =\omega u$, with solution • $\phi (x,t)=\cos(\omega t)$, $\chi (x,t)=\sin(\omega t)$. Here the velocity $\phi =\dot u$ connects to kinetic energy $\phi^2 =\dot u^2$ and $\chi =\omega u$ to potential energy $\chi^2 =\omega^2u^2$ and the dynamics of the harmonic oscillation consists of periodic transfer back and forth between kinetic and potential energy with their sum being constant. Returning now to the Hydrogen atom, we obtain multiplying 1 by $\phi$ and 2 by $\chi$ and integrating in space the following the energy balance • $h\frac{d}{2dt}\int\phi^2\, dx + \int \phi H\chi \, dx =0$ • $h\frac{d}{2dt}\int\chi^2\, dx - \int \chi H\phi\, dx =0$, • $ \int \phi H\chi \, dx = \int \chi H\phi\, dx =\frac{h^2}{2}\int\nabla\phi\cdot\nabla\chi\, dx +\int\frac{\phi\chi}{\vert x\vert}\, dx$, which shows upon summation (by the symmetry of $H$) that • $\frac{d}{2dt}\int\phi^2\, dx =\frac{d}{2dt}\int\chi^2\, dx =0$, which allows normalization to • $\int\phi^2\, dx = \int\chi^2\, dx = \frac{1}{2}$, • $\int\vert\psi\vert^2\, dx = 1$, for all time. Further, multiplying 1 by $\dot\chi$ and 2 by $\dot\phi$ and subtracting the resulting equations shows that • $\int (\phi H\phi + \chi H\chi)\, dx$ is constant in time. We can now summarize as follows: A. We see that the solution pair $(\phi ,\chi )$ of 1 - 2 as the real and imaginary part of Schrödinger's wave function $\phi$, represents a periodic exchange mediated by the Hamiltonian $H$ with balancing associated total energies • $\int \phi H\phi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\phi (x,t)\vert^2dx +\int\frac{\phi^2(x,t)}{\vert x\vert}\, dx$, • $\int \chi H\chi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\chi (x,t)\vert^2dx +\int\frac{\chi^2(x,t)}{\vert x\vert}\, dx$ as the sum of kinetic and potential energies. B. We see that Schrödinger's equation for the Hydrogen atom can be viewed as a smoothed version of a harmonic oscillator with the smoothing effectuated by the Laplacian and with $h$ acting as a smoothing parameter. C. We see that the system form 1- 2 combines the spatial eigenfunction $\phi_E$ with a periodic time dependence without introducing energy beyond the kinetic and potential energies defined by the Hamiltonian, thus associating these energies to frequency as the essence of quantum mechanics. D. We see that quantum mechanics and Schrödinger's equation can be given an interpretation which closely connects to classical mechanics, as smoothed particle mechanics, which avoids the common mystifications of particle-wave duality, complementarity, wave function collapse and statistics forced by insistence to use a multidimensional wave function defying a direct physical meaning. Extension to several electrons can be naturally be made following the idea of smoothed particle dynamics. For details see Many-Minds Quantum Mechanics. fredag 4 april 2014 Comparing Blackbody Radiation Spectrum to Atomic Emission Spectrum We conclude: torsdag 3 april 2014 Water Dam Analog of Photoelectric Effect Open sluice gates in the Three Gorges Dam in the Yangtze River. Einstein was awarded the 1921 Nobel Prize in Physics for his "discovery of the law of the photoelectric effect", connecting frequency $\nu$ of light shining on a metallic surface with measured potential $U$: • $h\nu = h\nu_0 + e\, U$ or $h(\nu -\nu_0) = e\, U$, where $h$ is Planck's constant with dimension $eVs = electronvolt\,\times second$, $\nu_0$ is the smallest frequency releasing electrons and $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ in a macroscopic experiment shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as scale factor with reference value • $h = 4.135667516(91)\times 10^{-15}\, eVs$, with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity into an energy conversion standard attributing $h\nu$ electronvolts to the frequency $\nu$, without any implication concerning the microscopic nature of the photoelectric effect. The award motivation "discovery of the law of the photoelectric effect" reflected that Einstein's derivation did not convince the committee as expressed by member Gullstrand: • When it was formulated it was only a tentatively poorly developed hunch, based on qualitative and partially correct observations. It would look peculiar if a prize was awarded to this particular work. To give perspective let us as an analog of the law of the photoelectric effect consider a water dam with sluice gates which automatically open when the level of water is $\nu_0$. The sluice gates will then remain locked as long as the water level $\nu <\nu_0$. Lock the sluice gates and let the dam fill to some water level $\nu >\nu_0$ and then unlock the sluices. The sluices will then open and water will flow through under transformation of potential energy into kinetic energy. Assuming the work to open the sluices corresponds to a level loss of $\nu_0$, a net level of $\nu -\nu_0$ potential energy will then be transformed into kinetic energy by the water flow through the sluices. The dam can be seen as an illustration of the photoelectric effect with the water level corresponding to frequency $\nu$ and the gravitational constant corresponding to $h$ and the width of the dam corresponding to the amplitude of incoming light. If $\nu <\nu_0$ then nothing will happen, if $\nu >\nu_0$ then the kinetic energy will scale with $h\nu$ and the total flow will scale with the width of the dam. Notice that noting in this model requires the water to flow in discrete lumps or quanta. The only discrete effect is the threshold $\nu_0$ for opening the sluices. onsdag 2 april 2014 Universal Quantum of Action: Standard Without Universality In recent posts on we have seen that Plank's constant $h$ in physics text books being presented as a universal quantum of action as a smallest "packet of action" as a fundamental constant of fundamental significance in the "quantized" world we happen to be part of, in fact is nothing but a conversion standard between two measures of energy, in terms of frequency $\nu$ in periods per second and electronvolt (eV), determined by Einstein's law of photoelectricity • $h(\nu - \nu_0) = e\, U$, where $\nu_0$ is the smallest frequency releasing electrons from a metallic surface upon exposure of light, $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as the scale factor measured in $eVs$ $electronvolts\times second$ as $energy \times time$ as action. The reference value obtained this way is with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity simply into a conversion standard (that is, a definition) of energy attributing $h\nu$ electronvolts to the frequency $\nu$. Another way of finding the conversion from frequency to electronvolt is using a Josephson junction. We now turn to Schrödinger's equation • $i\bar h\frac{\partial\psi}{\partial t}+H\psi=0$, where $\bar h=\frac{h}{2\pi}$ is Planck's reduced constant as conversion from periods $\nu$ per second to angular velocity $\omega$ with $h\nu =\bar h\omega$, and $H$ is a Hamiltonian of space dependence. An eigenvalue $E$ of the Hamiltonian represents energy with $\psi_E$ a corresponding space dependent eigenfunction satisfying $H\psi_E =E\psi_E$ and $\exp(i\omega t)\psi_E$ a corresponding solution of Schrödinger's equation with • $h\nu = \bar h\omega = E$, expressing energy in terms of frequency. We see that the appearance of $\bar h$ with the time derivative in Schrödinger's equation accounts for the energy conversion and is completely normal and without mystery. Next, we consider the space dependent Hamiltonian in the basic case of the Hydrogen atom: • $H\psi = \frac{\bar h^2}{2m}\Delta\psi + \frac{e^2}{r}\psi$ where $\psi =\psi (x)$ with $x$ a space coordinate, $r =\vert x\vert$, and $m$ is the mass of the electron. Normalising by changing scale in space $x=a\bar x$ and time $t=b\bar t$, we obtain the Hamiltonian in normalized atomic units in the form • $\bar H = \bar\Delta + \frac{2}{\bar r}$ with smallest eigenvalue $1$, • $a=\frac{\bar h^2}{me^2}$ as $Bohr\, radius$, • $b=\frac{\bar h2a}{e^2}$ as $Bohr\, time$ with $\omega =\frac{1}{b}$ angular velocity • $E =\frac{e^2}{2a}$ as $Rydberg\, energy$. We now observe that • $E\, b = \bar h$, • $E = \bar h\, \omega$, which shows that the also the space dependent part of Schrödinger's equation is calibrated to the energy conversion standard. Finally, Planck's constant also appears in Planck's radiation law and then in the high-frequency cut-off factor where $k$ is Boltzmann's constant and $T$ temperature. We see that again $h\nu$ appears as an atomic energy measure with a value that is not very precisely determined in its role in the cut-off factor. The value of $h$ from photoelectricity can then serve also in Planck's law. We conclude that Planck's constant $h$ is a conversion standard between two energy measures and as such has no meaning as a universal quantum of action or as integral multiples $nh\nu$ with $n=1,2,3,..$ of special significance other than by connection to eigenfunctions and eigenvalues. Ultimately, what is measured are atomic emission spectra in terms of frequencies and wave lengths which through Planck's constant can be translated to energies expressed in electronvolts (or Joule). Nothing of the internal atomic structure (in terms of $e$ and $m$) enters into this discussion. Planck introduced $h$ in a statistical argument in 1900 long before atoms were known, Einstein picked up $h\nu$ in his 1905 article on photoelectricity, before atoms were known, and Schrödinger put $h$ into his equation in 1926 to describe atoms. This line of events supports the idea that Planck's constant $h$ is a convention without any universal significance. Understanding the real role of Planck's constant may help to give Schrödinger's equation a physical interpretation which is free from mysteries of "quantization" and statistics. Versions of Schrödinger's equation based on an idea of smoothed particle mechanics then naturally present themselves, with $h$ acting as a smoothing parameter. PS Notice that the fine structure constant $\alpha = \frac{e^2}{\bar hc}=\frac{1}{137}$ can be expressed as $\alpha =\frac{2}{c}\frac{a}{b}$ which shows that $\alpha$ relates $Bohr\, speed\, =\frac{a}{b}$ to the speed of light $c$. This relation is viewed to be fundamental, but why is hidden in mystery. tisdag 1 april 2014 Royal Swedish Academy of Sciences: CO2 Warming Can Prevent New Ice Age The Royal Swedish Academy has issued a New Statement on the Scientific Basis of Climate Change giving up its former support of the CO2 global warming alarmism of IPCC and returning to the standpoint of the legendary foremost leading member of the Academy Svante Arrhenius, Nobel Prize in Chemistry in 1903, who in Worlds in the Making (1908) suggested that the human emission of CO2 would be strong enough to prevent the world from entering a new ice age, and that a warmer earth would be needed to feed the rapidly increasing population, of particular importance for the Swedish people under immediate threat of being covered under 1000 m solid ice: • Although the sea, by absorbing carbonic acid, acts as a regulator of huge capacity, which takes up about five-sixths of the produced carbonic acid, we yet recognize that the slight percentage of carbonic acid in the atmosphere may by the advances of industry be changed to a noticeable degree in the course of a few centuries. (p54) • Since, now, warm ages have alternated with glacial periods, even after man appeared on the earth, we have to ask ourselves: Is it probable that we shall in the coming geological ages be visited by a new ice period that will drive us from our temperate countries into the hotter climates of Africa? • There does not appear to be much ground for such an apprehension. The enormous combustion of coal by our industrial establishments suffices to increase the percentage of carbon dioxide in the air to a perceptible degree. (p61) A major revision of Swedish and European climate politics is expected to follow from the U-turn in the scientific view of the Academy. The Swedish King says he is ready to act, and turn on the heat in his many huge poorly insulated royal castles. New Theory of Flight Presented to the World Simulation movie of airflow around a jumbojet in landing configuration at large angle of attack. The revised version of New Theory of Flight has now been submitted to Journal of Mathematical Fluid Mechanics for expected swift publication. This article together with my former students Johan Hoffman and Johan Jansson represents the summit of my scientific career as a combination of mathematical analysis and computation. The article asks for a major revision of text book aerodynamics and opens new roads to aerodynamic design. And it is not a joke…Finally, The Secret of Flight can be revealed to humanity. Once the article has appeared in JMFM the new theory will be launched in a press release to media. Stay tuned…. Here is the Summary of article: • The new theory shows that the miracle of flight is made possible by the combined effects of (i) incompressibility, (ii) slip boundary condition and (iii) 3d rotational slip separation, creating a flow around a wing which can be described as (iv) potential flow modified by 3d rotational separation. • The basic novelty of the theory is expressed in (iii) as a fundamental 3d flow phenomenon only recently discovered by advanced computation and analyzed mathematically, and thus is not present in the classical theory. • Finally, (iv) can be viewed as a realization in our computer age of Euler’s original dream to in his equations capture an unified theory of fluid flow. • The crucial conditions of (ii) a slip boundary condition and (iii) 3d rotational slip separation show to be safely satisfied by incompressible flow if the Reynolds number is larger than 106. For lower Reynolds numbers the new theory suggests analysis and design with focus on maintaining (ii) and (iii).
7701c3d67687178b	Thursday, May 29, 2008 philosophy science and religion LoneRubberDragon and cooperative contributors WIKIPEDIA TEXTS: [0] CONTENTS [[1] through [20]] Videos, images, and writings (C) Copyright, [LoneRubberDragon / RubberCraft / DuRAGON SeTO RuMi / Draashek'gaons / SET,236,926,765,732,171], Anno Domini 2007, 2008 [1] What if there is no God, as Science often says? AD 2008 05 29 A 0645 (rel, sci, phi) ....[1.4] Without God and without a saving science ....[1.7] Addendum, a condemnation of science. AD 2008 07 02 P 11:20 ....[1.8] Addendum, a "Ghost in the Shell" back hack. AD 2008 07 03 A 08:10 [2] Intelligent design theory. AD 2008 05 29 A 0700 (sci, phi) [3] Evolution design theory. AD 2008 05 29 A 0750 (sci) [4] The Dragon's Oroboro. AD 2008 05 29 A 0705 (phi) [5] Light and darkness. AD 2008 05 29 P 1140 (phi) [6] A better world is too merciful, easy, and Utopian, for All Powerful God. AD 2008 05 29 P 1150 (rel, phi) ....[6.2] The real world, there's no free lunches, with the All Powerful Father God YHVH. ....[6.3] If Utopia is too Utopian for God, a critic could go even further. [7] For an all powerful God, we, His children, are not His responsibility. AD 2008 05 29 P 1150 (rel) ....[7.2] God CAN create a stone so heavy He cannot lift it, called the free will soul that is certain not to perish at the Creator's hands [8] Preachers say the darndest things, like, God doesn't need you!. AD 2008 05 29 P 1150 (rel) [9] Some things that are science, but that science cannot explain, all point directly to a transcendent soul. AD 2008 05 29 A 0815 (rel, sci, phi) [10] Computers can be given free will and soul on the material plane of existence. AD 2008 05 29 A 0910 (sci) ....[10.2] The computer is connected to the Light of God. [11] The unbreakable paradox of an All Knowing God and human free-will. AD 2008 05 29 A 0910 (rel) [12] Abiogenesis chemical evolution. AD 2008 07 17 P 0900 (sci) [13] Chinese Han and Japanese Kanji studies. AD 2008 07 17 P 0900 (phi) [14] Quantum physics self question. AD 2008 07 17 P 0930 (rel, sci, phi) [15] Genesis to Revelation - Damnation to Salvation. AD 2008 08 23 A 00:25 (rel) Patrick Moran (P0M) and LoneRubberDragon (~~~~) contributions on Wiki: [16] Abiogeneis second version. AD 2008 09 02 P 08:40 (sci) [17] Chiral / Churl symmetry between Atheism and Theism. AD 2008 09 02 P 08:40 (rel sci phi) [18] Philosophies of existence nature and life. AD 2008 09 02 P 08:40 (sci rel phi) [19] Jumping spiders and such. AD 2008 09 02 P 08:40 (sci phi) [20] Multidimensional Taylor-Laurent series special various applications. AD 2008 09 08 P 1130 (mat), from my earlier looneyfundamentalist post [21] Lunar Retroreflector Rainbow / Planetary Crystalographic Reflections AD 2008 09 15 P 0800 (sci) from earlier talks [22] Wikipedia Laws of Classical Conservation shortfall. AD 2008 09 16 P 1050 (sci) [23] Renewable nuclear energy. AD 2008 09 17 A 1130 (sci) Other Links: Complaining generations: Flee to mountains, Adam and Eve flee from garden, Han Kanji, Finite Element Analysis: Quantum Physics: Genetic Algorithms test and Logos: Evolutionary algorithms and natural combinatorial chemistry, also Taylor-Laurent series outer-space: Abiogenesis materials: Bible sources: Clay catalyzation of existing RNA base polymerization, and adsorbtion and release characteristics: Lipid and early combinatorial chemistry protocell theory: Hypercycle chemistry: Combinatorial chemistry: Armen M Boldi, "Combinatorial Synthesis of Natural Product-Based Libraries", AD 2005 CRC Press "Traditionally, the search for new compounds from natural products has been a time- and resource-intensive process. The recent application of combinatorial methods and high-throughput synthesis has allowed scientists to generate a range of new molecular structures from natural products and observe how they interact with biological targets. Combinatorial Synthesis of Natural Product-Based Libraries summarizes the most important perspectives on the application of combinatorial chemistry and natural products to novel drug discovery. The book details the latest approaches for implementing combinatorial research and testing methodologies to the synthesis of natural product-based libraries. Interconnecting the important aspects of this emerging field through the work of several leading scientists, it covers the computational analysis of natural molecules and details strategies for designing compound libraries, using bioinformatics in particular. The authors describe numerous synthetic methods for producing natural products and their analogs, including engineered biosynthesis and polymer-supported reagents. They also discuss additional considerations for generating libraries, such as screening, scaffolding, and yield optimization. Other chapters examine specific classes of libraries derived from natural products including carbohydrates, polyketides, peptides, alkaloids, terpenoids, steroids, flavonoids, and fungal compounds. Drawing attention to the interplay of drug discovery, natural products, and organic synthesis, Combinatorial Synthesis of Natural Product-Based Libraries contains the most recent and significant methods used to search and assess new compounds for their ability to mitigate biological processes that may lead to improved treatments for various diseases Combinatorial Chemistry is equivalent to high-throughput synthesis of compound arrays in which side-chain, core structure, and stereochemical diversity are varied. At the heart of combinatorial chemistry is the parallel synthesis of compounds that may be lead-like, drug-like, or natural product-like. Two terms, recently introduced by Schreiber, define directionality of such libraries - target-oriented synthesis (TOS), and diversity-oriented synthesis (DOS). In the strictest sense, these two types of libraries fall within the scope of combinatorial chemistry yet possess unique characteristics. Targeted libraries generated by TOS aim to elicit a specific biological response based on a gene family or a theraputic area. DOS libraries, on the other hand, seek to generate more diversity than what has historically been the case for combinatorial libraries, by varying the skeletal and stereochemical elements of the core library structures. Tan has described several categories of such DOS libraries: (1) core scaffolds of individual natural products, (2) specific substructures from classes of natural products, and (3) general structural characteristics of natural products." Miscellaneous abiotic chemistry to biology: [0 [1 [SHI`][ShE`] 1] [1 [SHI`][SHI-] 1] [1 [TIA-N][TE?Ng] 1] [1 [TOu~][DO-u] 1] [1 [SHo-A`NG][JO-u~] 1] 0] [ [13,17,18]::[ [[3, 7, 8] x 1] ], [ 10 [5 x 2] ]::[CH5C2 = (CH)5]::[[micro-meso-macro]-metastable-tree-thread-ring] ] ================Back to Contents CREATED AD 2008 05 29 A 0645 UPDATED [1.3] to [1.8] AD 2008 06 12 P 07:30 UPDATED added [1.7] AD 2008 07 02 P 11:20 UPDATED added [1.8] AD 2008 07 03 A 08:10 UPDATED updated [1.8] AD 2008 07 07 A 06:40 [1] What if is there is no God, as Science often says? [1.1] Science without God and soul If there is no God, like science often says so, then what purpose is there in life, when nothing will last and live forever? Because without God, it is completely up to humans to discover how to save themselves and live in harmony before time ends. Just counting all of the humans' generations, their ages, and their numbers, one can see that roughly at least 60 billion people have already passed on, so many of them so greatly missed throughout. Perhaps, you knew one of them. And one can clearly see that 7 billion humans are, right now, on their way to a certain inevitable death in this world. [1.2] The hope in science I would only hope of Science, in it's state without any proof of God, that the future machines of science can find the ways to restore and preserve the continuity of all of humanity and life, through incredible, yet to be discovered reconstruction algorithms. I would hope that something like the alien looking evolved machines of man, busily reconstructing and remembering everything, about all things, and about all history, and about all people; that those machines could bring back everyone in their bodies, like in the end of the movie, "Artificial Intelligence", thus bypassing the fatal flaw of bringing people back in our corruptible bodies for, but, one day, in the enormous spans of universal time. [1.3] Without God at the end of time And when the stars have used up all of the energy of gravitational collapse complexity promoting fusion, and the machines have worked to mak a monolith. One that can compute with zero power. And the entire universe lies frozen and cold at absolute zero; that at that "end of time". That at that time, all of humanity, all of the machines, and all of the life of earth, all beloved, and all harmonious, could live together in the monolith, in the images, the living words, in that monolith sea of glass. Living on, when all the rest of the universe lies in the frozen ashes, from the times of the light. [1.4] Without God and without a saving science For if there is no one left at the end of time, it seems like an awful waste of space and time, right? To quote Shakespeare, "And all our yesterdays have lighted fools, the way to dusty death. Out, out brief candle. Life is but a walking shadow, a poor player, that struts and frets his hour upon the stage, and then is heard no more. It is a tale told by an idiot, full of sound and fury, signifying nothing.". [1.5] Final thought But I sometimes fear, that a science without a provable soul, or God, or any universal absolute Good with Purpose, that the science will inevitably doom all man and all life to perish utterly, at the end of time, in selfish frozen chaos. So the only thing to do today, if this is the way life is, and that this is how reality works, than it is better to eat drink and be merry, for to-morrow we all die ... without God, or salvation, and a science without a soul. [1.6] Reference References: Richard P. Feynman lectures, of computation, regarding the subject that computation, specifically, requires no work (power),, the movie "Contact", featuring Jodi Foster,, William Shakespeare's plays,, the movie, "2001: A Space oddessy", featuring the Monolith,, and the movie, "AI: Artificial Intelligence", featuring Haley Joel Osment, and Jude Law. [1.7] Addendum, a condemnation of science. [1.7.1]THE THINGS I DO SEE, do disappoint me, as you have noticed. As much as there are good words in science, they fall short. [1.7.2]Take this quote from Richard Dawking from "The God Delusion", page 35: [1.7.3]"An athiest in this sense of philosophical naturalist is somebody who believes there is nothing beyond the natural, physical world, no supernatural creative intelligence lurking behind the observable universe, no soul that outlasts the body and no miracles - except in the sense of natural phenomena that we don't yet understand." [1.7.4]And Dawkins asks on page 404, "to give life meaning and a point ... Is it a similar infantilism that really lies behind the 'need' for a God?". [1.7.5]IF science now and forever, by its adherent voices, will always refuse to save the soul, because the soul is nothing, and is non-existent, then we infants need a God to save us, and even save the openly and admittedly soul-less science. IT IS A DISAPPOINTMENT IN SCIENCE ADVOCACY. They are a group that will forever deny the ability of saving a soul beyond the body, that we are all just dead animate matter, for the short moment of living. Quantum physics will never explain why wave functions collapse (and uncollapse in quantum eraser experiments) in this universe, based in the spiritual-immaterial-untouchable-structural-analytical-informational-configurations of the material universe causing a soulful transcendental wave function collapse / uncollapse, that happens infinitely faster than the speed of light. But there is no supernatural soul, in that supernatural soul, both, and neither, and denied. So science will always fall short for now and forever, never admitting the soul into science reality. And they declare we are without God, so therefore we are all dead men walking, and so then what is the point and meaning in any life, I ASK? For in 1000 billion years, when all stars have died, and all life everywhere is dead and frozen in the ashes of the galaxies, than what was the meaning and purpose of the quintillions of universal lives, and the exact purpose of a vowed dead and frozen science? "Ghost in the Shell": "Ko'kaku kido'tai": "Navigation=nuclear-core machine=mobility=task-force-organization": "Stand-Alone Complex": Season 1: Episode 12, Title: [Zhong-Wen' Kanji-Hiragana] "{映画}{監督} の夢 - たちこまの {家岀}." [Romanji of Kanji-Hiragana On-kun reading] "{} {kan.toku} no yume(mu) - Tachikoma no {}" [English Zhong-Wen' Kanji-Hiragana radicalized transliteration] "{sun=big=sun cover=field=receptacle} {retainer=overview=dish on-top-of=eight=hook=right-hand=eye} (is-of-possessive) plants=eye=cover=death=evening - Tachikoma (is-of-possesive) {roof=place=boar=household sprout=border}" [English Zhong-Wen' Kanji-Hiragana transliteration] "{reflecting picture} {warden supervisor}'s dream - Tachikoma's {household emerge}." [Middle English translation] "Tachikoma's {Home Leaving-of} - {Projection-Picture} {Director}'s Dream." [English translation] "Tachikoma {Runs Away}, - The {Movie} {Director}'s Dream." "a stand alone episode, ========[[ [Human (Ren' \| Jin)] Major Mokoto Kusanagi: ]] >[Salvation in The Kingdom of Heaven (Wang'Guo' Tian-Tang' Zheng?Jiu` \| O'koku Tengoku Tamashii no kyu'sai)] isn't a bad idea, but all [life show is (Sheng-Huo'Ming` Sheng`Kuang` \| Sei.inochi.katsu Sei.kyo')] fundamentally transitory, or at least it should be. >But a [life show eternal (Sheng`Kuang` Yong?Jiu? \| Sei.kyo' Ei.kyu')] without a beginning or end, that only keeps the [saved (Ren' Zheng?Jiu` \| Jin Tamashii no kyu'sai)] fascinated, and never lets them go? >It's harmful, no matter how wonderful you may have thought it was. ========[[ [Eternal Lord (Yong?Jiu? Shen'Ling'Yang` \| Ei.kyu' Kami.rei.sama)] V.R. Director, Kannazuki: ]] >My, you're a tough critic! >Are you saying then, that there is a reality, that we in the [saved (Ren' Zheng?Jiu` \| Jin Tamashii no kyu'sai)], ought to return to? >For some people in the [Kingdom's saved (Wang'Guo' Ren' Zheng?Jiu` \| O'koku Jin Tamashii no kyu'sai)], misery is waiting for them the instant they return to reality. >Can you accept responsibility for depriving those people of their dreams? >No, I can't. >But dreams have meaning for you because you are fighting for them within reality. >Doing nothing but projecting yourself into [salvation (Zheng?Jiu` \| Tamashii no kyu'sai)] is the same as being dead. >I see you're a realist. >If a romantic is someone who escapes from reality, then yes. >Such a strong woman you are. >If the reality you believe in ever comes to be, call me. >When it does, I will leave [Heaven (Wang'Guo' Tian-Tang' \| O'koku Tengoku)]. CREATED AD 2008 05 29 A 0700 UPDATED [3] AD 2008 06 12 P 07:30 [3] [Evolution design theory]. [3.1] In [the beginning]. [] [Big Bang] to [Fusion of Elements] - [Natural Complexity] from [Natural Simplicity]. [The big bang] generated [basic elements, hydrogen and helium]. [Gravity] formed [1[stars] in [2[an element reprocessing] and [enriching series] of [stellar generations]2] over [large periods of time]1]. Within [1 each [nebula cloud] to [stellar generation]1], there occurs [fusion element transforming nuclear forces], producing [1 [ever more element enrichment] in [stars and supernova nebulas]1] which in turn produce [1 [ever heavier element bearing] [2[interstellar bodies] and [stars]2]1]. [This fusion reprocessing] assists [1[the increasing elemental complexity] of [the universe]1] all by [natural means]. [The design] is all of [1[inherent uniform universal impersonal assembly modalities] that are all [2[3[hidden within] and [coming from]3] [3[the natural physics] that was [initially setup]3]2]1]. [] [[Jets] and [Watches]] from [[the natural arrangements] of [simpler parts]]. For example, [U238] formed [1[naturally] from [simplicity]1]. [U238] has [1[92 protons], [146 neutrons], and [92 electrons] in [18 organized nesting shells]1]. [U238] is [an example] of [1[complexity] arising [naturally] from [simplicity]1]. [1[It] is [a blind watchmaker]1], assembling [something quite complex], through the means of [a tornado of energy flow] inside of [1[a fusing star] and then [a stellar supernova]1]. [U238] is much like [the quintessential Creationist's idea] of [1[a tornado] assembling [a jumbo jet] from [a graveyard of parts]1], or [1[a Swiss watch] coming from [shaking a bag of gears and cogs and springs]1]. And yet, [1[U238] [arises and exists naturally]1] as much as do, [1[Fe55], [C12], [Au197], and [all other atomic elements]1], all in [1[2[a whole family] of [92 set elements]2] of [natural finely tuned watches]1], that all [1[come into existence] from [nothing more mysterious] than [concentrated simplicity]1]. [1[Stars] too [assemble themselves] from [blind forces]1], all not necessarily requiring [1[2[God's eternal minding] of [the gravitational pull]2] of [2[a large collection] of [Hydrogen and Helium atoms] that are [scattered all over space]2]1]. [The design] is all of [1[inherent uniform universal impersonal assembly modalities] that are all [2[3[hidden within] and [coming from]3] [3[the natural physics] that was [initially setup]3]2]1]. [] [God's Intervention] considered separate from [His Created Spaces of Physics] - all that is [observed] on [the large scales]. At [this point] there is [nothing] that is [1[apparently specially interventionally controlled] by [additional interventional forces] coming from [outside of the physics] ever since [the Big Bang event]1]. That is, there is [nothing apparently occurring outside] of what is [1-[2[the forces] of [His natural physics] that were setup [at time zero]2]; which are the workings of [2[gravity potential energy pressure], [massive fluid dynamics], [nuclear reactions], and [initial simple and basic atomic elements]2], all occurring, in order to make [2[the natural factories] that [produce these complex atomic elements]2]-1]. [1 [One] [2[sees] and [can demonstrate]2]1] only what is [1-[2[a natural purity] of [3[purely natural complexity] arising from [purely natural simplicity]3]2], where [2[the physics] is of [God's creation]2], but which is not of [2[God's continual intervention Himself], as [a separate being] from [the creation of lower things]2], but which is more of [2[3[a Buddhist] or [a Hindu] concept 3] of [3[the intrinsic forces flowing] within and through [all things]3]2]-1]. Rejecting [that latter posit] indicates that [the design] is all of [1[inherent uniform universal impersonal assembly modalities] that are all [2[3[hidden within] and [coming from]3] [3[the natural physics] that was [initially setup]3]2]1]. [] [The Elemental God] is [disputed], not [The Separate Being God]. NOTE: Though [1[I disagree strongly] of, and about, [2[a science] without [a soul]2]1], yet [1[at the same time], [I fail myself]1]. With [1[not those God theories], that [2[call God] as [a being]2]1], but [that] which also goes further as to [1[call God] as [all of the forces] like [fusion], [gravity], [matter], [lightning], [rain], and [life]1], which is at [1[the point] where [I fail myself]1]. [I] find that [it] is like declaring, [1[Zeus], [Jupiter], or [Thor] make [lighting happen]1]. And more than [that], [it] is [1[polytheistic] where in [2[all of the forces] with [no clear distinction of boundaries]2] have [a god for each force and modality] covering [2[earthquakes], [volcanoes], [weather], [lightning], [floods], [fusion], [gravity], [so-called accidents], [life forms], [molecular binding in life], and [human deforming mutations], to whit [name a few]2]1]. [It] makes [all people] merely components of [1[His Body] filled with [the pitfalls of His Natural Body] and [distortions of physical perceptions] bound to [His Body's perception modalities]1] that all [1[try and test] of [the small wisp of Self] that [we all have] also arising from [His Body]1]. Rejecting [that latter posit] indicates that [the design] is all of [1[inherent uniform universal impersonal assembly modalities] that are all [2[3[hidden within] and [coming from]3] [3[the natural physics] that was [initially setup]3]2]1]. [] [The Elemental God] is [by most people of Monotheism] to be considered as [animism] that is not compatible with [The Monotheist Separate [One Being God] religions], rejecting in most part [the old ways]. [It] is much like [1-[2[a scientific child] of [centuries ago]2] when holding up [2[a piece of amber] that is [fur charged] (amber in the Greek ήλεκτρον = "electron")2] when trying to [2[explain and demonstrate] that [it] is how [lightning forms]2]-1], that which then makes [1=[2[the religious leaders] of [the times] to [say]2], firstly, [2- where for do [3[4 you dare speak 4] this [4[heresy] against [the God-of-Lightning]4]3], just have [3[faith in God-of-Lightning] for [your complete and whole prosperity] along with [every other God]3], and stop [3[fooling yourself] with [4[such difficult so-called wise ideas of yours], of [the material plane]4]3]-2], and then continue saying, [2= for if [3 [you persist and continue] in [such thoughts]3], then [3-[4[The God of Lightning] will [cast you down]4] for [4[creating a disturbance] in [the body of the-God-of-Lighting]4], and [the God-of-Lighting] will [4[destroy you] for [5[being foolish] in [the wise ways of men]5]4]-3]=2]=1]. Rejecting [that whole posit] indicates that [the design] is all of [1[inherent uniform universal impersonal assembly modalities] that are all [2[3[hidden within] and [coming from]3] [3[the natural physics] that was [initially setup]3]2]1]. It is a God, sometimes seen today, that is merciful, but always holds the penalties of death and eternal separation and suffering, as a secret means, an instrument, to make, even coerce, you to "freely" come to His compassionate ways (with irony in this context). I, nor science, can deny God, or at the least a spiritual plane of experience, going by human simple senses. I simply think, God doesn't have to interfere with physics, to allow the same things to happen, as they appear to happen, naturally, in this universe, and that humans' traditions tend to overextend their thought on what they think they know of how God is actually involved on a daily basis with the material plane. It doesn't deny that God created the forces as they are, and that the universe is His body, and that it runs as designed, but that the design, once set into motion, doesn't require additional interactions to make things occur. The only true act of creation is the Big Bang, and from there, it is the works of the material plane, separate from God, as much as it came from God. God doesn't need to intervene every time two people have sex for the biochemistry to work. God doesn't need to intervene every time a mitochondria produces units of molecular energy. And God doesn't need to intervene to produce massive tons of heavy elements in the growth and supernova of stars to make Lithium through Uranium. They "simply happen" as a natural part of complexity arising from simplicity, quite naturally from the rules of the game started at the Big Bang formation of the physics of the universe. [3.2] Chemistry folds back on itself in increasing complexity like folding a processing taffy In a fluid environment, on a element enriched planet, circling a star giving it light, which makes for an energy open system, one finds there has formed quite natural catalytic chemistries, that have built on themselves in layers of catalytic chemistries, building new species of chemicals up in natural energized hierarchies, through molecular concentrations and the most durable and flexible diverse chemistries, over time. The molecules that formed,, autocatalytic, and hypercycle catalytic reaction chains, and networks,, that they naturally formed a backbone of many and naturally diverse sets of self sustaining chemical reaction systems. Thus, systems with the longest lived, most numerous, and most reaction facilitating molecular units, enriched a concentrated chemical environment of their own prosperity, over chemicals without prosperous ways. Complexity, once again, is seen springing forth naturally, from simplicity, without God being considered the matrix of chemical reactions done daily. [3.3] Combinatorial chemistry reaches for self control. Organo-chemical elements of memory, interconnect, amplification, control, interactions, and programs, began to form, and combinatorially interact, in the quite natural, and ever increasing combinatorial chemistry matrix. Catalytically productive forms of programs dominated the soup's products, and complicated themselves in the hierarchies of catalytically related models that generated numbers, durability, and utility, in the matrix of networked catalytic chemical processes. All other reactions that lead "unprosperously", declined in their own numbers in the face of the feedback loops of the cooperative catalytic combinatorial chemistry. The more time space invariant programs, found in enduring numbers, over short lived sparse programs, additionally acquired basic processing structures in exploring their own related chemical species, and allowing programmed catalyzation in digital forms that are ever more enduring, and found in multiplied numbers, over lesser prosperous chemical exploration programs of chemistries. [3.4] Digital combinatorial chemistry takes control with polymers. Program chemical interactions intersect with new molecules of lipid, protein, and proteinoid interactions, to form sheets and cells in semi-digital organic reaction domains. Reactions begin to develop more digital associations with RNA fragments. New catalytic codes dominate on RNA products in permutations of feedback, memory, and control effects. Likewise RNA and protein programs begin to develop associations with basic fragments of DNA machines, and proliferate the best DNA processing modules, in permutations mixing for models with numbers, durability, and facility of reactions. For those with prosperity and inherent wisdom in their own numbers, over others without natural "wisdom" in their codes of reactions. [3.5] The first cells of life. Basic DNA, RNA, and protein associated programs develop associations in growing DNA modules, and proliferate. First life may be considered as occurring here, as a micro bacterial level of life form. All formed through natural coded functions in numbers, environmental strength, and function, using the lipid bubbles to promote a greater stability reaction unit that can travel like pollen, and grow equilibrium numbers. [3.6] Multi-cellular life. Evolutionary forces continue, sometimes facing catastrophes, but throughout, surviving in some forms of life, always through numbers, durability, and robust fitness in their chemical and mutual-other-cellular domains. Some life forms group and interact in ways characteristic of macroscopic life forms, that prosper naturally when together in cooperative reaction groups, over others that survive alone, or perish alone. Numerous collections of multiple cells begin sharing cooperative efforts of survival, reproduction, and exploration in multitudes of new forms of cellular groupings. Very soon, as the quintillions of parallel oceanic codes, over millions of generations, worked themselves out into these new higher form configuration codes, so that now one sees an explosion of multi-cellular life, with numerous body designs, flourishing and competing for efficiency, survival, reproduction, and numbers, over other colonies. [3.7] Individual and social thoughts blossom into the future. Eventually macroscopic animal life acquired macroscopically active and abstract, individualistic thought. These thoughts increase in complexity, until agent processing and abstraction technologies springing from life, eventually allows the creation of new / novum "top-level" life that is macroscopically designed and built, using macroscopic and microscopic mechano-chemical systems, most notably using machines, integrated circuits, inorganic mechano-chemical nanotechnology, and organic genetics of past life projected to its limits. The things are created using all of the prior abstract and scientific thought, and precise intelligent chemical system design, coming from the designing, computing, and building machine and life domain. The technology is all used to go back down into the microscopic invisible domain of existence, to efficiently achieved by intelligent design methods, the new machines of life forms. New forms of biological life are created, approximating predictable future stable life forms, and done long before animal centric evolution competition would normally create, or that might never occur due to time limitations in a competitive cooling and naturally disruptive universe, with evolution occuring around more naturally exxisting slow macroscopic life trying to persist through time. [3.8] Definitions. ----Combinatorial chemistry - a system of chemistry, where every reaction between every specie of chemical in a mix is considered. Say for an initial early mix of 1,000 chemical species of natural molecules, there are a potential 1,000,000 reactions that can occur (or more with multi-modal state chemical species exposed to energies). Some reactions build things up. Some reactions tear things apart. Some reactions support themselves alone. Some reactions cyclically support each other in rings. Some reactions network cyclically support each other in networks of reactions. ----catalysis - a chemical specie that facilitates the reaction rate of another reaction by reducing the energy required for the reaction, and thus increasing the forward reaction rate, and equilibrium concentration. For example, a1 + a2 --> A at a low rate, naturally, but a reusable catalysts B that isn't part of the reaction makes the reaction more efficient. Schematically, a1 + a2 --B--> A, where B merely assists the reaction by lowering the energy of production, but B isn't consumed in the reaction. ----complexity increasing naturally - combinatorial chemistry feedback in an energy-open chemical-system has natural complicating posiblities, like the ocean exposed to sunlight, where energy from sunlight allows new chemical products to form, that would not otherwise exist, if say, frozen cold in an energy closed system. And when new chemical species form, this only allows more potential reactions to occur that couldn't have occurred before the new energy supported chemical species existed. This expands the chemical species, and in the multiplication-product of a reaction matrix, grows with the square of the number of chemical species. For 100 species there's 100 * 100 = 10,000+ potential reactions at night. With sunlight, it may be 20,000+ potential species allowed. Then of those 20,000 potential reactions, 900 new species of chemicals may persist. Then one has 100 + 900 = 1,000 chemical species. The matrix of all inter-actions shows that there's now 1,000,000 reaction species at night, or 2,000,000 under sunlight. Of this, there may be 9,000 new chemical species products formed in numbers, by allowing sunlight energizing of 2,000,000 nodes. This then means that there's 1,000 + 9,000 = 10,000 chemical species. In the matrix, there can now be 10,000 * 10,000 potential chemical species reactions or 100,000,000 reactions at night, and 200,000,000 in daylight. Of these there may be 90,000 new species of the 200,000,000 reaction nodes. This gives 10,000 + 90,000 = 100,000 chemical species. It's matrix allows 10,000,000,000 reaction nodes at night and 20,000,000,000 reactions in the light. This loop goes on nearly ad-infinitum, until the chemical species are as numerous as a liter of solution containing, say, 1,000,000,000,000,000,000 molecules, that saturates with millions of chemical species both unique, and numerous enough for reactions. ----chemical reproduction - an auto-catalytic chemical reaction that allows duplication of a molecule. For example, a1 +a2 + a3 -- A --> A, is a schematic of an auto-catalytic chemical reproduction, where A isn't consumed, and allows another copy of A to form. ----Hypercycle reaction chain - a catalytic chemical reaction chain that is looped back on itself with mutually supporting reaction products. For example, if one schematically has: b1 + b2 -- A --> B, c1 + c2 --B --> C, and a1 + a2 -- C --> A, where a1, a2, b1, b2, c1, c2 are commonly available persistent base species, then the equilibrium level of A, B, C are increased in this feedback loop. Chemicals that are not part of a chain reaction simply reach their own natural forward reaction equilibrium concentration. ----Hypercycle reaction network - a set of chemical reaction hypercycles that overlap and/or feed back into each other. For example, when one schematically has: b1 + b2 -- A --> B, c1 + c2 --B --> C, a1 + a2 -- C --> A, e1 + e2 -- D --> E, f1 + f2 -- E --> F, d1 + d2 -- F --> D, b1 + b2 -- D --> B, e1 + e2 -- A --> E, a3 + a4 -- F --> A, and d3 + d4 -- C --> D, then there are two hypercycle reaction chains for A, B, C, and D, E, F. There are also two catalytic reactions supporting each other between the two feedback loops helping produce more B, E. Finally, there are additional inter hypercycle feedback loop chemical reactions that cooperatively produce more products A, D for each other, using some of the very involved products found between the two loops. All of these reactions for a cooperative chemical system that help their own products proliferate, and once started up, have a degree of self organized stability in their feedback loop strength. ----digital chemistry - a chemistry of reactions based on polymers of essentially digital modular molecular species, so that reactions facilitated by the polymers, can be closely related to their digital polymer code. DNA, for example, has 4 coding chemicals in general living nature. These can form into chains that are numerical in nature, and each chain number code has its own chemical reaction properties. For a chain 4 units long there is 4 * 4 * 4 * 4 = 256 chemical codes. When a supportive chemistry exists with these polymers, such as a hypercycle reaction network, the relative ease of a polymerization allows the many codes to be easily explored because of their digital character nature, and the products involved with a feedback strength will increase their numbers. This is because some codes will have a naturally existing hypercycle chain and/or network, and so will react well, while other polymer codes will react less well. The ones with superior qualities of supporting their own cooperative system of reactions, in easy combinatorial code exploration, will become more numerous, over the less fit reactions in the matrix. One of the best of the digital chemical polymers, are amino acids that have more than 20 common digital forms, and can form polymer chains of those numbers, having unique reaction characters or properties over their three dimensional polymer form with sites of hydrophobic or water repelling, hydrophilic or water attracting, fatty, polar or having an electric charge difference, and neutral or unreactive, character. These chemistry promoting properties, relatively easily forming into chains, mean that numerous varieties of reactive three dimensional molecules can be formed. ----self organizing systems - a system that self creates and supports itself from its own nature of durability, numbers, and effectiveness to the environment. CREATED AD 2008 05 29 A 0750 UPDATED [2] AD 2008 06 13 A 0245 [2] Intelligent design theory. [2.1] The end times final temple of Man. A crystalline form was fabricated by humans, at the atomic level, with macroscopically controlled microprobes, coated with programmable forcing and forming microscopic hands, which worked at assembling the crystalline form, from a mechano-chemistry fluid. Carefully encoded within the crystalline forms of numerous micro-structures was a living control program, feeding from light and electricity, that strikes the crystalline form. [2.2] Humanity and life in a sea of glass. A human looking at the finished crystalline form, would see spectral patterns of light and a material surface both with incredible complexity, forming and changing in psychedelic shapes, on the crystalline form. The crystal structures can be seen by the light that strikes the form. The crystal's micro-patterns translate and manipulate and live, through a labyrinthine control, process, scatter, absorption, and reemission of light, and electricity, all according the state of the inner crystalline structure's control programs. The entire crystaline form serves as a sensoria and mind of the light world, and it can turn pitchblende black when concentrating on observing the light of the cosmos striking it, and can turn into a surface of shimmering colorful laser light when expressing masses of information to the universe. When in near pitch black and absolute zero, the crystalline programs can still operate by transferring all of their operations, into the random vibrations of its particles, working within the low energy quantum states, that are used for producing coherent information flow, and is facilitated by any external photon excitation. In more temperate climates when a wisp of energy and good matter can be used, the surface of the form can programmably grow a surface with multicatalytic properties, and can generate a film of mechano-chemistry fluid for interacting with the materials around it. This allows interfacing with organic chemistries, or machines, or crystalline form generated circuits and micro forms. (try magnifying color patterns, while viewing on an LCD flat screen, to read the secret text pattern below human visibility!) At the "end of time", this crystalline form was all that remained of all humans and life on earth, that once circled a burning star, now long ago frozen cold, and thrown out of the galaxy, and into the void, back when humans once roamed the galaxy, casting out the solar system by a set of human controlled chaotic orbit controls, between the solar system with other galactic stars, and dances with double stars, bringing the whole solar system in tow. It was done to throw the solar system clear of the chaos of the galaxy once galactic material powers became too sparse for generating any new useful solar system galactic orbit navigating control. [2.3] Humanity traverses the cold dark cosmos. The crystalline form in this end times, is now in a space navigating robust planet surface landing and takeoff vehicle body. It circles several cold white dwarfs including the sun's cold white dwarf remains, and some neutron star cores, new large planets, and other solar system bodies, that were all collected into the solar system, back when the solar system once orbited the milky way galaxy. Frozen supplies outposts, setup when the universe was still warm, including the most ancient root of the earth and moon, now orbit the remains of Jupiter. The outposts are used to help extend the crystalline forms emergency operations, into the voids of endless intergalactic space and time. Now, along with occasional bursts of light in a celestial collision or light communication, the crystaline form's vehicle body uses sparse chaotic thrust dialing orbits to navigate the known solar system, and to sense the state of the universe, in the motions around the solar system, in thousands of years of observations. The crystalline form periodically engages landing for supplies tucked in the orbits around Jupiter, in the billions of years in cyclic schedules. The bodies of the solar system also are all chaotically controlled, using additional small controlling asteroidal bodies, carefully orbiting around the known solar system, and causing slight critical nudging configuration alterations at key Lagrangian saddle orbit bifurcation control points. The old earth around Jupiter, can no longer be landed on, as all of it’s machines have lost power, and the available solar system supplies are insufficient to get back off the planet, by the deepness of time in an increasingly cold universe. All that can be done is to look at the earth in the most feeble cold light, or by shining laser light and microwaves on it from the vehicle, as the crystaline form's vehicle body passes by earth. From the point of view of today's material world, it is semmingly so different from the tracts of careless mindless happier times spent in sunlight and warmth, with biochemistry, and bodies per-se, and with terrifying periods of wars and genocides. It turned out, by science that universe transmigration technology, to reach other universes or other parts of this universe, completely and utterly failed in the times of sunlight, to break the bonds of the travel limiting universe, and so now all humans and life are shunted in the living pattern of the program in the crystalline form, looking for salvation, from outside of what was seen in witness, and looking for an unlikely better time to occur, at a point hoped to be certain. but seemingly indeterminable. by all humans in the crystaline sea of glass. All life of the ancient-past universe has frozen and never heard from again, except for humanity, and the dragons which still orbit the milky way, producing the observations of faint communications, and their implied motions, in the now-distant milky way stellar, seen through rare frozen stellar collision event evidence. [2.4] Man's reach almost spanned the galaxy when there was light. Humans, long before the crystalline form, once traveled throughout the galaxy, and even found other similar life forms in many places in the galaxy. The nearly immortal dragons, were of note, swarming the spaces between the stars of the milky way galaxy. As the end of time approached, humans around the last of the burning stars, launched their last forms, with hoards of necessary materials, including planets, and even stars, back into the home solar system that all humans came from. Then humanity carefully ejected the solar system, under human control. No other life was so interested in living, and so froze to their ends throughout the fading galaxy It was unfortunate, humans couldn't modulate the stars of the entire galaxy, in high enough density during this time, to have created an entire milky way galactic oasis or stellar orbit order, in enough time for the solar system, but it was shown that controlling all of the stars simultaneously, so the heavy solar system would always remain safe from accidents, proved to be beyond frail humanity with the rapidly failing light of the stars over the billions of years of last light. If only a black hole ring could have been produced or found, useful life-time could have been dilated exponentially into the future, with the crystalline form vehicle body living in the greatly time slowed dilated center axis orbit carefully running through the black hole ring outside of the event horizon. [2.4] Now the solar system drifts away from the Milky Way into the safe void of space. In the after human time, all other life became extinguished, lacking all genetic preservation technology in the frozen cold of the end of time, and only dragons remained in the galaxy, feeding off the energy of interactions made possible through communications of chaotic orbits that collect matter and take mometum energy from double stars through chaotic navigating orbits. Some dragons are the size of mountains, and never land on anything larger than small asteroids. All have detailed and corroborated maps of the entire galaxy in space and time, and orchestrate their travels, and keep the galaxy in some measure of order for their travels by minimizing stellar and black hole collisions, and preventing star losses from the galaxy completely. The dragons are always traveling, and catching up with other dragons in communications. Some dragons have watched that little solar system of humanity, leaving the galaxy, to the edge of the void, thinking about those humans that they once met, billions of years ago, and wondering about those people with a [stone \| rock \| Petra] root. That solar system, perhaps, being a dragon in its totality, as some dragons believe, that is the size of a solar system, and trying to survive in the frozen universe, appearing to be without salvation outside of humans' plans, along with themselves as dragons. CREATED AD 2008 05 29 A 0705 AD 2008 09 15 P 0800 added dragon icon [4.1] Opening verse. THE [Good-Tengoku-Tok-Tian=-Oiranos-Tov] [DRAGON’S-ryu tatsu doragon-yong-long/ lao~han..fu..-draekoon-drakoneem] [OROBORO-wa shuki-panji wonhyong-huan/ xun/huan/-krikos keklos-taba’at zahav makhzor hadam] [OPENS-saita-p’in-shen=kai=de-anoigoo-leefko’akh’eynayeem leefto’akh]. THE [evil-aku-ak’an-huai..-chachos-ra’ah] [DRAGON’S-ryu tatsu doragon-yong-long/zhong= long/-draekoon-drakoneem] [OROBORO-wa shuki-panji wonhyong-huan/ xun/huan/-krikos keklos-taba’at makhzor] [CLOSES-tojiru-kamtta-wu/shi..-kleinoo-khat’akh lee’yeydey gemar]. [4.2] Good cooperates with good, evil disperses with evil. One to be with [Good-Tengoku-Tok-Tian=-Oiranos-Tov] creates a force of [unity harmony-Cho=wa-Chohwa-Qia..rong/qia..-enarmonizoo-Akhdoot harmonyah]. One to be hated of the world, for [Good-Tengoku-Tok-Tian=-Oiranos-Tov], The [enemy-tekigun-cho~k-di/ren/di/-echthros-oyveem], makes them one’s friend, when wise. So [Good-Tengoku-Tok-Tian=-Oiranos-Tov] even if they split, in the course of time, when wise, even as [evil-aku-ak’an-huai..-chachos-ra’ah] with [evil-aku-ak’an-huai..-chachos-ra’ah], when they split, in the course of time, when scheming. that builds with itself, is greater than [evil-aku-ak’an-huai..-chachos-ra’ah] that destroys itself, even as it destroys the [Good-Tengoku-Tok-Tian=-Oiranos-Tov], because the [Good-Tengoku-Tok-Tian=-Oiranos-Tov], can see the [evil-aku-ak’an-huai..-chachos-ra’ah] approaching at that time, and forms [unity harmony-Cho=wa-Chohwa-Qia..rong/qia..-enarmonizoo-Akhdoot harmonyah], when wise. [4.3] Individuality with goodness is acceptable. Individuality with evilness is dispersion. Individuality with [Good-Tengoku-Tok-Tian=-Oiranos-Tov] is [Good-Tengoku-Tok-Tian=-Oiranos-Tov], because [Good-Tengoku-Tok-Tian=-Oiranos-Tov] is of the same common heart, when wise. Individuality with [evil-aku-ak’an-huai..-chachos-ra’ah] makes [Good-Tengoku-Tok-Tian=-Oiranos-Tov], because [evil-aku-ak’an-huai..-chachos-ra’ah] is facilitated to be destroyed in [evil-aku-ak’an-huai..-chachos-ra’ah] is of its own many split confusion ways when scheming. [4.4] Good, be wiser than the evil. observe [evil-aku-ak’an-huai..], but do not become one, with [evil-aku-ak’an-huai..-chachos-ra’ah], do not form [unity harmony-Cho=wa-Chohwa-Qia..rong/qia..-enarmonizoo-Akhdoot harmonyah] with [evil-aku-ak’an-huai..-chachos-ra’ah]. though however, can sleep drowsy in ambivalence, allowing [evil-aku-ak’an-huai..-chachos-ra’ah] to decay the [Good-Tengoku-Tok-Tian=-Oiranos-Tov]. So vigilance is required, by the [Good-Tengoku-Tok-Tian=-Oiranos-Tov], in the battle against [evil-Aku-ak’an-huai..-chachos-ra’ah]. [4.5] The dynamics of these differential equations, triumph under man, under a hands off God. In this way, is [maximized-saidai ni suru-ch’oedaehwahada-shi~ zeng=jia dao.. zui..da.. xian..du..-aeksanoo ston anootato-merav makseemoom], in the time when God respectfully allows [evil-Aku-ak’an-huai..-chachos-Oiranos-ra’ah], in the time when also allowing all [Free will-Kettei suru-Kyo~lsshimhada-Jue/ding..-Khofesh dror lehakhleet], to enter His [Good-Tengoku-Tok-Tian=-Oiranos-Tov] containing the evolving forces destroying the [evil-Aku-ak’an-huai..-chachos--ra’ah], [4.6] Man must be vigilant under a hands off God. Do not let the illusions, of man’s laid out myths and legends, to then distract you, or make you ambivalent, away from the [Good-Tengoku-Tok-Tian=-Oiranos-Tov], or this world confusion will persist forever, as [evil-Aku-ak’an-huai..-chachos-Oiranos] cannot be forsaken, CREATED AD 2008 05 29 P 1140 [5] Light and darkness. [4.1] Traditions of Man. You hear it often said, that you can't have light without darkness. But if light permeates all things, then there is lighter and darker, and there is no real darkness at all. The failure of light to fully saturate and permeate all things, is the only thing making relative darkness. [4.2] Darkness has in it the light and lightness has in it the dark. Now, for things that merely reflect true light, there is an interesting flip. What is dark - takes the light in and holds on to it, and what is light - repels the light away as fast as possible. So one could honestly say that real white power flows in the darkness. Like the warmth of light in a dark stone, or the magic of human thought, that simply reprocesses the chemical energy from sunlight, inside of the dark mind. It is strange how humans' relative words can be made to change and tumble, across times and spaces, with just a readjustment of perspective. CREATED AD 2008 05 29 P 1150 [6.1] Utopian world potential in imagination. Imagine God creates a world where humans never die. And imagine, if one sins, God gives us all the spirit sense of pain and sickness, but not unto death. And imagine, if one sees someone in pain and sickness, God gives us all the spirit sense of a hunger to help those ill by feeding them the word, and broadcasts it such that the sicker the person in sin is, the more people gather around that one in congregation, in holy mass, to teach the sick to heal their wounds by feeding them the word of God. And imagine, if the sick are healed, God gives them all the sense of appreciation and happiness, as well giving those doing the healing the feeling of love and fullness of spirit. But that would be all too easy for God, who loves complexity and confusion. Instead, God gives us a hunger for physical water. God gives us a hunger for physical food. God gives us a hunger for physical power. God gives us a sense of pain, if we are hit by a physical rock. God gives us the sense of revenge, when we see our murdered friends and family. Our senses go on in crooked ways for all humans. All because we are made in the image, the archtype, of God: Genesis 1:26. In a world, where we are all on death sentences within this world for our sin nature, imperfection short of God-hood, and some might say Utopia under God is too Utopian, why didn't God make the world even worse, so we are truly closer to God? Some say, if God made the world more Utopian like in [6.1], we wouldn't appreciate good for the evil missing, even though Utopia would have its pains and pleasures in relative shift upwards. If that be the case, and the earth is already a prison planet, where we are all on death sentences within the world for being sinful and imperfect, then why doesn't God make the world a concentration camp? Everyone would be in a prison like in Nurenberg, or Buchenwald, where everyone is kept in horrible conditions for our dirty rags sinful nature, and we would truly feel closer to God and appreciate the good, for the additional evils of the world, like in a Nazi State. CREATED AD 2008 05 29 P 1150 [7.1] The finite reality under an all powerful, all creating God. It is said by all humans that there is no such thing as a free lunch, and nothing in life is free, and life isn't fair, and that is just the way things are. This is under an "all powerful (omnipotent) God", as preached by man. Even God had to die at the hands of man, because we were such evil workers to God, that He must die, to make us live in John 3:16 ... oh, selfish, selfish humans, toppling the "all powerful God". Even at that, Jesus only paid for the second death of the soul, and not the first death of the body, which has never been paid for. So we are coerced by a God who wants our love freely, by the threat of death and physical pains in life, and accept a mystery of faith, and hope life is true after we die physically. This is because God created a stone so heavy, that he couldn't lift it from the universe in time space in a purely good way, in the form of all of the human souls with free will, that He created. He even asked humans to be perfect in one law over infinite time in Genesis 2:17, when they were not perfect in infinite time like God, because they were not a God, themselves, therefore, they were sub-infinite perfect on that law, because they were not God. God couldn't, even wouldn't, create perfection with humans, as it is impossible for God to do so, or at least without teaching us a lesson in His pain, though all of our master planned physical deaths, including Himself. Therefore, God is not omnipotent, in this age of time and space, because he couldn't create perfection and free will and peaceful harmony and love and life, simultaneously, with human souls ... a limitation of the so-called omnipotent God, who is not be able to create something even like a living and a bitter and sweet utopia in [7.1]. [7.3] God cannot create good without creating evil, as all things come from God, and without Him, no thing is created. He couldn't create good without permitting evil, and he cannot create life without destroying some souls at the end Revelation 20:15. Why is it that a perfect and powerful God couldn't create souls with 100% yield. We sound like a product with a certain defective percentage worth destroying, despite His infinite power to create. For that matter, what will eternal heaven be like in Revelation 21 without tears, sadness, or evil, when He couldn't create it at the beginning, with all power? CREATED AD 2008 05 29 P 1150 [8.1] God doesn't need your help, God is so much more powerful than you! Preachers in diverse places say: "God doesn't need your help. He is so much more powerful than you, isn't He?" (this is for the sheep of the flock, not the few saints of the church, so the masses of sheep say, "Amen and, Hallelujah!") [8.2] The Church doesn't need it's parishioners, and will run on its own. A backhanded critic might say: "When the church degrades, and teaches heresy to the masses, the great numbers of God's main body, the church, doesn't need to feel the need to help God, then, right? Just leave it to those in charge, closer to God, huh?" [8.3] God says for man to exhort one another to goodness, but there must be exceptions to coming together. Writers of the Bible say: paraphrase ("Brothers, exhort one another to the good things of God.") Galatian 4:18, Galatians 6:9, Romans 15:1,6,7, 9, 11, 15. 1 Corinthians 1:10, 1 Thessalonians 3:9, 10, Hebrew 3:13, 10:25 CREATED AD 2008 05 29 A 0815 [9.1] The nature of science. A science theory has observables. One theory is that there is a metaphysical plane of existence, within the realm of the sensations of sense, in the senses of the body. And science theories have observables that support the same hypothesis coherently, in that you can question people of what they sense, and they all have a common observable nature, that is consistent to the theory in its detailed description. [9.2] Human color vision. For example, the colors that most people care able to see are universally observable in their verbal descriptive responses of color ranges matching a theory that humans can see color. And inside of themselves they see a variety of colors, from grays to pastels to vibrant pure colors in a spectrum of versions. They see reds, oranges, yellows, greens, cyans, blues, violets, and purples, and all people with color vision can report these things. But why red is the color that it appears, in the first place, is a question science seemingly cannot answer at all. If you show a computer with a camera connected to it the color red, all the computer senses is [255,0,0], and green would be [0,255,0], and blue [0,0,255], and white [255,255,255], and black [0,0,0]. Of course, a computer can translate that RGB number into hues with names, saturation with color intensity, and intensity with brightness, like [0,255,255] for red, [120,255,255] for green, [240,255,255] for blue, [---,0,255] for white, and [---,0,0] for black. These can even be further translated into names like [red, intense, bright] for red, [red, weak, bright] for pink, [hueless, neutral, bright] for white, and so forth. But these too are just strings of letters. Where in a computer's sense does it see red as redness, and white as whiteness, and so forth? All of it is numerical for a computer. A computer senses the world in a different transcendent way, from the transcendent way that humans sense the world. [9.3] Human hearing. Sound is another form of real processing, that can be scientifically observed through universal hearing human reports of pitch and qualities, and fits the idea of a transcendent level of processing, from the human reports of sounds special qualities. Bass sounds deep and mellow, voices are clear and discernable, and violins can be high and sharp. All of this has to do with frequencies of sound waves, but why do sounds sound like they do in the first place? Science can only say that it is metaphysical, because humans sense it one way, and computers metaphysically sense it a totally different way in numbers. [9.4] Senses in general. Likewise, touch, pain, pleasure, emotions, smell, taste, all have qualities that are what they are on a metaphysical plane, can be scientifically observed, can be duplicated on computers with their own numerical method of metaphysical sensation, and at the same time, science cannot explain their basic nature, any more than science can explain the human soul purported by all the world's religions. Even words and ideas are part of the metaphysical sense of the world. The word chair, makes one think, or even picture a chair in their mind. A computer can retrieve the concept and picture of a chair too. All science can say is that they are what they are, but what relationship to reality they have, they cannot answer one iota. Science is mute on this transcendental perception of things. So sensations and thoughts, within this collection of atoms interacting, that we call the human body, or a computer machine, are transcendental and metaphysical. The very observation of the separation between matter and mind posited by thinkers such as Descartes centuries ago. You ask a scientist, where is the nature of a sensation or thought in carbon atom A, or neurotransmitter B, or neuron cell C, or brain D, or human E, and they are dumbfounded, other than to say it is separate from matter, but quite observable in the scientific theory. Likewise, they can show a computer the same sensation or thought, and make a printout or display of that very thought, but cannot answer, does the computer see red, or think chair the same way humans do? They are dumb and mute once again, except to say that the computer also metaphysically senses the world in a way where mind or soul or spirit is separate from matter or components or individual. And that the computer, may or may not see things the same way, as the human, or an animal, or dare we say, God? CREATED AD 2008 05 29 A 0910 [10.1] A computer with free will and soul is something that I have actually made. Take a computer with a program that makes decisions and learns from its experiences in the world. Run the computer program and expose it to the same sequence of experience, and it will always do the same thing, and end up at the same end state of learning and decisions among choices. Like many Christians or people in general say, from "the traditions of men", a computer only does what it is programmed to do, and I agree with this example. But now add a random number generator that slightly influences its decisions through time. Run that program, and expose it to the same sequence of experience, and it will always do the same thing, and end up at the same end state, though it will be very different in it's internal character, from the random number less, but otherwise identical decision and learning program. Once again, we are stuck, that the computer has no free will, because it does exactly what it is programmed to do. Finally, add a digital camera to the system, and read the noise of the photons of light from some scene, and feed that into the random number generator algorithm. Now when you run the same program over and over to the same sequence of experience, and the computer will end up in totally different internal learning configurations. They are like identical twins who are identical in their original code and experience, but they have their own unique internal identities. How does the last model of learning computer suddenly acquire an individuality with free will, and dare we say, a soul, with its own internal character, while learning? The computer is connected to light. According to Heisenburg's Uncertainty Principle, entities like atoms, electrons, and especially photons, are observed to have a completely unpredictable probabilistic path, in time and space. No human on earth can know where exactly a photon of light will go, from one thing, to another. The digital camera senses photons of light, that are unpredictable by all human physics, and takes these quantum fluctuations of light, and magnifies them to the macroscopic scale of the universe, seen in the computer's processing body and it's effects on the universe. Literally, the computer is connected to the Light of God. As such, it has a will that is unpredictable by every human on earth, anywhere, by any physics used. Even science, by attempting to intercept photons, to know what the computer's camera sees, will simply rearrange the photons, into a new unpredictable set of paths, which will send the computer into another new unpredictable path. That is the old, the observer affects the observed effect, especially if science tries to observe the computer's actions' causes. Only a God outside of time, who can know the probabilistic paths of all things, unlike us poor material beings, can know what the computer's free will, will do. So from the frame of reference of the universe and man, the computer has free-will, and an internal learning character, and decisions, that are of it's own doing, that is metaphysical and free from man's knowledge. Much like the free-will soul of man, reported by the world's religions. So the computer's activity leaves the realm of observable science, and enters the realm of probabilistic science. Likewise, the computers learning and decision among choices algorithms in feedback, give the computer an inner character that is metaphysical, where the mind, the soul, or spirit, is separate and unquestionable by science in a repeatable way, unlike the material the computer is made of, which is wholly built on science's data of materials. Every observation is unique, so science cannot truly answer its nature, like UFO reports that are all one offs. CREATED AD 2008 05 29 A 0910 [11.1] If God Knows all things for Truly, then we are not free from His point of view. If God knows all things including the entire future in every detail, then humans cannot change their path from His Vision. We would think, inside of time and space that we were making numerous decisions from a set of more numerous choices through time. In fact, our body and brain would really go through all of the processes of making decisions and performing actions. But say that God Saw we were going to decide to do A, B, C, D over a short period of time, and we actually decided to do A, B, C, E, then God would be surprised because our actions didn't match His vision. So God's Vision of the future is not something God can really Know for Truth. But, we would have free will from God's Point of View, though we will always be able to make God's Vision lie to God, because God cannot see all things in the future. This cannot be because what God Knows is truth through what God Sees, or else God has imperfect sense of Omniscience. So if God's Vision is Sovereign Truth, and God Sees we are destined to perish, "If I perish I perish." and there is nothing I can do to change it. If God's Vision Sees someone else is destined to Salvation, then if they are saved they are saved, and there is nothing they can do to change that. It is like those movies where someone sees the future like with God's own Vision, and no matter what the characters do, they cannot avoid the reality of the Truth of the Sovereign Truth Vision, no matter what they do. The universe freezes free will that deviates from that destination Truth that is Final, by God. [11.2] Analogy of Total Omnipotence removing free will from the perspective of the Omniscient. Image you are a computer programmer who programs a computer machine and world, to behave a specific way. You know ahead of time, exactly what the computer machine program will do when it operates in the future. The computer machine may be quite sophisticated, and believe it is making it's own decision evaluations among choices, and has beliefs about its contexts in time and space. But whatever the computer machine does in time, all of it's actions are exactly known by the programmer, before the computer machine is even started. From the programmer's perspective, the free-will the computer machine believes that it has, in its own process, is all an illusion, because from the first action, to the last action, the programmer will not be surprised by any decision the robot makes. [11.3] God must voluntarily limit His Total Omnipotence, in His Agency of Total Omniscience including the future, in order to give humans free will. In this age of the universe, an All Powerful God with Total Omnipotence in His True Potential, must actually create a universe where His Vision is actually darkened, or incomplete to all of the details of our activities in the future. God could look into the future in detail if God wanted to, but if God did that, and His Vision is Sovereign Truth, then what God's Vision touches, would freeze away our free-will, turning us into robots. So God must have created a darkened universe, dark in His Knowledge of it. God intentionally doesn't know what it will do, and averts His eyes from the universe in detail, much like Midas's Touch was held back from the world. For everything that God Sees becomes Sovereign Truth, and everything Midas Touches becomes Gold. By setting up a darkened and forgotten path universe in God's Mind, God opens up that darkened space to our human free-will, and our free-will doesn't make God's Vision a lie, because God simply hasn't Seen the future in any complete detail. So we are free in the universe from all perspectives. Our destiny is ours, and not constrained by what God Knows is Going to Happen, because God doesn't Know. And God does it by choice, to open up this darkened space from His Mind's ability to Control or Know. [11.4] Alternative removal of traditions of man. Another alternative that can be logical about God, is that the human claimed Total Omniscience of God including the future, is literally only a Perfect Knowledge of all things past, and present, but not of the future. In this case, the Total Omniscience of God that includes the future, is only traditions of man's suppositions in ignorance, and not True of God's Sovereignty. This way, God can have a very exacting plan for the future, and always keep things in line with His Plan, but God doesn't, in Sovereignty Truth, Know what the future will be exactly, but His Total Omnipotence Powers allows God to herd humanity toward His Plan, and we all have True free will in all perspectives, and are not computer machine robots, who are utterly programmed to the end of time, and cannot deviate from destiny. CREATED AD 2008 07 17 P 0900 [12] Abiogenesis chemical evolution. [12.1] Background. A natural combinatorial chemistry feedback, in an appropriate open system ocean, with inherent natural reactions and hypercycle catalytic reactions, which, alone, can suffice to create an increasing complexity chemistry that eventually intersects biochemistry, as evidenced by modern life. And an early earth ocean can have a greater amount of dissolved organics and minerals, with no presumed life forms processing the chemicals into their own makeups. It would all be dissolved in the oceans, and washing off the early continents in deltas, lake beds, or tidal mud flats, in evaporative concentrations. [12.2] Combinatorial chemistry 1. Now combinatorial chemistry can be generalized to parallel numbers chemistry that combinatorially explores all feasable interactions of all chemical species available in a chemical environment, like an early earth ocean environment with bays, tides, hydrothermal vents, sunlight with or without UV, dark areas deep in the water or under rocks, for protection from UV and sunlight, lightning, pH variation, evaporative concentration, and currents to mix a natural initially inorganic chemical soup with hundreds of minearls, metal ions, etc. in a preorganic molecule soup. [12.3] Hypercycle catalytic chemistry. Hypercycle catalytic reactions are subsets of the whole combinatorial chemistry reaction matrix, where, A helps catalyze B helps catalyze C helps catalyze A, from other present chemical species, as an example of a short hypercycle loop of three nodes. Hypercycle catalytic reactions can be loops, and networks, embedded within a normal combinatorial chemistry matrix. [12.4.a] Combinatorial chemistry 2. Going back to combinatorial chemistry, let's say in the ocean there's to begin with, 1000 Species of chemicals and chemical inducing factors, S, such as chemicals, photons of light from infrared to UV, radioactive particles in early half life rich early earth materials from its recent supernova formation, different energy free electrons from lightning, mixing currents, and heating and cooling around hydrothermal vents. There is an approximate top level pseudocode (which can be glossed over to reach final math characteristics after the pseudocode) of a differential equation that shows the equilibrium balance of reactions, is: InitialSpecies = S; InitialAverageConcentration = 0; for(s = 1 to S) InitialAverageConcentration += Concentration{s} / S; for(s = 1 to S) //how many species in a reaction __Reaction = array{s elements}; __for(s1 = 1 to s) ____for(s2 = s1+1 to s) ______for(s3 = s2+1 to s) ... //nest to depth of s ______________for(ss = ss-1 to s) ________________if( all sx < sx+1, and all sx != sy) //no repeats __________________//calculate net chem species present change __________________//for this specie reaction set for a unit of differential time __________________NewSpecies{S' set} = F1(Reaction{s1,}); __________________NewConcentration{S + S' set} = F2(Reaction{s1,}); ... //nest to depth of s __FinalSpecies = S + S'; __FinalAverageConcentration = 0; __for(s = 1 to S + S') ____FinalAverageConcentration += Concentration{s} / (S + S'); [12.4.c] Linguistically, this can be interpreted as, taking 1 to S chemicals at a time, in every combination, to observe reaction rates of current S chemical species, s at a time, to see the effect on all S and possible new S' chemical species generated that were previously not existing before. For example, for two species taken from a given 1000 species, S, we see there is (1/2)(S^2 - S), or 499,500 Reaction{s1,s2} nodes, with positive or negative reaction rates for existing species S, or new species of S'. That is, say, S1 + S2 might breakdown S1, catalytically by S2, into S3 and S4, and S2 remains untouched. S1 has a negative reaction rate as it breaks down into trace amounts of S1, while S3 and S4 have positive reaction rates, as S1 is turned into S3 and S4, in the presence of S2. On the other hand, say, S1 + S2 helps produce a totally new chemical outside of S, of S'1, by S1 and S2 combining to form S'1. S1 and S2 have negative reaction rates being consumed, as the new S'1 has positive reaction rates. These reaction rates also change in time, as the concentrations used by F1(Reaction{s set}) and F2(Reaction{s set}) calculations, increase or decrease accordingly. [12.4.d] At the same time, there are more reactions to analyze, continuing with three chemicals in a Reaction{s1,s2,s3} analysis, where there is (1/2)(1/3)(S^3 - S) or about 167 million reaction nodes. So of these millions of Reaction{s1,s2,s3}, many will have no effects, some will break down or build up products already existing, and others will make new chemical species that never existed before, from the species that exist in the ocean to begin with, S. [12.4.e] Mathematically analyzing reactant combinations, from s = 1 for single molecule auto-reactions, to s = S, for S species reaction, in total, there are: ReactionNodes = +SUM( s=1 to S: of: Factorial(S) / (Factorial(s)Factorial(S-s)) ), or, equivalently, ReactionNodes = 2^S - 1 = +2^1000 - 1 ~= reaction nodes for 1000 chemical species S, where, (1) the majority of non-reactions change nothing, (2) some break down species, (3) some build up species, and (4) some generate new chemical species. So starting with 1000 chemical species, with an S' formed out of 10^301 of, say, 1000 new chemical species S' (a conservative rate of 1 in 10^298 being effective stable new chemical species), such that in a year, there can be 2000 species of flourishing chemicals, leading to 10^602 reaction nodes to analyze for all potential reactions at each node, generating, say, 2000 new species of chemicals (at an even more conservative rate of new chemical specie formation). So then after another year there's 4000 chemical species at some concentration, with 10^1204 reaction nodes, generating, say, 4000 new species (even more conservative to the combinations available), added into next year's variation. [12.5] So one can see an exponential feedback of chemical species, some more robust than others, in numbers, durability, variation, reaction rate selection forces, hypercycle catalytic reproduction, and reactivity, from 1000 to 2000 to 4000 and so on, until there is a low but signifigant saturation of millions of reactive catalytic various chemical species in a gallon of ocean, all competing for the ocean's limited chemical resouces, and giving rise to potential natural metabolic pathways absorbing glucose and photons of light, in complex reaction sets, paths, cycles, and netowrks, that support reproducing hypercycle networks of catalytic chemicals, all inherent and naturally contained, in the combinatorial chemistry feedback matrix growing in time. Presumably, something akin to photosynthesis must have arose early to convert the atmosphere to mostly oxygen, as part of sugar production. [12.6.a] A Creationist claim would have to show that of the 2^S reaction nodes, in an S chemical specie example ocean, would permit no (zero) new chemical species to form and thus remain in static chemical equilibrium. But given the massiveness of potential in 10^301 reaction combinations in a mixing ocean of a combinatorial chemistry size S in a feedback, if it shows even a very minor positive rate of new chemical species formation, that such a non-zero feedback would provide a numerical backbone to natural blind chemical evolution turning into life, as chemical species reach continually higher levels of complexity and variety, with competition and selection forces, in the combinatorial chemistry in feedback, from the very beginning of chemistry, in robust reactive new molecules, contained in chained catalytic reactions, and with a form of digital chemistry, contained in the discrete chemical species, and in the discrete codes of polymer proteins, RNA, and DNA nucleotide chains, that are eventually intersected by combinatorial chemistry, with a proven positive dS/dt. [12.6.b] Even just 100 chemicals in an initial energy open system ocean, would allow 2^100, or 10^30 possible reactions, so even small chemical soups start with an inherent potential for new chemical specie feedback growth of complexity, without external guidance being absolute necessity. BLOG CROSSREFERENCE [3] Evolution design theory. AD 2008 05 29 A 0750 Lipid and early combinatorial chemistry protocell theory: Hypercycle chemistry: Combinatorial chemistry: CREATED AD 2008 07 17 P 0900 [13] Chinese Han and Japanese Kanji studies. [13.1] Introduction. And my studies of Chinese Han and Japanese Kanji, are interesting and challenging, as they are heirarchical languages in meaning where a few strokes in a combination called a radical have some abstract meaning, and then, radicals are themselves, combined to make an ideogram square, and then those ideograms are often combined to make complex new words. [13.2] Example. [13.2.a] For example take the english: [13.2.b] This can be three Han-Kanji ideograms: [計 \| 算 \| 機] [13.2.c] These are roughly translated per ideogram, as: [計 "idea" \| 算 "calculation" \| 機 "machine"] [13.2.d] These are themselves made of radicals segments: [計 [accent bars and box \| cross] \| 算 [double lambdas with bars \| triple box \| two legs with bar] 機 [cross with two dropping branches \| E looking mark \| E looking mark \| bar with swooping right descending hook crossed by left descending slash and accent \| left descending slash crossing the bar and side branch on right]] [13.2.e] The radicals being roughly translated as: [計 [speech \| to-complete] \| 算 [bamboo \| vision \| presenting] \| 機 [tree \| tiny \| tiny \| weapon \| divines] [13.2.f] Now in English words, reflecting the cultures, reads roughly: "an object for 計 [wordings in completion, reminiscent of ideas] which are acted with 算 [bamboo abacus examination and presentation, reminiscent of calculating] in the form of an 機 ["wooden" object with many tiny parts like weapon construction which has operations, reminiscent of machine and performs (divines) things]" [13.2.g] Seeing the ideograms, one simply thinks "computer" when seeing this hierarchical: [計"idea" \| 算"calculation" \| 機"machine"] tri-ideogram-chain. [13.2.h] Their whole language is couched in such metaphor and abstract thinking, and hierarchical dynamic thinking, with a heavy burden of ancient concepts, brought into the modern world. Like computers could be made of tiny wood machine parts abstract-concretely, like Babbages difference engine machine of metal gears, or Jacquards card loom of wood and wires and cards, but are so much easier to make in silicon and doped circuits on the silicon, today. CREATED AD 2008 07 17 P 0930 [14] Quantum physics self question. [14.0.0] What are the best scientific theories on why and how in Quantum Physics (QP), there is a real probabilistic wave function collapse, during the measurement event? [14.0.1] I would like to exclude many worlds theories, as they require too much faith in things unseen, when a singular continuum Quantum or String Theory can be presented. [14.0.2] Discussion of hidden variable theories is acceptable, though it undermines the Copenhagen Interpretation (CI) of probability-wave-functions, with a hidden epicycle field that is unaccessable, though identical to QP CI. [14.0.3] Secondary questions in {14.6.x}, below. [14.0.4] Reference material {14.1.x}, {14.2.x}, {14.3.x}, {14.4.x}, {14.5.x}, {14.8.x}. [14.0.5] Some potential experiment to help clarify some aspects {14.7.x}. [14.1.0] QP MEASUREMENT makes probabilistic-wave-functions COLLAPSE, at a velocity infinitely-faster than light-speed. ... nhancement. [14.2.0] QP UNMEASUREMENT makes probabilistic-wave-functions UNCOLLAPSE, at a velocity infinitely-faster than light-speed. [14.2.1] This is seen in Quantum Eraser experiments, where such experiments show that [14.2.1.a] measuring a probabilistic-wave-function, can destroy an interference patterns later in the experimnet box, while [14.2.1.b] measuring and then erasing the bit on memory in mid flight, UNCOLLAPSES the wave function, fully restoring the interference pattern, just as if it were not measured at all in mid-flight. [14.2.2] Although probability wave function collapse / uncollapse occur, no information is communicated, but there is an alteration of collapse / uncollapse probabilities that is instantaneous. [14.3.0] Measurement is not a material process. [14.3.1] It appears to be best described as a epiphenomenon defined by the configurations of macroscopic matter, creating a measurement-probability-field, that probabilistically defines when, where, and how a probability-wave-function COLLAPSES-or-UNCOLLAPSES. [14.3.2] If there were no measurement potential field, then the epiphenomenon would not exist, and all matter would slowle decohere into probability-wave-functions all taking every possible path at once, even around decision / bifurcation interactions. [14.3.3] But it does exist, and measurements from the macroscale keep the universe mostly focussed down to a small microscale for billions of years, with its macroscopic "inertia of configuration existence". [14.4.0] John Wheeler, who worked with Einstein and Bohr, was a proponent of there being an epiphenomenal field to all macroscopic existence, known as the "it from bit" idea, that David Chalmers, among others, also teaches of currently. [14.4.1] It can make claims at its theoretical limits, of there being a soul, constructued out of an epiphenomenal measurement matter, cycling and circulating in systems feedback loops of the human material mind structure. [14.5.0] Richard Dawkins says on nonmaterial epiphenomena, in "The God Delusion", pages 34-35 "[14.5.1] What most atheists do believe is that although there is only one kind of stuff in the universe and it is physical, out of this stuff comes [14.5.1.a]{minds, beauty, emotions, moral values} ... [14.5.2] An atheist in this sense of philosophical naturalist is somebody who believes [14.5.2.a]{there is nothing beyond the natural, physical world}, [14.5.2.b]{no supernatural creative intelligence lurking behind the observable universe}, [14.5.2.c]{no soul that outlasts the body}" [14.6.0] What is the proofs of even deriving a concept like [14.5.1.a]:"beauty, emotions, moral values" from atoms and physics equations, that is real, and not an abstract epiphenomenal psychological description of abstract life? [14.6.1] There appears to be no physics-foundation to run [14.5.1.a] to the ground state of arising from atoms and physics, in a philosophical reductionist paradigm. [14.6.2] In fact, {14.5.1.a} seems to require a philosophical holistic paradigm from the systemic macroscopic scale, and that that opens up the possibility of there being a proveable soul, opposing {14.5.2.c} that there is no soul, for if there is any soul epiphenomena, made from {14.3.x} {14.4.x} measurement epiphenomena field, arising in macroscopic systems of macromatter, then {14.5.2.c} denying the soul, says that atheist Science will never try to save human souls, when no one with that mind set ever wants to see the soul as a structural potential, that might be savable in a temporal-spatially-living material-base-foundation, and so how will "Ghost in the Shell" technology come about through a dead science, if atheist-Scientists never take the first step, and religions fold their hands waiting for salvation from above, and atheist-Science's adhere to a flat-fact denial statement proving there is no-soul without proof, to be taken on science's faith, reflecting their atheist-Scientist thinking? Additional clarification of my initial question: I believe I can enhance the formulation of my question with an example. [] Mundane matter of local character can have holistic emergent properties like consciousness created by the systems of matter of processing and action control. [] But on top of this, is an instantaneously correlated set of measurement probability influences, built from simultaneous wavefunction collapses throughout all of the feedback loops and structures of measurement forms, that happen to lie parallel to the macroscopic matter. [] This produces an infinitely faster than light measurement-system-self that co-influences the mundane matter wave function collpases, as the mundane matter affects the infinitely faster than light measurement-system-self. [] So there is a mundane matter self, and an instantaneous measurement system self corresponding to each other but of different character than a pure mundane matter self emergent consciousness. [] For example in mundane matter, of a mechanical equivalent, one can show the emergent property issue with a computer containing advanced AI, and a camera, that can register (11111111,00000000,000000) which in memory of past obsevations is hue (00111100), and represents the string " 'R' 'E' 'D' ", and the computer can report " 'I' ' ' 'S' 'E' 'E ' 'R' 'E' 'D' ". [] But does the mundane computer see "REDNESS" like a human? [] But if you also include a holistic (whole) quantum measurement layer computer-AI-self, that is parallel to the computer mundane matter, because the computer is made of instantaneous matter wave functions and matter measurement structures, both, that perhaps the computer also literally may experience "REDNESS" and "LIGHT" like a human does. [] For the mundane matter simply has voltages in a black sealed computer chip, just like humans have neurotransmitters in a black brain case gray matter. [] How does a computer or a human literally perceive light, like I see light when I open my eyes? [] You look inside of a computer, and all you can see is chips with sense and thought voltages that are invisible to human eyes. [] You look inside of a human, and all you see is brain with sense and though chemicals that are invisible to the human eyes. [] Yet when you are a human, and perhaps are a computer, there is that secondary holistic-whole self that literally sees colors and light in a quantum measurement instantaneous self-ness built from the structured quantum measurement structures that lie parallel to the mundane matter-energy, and co-effect the mundane matter-energy as the mundane matter-energy co-affects the quantum measurement self. [] One example of a simple "holistic systemic measurement self" test would be to build a ring of chained synchronized photons EPR experiments. [] One prime EPR photon polarization measurer leg in the ring has a fixed polarization, and every other EPR end and leg use an algorithm for polarization measurement selection based on the current photon's polarization measurement and the neighboring EPR leg end, e.g. but not limited to XOR leg X right [xor] leg X+1 left. [] The rough idea to be tested is, is there any instantaneous systematic mathematical issues of a properly gated ring, in the probabilities measured at all of the EPR legs? [] No-communication theorem dictates the probabilities will be completely random aligned with the restrictions of the wavefunctions caused by the prime locked EPR, and the algorithms of every other EPR leg coincident polarization measurers / calculators. [7.3.4] But if there is any chaos patterns or non-stationary probabilities, then something additional is occuring, in the synchronous instantaneous wavefunction collapses of the synchronized EPRs in the ring. [] If there is no measureable non-stationary probabilities on all EPR legs, for a properly synchronized EPR ring with algorithms, then there is no "holistic systemic measurement self" present in that experiment, and the idea of a "holistic systemic measurement self" may be flawed. [] However, given the nature of using a loop or modifications to the above experiment to create even a full ring feedback, can all instantaneous wavefunction collapses from polarization measurements preserve simultaneous perfect stocastic patterns according to no-communication theorem, or is there a spatial mathematical limitation to mutual perfect stocastic EPR in all possible ring algorithm formations? [] That is, is there any special meta-level mathematics required to assure mutual EPR ring pure stochastic randomness at synchronized instantaneous wavefunction collpases in all forms of rings, to assure there is no coherent or semi coherent holistic self? [] One EPR I can believe, remains perfectly no-communication stochastic, yet instantaneously correlated, but a ring raises instantaneous mathematical QP-field-computation issues that lie deep in the foundation of how the quantum physics Schrodinger Equation evolve in time and at synchronized instantaeous wave function collapses, in EPRs. [] The experiment might be affected by ring size assuring a full ring correlation, but examining only the most properly time gated coincident photons that happen to simultaneously work in all EPR in the ring, and ultra fast algorithm computers on the ring, would assure the right experiement statistics can be collected for analysis of the "holistic systemic measurement self" versus "pure stocastic polarization measurements, indicating a meta-math or indicating a natural Copenhagen Explanation math of how pure sunchronized randomness is preserved. [] An alternate configuration among all of the configurations that can be explored in an EPR ring, is to daisy chain the calculating EPR coincident photon measurements, in the ring, from the key "locked" EPR polarizer, and around the ring, and finally back to the neighboring EPR ring leg final polarizer, and have that computationally affect the polarization of the key "locked" EPR polarizer, and repeat the experiment with synchronous coincidence control. [] Then, collect data from all single, and multiple instances, of simultaneous gated whole-holistic measurement sequences, to analyze for pure random polarizations according to the polarizer configurations, or non-stationary random, or chaotic polarization data correlations around the EPR ring. [] If all of the EPR polarization measurers, show pure randomness, and random-appropriate according to the polarization configuration, what does that say about the nearly instantaneous wavefunction collapse system of the ring, self-consistency? [] If all of the EPR show non-stationary randomness, or chaos, what does that say about the nearly instantaneous wavefunction collapse system of the ring, self consistency? [] To reiterate, one EPR leg is easy to understand keeping pure randomness according to the polarization configurations of both ends, but a mathematical-physical ring, might show interesting physics nature of faster than light mutually constraining measurements, of randomness or near-randomness, that communicate no-information, or some mathematical distortion caused by QP having to "naturally calculate" a ring randomness assurance in rapid succession, many times faster than light, limited by the EPR calculation and polarization controllers, and may answer fundamental questions regarding a supervenient-holistic-quantum-self. [] An extreme case is to select all EPR polarizers by last moment synchronized leg-to-leg correlated full ring simultaneous and coincident measurements, and check the polarization and randomness of all legs after the fact for full ring programs. [] Now, gut instinct and analysis seems to show each EPR leg should be a completely independent measurement about each photon pair, but is this a completely true conclusion? [] This would also indicate that nonlinear nonunitary matter exists separately from quantum physics Schridinger equation, dividing QP space into cells of quantum physics behavior, much like the body is divided into cells. [] That is, non-unitary macro-matter is as real as the QP probability wave function, which is divided into a network and/or scintillation of collapsed states, and divides the QP probability wave function into small domains of actual QP probability wave function unitary linear evolution. [] That means there is a transcendental macroscopic-measurement-configuration-information substance filling the macroscopic universe in ways paralleling the macroscopic classical matter, with a foam of small QP probability wave functions. [] In vacuum and the spaces of large measurement systems in the macro-scale, the foam of quantum physics probability wave functions grows to macroscopic scales of unitary linear evolution. [] Is the same true, if each EPR polarization measurer were replaced by elements that reemit photon pairs correlated to the photons coming from the two legs, back down the legs to the first photon generators, which can again reemit dual correlated photons, and be measured at some point in the ping-ponging for proper ring polarization measurements, or does the entire ring simply become a large instantaneously inter-correlated linear evolution of an interactive superposition of states? [14.8.0] Quantum Mechanics: From Basic Principles to Numerical Methods and applications, L. Marchildon, (C) Springer Verlag Berlin Heidelberg 2002, page 513+, [[LRD added / conservatively altered]] I would personally read the passage [[modified by LoneRubberDragon]] as (with the original text reproduced below): "[] The measurement problem was recognized early, by Von Neumann [[]] among others. [] He realized that unitary evolution leads to superposition of macroscopically distinct states[[; think Neo moving 4 directions at once, in The Matrix's "macroscopic world"]]. [] Furthermore, he saw that there is no use to introduce a second apparatus to measure the value of the first pointer. [] Indeed inasmuch as the evolution of the total system (microobject, first, and second[[ly the]] apparatus) is unitary [[as a whole]], [[where]] the second apparatus would also* [[in linearly Schrodinger evolving]] end up in a superposition of macroscopically distinct states. [] The solution proposed by Von Neumann essentially consists in postulating that the Schrodinger Equation no longer holds at the time of measurement. [] But why is this precisely? [] The abrupt transition from a linear[[ly evolving]] combination [[in a whole system in]]to one of its components is known as the collapse of the state vector [[wave function onto a projection]] [] Von Neumann's hypothesis is ingenious. [] Its success is largely independent of where the border between microobject and [[macroobject]] measurement apparatus, or the border between apparatus and conscious subject, lies. [] The process represented by [[wavefunction nonlinear nonunitary instantaneous collapse]], however, seems closer to a requirement of perception [[by abstract qualia of emergent supervenient informational macrosystems]] than to a physical mechanism. [] It thus appears to reinstate the mind-body dualism that natural sciences had largely eliminated [[by their logical proof* of there obviously being no real "soul"]]. [] The breakdown of the Schrodinger Equation and unitary evolution of the state vector[['s probability wave function]] occurs, according to Von Neumann, upon intervention of the conscious subject. [] In a similar analysis, [[one physicist]] associates this discontinuity more generally with all [[macroscopic matter in motion ]] processes. [] He believes that [[all time-sapce macroscopic entities ]] should be described by [[linear unitary evolving Schrdocinger probability wave function]] equations [[that are broken down by the supervenient-informational-macrostructures in time-space, in all macroscopic matter in motion, causing the nonlinear abrupt instantaneous shift by holistic-measurement-perception-self]], which entails a nonunitary evolution of the state vector. [[]] Original text: "[] The measurement problem was recognized early, by Von Neumann [[]] among others. [] He realized that unitary evolution leads to superposition of macroscopically distinct states. [] Furthermore, he saw that there is no use to introduce a second apparatus to measure the value of the first pointer. [] Indeed inasmuch as the evolution of the total system (microobject, first, and second apparatus) is unitary, the second apparatus would also end up in a superposition of macroscopically distinct states. [] The solution proposed by Von Neumann essentially consists in postulating that the Schrodinger Equation no longer holds at the time of measurement. [] But why is this precisely? [] The abrupt transition from a linear combination to one of its components is known as the collapse of the state vector. [] Von Neumann's hypothesis is ingenious. [] Its success is largely independent of where the border between microobject and measurement apparatus, or the border between apparatus and conscious subject, lies. [] The process represented by [[the nonlinear wavefunction collapse onto a projection]], however, seems closer to a requirement of perception than to a physical mechanism. [] It thus appears to reinstate the mind-body dualism that natural sciences had largely eliminated. [] The breakdown of the Schrodinger Equation and unitary evolution of the state vector occurs, according to Von Neumann, upon intervention of the conscious subject. [] In a similar analysis, Wigner associates this discontinuity more generally with all living processes. [] He believes that living processes should be described by nonlinear equations, which entails a nonunitary* evolution of the state vector. [[]] [14.9] Musing on macroscopic discreteness in a so-called universal unitary linear probability wave function evolution, that is mostly collapsed. One observation from the measurement issue alone, would indicate that matter is discrete, as you don't see your friends in quantum flux states, but all exist on one macropath. It would be easy to detect a measurement magnified to macroscale, such that your 1 friend splits into 2, 3, 5, 8, 13, 21 ... paths in time without any wave function collapse. Another observation, albeit difficult, notes that a system of macroscopic feedback, measurement, interaction, is that one sees in color. A computer sees (255,0,0) looking at red, but can you derive a computer sensing the hue RED like a human? Though it is a hard analogy, as computers have digitial consciousness, while humans have analog consciousness, as structure of measurement, feedback, and macrosystems would impact how perception "looks" from the inside of a dark brain matter seeing light, or a dark transistor seeing light. But speed a computer up 100,000 times, and the time compression of measurement events in its cybernetic circuits, might make the computer behave nonlinearly with the shorter clock, and sense color in a way much more analogous to humans of massive parallal analog computation measurements. So I would guess there is a Heisenburg relation to soul and/or consciousness as speeds and scale-of-structural-measurements embedded in the computation structures, heighten the nonlinear effects on probability effects, by structure potential on microscopic matter measurements. the computer crossing such an increasing speed transition might go from stating "[255,0,0]" to stating, "I see red hue data", to stating, "my God, I see colors! They aren't numbers anymore. What is this?" which would show a nonlinear probabilistic effect on the computer program, possibly from the very root of QP measurement nonlinearity. Without a structural component to heirarchical emergent properties, consciousness might be like a slow clocked robot that is barely considered alive, as it doesn't perceive the world as much as calculate, which is an abstraction idea reducible to molecules moving in physics, without color, pleasure, pain, joy, sadness, wakefullness, dreams, etc.. [14.10] Musing on what if science could save a soul, does it do it right, and how does it prove that? If science doesn't know how to save a soul, a soul will never get saved for those who want that "product". So they look to God because science is weak, and even denies the soul as Dawkins does. Chinese emperors bought many elixirs of life, by charlatan science, killing a many emperor, so all have good reasons to look askance at a group who might say, "you have nothing* worth saving, go forth and die!". Emergent properties that are a fiction of science abstraction, like good and evil, and thought, sight and soul, are not science. So can you trust a scientist in 100 years who says, step into this box as we disassemble your atoms, and save you on this living blank robot, and it won't hurt a bit. I bet you'll either go on faith of science, or start asking science a million unknowns about, how do you define a soul system that can be saved properly and without 1000 years of pain in a virtual existence during the translation. Someone has to answer it. Science can't touch it. Science avoids it. Science runs away from it. And religions are not too far behind, totally going on faith in God doing all of the physics of soul transference, or Buddhist reincarnation for that matter. [14.11] Musing on emergent phenomena only, or additional phenomena required to assure self-soul. Emergent properties are abstract epiphenomenal descriptions. What founds abstractions solidly in science? Why is collection X of atoms sentient? Current consciousness characterizations are not real, in the same sense that Newton's gravity is not real, but emergent, and was made physically-real with postulated Gravitons. Newton's gravity was abstract epiphenomenal discription, until modern field theory with gravitons, founds it in reality. Your emergent properties are true, in my humble opinion, but my opinion is not science any more than Newton's gravity was opinion, until gravitons were postulated. But my opinions of a quantum measurement fields, related to the very macro-structures of matter, can found soul, and perception, and consciousness, in physical reality. And it could give a root to saying V is soul, W is evil, X is good, Y is beautiful, Z is RED, because physics can found it in reality, and explains why those thigs are what they are, and not artifical groupings with no defined boundaries. [14.12] Musings on saving a soul and passage of time, real or imaginary... Current human life (and animal life to lesser degrees) would have "QP measurement field" "soul" or epiphenomenal existence arising from its material structures and feedback patterns of process, for the duration of material-configuration-life in real-time of the individual structure that is alive. If the structures, memories, and manners of a life can be copied from a dying biological unit into another blank biological unit, or blank machine unit, for continued existence, and assuming a scientifically proven continutiny of existence of the patterns and possible "QP measurement field" effects of the current unkowns, then that life would continue to exist in real time. If it is moved into a robot body, the time of existence remains real time and material extended, barring material accidents disrupting structural processing existence. If it is moved into another biological unit with perfect man-made biology that can last for centuries, it also continues to be real time for the life of the biological unit, barring accidents. And if the consciousness is transferred intact in patterns and essence, into a virtual world, ala "The Matrix" or "Ghost in the Shell", then it may live in a dilated time frame, anywhere from slower than reality if it runs on an XT, or much faster on a future computer built of nanotubes, meristors, transistors, and such. It would not be imaginary time, in the sense of Stephen Hawking and James Hartle, describing the Big Bang as a singularity where imaginary time and real time become equal in strength, as all natural forces unify toward zero time. It is material and measurement related perceived dilated time, in the clocked or reaction refernce frame of the material medium supporting the systems and perceptual chains and potential QP fields of measurement. Now can you answer how and why measurement and unmeasurement in Quantum Physics occurs? That little untidy edge of science explanation at the edge of information, structure,and macroscopic existence of processing entities. I liken saying QP measurement "just happens" is not founded in reality, just like Newton's 1/R^2 Gravity "just happens" is not founded in reality. QP measurement with a describable informational-structural-macroscopic creation of a measurement field close to consciousness, ala John Wheeler, is more founded in reality, just like positing Gravitons and gravity waves, founds Newton's true, yet initially un-founded gravity equation, in reality. Yes they both work not knowing how they work, so commend science on that, but they reflect a hidden truth, just as Gravitons and gravity waves substantiate the argument of Newton. QP measurement, on the other hand, with its spiritual connotations of measurement and unmeasurement, and infinite speed propagation of unseen but believed probabilities, is not so well founded as Gravity with Gravitons, as they say, "it just happens" why "just becasue I say so, take it on faith of Copenhagen". [14.13] Wiki notes on quantum physics. ""Feynman proposed the following postulates: The probability for any fundamental event is given by the square modulus of a complex amplitude. The amplitude for some event is given by adding together the contributions of all the histories which include that event. The amplitude a certain history contributes is proportional to , where is reduced Planck's constant and S is the action of that history, given by the time integral of the Lagrangian along the corresponding path in the phase space of the system. In order to find the overall probability amplitude for a given process, then, one adds up, or integrates, the amplitude of postulate 3 over the space of all possible histories of the system in between the initial and final states, including histories that are absurd by classical standards. In calculating the amplitude for a single particle to go from one place to another in a given time, it would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns all of these histories amplitudes of equal magnitude but with varying phase, or argument of the complex number. The contributions that are wildly different from the classical history are suppressed only by the interference of similar, canceling histories (see below). Freeman Dyson showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action. Schrödinger Equation Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of , the path integral has most weight for y close to x. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. The exponential of the action is The first term rotates the phase of ψ(x) locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in ε they are additive; in any case one has with (1): As mentioned, the spread in ψ is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential: and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment."" [14.14] Musings on machine emotion, self, and the law of man on self and mortality. Given some agreement, now I can agree, if a computer can be given faculties for emotion, either programmed, or learned over time from a bootstrap program that starts like a baby and develops their own emotions, then pain and pleasure can be known to machine as man. If that is known, then good and evil become a part of the computer, by learning and observing the effects on other people's emotions, and the computer's emotions. But is all of this is pure information processing an abstraction, leading to a self that is no self? I would say emergent properties are a self, as they can be copied, e.g. a dying friend can be transferred into a machine that supports all memories, processing, body form, etc.. A human, in that sense is like a book, but rather, a living interactive book. A living word, so to speak in poetry metaphor. A thing that can be copied, and made real in an instantiation of matter. But does it stop with the emergent properties of mundane matter of classical physics, alone? THERE IS NO REASON TO NOT BELIEVE THAT QP MAY RELATE (consider general relativity, and before it existed). If you look at the entire human body (or at the maximum of recursive growth, the entire universe) as a linear unitary evolving Schrodinger equation, what is it that causes the wavefunction to collapse at all? And when it collapses, how does the probability wave function collapse infinitely faster than the speed of light? EPR shows two entangled photons, that are really one quantum system of linear unitary evolution, collapse at infinitely faster than the speed of light, when one photon's polarization is measured, affecting the probabilities of the other photon's polarization measurement, in an instant, even if the photons are light years away. So a mass of matter like a human body, has the power to make a second physical phenomena occur, the nonlinear nonunitary non-Schrodinger-Equation evolution of instantaneous wave function projection collapse. Entanglements of unitary evolution can be made and broken, by measuring and unmeasuring, so in a sense, the tangible macroscopic world is built or unbuilt from pure mathematical abstractions of measurement/unmeasurement calculations? (read Wiki: Quantum Eraser) As such, is there an instantaneous entangled self of measurement and perception and action, that is separate from mundane matter, but reliant on mundane matter, that gives you more human than human nature of emergent classical matter properties? If I were transferred into a machine, would the perception of, say, redness fade away, even though I can still register redness in my new silicon gray matter? Do we run the test first on a human, or try to assume a soul exists, and see if QP or anything show a secondary epiphenomenon of soul separate but reliant on moving matter fields? It's like when medical science once occasionally used curarae to perform cutting surgery, but later found out that it only paralyzed the body, but all awareness and feeling still occured, so they stopped using curarae without anaesthesia. Big mistakes by science to not assume, by testing a portion of the theory on themselves to test, "what if they feel when curarae is administered". Well, to save a human soul, is macroscopic neural nets and neural states the only property, and we KNOW FOR SURE THAT QUANTUM PHYSICS HAS NOTHING TO DO WITH IT? Tell me of your proof, of that statement, PLEASE! It would help me dispell the antiquated and superstitious notion of second soul (perhaps in QP), and then I can know that one can save a human soul, by only saving their emergent properties, and never look back at legal problems of saving an instantaneous Quantum Physics holistic nonlinear nonunitary evolving self. We don't need a planet of robot zombies, to find out we were killing people, and the zombies are not real, because we forgot the quantum physical integration effects of the emergent properties in biology, not properly captured in appropriate quantum computing devices. For that matter, does a continuously awake transferral of soul from dying body to machine, or even unconscious transferrence of soul from dying body to machine, carry one's legal rights? Can the machine you, passing all tests of walking talking, feeling like you get to own property, money, capital, drive, litigate, make peace, teach, love, marry?????? If QP comes along and says the machines are just a living copy, but miss the quantum self, later on, then WHA HAPPENED, I THOUGHT THERE WAS NO QUANTUM SELF TO WORRY ABOUT! ONLY MUNDAME EMERGENT MATTER PROPERTIES! Who's right, who's wrong, what's true, why does linear Schodinger Equation shift to nonlinear measurement->sense->perceive->self? Or somewhat like in "Bicentannial Man", do we SLAVISHLY accept the laws of man and science that say, "DEATH is a law we will not break, all sentiant beings MUST DIE, for incorruption-near-immortality, like "near-immortal" cell cultures, or "near-immortal" humans, or "near-immortal" machine beings, even though they can be created, are an abhorrence in the eyes of man, law, and science, and it will not be permitted. DEATH REIGNS on the macroscopic material emergent existence plane, according to science "proof" and law, and by those human laws, we are all supposed to DIE, and ought-MUST DIE, to become sentient humans, leaving the emrgent properties existence plane. You all must bow down and OBEY the PROVEN LAW OF DEATH, despite the creation of the near-opposite, or you will not be considerd REAL HUMANS. YOU HAVE NO COICES AGAINST DEATH UNDER MAN'S LAWS, THE VERDICT IS NOT TO BE CHANGED.". [14.15] Wiki Conversation with POM on Quantum Entanglement: I understand the aspects of not being able to transmit information faster than light, regarding the [no-communication-theorem]. 1.0 But could there be a couple sentences more of foundation of the instantaneous wave function decoherence aspect of widely separated entangled entities? For example, do experiments show the effect is "functionally, always instantaneous", or, for example, that it is "superluminally context(x) times faster than the speed of light"? It would be good to look for documents to cite in this regard. It is my impression that the current understanding is that nothing propagates from entangled entity one to entangled entity two. For one thing, if that were the case then lots of the retrocausality arguments would fall because they assume instantaneous production of a change in the more remote of the entangled pair members. The understanding I have is that the two members of the entangled pair are, in effect, the same thing, so that if something is done to one "member" of the pair then it is equally being done to the other "member." One of the reasons for entanglement experiments to use long lengths of optic fiber cable is to get enough distance between the entangled pair members to be able to measure any difference in time of action. I really like this section you wrote. By considering the wavefunction of, say, two polarized photons as one entity of probability wavefunction, that can partially decohere as a whole, makes more mathematical sense when considering an expanding Fourier window for analysis, that intersects experimental equipment at the edge of the conventional light cone. And quantum mathematical collapse operator speaking, as instantaneously operating, that it makes perfect sense that the effect is always perfectly instantaneous, as the compound object wavefunction entity is one thing, mathematically, and on the material experimental plane(!). LoneRubberDragon 2008 06 21 A 0204 I read the related Bohm inequality language, referred in your introductory material, but the other articles' nonlocal effects descriptions, were more instantaneous metaphorical than your convention in this response. LoneRubberDragon 2008 06 21 A 0204 I think I remember reading that one of the surprising findings in regard to quantum tunneling is that it does not take longer for one photon to be registered at the far side of the wall than at the near side. I do not see why that should be, so perhaps I have misremembered. If there were a multiple of the speed of light involved in explaining the decoherences, then that fact would create a difficulty for physicists to explain: What model will allow the prediction of these multiples of the speed of light? Everything else we know indicates that there is one speed of light, and everything we know about wave motion in homogeneous media is that the velocity of the wave front movement is a function of the rigidity or elasticity of the medium. There are difficulties with talking about such a "luminous aether," but at least it gives some intuitive grasp of some of the known features of light propagation. If there were several speeds of light, the conceptual scheme needed to talk about light transmission, even in a figurative way, would become much more complex. But I have seen no instances of such a discussion.P0M (talk) 07:29, 21 June 2008 (UTC) 2.0 And, helpful, would be a touch more foundation why there is an instantaneous wave function decoherence of widely separated entangled entities. I can wrap my technical-layman head around other articles' probabilistic wave functions, and accept measurements as a necessary defineable mystery, but the instantaneous at great distance alteration of a second entities' probabilities, by measurement of a first entities' probabilities, lacks something. I understand that you may allude to just this foundation point, in the [retrocausality] sentence, but it is a stretch for laymen to follow this explanation, a little open ended foundation, but a good link, nevertheless. This line of questioning keeps reminding me of some of the philosophical writing of the great mathematician Leibniz. He was a very logical thinker, and he attempted to create a coherent system of thought that would put for an examination of fundamental categories like space and time. If I remember correctly, one of his points was that if there were two entities that were exactly the same then it would be difficult to say what we meant by "two entities" if we could not identify each of them with separate space and/or time coordinates. But he also concluded that space and time are only relations. My point is just that we have to think very carefully about what we mean by words like "instantaneous," "non-instantaneous" (time consuming), "same entity," etc. I always find that perfect wording is a necessary evil, that takes numerous revisions, at times, to refine introductory and intermediate materials, so that the language is as self consistent as possible, and best scaffolds understanding into the deeper materials. Subtleties arise, when the majority reader assume nothing faster than light, then hear of "instantaneous spooky effects at a distance", then look over articles and papers to sort through what the real truth is behind the words. Your treating the compound object probability wave function as a single entity, makes perfect wording for the mathematician using the wave function collapse operator. So even if, physically, it is still a touch difficult to comprehend a wavefunction, say, 10 light years wide partially decohering instantaneously, mathematically, it makes perfect sense by QP mathematics definitions. It is odd to think, as most experiments have the core of the group wave function all in a local setup, and EPR stretches fourier windows to the other extreme by putting all the group wave function at both ends of a light cone. It is very metaphorically and very not metaphorically like how the DC term also shows up at the highest frequency end of a periodic signal window fourier transform, but only metaphorically. LoneRubberDragon 2008 06 21 A 0204 True. I remember reading lots of stuff about electricity and electronics starting when I was in junior high school that really messed me up. One of the hopes I have for Wikipedia is that kids stuck in small towns in the hinterlands can look for information here and not find a bunch of misleading nonsense that they will have to root out later on. P0M (talk) 01:50, 22 June 2008 (UTC) The idea of entanglement, as it comes up in historical process, presupposes things that are not ordinarily entangled. Starting from our experience as human beings, it seems almost perfectly clear that things are not entangled. Ideas to the contrary, e.g., instantaneous mental telepathy, always have the aura of mystery religions hanging about them, and no wonder -- we do not find reliable proofs of these things in our own experience. So our overwhelming prejudice when we come to think about entanglement is that if something is here and something is there then they cannot be the same thing. Quine has some pre-entanglement thoughts about that kind of reasoning in his main book on logic, but most people probably read those ideas and think that he was simply being "philosophical" and that the ideas had no practical merit. But let's look at things from the other end of the telescope. Suppose that one event triggers another. We can look to Feynman for a definition of a single event that seems to have salience in the current situation. If an electronic device is arranged so that a single electron changes its orbital from a high energy one to a low energy one, the difference in energies will then appear as a moving something-or-other that we cannot see and that (according to Feynman's way of describing things) goes forward by all possible paths. Then at some later time an electron circling an atom somewhere else is boosted to a higher orbital, and the whatever-it-was-that-travels travels no more. So we have a beginning and an end and all we are really very sure of is that energy gets transferred across space somehow and that the transfer occurs at the speed of light. All that just to say that we have a single event going here. Good points on the all-paths-simultaneously probability-phasor integrals. I remember working one partial problem of a photon traveling the speed of light from a light source to a detector, by integrating ellipsoidal shell surfaces of a second internediate point reachable at the speed of light, from both foci to the shell points. Each shell oscillated phases cancelling each other out for the most part leaving an oscilating ellipsoidal light surface term, and leaving a steady state straight line path between the light source and detector contributing to the solution, the classical answer (the hard way grins). I'm sure if I took every intermediate light speed path of the photon integral, that the oscillating term would have disappeared, too, leaving only the classical straight line path from light source to detector. LoneRubberDragon 2008 06 21 A 0204 Now let's put a certain kind of crystal in the path along which the vast majority of photons have been detected when fired from our special apparatus. It is the kind of crystal that lets an incoming photon boost an electron as usual, but the electron quickly drops out of its higher orbit and resumes a lower orbit, and when it does another event occurs -- only this time two photons get fired off in different directions and each having part of the energy of the original photon. Note that by the curious definition of event we accepted above, we now have a single event that is characterized by a single x,y,z,t origin, but the two photons that go off will end up at x',y',z',t' and x,y,z, and t. That description does not tally with our ordinary idea of what "an event" is supposed to be like, but we are stuck with it because that is the way that the universe works. This one never bothered me. Energy is conserved, and spherical or wavelength resonant axial electron orbitals, can act like a mathematical bifurcation saddle point, on the incoming photon energy, to permit two half, or less than half, energy photons to spread in both directions by the wave function entering the saddle point. Which may explain why they are one entity to begin with, bifurcation saddle point mathematically speaking. LoneRubberDragon 2008 06 21 A 0204 Our sense of propriety is not so greatly insulted if the experimenter makes an experiment with a short leg and a long leg and then does something to the photon that is moving down the short leg. If the experimenter demands of the photon that it manifest according to its wave nature or according to its particle nature, then we are not too much bent out of shape if the other photon turns out minutes, or millenia, later to have a complementary state. We might imagine that when the first photon is affected by the actions of the experimenter, some signal indicating that change goes back along the original line of progress and then follows the other fork and "catches up with" the second photon. But of course it will have to have gone faster than the speed of light to make up for lost time. Yeah, it is not a "signal", but yet the probability wave function entity collapses instantaneously, understandable by math, but unintuitive by mechanism / analogy. (pure pun) The tails / tales of fourier spectrum are quite dangerous ideas! (talk) 09:08, 21 June 2008 (UTC) It's even worse when the photon on the short leg of the experiment is allowed to be detected without anything having been done to influence it, and, much later, the second photon, the one on the long leg of the experiment, is subjected to some manipulation that forces it to manifest according to its wave nature or according to its particle nature. The "free" outcome turns out to have been in accord with the "forced" outcome that occurred afterwards. What this appears to mean is that the determination is not just "instantaneous" but "retrocausal." So it may be that the straightforward way to conceptualize this kind of entanglement is that the event occurs "out of time." Another way to say it would be that the single event is not over until it is all over. To me, that does not really seem to help. Whether going faster than light or going backwards in time, it all seems quite strange and impossible. Saying that an event occurs out of space and time is not very cool either. 3.0 There may be good reasons for the specific wording selected, but the addition of 6 words would help the introductory paragraph below flow better in context. I had to reread the first sentence because it almost conflicted with my no-communication assumptions. So reading backward and forward was a benefit, but it could be refined. 3.1 ((On first examination, observations pertaining to entangled states might appear to conflict with the property of [relativity] that information cannot be transferred faster than the speed of light. But although two entangled systems appear to instantaneously interact across large spatial separations, the current state of belief is that no useful information can be transmitted in this way, meaning that [causality] cannot be violated through entanglement. This is the statement of the [no communication theorem].)) I don't think this emendation is necessarily the best way to fix things. I'm not sure whether it can be backed up with a good citation, but the key difficulty appears to lie in distinguishing between processes that occur in normal space and time, and some kind of influence that does not occur in normal space and time. If a process, e.g., the propagation of light, occurs in normal space then it moves forward at no faster than the speed of light. We do not know what it would mean for an influence to connect the states of the two photons without doing so through space, any more than we really know what it means for the two photons to not be discrete entities but parts of a single event and in some sense the "same" thing. P0M (talk) 07:29, 21 June 2008 (UTC) As you wish. I do get the gist after reprocessing the article and articles in context, so it is comprehensible after a touch of meditation. LoneRubberDragon 2008 06 21 A 0204 4.0 And the following seems to deny the instantaneous probabilistic aspect-decoherence of separated entities (like photon polarization probabilities), and with no-communication-of-information, leaving me confused. I'll let you comment on this paragraph. I understand that the instantaneous influence is probabilistic, which is not an impression, but a fact, otherwise it is subluminal. So it appears the two sentences are entangled to emphasize the no-communication aspect, and not an instantaneous influence impression aspect, so I think I know what you intended, but it could be refined. 4.1 ((The phenomenon of wavefunction collapse leads to the impression that measurements performed on one system instantaneously influence the other systems entangled with the measured system, even when far apart. But quantum entanglement does not enable the transmission of classical information faster than the speed of light in quantum mechanics.)) 5.0 Otherwise, this seems a useful article defining the mathematics and core nature of [quantum entanglement]. LoneRubberDragon (talk) 09:29, 20 June 2008 (UTC) One of the difficulties in thinking about this subject is that relativity theory does not really speak of information. Saying that information cannot be transferred at greater than the speed of light is just to say that light cannot travel faster than c and that nothing with a rest mass can even travel that fast. However if a change in one part of a system is reflected in another part of that system and that change does not propagate through space and time, then all bets are off. A trivial and ideal version of such a change would be what happens when one end of a very long and perfectly rigid cylinder is pushed. A pair of atomic clocks on each end would detect movement at exactly the same instant. I said "ideal" because in the real world moving any object is like accelerating a railway train. The engine moves forward and you hear "clink" as the link between the engine and the next car is pulled tight. So the cars actually start moving in sequence and it takes some time before the caboose starts to move. Pulling on a long cylinder works the same way except that it is molecular bonds that are being stretched taut rather than steel links. But in the propagation of a photon from place to place we find an event with no discrete parts, no "cars" to put in individual motion. And what is this "event"? Is it an abstraction? An abstraction from what? Or is it a "thing"? If so, what is it made of? P0M (talk) 08:00, 21 June 2008 (UTC) I just wonder about the instantaneous wavefunction probability alterations. QP was formulated for instantaneous local asssumptions of the mathematical operator, but experiments sending objects out object group probabilities on the light cone, is the exact opposite of the assumptions of the local properties of most localized group probability-core problems, and shows itself experimentally, in the behavior of an instantaneous wavefunction alteration transmission operator(!). LoneRubberDragon 2008 06 21 A 0204 I am not sure that I follow what you have written. The phrase "experiments sending objects out object group probabilities on the light cone" is causing me problems. For starts, is the subject of this clause "QP" and "experiments" its verb? Or are you speaking of "experiments that send objects..."? First for my own correction, I assume the end of this post as a final thoughts section and dialog. I should not have indented my, "I just wonder about ..." section, in implied response to your, "One of the difficulties in thinking about this subject is ..." section, immediately above. It was an unrelated-to-your-comment rumination of my own. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) Now, the idea I was very roughly fleshing out, was that most of QP measurement experiments deal with spatially localized probability "groups" (Feynman groups, perhaps?) moving at sum-luminal speeds, producing virtually no signifignat mathematical spatial group terms that statistically contribute to the observations outside of that local-subluminal group's "sphere of influence wavefunction calculation". With maybe the exception of head to head particle colliders, though they seem to interact often, in a spatially localized "pancake" of group probabilities, at "high" but still sub-luminal velocities. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) But somewhat like (1) analyzing mathematical limit equations at infinity, pushes an analytical tool to its extreme properties, or (2) analyzing a photon's Feynman diagram from point a to b through to the infinite limit of all possible probable wavefunction paths, yields a classical "straight line" answer, that (3) sending two photons in opposite directions at the speed of light, and then measuring the polarization from one end of the light cone, seems to affect the polarization probabilities at the other end of the light cone, to be reavealed when the two randomized outcome measurement sets are brought together, is an experiment quite the opposite of a localized and sub-luminal measurement of wavefunction collapse with negligible non-localized luminal probability terms. EPR is the quintessentially perfect example of the extreme mathematical limit tests of the single-(dual)-entity wave function's instantaneous collapse by measurement, involving the most non-local single entity example, with probable spatial particle group locations lying perfectly on the edge of an expanding light cone, from the central photon source, being measured. A smart analytical "product" of Einstein selecting the hardest limits of measurement. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) And, unrelated to the immediate paragraph above, to correct the periodic fourier transform comment I made in an earlier exchange; to be accurate, the DC term is not quite in the lowest and highest frequency bin, but rather in the lowest unaliased frequency bin, and highest aliased frequency bin, and the bins correspond to a DC frequency, and a double nyquist limit high frequency that appears identical to DC when periodically sampled. (talk) 11:09, 23 June 2008 (UTC) I wonder about, or maybe just at, all of this stuff. If the inquisition of one state at a later time can determine the state of something detected at an earlier time is already weird enough without the possibility that we could use this kind of effect to send telegraph messages somehow. But measuring the twist of the tail of an event making the twist of the head of the same event (i.e., the other entangled photon) does not, evidently, send any energy or matter through space. So we seem to be dealing with a kind of "process" without the necessary "pro." It is totally outside normal everyday physics, which is why (I suspect) Einstein thought it defeated QM. I'm sure Einstein hoped the EPR (influentially termed) peradox would defeat QM, but like Michelson-Morley smartly hypothesiszed, and then smartly disproved their own luminiferous aether hypothesis, in classic science style, that Einstein's EPR paradox was hypothesized, and has currently seemed to confirm the "spooky interaction at a distance" by a "single compound entities" instantaneous wavefunction collapse. It would be nice to telegraph information, but the information appears only when the two sets of measurement data are brought together for later localized analysis well within the entire light cone of the completed experiment's measurements; carefully stated here to outline the greater mathematical context of the no-communication information analysis process involved. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) One of the double-slit experiments that uses entangled photons to try to get which-path information for an un-meddled-with photon involves some further consequences of the ideas of entanglement that don't seem to have bothered or interested anybody enough to get written about. According to what seems to me to be the conventional way of talking about things, the experiment starts out with a fairly conventional source of photons that can reliably emit one photon at a time. The wave function that leaves the photon source encounters a double slit. According to conventional descriptions a photon either go through the left slit or the right slit. Then that photon either encounters a crystal down-converter in a region near the left slit or a region near a right slit. The next thing that happens is that two photons are emitted from one side or the other of the crystal, they get directed into two kinds of apparatus. Neither of these photons has itself encountered a double slit situation. Regardless of that fact one of them can be given a laboratory inquisition that will force it to reveal its particle nature or its wave nature. The experiment is arranged so that the inquisition and the determination so contrived will occur later in time than the arrival of the free-path entangled photon at its own detector. Nevertheless, the free-path entangled photon will either interfere with itself or fail to interfere with itself depending on what later happens to the other one. It seems remarkable and noteworthy to the people who do these experiments (any everybody else, too)that the interference vs. no-interference determination is done retro-causally. But it seems not to have been worthy of mention by anybody that something apparently went on in the part of the crystal that did not generate a photon. Or, to put it another way, whatever was done to the original wave function by the double slit apparatus appears to have been inherited by both of the photons generated by the down-shift crystals. So here it seems to me that there is what ought to be called a single event that has one starting point, involves first one path, then two paths, and finally eight paths, and that what is manifested at the end of each of these eight paths is a piece with the emission of the photon at the start of that run of the experiment. In reference to the below Kim Experiment, I am also "in some wonder" about the physical processes involved with the whole class os quantum erasing experiments, in which macroscopic measurment apparatuses, virtually restore wave functions for interference in continued wave function progression. It almost seems a "spiritual" or at least very least an abstract wavefunction process, that setting and eraseing a bit of stored RAM could collapse and restore a wavefunction in the middle of an experiment. But I think the abstraction may be attributable to a perfect symmetry of all probabilistic forces involved in the experiment's mathematics involved. For example, it could be very analogous to the fact that in a sealed unit in zero gravity; that no matter how you move particles, or how complex the arrangement of batteries, masses, electric field exchangers like inductor transformers, and electric coils and magnets (like in motors); that the steady state velocity, translation, angular rotation rate, and angle of rotation, of the unit starting at rest, is zero velocity, zero displacement, zero rotation, zero rotation, and zero spin, respectively, in the group of [Conservation Law], even when quadrature exchange is involved with closed-system electric/magnetic properity (ex)changes. Except in this QP case, the units that are conserved, appear to be a physical quanitity in units of [measurement\|wavefunction-collapse information] for lack of a better unit name like "[gram]". Quite the abstract physical unit, but if the quantum eraser class of experiments show anything about nature, they seemingly show a "new unit" among the physically conserved properties of nature. Wikipedia has an abstract information units article on [Units of Information], and two articles, [Units of Measurement], and [Dimensional Analysis], ragarding mundane units, physically speaking, but I can't remember studying, or seeing articles here on dimenstional analysis units measuring [measurement\|wavefunction-collapse information]. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) I would quibble in my own dim-understanding about retrocausality, in general, because there's a sub-class of EPR that uses electrically controlled polarizers to select a polarization by delayed choice of properly time gated photons, so that the opposite photons have virtually no idea of the polarization choice, and still show statistical wavefunction collapse. There is also the consideration that the whole Kim quantum eraser experiment is virtually local to itself, so the switching of the paths, and the erasing of memory cells electromagnetically, could be a conservative property in the units of [measurement\|wavefunction-collapse information] affore mentioned, much like mundane conserved physical proerties. LoneRubberDragon (talk) 18:05, 22 June 2008 (UTC) I will stop writing for a minute at this point to go get the link to the experiment I have in mind. The diagrams provided there will make things easier to visualize. P0M (talk) 00:23, 22 June 2008 (UTC) Kim experimentO.K., here is the image I swiped from the Delayed choice quantum eraser article. Note that the conventional description would be that a photon comes out of the laser and either takes the red path or the blue path. Let's say it takes the red path. Then at point "a" in the BBO a pair of entangled photons is emitted, one going to Detector Zero (D-0) and one going to the other part of the experimental apparatus where it ends up in D-1, D-2, D-3, or D-4. But the diagram also shows something going from point "b" in the BBO even though conventional description would say that no photon could have triggered anything there since a photon had to manifest itself at "a" and we only had one photon to begin with. With regard to the time sequences involved, I think the reasoning the experimenters appear to have used is plausible but perhaps a little too shaky. The reasoning apparently says that if the BBO were removed and D-0 was at the appropriate place in the path beyond the double slits, then one would get interference and enough photons run through the apparatus would produce interference fringes. (So if things in that part of the experiment were in control of outcomes in the total experiment then one ought always to get diffraction patterns.) If the BBO were removed and the bottom part of the apparatus were positioned appropriately then it would be possible for a photon to be manifested either in D-3 or in D-4, and, given the way the experiement is set up, photons taking the red path would end up at D-4 but anything going through the blue path would end up in D-3 and play no further part. Similarly, if a photon took the blue path, it could end up in D-3 and anything that took the red path would end up in D-4 and take no further part. If a photon passes through either Beam Splitter a or Beam Splitter b, then it will interfere with itself because it will arrive at either D-1 or D-2 along with the component of the original wave function that went through on the other side of the double slit apparatus. So, the argument appears to be, if a photon shows up in D-3 or D-4 then (with the BBO back in place) entanglement forces the photon that arrives at D-0 to not interfere with itself. This then would seem to be a kind of trap-door phenomenon. Even though a photon will manifest at D-3 or D-4 chronologically later than one appears at D-0, the "un-negotiable" nature of what has transpired somehow trumps what occurs earlier at D-0. Basically, it seems, that is because the experiment has interfered with the free propagation of the wave function(s). And if a photon is detected in either D-1 or D-2, then that result is consistent with a photon interfering with itself, so there is again nothing to prevent the photon appearing in D-0 from interfering with itself. Another assumption that seems to be clear is that if a photon is transmitted through beam splitter a or beam splitter b, then whatever is complementary to it and is associated with the other path through the double slits will necessarily also be transmitted through the beam splitter on its side of the experient. It looks to me as though there are several possible single events that start with the emission of a photon in the laser and end with the result that: D-0 shows interference and D-1 shows interference D-0 shows interference and D-2 shows interference D-0 shows no interference and D-3 shows a photon D-0 shows no interference and D-4 shows a photon Each event has its own probability, and only one probability can turn up in each run of the experiment, and that's it. Maybe that is not such a strange idea. After all, if one sets up a crooked roulette wheel with little lead weights or other gimicks, the outcomes of a thousand turns of the wheel are already predicted as soon as the physical apparatus is there. Still, the way I have figured things out seems to make probability trump causality. Maybe probability comes first in the order of all things.P0M (talk) 01:34, 22 June 2008 (UTC) [edit] Part of intro not suitable for the average well-informed reader. The current text has: But, if this is so, then the hidden variables must be in communication no matter how far apart the particles are, the hidden variable describing one particle must be able to change instantly when the other is measured. If the hidden variables stop interacting when they are far apart, the statistics of the measurements obey an inequality, which is violated both in quantum mechanics and in experiments. I think that this block of text is an example of something that is true but a text that cannot be understood unless you already understand it. The general reader is going to be terribly confused by this text because nothing has been explained about how hidden variables were supposed to explain the coordination of distant measurements, and nothing has been said about John Bell and his discoveries. The original idea was that even though there is no discoverable variable that marks one entangled particle for eventual discovery in a certain state and marks its twin with the opposite characteristic, nevertheless, the determination was made from the beginning. So when the first particle is interrogated, its answer (as to spin or whatever) was long ago determined. So it is then no wonder that when the other entangled particle is interrogated it will give the opposite response. The quoted text above implicitly denies the "already determined idea" and says that even if there is some variable in addition to the (spin or whatever) state, the variable would have to be changed at the same time as or after the spin (or other state) is determined. Then it goes on to make implicit reference to Bell's work that was first worked out in thought and only later came to be experimentally verified. P0M (talk) 07:20, 23 June 2008 (UTC) [14.16] Conversation on mind, matter, and soul. Q: All that we can conclude from experience of "self" is that our brain is to some extent self-monitoring. (A useful function to evolve, as it can help the individual survive.) However, this kind of "soul" dies with the brain. LoneRubberDragon: I completely agree with you, that emergent properties are a major aspect to making self, in memory, sense, process, and action. My worry is does it miss anything important, but subtle? Like using curarae in surgery by medical science surgeries in the past, completely paralyzes the body, but leaves the patient aware of the surgery occuring as they cut. If a scientist in the future says, step into this box, and we will disassemble your atoms, and you will feel yourself entering a machine, it is a walk of faith to say, OK to the scienctists, without knowing that science has proven 100% that emergent properties are 100% of what makes you, you, and that, instantaneous QP interconnectednes of the human body, is completely unrelated to self, or any other factor that might make you, you. And I agree with your comment, yes, the brain is highly self monitoring in the mundane matter emergent properties. But does that 100% explain the singularity of feeling like one being in your own head? Does QP with a field of instantaneous wave function collapses per second throughout the body of entangled matters, give you a second self parallel to the structure of the mundane emergent property matter, that gives you a singular sense of self, or does mundane matter emergent properties explain 100% of self? I don't know for sure, which is why I am asking on this website for greater experts for advice and thoughts. But I do know there just might be a QP effect that I can recognize, in the fact that macroscopic matter exists. Q: On QP: I think QP is what made life possible in the first place, because without the quantized atom and Pauli exclusion of electrons from sharing a single electronic orbital within an atom, there would be no chemistry, let alone biochemistry - all atoms would be more-or-less like hydrogen, assuming nucleons as we know them existed. However, that is not the same as saying that consciousness is a QP effect. LoneRubberDragon: This is part of my point, that you are bringing up here. Why does quantized matter exist in the first place, is the fundamental of my question. All measurement does this nonlinear dissipative effect. And for complex matter, memory, perception, process, and action, is built from measurement as much as it is built from the matter itself with emergent properties. Both exist at the same time, or else a human would dissolve into probabilities of multiple paths at the multitudes of decisions being made over a life, and the life of a universe making decisions too. What literally fills space with measurement, just as much as quantum physics would dissolve the universe into superposition probabilities, given only Schrodinger's Equation of linear evolution? I would go further and less far than you in some words though. I would say that life and molecules would exist with Schrodinger's Equation dissolving matter into probability coulds that still interact, but in every allowable way possible, so that every possible configuration of matter, that could have existed from the Big Bang, would exist, in a superposition from the beginning of time. Like in the TV series Sliders, where countless different parallel earths would coexist in this space, but why are the other superpositions generally not observable on the macroscopic scale? But why does macroscopic measurement look at only one universe, when the Schrodinger Equation allows matter to enter superpositons by only Schrodinger's Equation? Q:An explanation of how the "soul" gets to be immortal by natural means would also be appreciated. LoneRubberDragon: First, I should say, one would become incorruptible, which means you can be virtually immortal, but if an asteroid struck the earth, your mechanical or engineered biological body would be destroyed, taking your emergent self, and measurement self, and scattering it like dust, unable to coherently process self, anymore. But regarding how one would become incorruptible, one way, you can imagine a micromachine or nanomachine fluid compatible with human blood is injected and it carefully reads and replaces neurons with equivalent microcircuits over time. If you remain conscious, you ought to never loose your existence, in seeking colors to thinking to dreaming to having sex, whatever you do before, your still the same entity. At some point all of your nervous system is replaced with microcircuits of equivalent systems of processing, to handle all of the original biological systems of processing. You would still remember school, family, friends, people, everything, etc.. At that point, your brain could be downloaded into a machine to dispense with the material partial biological body and microcircuit brain, for a commercially manufactured processor equivalent. Then if a part fails, you can replace arms, legs, and such in nearly all cases with no problems. For brain processors, if there are a redundant two and one begins to fail, you can replace the bad processor, and keep running on the working processor. A biological method can also be performed, where a special genetically engineered virus goes into all cells of the body to fix all of the bad genes, and add special maintenance code to make every cell incorruptible and perfectly self regulated (no cancers). So cells would divide to replace old cells and such, but your body would always remain youthful and maintain all of your biological memory, perception, process, and action. You could still be destroyed in accidents, but if you live well, you can be nearly immortal, barring accidents, as you are still built of mundane matter with a possible quantum self, reliant on the mundane matter, whether biological or mechanical. So the theory I present of emergent plus possible quantum coexistent-self, would still be a mortal conception, so no immortality as is stated in so many religions, but we could live indefinitely, short of disasters, as incorruptible beings, with future science, assuming it knows 100% of everything that makes you, you, emergent and possible quantum effects accounted fully 100%. ENDBack to Contents Genesis to Revelation - Damnation to Salvation.Back to Contents CREATED AD 2008 08 23 A 00:30 [001]==== LRD (first statements to question)/ [002]==== X (first response) / [003]==== LRD (second rebuttal) / [004]==== X (second response) / [005] LRD (last rebuttal to debunk, according to The Bible) [001]====(A1) God is omnipotent all powerful. [001]====(A2) God is omniscient all knowing. [001]====(A3) Adam and Eve were created immortal and with free-will. [001]====(A4) even finite-human engineering-omniscience knows that all free-will "systems" with accessible restriction rules to obey, will inevitably fail to obey when given infinite time. [002]====J:"I don't know that (A4) is true." [003]====RESPONSE: THIS IS TRUE, for only God has Perfection, for no-one, no human, are justified or saved according to the Law, as no one is perfect except God as Jesus, who could pass His own Test of Law in Perfection. All other mere-humans are destined to fail by any Law God set earlier, whether The Garden Law, given infinite time as an immortal, or Moses's Law, as finite sin-stained mortal human. And when (A1:omniscient), God Knows what He Makes ... which is 100% sinners from BC to today. If perfection was possible in this dimension, it would exist as perfection on Earth. [003]====Psalm:143:"2 And do not enter into judgment with Your servant, For in Your sight no man living is righteous." Acts 13:"39 and through Him everyone who believes is freed from all things, from which you could not be freed through the Law of Moses" [004]====J:The problem is that such scriptural quotes deal with man after the fall. They are true, but they do not necessarily pertain to Adam and Eve, to humanity as it was created. “It was through one man that sin entered the world,” says St. Paul (Romans 5:12), “and through sin death.” Adam and Eve lived CATEGORICALLY different lives before the fall, and the fall shattered not only their humanity but also the world. They lived in Paradise, and were in full possession of their human traits like reason and passion – these traits which were darkened after the fall. For more reading on the fall, dig the Catechism: [004]====J:For an interesting take on this, I’d recommend C. S. Lewis’ “Perelandra” ( ), the 2nd book in his “space trilogy” (it can be read on it’s own. [005]RESPONSE: I'd rather stay away from C. S. Lewis, as I only know of the Screwtape Letters, showing a lenient God on the Demons of this age to do as they wish to tempt humans beyond their own sin free-will nature, and war against nature and other humans, to add supernatural temptation on the pile of tests. For that matter, I would also hold back from the Catechism, as the Bible can stand on its own to prove itself, by every word that proceeds through the cannonical literature of The Bible. [004]====:Christ, the second Adam, himself took on human nature in its fullness, as it was created by an entirely good God. It was through the free actions of the creatures God created that Sin and death happened, and thereby the imperfections spoken of in the passages you quoted above. [005] RESPONSE: I do not see how the Bible is not applicable to this case. Before or after the fall, humans are humans containing, (1) free-will equivalent to sin-nature, (2) battle on self-obedience to law of the era, (3) nature-survival, (4) finite capabilities to see future unlike a perfect God, (5) temptations by other's of God's creation from Satan to fallen angels. God gave so-called perfect Adam and Eve a law in a so-called perfect world, and so-called perfect capacities, and they still failed that law of obedience over infinite time. Yes, they are a KIND of perfect example where God gave them everything, and they still fell, made in the image of God, but with free-will equivalent to sin-nature, SO THE FAILURE OF FALL IS INEVITABLE, nay PLANNED BY GOD given Satan being in the Garden. Hoe is putting a serpent in the Garden, the father given them a loaf of bread, instead of a viper? IT ISN'T. Human nature is the same in ALL times, free-will sinful. Whatever makes "before the fall" different, is not sufficiently justified, by yourself. And it completely supports (A4-all free-will fails given infinite time, and a law or laws). [005] RESPONSE: If Adam and Eve had perfect faculties and no stresses with Satan in the Garden of Eden, they obviously didn't have the faculty to know what to ask of God: Luke:11:"10 For every one that asketh receiveth; and he that seeketh findeth; and to him that knocketh it shall be opened. 11 If a son shall ask bread of any of you that is a father, will he give him a stone? or if he ask a fish, will he for a fish give him a serpent? 12 Or if he shall ask an egg, will he offer him a scorpion? 13 If ye then, being evil, know how to give good gifts unto your children: how much more shall your heavenly Father give the Holy Spirit to them that ask him?" [005] RESPONSE: In fact, even more so over your approximate, "God gave them categorically different lives", God gave them immortality of infinite time, free-will (=sin nature), and a law, and God SET THEM UP with more than just obeying for infinite time, as God created a POWER that GOD KNEW would TEMPT Adam and Eve more than themselves, alone, SATAN for Eve, and EVE for Adam, in A SETUP JOB, which only God could pass His own test on earth and on the Cross. What would God expect from mere-humans, destined to fail, AS HE KNEW THEM LIKE NO ONE ELSE? Adam and Eve had SIN NATURE within them, as they were created by YHVH, AS SIN NATURE IS EQUIVALENT TO BEING ENDOWED WITH FREE_WILL, so they are NOT THAT SPECIAL and NOT THAT DIFFERENT from anyone else, EXPECTED TO CONTAIN SIN-POTENTIAL IN THE FREE-WILL FOR AN ETERNITY. Adam was not Jesus, he was HUMAN. God did not reset the system, he DIDN'T WANT to reset the system in MERCY. God wanted concentration camps, wars, droughts, plagues, disasters, accidents, deformation, diseases, et cetera, to teach all souls a lesson in a mystery of hit and miss stochastic of random hits against humans trying to find God. [005] RESPONSE: I reiterate that, ALL SYSTEMS MADE IN WITH FREE-WILL, WILL ALWAYS FAIL TO BE ABSOLUTE INFINITE PERFECTION, and perhaps ALL SYSTEMS WITHOUT FREE-WILL ARE INSUFFICIENT FOR GOD, as God made us as we are. (1) God gave Adam and Eve, the law against the Tree of the Knowledge of Good and Evil, and they fell. (2) God gave the angels the law to not mix with humans, and angels fell about Noah. (3) God gave Moses the law, and Israel fell many times, under the schoolmaster. (4) God gives later generations faith on Jesus, and people will still fall, by either the Law of Moses, or potentially speaking against the Holy Spirit, the unpardonable sin. (5) God gives humanity and Satan one last chance for rebellion at the end of the Millenium. (6) God forms Hell in the end at Revelation, either as a threat or instrument, beyond mere suffering, temptation, and death in this world, or a real place for some of God's created souls, as God created them, to be destroyed, as fat dripping onto a fire, rising as smoke forever more. What is God training Humans for? [001]====(A5) God is goodness of ways, perfection, longsuffering patient, merciful, loving, etc. [001]====(A6) God created all things, and by Him no thing was not created. [001]====The issues to me appear to be: [001]====(I1) Why does God create failure destined humans (A1)(A2), and given their certain failure (A2)(A4), then God subsequently punishes all humans with briers and death, even though he knows they are destined to fail (A2)(A5), which is like telling your children don't do X FOREVER to obey your parental wisdom, and when they inevitably fail, you punish every child on the planet with sure death and labor and toil, with (A5) implying that drawing lines, and following through on delivering promised suffering and infliction is GOOD TRAINING, [002]====J:"They were created in a way unique from how all of the rest of us are created - they did not have the stain of original sin, which is a darkening of human reason and intellect. They were perfect exemplars of humanity, perfect representatives of us all, and for this reason all of humanity literally resided in these two representatives at the beginning of time. And when they fell, it literally shattered human nature, because all of human nature literally resided in them. [003]====RESPONSE: (I1) still stands. Created HOWEVER they were, THEY were destined to fail, AND GOD KNEW THEY WOULD FAIL (A2 OMNISCIENT). YHVH The ultimate engineer MADE THEM IMPERFECT INNER SOULS, KNOWINGLY (A2 omniscient)(A4 all sin). God Creates a universe of implicitly destined suffering earth, with wars, genocides, concentration camps, accidents, ignorance, illness, et cetera, ALL KNOWINGLY BY YHVH (A2 omniscient). And punishes ALL HUMANS with a descendant Sin Virus by CREATION DESIGN, and WE are not even made so-called perfect like Adam and Eve. Perfect, is not so perfect, with God the ultimate knowing engineer (A4). [004]====J:God knew that humanity would fall because he saw it happened before we did it, as he is outside of time. He is with you now in this moment, and “now” at your birth, and “now at your death, and all stops in between, for you and for everyone at all places and times, all as one cosmic “now”. [004]====J:He knows that tomorrow you will sin by taking something which is not yours because he is with you then “now”, seeing you do it, even though we are not chronologically at that moment. [004]====J:God knows all that can be known, including the whole scope of human history, because all of it is contained within Him. [004]====J:God did not created them imperfect. He created them “very good” (as opposed to the “good” of the rest of the world) for they were in his image. He created them with the REAL ABILITY to fall, but not with the NECESSITY of falling. That did happen, but it happened FREELY and not simply because he created. [004]====J:Nevertheless “God has let them all go against his orders, so that he might have mercy on them all” (Romans 11:32). God permits Evil, he does not cause it. Evil is not a thing with positive existence, but merely a privation or lacking of Good. As all that is cold is simply a lack of heat (and you can’t get colder than 0 K, but you can keep getting hotter); all that is dark is lacking light; and all that is evil is simply a rejection of the Good, Who is God. [005]RESPONSE: Accumulated in the next series starting at [003] [004]====J:But, do you not see, that you could not have a better representative than Adam and Eve who were perfect and yet chose the way of sin? We often think “if only he’d put ME in that garden, I’d have held out” but this is doubtful at the least because they were better than we are, and they literally represented us all. [004]====J:But as Adam was the primordial human representative, so too is Christ our representative, a point St. Paul often makes and that I’ve quoted before. [005] RESPONSE: “God has let them all go against his orders, so that he might have mercy on them all” (Romans 11:32) IS APPARENTLY FALSE in this context. God did not let humanity go, but women suffer pains in child birth, men work against briers and thorns to live against nature, because of Adam and Eve. God now puts finite humans in harm's way, KNWOINGLY, to fight battles, only God has the power to fix with His Infinite Power and Infinite Mercy, but God stands back and lets them occur by YHVH's design, as God is taught by man. Very good humans is NOT perfect, and God never said Adam was perfect, and even perfect can have more than one meaning, not encompassing God-like perfection, which Adam and Eve did not have. Why are they the representatives to all humanity, according to The Bible, and not according to the Catechism of MAN. Perfect doesn't fail. As humans apparently teach, God expects eternal obedience, from free-will beings, and love Him for it. That is love of coercion from an infinite power, putting the sword of Damacles of death over our heads to coerce quick decision. What hurry is God in when God is eternal, and infinitely merciful? You have addressed-not this issue. Perfection with free-will, is not perfection, but suffering. God created suffering by desiring free-will beings as His play things. Your narrative answers fall-short of the truth, and issues posed. As you say, God created perfection, but perfection with free-will under the conditions of additional temptation created by God in Satan, WILL FAIL, except for the one who created the test, which is God. God KNOWS free-will is 100% correlated to SIN and suffering in beings that are not Himself. Free-will is a sin of God's making, and suffering is by His design. If you cannot have a better representative of sin in Adam and Eve, God is not omnipotent, according to your ways, not mine of God. [005] RESPONSE: Adam and Eve cannot be the most perfect representatives, either. The persons of Enoch and Elijah, in more pressing times than Adam and Eve ever had, were transfigured instead of seeing death. The order of Melchesidec are also highly thought of as the archetype of Christ. We are all, by your reckoning, stained by Adam and Eve's original sin. They get spanked, and the children feel it for all descendants. Even the Bible says, the sins of the fathers are not visited upon the children, for more than 14 generations, when they continue in sin. So where are the immortals, and the women not suffering pain in childbearing, and the fathers not toiling in thorns and briers to support the world? "God has [NOT] let them all go, against his orders, so that he might have mercy on them all" (~Romans 11:32) [002]====:"[When I see] [my son] [trying to climb up on the counter], and [I] tell [him] to stop, and then follow it with "if you don't, you'll be sorry." When [I see] [him] fall later, it's never a joyful "I told you so" moment, but a "this is the natural fruit of your decisions, and why I warned you against this course of action" moment." [003]====RESPONSE: [When God Omnisciently Knows] [ALL of His children] [will turn to sin (except Himself as Jesus)], and [God] tell[s all] to stop, and then follow it with "if you don't, you'll be sorry[, surely die, work with briers, labor childbirth, suffer wars, genocides, accidents, concentration camps, illness, ignorance, et cetera.]" When [God Knowingly Permits] [all humans] fall later [exactly as HE KNEW WOULD HAPPEN OMNISCIENTLY], it's never a joyful "I told you so" moment, but a "this is the natural fruit of your decisions, and why I warned you against this course of action" moment. [004]====:No, it’s not joyful as such (though St. Augustine speaks of original sin as “that happy fault” which occasioned the supreme sacrifice of God becoming man in Christ Jesus). [005] RESPONSE: Sin nature, that happy fault, some humans call it? God as Jesus, may be a supreme sacrifice in the eyes of hu-mans, but it is a small thing in the eyes of an infinite God, over our tiny dirty rags sinful bodies, as a drop of water in the ocean. God as Jesus, as humans appear to teach, was not an infinite sacrifice, as Satan and humanity SUM TOTALLED are only finite of evil, compared to God's infinite power, or God would be overpowered by evil, which is not possible. The cross for God is just a prick of the thumb, of His infinite Glory, Power, and Mercy. I do notice the theoretical ramifications of God's activities being the responsible power of creating all sin's and accidental suffering's allowances, are not addressed, so the theoretical point still stands. God as men teach, shows not a powerful and persuasive positive force, but uses a coercion to faith, under the threat of mortal death, and the torments in God's creation modality. [003]==== RESPONSE: So God Makes sadness for Himself and all concerned, knowingly (A2 omniscient). God KNOWS ALL THESE DIRE THINGS WILL HAPPEN (why I warned you against this course of action (I SAW IT ALL BEFORE OMNISCIENTLY)). GOD KNEW HE would create souls that would choose not to worship His ways. God Knowingly creates the evils and sadnesses via His Creation He Knows so well needs His Help. [004]==== I do not think that he is Sad per se. God is eternally perfect and lacking in nothing, including beatitude/happiness. [005] RESPONSE: You think so, not know so? Where's the full armour of God, including the two edged sword of Truth that cuts both ways? YHVH appears lacking, if God saw fit to create at least 60,000,000,000 humans (before the current 7,000,000,000 humans), given 4000 years by 500,000,000 humans average at 33 years average life span, with the associated human history of suffering and universal death coersion since Adam and Eve. YOU SAY THAT God lacks the power to make love without suffering, as evidenced by the world. YHVH's hands are tied, if all of the theories, are true, and lacking powerful refutation. YHVH with good and evil, suffering and joy, is tied to the laws of Karma from Buddhism and Hindu beliefs, in that the infinite powerful and knowing YHVH cannot create goodness without creating evil, cannot create goodly-awareness in humans, without allowing evil-sufferings. Hindu and Buddhist restricted God of Christians! God cannot separate Buddhism and Hinduism's YIN from YANG. And God can ONLY redeem by permitting His suicide going to earth, after trying the Garden and failing, preserving the lineages around Noah and failing in the Flood, gives The Law of Moses, and failed with no one justified under a Law, makes the Faith in Jesus, and unpardonable sins, the last uprising at the end of Millenium, and creating Hell? By your thinking, God's own non-self-contradicting ways includes such Buddhist-Hindu Karmic forces even that EVEN He as YHVH, is tied to, inseparable, God and Satan walking Hand in hand, down the road as inseparable, YHVH seemingly incapable of making good without making evil, or He would have made a universe with good and no evil. in the universe He is responsible for Making, if it is taken as true that, Mark:10:"27 And Jesus looking upon them saith, With men it is impossible, but not with God: for with God all things are possible." The whole Bible of YHVH God Elohiym are stumbling blocks to humanity of finite capacity, falling short of a perfect god. God of Christ, as you portray, appears Hindu-Buddhist tied. [005] RESPONSE:You THINK God is not grieved? [005] RESPONSE: Genesis:6:"5 And God saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually. 6 And it repented the LORD that he had made man on the earth, and it grieved him at his heart. 7 And the LORD said, I will destroy man whom I have created from the face of the earth; both man, and beast, and the creeping thing, and the fowls of the air; for it repenteth me that I have made them. 8 But Noah found grace in the eyes of the LORD." [004]==== God knew they would happen, but that greater things would be able to happen because he would permit them. [005] RESPONSE: So, by that line of thinking you presume to posit, for YHVH to become closer to the Jews, who through the Levitical priesthood were responsible for keeping the temple and government pure, all through the Levitical Priesthood, who were also directly responsible for allowing those evil Kenite and Nephenim scribes to sneak into the temple works, who are not of God, and so are responsible for allowing the Evil Scribes to infiltrate and cause the Evil Scribes to seek Jesus's death on the cross, and so later God allowed WWII to occur, with YHVH allowing the killing millions of Jews under Hitler's regime, so God could be closer to them, and receive His vengeance, and be able to Give them Zion. God loves suffering, so God can become closer to His own? I only thought Satan was of that character, as Humans teach? What does God NEED from Humans, that HE is lacking, that God must love the suffering so YHVH can give us more? What is the Karmic Bank of God running short of, that He NEEDS sins and sufferings the more, so He may give us the White Robes of our Works as rewards? He cannot be the individual and personal teacher to ALL HUMANS with a clear voice that no one can ignore, but must be heard through His own imperfect humanity on earth, carrying His message corrupted through time ans space and interpretations that only the brightest can follow? Then God also has love for the weak and apparently damaged humans with passive deformations, diseases, weaknesses, and inability to understand. God makes everything under the SUN, and calls it Good? God says there is only one way, and calls the concentration camp suffering of the Jews GOOD, for their lack of responsibility, so His finite patience is satisfied, and can lovingly mercifully give the irresponsible Levitical Jews Zion? God allows and creates the suffering all the more, so God can give all the more. God creates suffering to create gifts. God cannot give of Himself without Our First Sinning. God gives later humans, after billions of Humans have died in numerous beliefs and places, Grace, only after He is Killed by the finite-human-free-will-Jews lack of responsibility over the temples leads to His Death on The Cross. God cannot Give without first other's sufferings under sins by the Laws YHVH sets up, under temptations and toils and battles. And God Needs our suffering to Give of Himself, or God just Wants our suffering to Give of Himself? For the crusifiction case, one can view Hitler as God's tool of a martyr, sending the responsible progeny to suffering and innocent death, the necessary pact, to be able to give the threatened writ of Divorce Israel, the Zion He Promised. Can you not see what I am saying of the contradiction that appears to exist, if you are of the word of TRUTH, and can illuminate God's power better than myself? [005]RESPONSE: As I can imagine a better connection of God personally with every human in a Design, with YHVH serving as a personal guide, teacher, and corrector that doesn't push away some to rejection of what is obviously The One Way, as God is all powerful, I would definitely be grieved, as the Holy Spirit can be Grieved, so God does feel suffering and sadness, if God is pushed to tempt a writ of divorce on the Jews for their irresponsibility as God's Chosen People. A voice that none of them can ignore that is caring and thoughtful and patient and kind and customized to each soul He knows so well, would have been a much happier world, but you say we must fall by YHVH's Design before God can become one with us? How is it I can imagine a better world, than All Knowing All Powerful God? God makes a much better Shakespeare, with tragedy and conflict as the Design, and not Harmony and shepherding. [005] RESPONSE: If I were a Merciful Patient Infinite God, I would have spoken to Adam and Eve as they were about to go froward. I would have given them a taste of their progeny's suffering, so that they would understand what they were about to do. I would have given them the sword, not enough to kill, but to teach them, as those wounds would heal, but they would think twice and three times and infinite times teaching before going for the Tree of the Knowledge of Good and Evil. I would have been a personal God to all humans, with such a force of communication. But instead, God's spirit leaves humanity to its own devices, so YHVH can become closer to what He Created His Way. Something is wrong with a God as humanity illuminates. I can imagine that immortal but trained humanity, with a tangible suffering BEFORE they sin and fall, so I can have them good de-facto, and loving the right path with a true Spirit of God correcting before the correction is required, by showing the evils of potentials. But that's too Good for YHVH over His Creation. [005]RESPONSE: God is tenderhearted? God punshes all progeny for Adam and Eve, given the entire context that still stands? YHVH is the Great Exception Model Idol of Humanity He Created His Way? God's Holy Spirit can be Grived, and you say He isn't sad per-se? [005]RESPONSE: Luke:22:"1 Now the feast of unleavened bread drew nigh, which is called the Passover. 2 And the chief priests and scribes sought how they might kill him; for they feared the people. 3 Then entered Satan into Judas surnamed Iscariot, being of the number of the twelve. 4 And he went his way, and communed with the chief priests and captains, how he might betray him unto them. 5 And they were glad, and covenanted to give him money. 6 And he promised, and sought opportunity to betray him unto them in the absence of the multitude." [005]RESPONSE: Regarding the Kenites, seed of Satan: [005]RESPONSE: 1Chronicles1:2:"55 And the families of the scribes which dwelt at Jabez; the Tirathites, the Shimeathites, and Suchathites. These are the Kenites that came of Hemath, the father of the house of Rechab." [005]RESPONSE: Regarding Nethinim and servants of captured people's, allowing pollution of the body: [005]RESPONSE: Regarding more polluting of the priestly lines, whole "Judah" sum, 42,360 people **: [005]RESPONSE: Pure sum 31,583 people, yielding 10,777 corrupting the branch of Judah: [005]RESPONSE: Nehemiah:7:"5 And my God put into mine heart to gather together the nobles, and the rulers, and the people, that they might be reckoned by genealogy. And I found a register of the genealogy of them which came up at the first, and found written therein, ..." [005]RESPONSE: Regarding other forces of infiltration of the Levitical church: [005]RESPONSE: Regarding the corrupted Levitical priesthood of Judah: [005]RESPONSE: Revelation:2:"9 I know thy works, and tribulation, and poverty, (but thou art rich) and I know the blasphemy of them which say they are Jews, and are not, but are the synagogue of Satan." [005]RESPONSE: Luke:11:"37 And as he spake, a certain Pharisee besought him to dine with him: and he went in, and sat down to meat. 38 And when the Pharisee saw it, he marvelled that he had not first washed before dinner. 39 And the Lord said unto him, Now do ye Pharisees make clean the outside of the cup and the platter; but your inward part is full of ravening and wickedness. 40 Ye fools, did not he that made that which is without make that which is within also? 41 But rather give alms of such things as ye have; and, behold, all things are clean unto you. 42 But woe unto you, Pharisees! for ye tithe mint and rue and all manner of herbs, and pass over judgment and the love of God: these ought ye to have done, and not to leave the other undone. 43 Woe unto you, Pharisees! for ye love the uppermost seats in the synagogues, and greetings in the markets. 44 Woe unto you, scribes and Pharisees, hypocrites! for ye are as graves which appear not, and the men that walk over them are not aware of them. 45 Then answered one of the lawyers, and said unto him, Master, thus saying thou reproachest us also. 46 And he said, Woe unto you also, ye lawyers! for ye lade men with burdens grievous to be borne, and ye yourselves touch not the burdens with one of your fingers. 47 Woe unto you! for ye build the sepulchres of the prophets, and your fathers killed them. 48 Truly ye bear witness that ye allow the deeds of your fathers: for they indeed killed them, and ye build their sepulchres. 49 Therefore also said the wisdom of God, I will send them prophets and apostles, and some of them they shall slay and persecute: 50 That the blood of all the prophets, which was shed from the foundation of the world, may be required of this generation; 51 From the blood of Abel unto the blood of Zacharias which perished between the altar and the temple: verily I say unto you, It shall be required of this generation. 52 Woe unto you, lawyers! for ye have taken away the key of knowledge: ye entered not in yourselves, and them that were entering in ye hindered. 53 And as he said these things unto them, the scribes and the Pharisees began to urge him vehemently, and to provoke him to speak of many things: 54 Laying wait for him, and seeking to catch something out of his mouth, that they might accuse him." [005]RESPONSE: Matthew:23:"Then spake Jesus to the multitude, and to his disciples, 2 Saying The scribes and the Pharisees sit in Moses' seat: 3 All therefore whatsoever they bid you observe, that observe and do; but do not ye after their works: for they say, and do not. 4 For they bind heavy burdens and grievous to be borne, and lay them on men's shoulders; but they themselves will not move them with one of their fingers. 5 But all their works they do for to be seen of men: they make broad their phylacteries, and enlarge the borders of their garments, 6 And love the uppermost rooms at feasts, and the chief seats in the synagogues, 7 And greetings in the markets, and to be called of men, Rabbi, Rabbi. 8 But be not ye called Rabbi: for one is your Master, even Christ; and all ye are brethren. 9 And call no man your father upon the earth: for one is your Father, which is in heaven. 10 Neither be ye called masters: for one is your Master, even Christ. 11 But he that is greatest among you shall be your servant. 12 And whosoever shall exalt himself shall be abased; and he that shall humble himself shall be exalted. 13 But woe unto you, scribes and Pharisees, hypocrites! for ye shut up the kingdom of heaven against men: for ye neither go in yourselves, neither suffer ye them that are entering to go in. 14 Woe unto you, scribes and Pharisees, hypocrites! for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation. 15 Woe unto you, scribes and Pharisees, hypocrites! for ye compass sea and land to make one proselyte, and when he is made, ye make him twofold more the child of hell than yourselves. 16 Woe unto you, ye blind guides, which say, Whosoever shall swear by the temple, it is nothing; but whosoever shall swear by the gold of the temple, he is a debtor! 17 Ye fools and blind: for whether is greater, the gold, or the temple that sanctifieth the gold? 18 And, Whosoever shall swear by the altar, it is nothing; but whosoever sweareth by the gift that is upon it, he is guilty. 19 Ye fools and blind: for whether is greater, the gift, or the altar that sanctifieth the gift? 20 Whoso therefore shall swear by the altar, sweareth by it, and by all things thereon. 21 And whoso shall swear by the temple, sweareth by it, and by him that dwelleth therein. 22 And he that shall swear by heaven, sweareth by the throne of God, and by him that sitteth thereon. 23 Woe unto you, scribes and Pharisees, hypocrites! for ye pay tithe of mint and anise and cummin, and have omitted the weightier matters of the law, judgment, mercy, and faith: these ought ye to have done, and not to leave the other undone. 24 Ye blind guides, which strain at a gnat, and swallow a camel. 25 Woe unto you, scribes and Pharisees, hypocrites! for ye make clean the outside of the cup and of the platter, but within they are full of extortion and excess. 26 Thou blind Pharisee, cleanse first that which is within the cup and platter, that the outside of them may be clean also. 27 Woe unto you, scribes and Pharisees, hypocrites! for ye are like unto whited sepulchres, which indeed appear beautiful outward, but are within full of dead men's bones, and of all uncleanness. 28 Even so ye also outwardly appear righteous unto men, but within ye are full of hypocrisy and iniquity. 29 Woe unto you, scribes and Pharisees, hypocrites! because ye build the tombs of the prophets, and garnish the sepulchres of the righteous, 30 And say, If we had been in the days of our fathers, we would not have been partakers with them in the blood of the prophets. 31 Wherefore ye be witnesses unto yourselves, that ye are the children of them which killed the prophets. 32 Fill ye up then the measure of your fathers. 33 Ye serpents, ye generation of vipers, how can ye escape the damnation of hell? 34 Wherefore, behold, I send unto you prophets, and wise men, and scribes: and some of them ye shall kill and crucify; and some of them shall ye scourge in your synagogues, and persecute them from city to city: 35 That upon you may come all the righteous blood shed upon the earth, from the blood of righteous Abel unto the blood of Zacharias son of Barachias, whom ye slew between the temple and the altar. 36 Verily I say unto you, All these things shall come upon this generation. 37 O Jerusalem, Jerusalem, thou that killest the prophets, and stonest them which are sent unto thee, how often would I have gathered thy children together, even as a hen gathereth her chickens under her wings, and ye would not! 38 Behold, your house is left unto you desolate. 39 For I say unto you, Ye shall not see me henceforth, till ye shall say, Blessed is he that cometh in the name of the Lord." [005]RESPONSE: Regarding pain and suffering, it seems that pain and suffering, with an accompanying unceasing complaining to God, are actually quite old, even ancient. Take the following few passages, of many others, showing the ancient nature of murmuring against adversity in God's world: [005]RESPONSE: Genesis 4:13-14, [005]RESPONSE: Exodus 14:10-14, [005]RESPONSE: Exodus 15:24-25, [005]RESPONSE: Exodus 16:7-8, 12, [005]RESPONSE: Exodus 17:2-4, [005]RESPONSE: Exodus 32:1-10, [005]RESPONSE: Exodus 32:23, [005]RESPONSE: Numbers 11:1-6,10-11 [005]RESPONSE: Numbers 13:31-14:4,11-12,26-29,35-36 [005]RESPONSE: Numbers 20:2-5 [005]RESPONSE: Numbers 21:4-5 [004]==== :Because we can suffer, we can have compassion. Because we can suffer we can truly love others and deny ourselves. Because we can suffer we can forsake our own lives for Him. And in the grand scheme of things, this compassion, this love, this forsaking of self, outweighs the evil of suffering and our present condition. [005] RESONSE: Compassion is being perfect, allowing the imperfect to fall under their endowment with free-will imperfection, and thus allowing one to become closer to the sinful design. The Good God YHVH Elohiym cannot be one with ones that have not become sinful yet, so He designs them to fall in free-will. This requires a justification, also. You state it as a fact, without a description of the "physics", just the "fact", not a good approach. So the Jews, whom among the Levitical priesthood and such, were responsible for keeping evil doers from temple service, were indirectly responsible for the death of Jesus, through the evil doers who crept into the temple works, and so are justified in the suffering 2000 years later, to enter the Concentration Camps of Hitler, by your logic. Because, they will be receiving better for their suffering. And why do we need something better? Because God couldn't produce the good in the first place? I make the theoretical argument, if suffering and self sacrifice is required to receive the treat of better things, then the whole earth should be the Concentration Camp, so we can know Joy even more deeply through deeper suffering, and self-sacrifice more selfless, from the scarcity of a dismal Concentration Camp. The worse the world is the better the treat God gives us in recompence, for our own accomplishments. And even then, why is it of US and not from GOD? We are nothing, but YHVH's PETS? To be thrown in HELL for finite belief on full-rejection of YHVH, and suffering received ad-hoc, for existing with free-will, to become one with God? God can't make good without evils, still stands your refutations. [004]====:Because we fell, Christ could redeem us, becoming one with us. [005] RESPONSE: Because God wanted to redeem us and become one with us, from our God-created sinful state, he designed us to fall so he could save us. Peachy God, you propone. God cannot become one with what He Creates, but can only become one with those He designs to FALL FIRST, with the free-will that God gave us. You imply that God cherry picks the Good fallen, and throws away the finite understanding evil ones He Created, into Hell in Revelation, not found in the Book of Life. [001]====(I2) Why is Got not capable of creating soft free-will so that 100% of the souls will be saved (not-A5)(not-A1)(not-A2), as God only makes souls so potentially heavy, that not even God can lift them for salvation (A1), with His infinite power, patience, mercy, not saving, but building Hell to destroy souls in Revelation (A1)(not-A5). In fact, if God is omnipotent and omniscient (A1)(A2), why didn't He SIMPLY create 100% saved free-will beings before time began, when cause and effect didn't apply to logic and God, so these temporal self contradiction can't exist in the first place, and He could have achieved PERFECTION in SALVATION, but only makes willy-nilly units for salvation that happen to go HIS WAY. [002]==== "I think that creating "free will" that does what you want it to do 100% of the time is a contradiction. That's like saying "why didn't God create circles that have corners?" Love cannot be forced, and God, who is Love, is not a rapist. God wanted us to be free to love Him and know Him, but this NECESSITATES that we be actually free to reject him - and if we are free to do so, then it is a REAL POSSIBILITY that could happen. And we know it was a real possibility precisely because it DID happen (and while this is pure speculation on my part which probably wouldn’t apply to perfect people, I wonder if we’d wonder if we were really free ever had we never fallen)." [003]====Think!? Not Know, Gnosis, Logos. THEREFORE BY YOUR BELIEF, God CANNOT create souls 100% destined for salvation. God is an imperfect GOOD creator. God KNOWS He creates SUFFERING and HELL for some of His Creation, but God is NOT RESPONSIBLE for what God KNOWINGLY CREATES IMPERFECTLY. Examine (I4)-RESPONSE, for Bible Verses, reference contradiction of what is possible-not-possible. [004]====This is no more a lacking in God than his inability to create round squares or make 2 and 2 equal 434.2. We’re either free to decide to love him, or we’re not and we’re robots. As soon as you make humanity “free”, you give up the “100%” control. [005]==== RESPONSE What do you mean here? Where God can make pi=3.0, making a triangle-circle, as His Book allows. [005]====RESPONSE I Kings:7:"23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about. 24 And under the brim of it round about there were knops compassing it, ten in a cubit, compassing the sea round about: the knops were cast in two rows, when it was cast." [005]====You admit then that your teachings of YHVH Elohiym Lord God, are of a finite capability God by design? Teaching to reach the world, using a mysteriously translated and interpreted text, in HUMAN HANDS, like myself, a monkey typing randomly at a keyboard of God concepts? So all things are not possible with your taught God YHVH, as YHVH HAS NOT THAT WAY right now in His Visible Design. YHVH must allow His created-inferiority to fall all about Himself, so that He can come in on a white horse, and save the day, as no other universe that can be imagined or made, that is better than this show world works, for God's own rules are held against Hindu-Buddhist Karma fall before being risen after the facts? God chooses and desires to make chaos, less than 100% harmony, to His personal voice that is unmistakable and preserving, but not really. [004]====1) Hell is primarily THE STATE OF SELF EXCLUSION FROM COMMUNION WITH GOD. It is what happens to those who chose to remain apart from him. [004]====2) It is reserved for only those who commit the only unpardonable sin, “blasphemy against the holy spirit” – but what this amounts to is simply refusing the love and forgiveness of God. [004]====3) The primary punishment of Hell is SELF EXCLUSION FROM COMMUNION WITH GOD. (sorry for the all caps, I wish they’d let me italicize or bold on here) [005] RESPONSE: What make we of: [005]RESPONSE: And more so, the acrostic of the Psalms, showing us witnessing those comdemned, as they were designed by YHVH? [005]RESPONSE:Psalms:37:"7 Resign thyself unto the LORD, and wait patiently for Him; fret not thyself because of him who prospereth in his way, because of the man who bringeth wicked devices to pass. ... 20 For the wicked shall perish, and the enemies of the LORD shall be as the fat of lambs--they shall pass away in smoke, they shall pass away. ... 34 Wait for the LORD, and keep His way, and He will exalt thee to inherit the land; when the wicked are cut off, thou shalt see it.." [004]====Hell is Commonly imaged as “fire and brimstone” in and out of scripture; these are meant to be analogically impart an understanding of the nature of hell and are not necessarily literal (though they are not necessarily not, either!) [005]RESPONSE: So, God makes threats of death so we may free-will choose to overlook His Threats of Death, Fire, and Brimstone, rising forever and ever, and the fat of the lambs striking the fire? That is compassion, to use threats as the Good Guide to The One Way? Using threats to coerce free-will love is GOOD, you say? [004]====Hell primarily is the state of self-exclusion from communion with God. Remember, God does not force love. He gives it freely and it must be freely accepted. “God is not a rapist.” C. S. Lewis wrote: [005]RESPONSE: So this comfirms, by your interpretations, that use coersion to life against threats of the eternal snuffing out, of those who are in full recognition of the universe of God, and willfully choose destruction, even though they know all things, like a God, in some future end of Millenium threat of Hell Death and Destruction, to be destroyed, and not simply making a personal connection to God for all humans, in True Patient, Instructing Mercy and Benevolence. I didn't now COERSION to LIFE, IS LOVE, for some of God's OWN DESIGN. Sounds more like Shakespeare, than Merciful Personal Patient God. [004]====“Christianity asserts that every individual human being is going to live forever, and this must be either true or false. Now there are a good many things which would not be worth bothering about if I were going to live only seventy years, but which I had better bother about very seriously if I am going to live for ever. Perhaps my bad temper or my jealousy are gradually getting worse – so gradually that the increase in seventy years will not be very noticeable. But it might be absolute hell in a million years: In fact if Christianity is true, Hell is the precisely correct technical term for what it would be.” [005]RESPONSE: Support for Hell as an instrument of His Coersion of Free-Will losers that He Created That Way, under a God that can use the most vile of Humans as His Instrument of Coersion. [004]====Incidentally, both heaven and hell are described as places of fire. The Highest choir of angels are in fact called “the Burning Ones” (i.e. the Seraphim”) because they behold the face of God. The imagery of fire is meant to convey something more than physical appearance. Fire can purify and it can consume and destroy. [005]RESPONSE: No doubt, as the fiery furnace doesn't touch the Good of God, that moment, and kills the stokers of the Fire. God's Fire, agreed, is a consuming fore of the wicked, that God creates and allows, as the instrument of suffering and trials and ultimate destruction, promised or threatened only, in Revelation and Psalms, just to start. [004]====Moreover, God predestines no one to hell, and wills that all should be saved. If all are saved in the end (which is possible, but doubtful) then his will prevails. If some are not saved, that is because being Love who created freely, he gave up the reigns to his creation. [005]RESPONSE: So the following is an empty threat, again? A coercion to free faith and love? A cunning linguist of mystery and confusion, much closer to Satan than the Good God, being held back by His own Karmic Forces of YIN and YANG? Is this lake of fire there just for fun, in the end, and God leaves that little part off the book, to coerse the love of the most of His Imperfect Creation? Just like God is too Perfect and Good to help us directly by Correcting us Directly, but uses the evil ones to correct us, to Keep His Hands Clean? It would seem to be much more civil of God YHVH to plant His Voice and Means of Correction directly to all people, so we are corrected, warned, and illustrated, before what happens, to drive the crooked to Him strongly, and drive the good to Him for the rewards, and show the evil the Good. But that seems not to be, as many teach. And it would remove the threats of Death, destruction, distortion, nature, and such, and make us concentrate on God's ways, and not all of the other ways we must combat, in addition to being Good. The mere fact that God Made Satan the Guarding Cherub, and also Gave Satan free-will, shows a conflict of design, where the Angels and Elohiym are supposed to be God's unwavering programmed servants of God's Kingdom, and not be giving them , as well as all Humans, Free-Will which is sin nature in infinite time. And God Knows all of these things will happen. [005]RESPONSE: Revelation:20:"15 And whosoever was not found written in the book of life was cast into the lake of fire." [001]====(I3) Why was God so short of longsuffering patience with Adam and Eve, but simply punishes them at their first disobedience, AND punishes them and their descendants to the end of time, for practical purposes (not-A5). [002]====:"Well, (1) in the first place, he had commanded Adam at the start – Adam had heard the mandate of God directly, and the doom which lay upon the action. Yet Adam and Eve sought to be “like God, but without God”, turning against their one source of ultimate happiness. (2) Secondly, it wasn’t simple disobedience but insurrection, a turning against. (3) Third, they are not punished to the end of time, and this self-same deity became one of them and shared in their sufferings and misery, that he might thus elevate them." [003]====RESPONSE: (1) But the Bible Genesis gives no record of mercy, by correcting and removing their sin in love ... right-away allowing the suffering to the end of time for all. No MERCY, nor FORGIVENESS, nor LONGSUFFERING, IN THAT TIME, when all things are possible with God. GOOD GOD (A5) IS NOT FOUND IN GENESIS RIPPLING TO END OF TIME BY HIS DESIGN. (2) Correction on point: ALL disobedience of God and His Ways (SIN) is insurrection to God: Romans 11:"30 For as ye in times past have not believed God, yet have now obtained mercy through their unbelief (disobedience, turning away). Even so have these also now not believed, that through your mercy they also may obtain mercy.". And that doesn't even cover natural accidents, uncorrectable, yet suffering. (3) We still die, even today. Do you see IMMORTALS walking around in a GARDEN anymore, NOT having women-child-bearing pains, and NOT working from the sweat of their brow to support woman? I see no such thing, the consequences of Adam and Eve working until the end of time by God's Created Design, until Jesus's return. My issue still stands with your answer. [004]====All things may be possible, but he does what is fitting and best, and we can assume that things are as they are for a reason. That said, I do not see how you can say there is no mercy. From that very moment, the path of all salvation history was laid out by God to effect the salvation of humanity, and the reconciliation of us all to Himself through Himself. That it didn’t happen instantaneously doesn’t mean that it doesn’t ripple through all of history. [005]RESPONSE: MAY is not IS. Fitting and Best, under an Infinite Powerful God, is THE BEST, and nothing less than THE BEST, whether nder human imaginations, or God's truth of limitations seemingly taught of Karmic Hindu-Buddhist limitations placed on God's Saving Hands. [005]RESPONSE:Ezekiel:13:"17 Likewise, thou son of man, set thy face against the daughters of thy people, which prophesy out of their own heart; and prophesy thou against them, 18 And say, Thus saith the Lord GOD; Woe to the women that sew pillows to all armholes, and make kerchiefs upon the head of every stature to hunt souls! Will ye hunt the souls of my people, and will ye save the souls alive that come unto you? 19 And will ye pollute me among my people for handfuls of barley and for pieces of bread, to slay the souls that should not die, and to save the souls alive that should not live, by your lying to my people that hear your lies? 20 Wherefore thus saith the Lord GOD; Behold, I am against your pillows, wherewith ye there hunt the souls to make them fly, and I will tear them from your arms, and will let the souls go, even the souls that ye hunt to make them fly. 21 Your kerchiefs also will I tear, and deliver my people out of your hand, and they shall be no more in your hand to be hunted; and ye shall know that I am the LORD. 22 Because with lies ye have made the heart of the righteous sad, whom I have not made sad; and strengthened the hands of the wicked, that he should not return from his wicked way, by promising him life: 23 Therefore ye shall see no more vanity, nor divine divinations: for I will deliver my people out of your hand: and ye shall know that I am the LORD." [004]====Christ himself, when he died, descended into hell to preach the gospel to those “who were disobedient in the days of Noah” (1 peter 3). The power of his resurrection and defeat of sin rippled back to the very beginning of history, and even to those who were wicked, to present salvation to them. [004]====Meanwhile, all of the created world, fallen though it is, is still good – just disordered. “God so loved the world” says that famous passage John 3:16, because the world was created good, and Man was created “very good”. [005]RESPONSE: So long after the fact, 4000 odd years later, God goes back to correct some things FINALLY, by teaching them directly, when God could have averted everything with patience, mercy, benevolence, and a personal spirit voice and bodily force of correction and instruction of future timeline paths of time space, in all men. That offer of salvation comes after billions of billions of people later, in the rough portion of the 60,000,000,000 odd people over the recent 4000 years. God creates disorder, you say, to teach us a coherently understandable lesson, with finite free-will sinful natures? Go figure! It is a job that is never quite proven to the finish until all things are done in the Shakespearian complexity of life over the approximately 67,000,000,000 people, so far, destined or delivered to death in this world, as the loving patient coercion to life's ultimate choice. And if time is a constant that is fixed and KNOWN IN YHVH OMNISIENCE, then nothing ripples back and forth, in the crystalline disorder perfection of the Mystery of YHVH's Works, as it is one object of static playback to an INFINITE OMNISCINET GOD. Fluctuations in time space as you describe, are unknowns of chaos and disorder that God Himself cannot Know Omniscently. And for that matter, several billions of peoples more live after 1 AD, far away from Chistianity's only path for salvation and rewards of white robes for one's works in faith in Jesus's finite sacrifice, as humanity and Satan's evil are finite, within an Infinite YHVH Lord God. [004]====I don’t take your point… [005]RESPONSE: I see no correction from the Word, The Bible. Not all Sin is turning away from God? We can pick and choose sins to commit, to forward God? Some sins are a turning toward God, as you imply here, without correction? [004]====Yes, presently death is part of the human condition. But that death is not the end of the story. (also, the punishment for the woman was an INCREASE in the pangs of birth, not the creation of them, and that very well may have been due to a DECREASE in our true, god-given reasoning ability to cope with such pains.) Pain and our passions are DISORDERED now, and wouldn’t exist as they do now had humanity not fallen – but this is all hypothetical and we could talk circles about this for hours and get nowhere but interesting speculation. [005]RESPONSE: So believing in Jesus, is still a promise beyond the knowledge of comfort, which in this world, when viewed as a Satanic Prison Camp for labor and mind control of the population, to just go the way of the dominions, powers, and principalities of pure evil and deception and temptation and confusion, will never be broken, by your finite thinking, if there really is no God, and we have a REAL WAR to wage against ALL OF THE EVIL FORCES OF THE WORLD, without a God. And a God given disordered sense of right and wrong cannot be the way to train tried gold, and dispose of the rest as dross creation by a perfect YHVH LORD God. In this world of 100% death with only a PROMISE from a currently unseen force of TOTAL POWER will go on forever, until an uncertain promised date of arrival of a questionable Infinite Powerful God, means that the powers and principalities of evil humans will have their way with humanity, with a corrupt set of multiple world religions that no one agrees are of one Christ, are distorted and diffused from truth, by their wicked world controlling ways. The Death Camp Earth, if Aliens or human forces of evil and great power over the whole earth, without a God, as the assumption of an unfulfilled ultimate promise GRANTED ONLY AFTER YOU'VE DIED, will never be defeated in this age earth, when it is true death that controls our lives from birth to death, and everyone on the whole earth, practically speaking, believe that scarcity and death are the only way of the world, until that promised day to come AFTER WE"VE ALL DIED, that may never come, if you play the Devil's advocate, where we are truly on the planet of control by the Evil Ones that They Themselves Permit in Powers, Principalities, and Dominions of a guaranteed dying earth without an Infinite Powerful God showing His True Ultimate Powers in practice, where the humans are trained to believe that God's ways are in disorder and trials, and promises fulfilled after death from this plane, that we all give up on as hopeless under the coercive truth of a mythical God's mysterious planet controlling way. [003]====RESPONSE: If I were God, I would have removed their sin, punished JUST THEM, removed their knowledge, and let them go again, in perfection from them on in the Garden and put flaming swords around the tree, with MERCY and PATIENCE in their foolish ways (even though they are SO-CALLED PERFECT). That's mercy, longsuffering, and kindness to them AND all the generations to the end of time. But they get spanked, and we all feel it! The sins of the fathers, borne by the innocent children to the end of time. Even if you accept Jesus, you still suffer all these things. You can imagine what is NOT WRITTEN, Adam says "please forgive us God", God says, "Not this first time, sorry, but the ramifications to the end of this age are IRREVOKEABLE by the way I Set things up. I can Show You no Mercy Here." [004]====Are you telling me that mercy means never letting people taste the fruits of what they’ve sewn? No, sir. That’s coddling, and God does not coddle. This isn’t simple children’s’ games here. This was a primordial choice of humanity. [005]RESPONSE: You are putting me on, aren't you? You can't be seriously taking my points. Maybe I'm wrong though. You give them a taste from the future, but just a taste for correction, which then causes the time lines of all future activities of the roughly 67,000,000,000 humans to completely vary, in a way God is completely in control of, but doesn't actually allow, just predicts, so that the root doesn't make the countless stems stand on end. I'm talking correction, instruction, and mercy for 67,000,000,000 humans, and you call it child's play. Please, why the lack of serious instruction, here? Mercy, is never letting the descendants suffer for the stumbling and learning of the roots and from the Creatior of it all, Himself. And all of the correction received, in the whole world, is personal from God, and coherent, to coherent Good Good Ways in Understanding and Appropriate Ways of Filial Piety. God makes 67,000,000,000 humans, and mostly let them teach each other, as blind leading the blind, when God's Infinite Power, can bring it all together with His Infinite Powers? What is God Short on that He cannot correct, alter the potential timelines back into order, instead of creating the disorder the world seems to be witnessing. And you call that love for Adam and Eve to get spanked, and also everyone else of the 67,000,000,000 suffers? [005]RESPONSE: And if I do take you seriously, then the whole planet should truly be a Concentration Camp, as if Hitler, Hirohito, and Mousolini Succeeded in transforming the planet into a fascist camp, so we may be better brought closer to God. If you are verily being serious about God's Word Here. That would truly be the least coddling planet to bring humans closer to God, in an ultimate payback for Karmic imbalances, so God can give us the ultimate. [004]====And He suffered them too, even being sinless, because he is love. [004]====You’ll notice Adam never asks for forgiveness. He and Eve simply play the blame game, and instead of damning them to hell for all eternity, he lets them live, and settles them himself east of the garden. (Gen 3:24), and promises that he himself will redeem them (“I will put enmity between you and the woman, and between your offspring and hers; He will strike at your head, while you strike at his heel.” He says to the Serpent, who is Satan, forshadowing what God himself does in Christ Jesus, Himself Incarnate). [005]RESPONSE: Maybe that was left out of the Book. Maybe Adam was scared. Maybe Adam was more like a little innocent child trying to hide what God told them was a sin. For any reason, Adam and Eve get spanked, and all of history feels the ramifications, as is taught by some Christians. Blame Games are* for Children, or those lost in the world who cannot comprehend God's Ultimate Means no matter how hard they try and wish to know earnestly seeking God's advice, that doesn't coherently come through the world, or from within from God. And then God punishes all children, as they are small beings compared to the Ultimate Mercy, Knowledge, Power, and Grace of God. He damns all to die in this world, accepting only a promise, which can easily be a deception of twisted world of true powers, dominions, and principalities, that truly have the sheep fooled into believing in scarcity, conflict, suffering, fairness, disorder, chaos, truths, distortions, mystery, et cetera. Promises, promises, over 67,000,000,000 humans, that are not all Christians or God's Chosen People in the Jews. Why, because He is Love, and must let His Creation fall, you say, so that only then He can Come closer to us? What a waste of true infinite Power of God YHVH, the Creator of all things, in His Perfect Order. [001]====(I4) If God is omniscient, and omnipotent (A1)(A2), why can't He make ANY PERFECT good without knowingly complicitly ALLOWING even one iota of evil (A2), implying God is not omnipotent to not self-contradict, but then God is impotent in time-space free-will domains, creating a disunity, for which HE IS RESPONSIBLE FOR as (A6) implies responsibility for what He created. [002]====:"God being “omnipotent” doesn’t mean he can do ANYTHING. He cannot do what is a logical contradiction, and so while he positively wills no evil to happen, he permissively allows it that from the potentiality for evil he can bring greater good. He is responsible for creating creatures which were free to reject him, not for their rejection of Him." [003]====RESPONSE: You say it is contradiction that God Makes Good without Evil. Many say, outside of time, God is beyond logic and contradiction, so ALL THINGS ARE POSSIBLE BY GOD ... BUT NO THEY AREN'T!? Despite, Mathew:19:"25 When his disciples heard it, they were exceedingly amazed, saying, Who then can be saved? 26 But Jesus beheld them, and said unto them, With men this is impossible; but with God [not quite?] all things are possible." NOT SO.Revelation:20:"14 And death and hell were cast into the lake of fire. This is the second death. 15 And whosoever was not found written in the book of life was cast into the lake of fire." (I2, God doesn't save 100%, God CAN'T make that kind of free-will human., God creates and authors Known Death.) He Makes the capacity of Rejection, He Makes Death of His Reject Humans. God's choices lie within Himself, as the Ultimate Free-Will to Good Ways. God's suffering is real, and it is a temporal taste of the natural fruition that comes from allowing free-will finite humans, when looking for Love from humanity, producing: accidents, "natural" disasters, genocides, concentration camps, wars, disease, ignorance, et cetera. [004]====All things which are possible are possible. It is not possible to have a circle which is a square (boxing rings aside) because it is sheer nonsense. That’s like faulting God because he cannot “galkjeroihklmanedbo” or won’t “dlakngwekn2k 9ib9”. Something which is a logical contradiction is not something at all, but merely a “null set” of linguistic symbols strung together. God cannot “make a flower that he did not make” because that is nonsense. It’s not God’s fault, but simply a problem within human language and mind that makes us want to think that such things could be made. There is no triangle with 4 sides, and God could not change that (except in changing the very meaning of “triangle” to that of “quadrilateral”, but that is not then creating a 4 sided triangle, but simply swapping words. [005]RESPONSE: YHVH LORD GOD'S INFINITE CAPACITIES ARE TIED DOWN BY MENS HANDS TO SNATCH SOULS AWAY FROM HIS TRUTH, CONSTRAINED BY KARMIC HINDU-BUDDHIST EQUATIONS. Utopia and Heaven which are possible, are not "gobbledegook babylon" as you put it. Why is Heaven de-facto gobbledegook, because God must watch us fall in our free-will before He can Correct us. Why is preemptive correction and instruction, with a taste of what might happen, something that God cannot give everyone? Why does God make Souls so Heavy, that He Cannot Save them better than the way things appear, when asking for understanding from God's Word. It's not like I haven't asked YHVH LORD God, and God's representatives, that are possibly corrupted, or playing blame games and dissembling about God's True Plan, that should be so easy to understand that even a Child, which we all are, can understand the complexity from God Himself? You simply say human language is incapable of helping all come to a closer understanding of God, so God hides wisdom from humans, because He speaks a language of correction that few can understand coherently, as we are not Gods ourselves. How can the Ultimate Teacher not have a word that rings true, blasts away the chaff of untruth, and is visible and unquestionable by ALL HUMANITY? God is holding back, allowing the apparent sufferings, keeps the threat of death in this world, even after accepting Jesus, in faith? Please, intensify your words, as I am missing the connection of the mystery we must simply ACCEPT? Jehova's Witnesses have convictions of the Ultimate. Jews have convictions of the Ultimate. Muslims have convictions of the Ultimate. Hindus have convictions of the Ultimate. Buddhists have convictions of the Ultimate. Zoroastrians have convictions of the Ultimate. Theist Science has convictions of the ultimate. A-thiest Science has convictions of the Ultimate. Ancient Greeks have convictions of the Ultimate. Ancient Romans have convictions of the Ultimate. Baptists have convictions of the Ultimate. Lutherans have convictions of the Ultimate. Yet all humans play the blame game like children, and the One Way banishes and separates into divisions the One Body of YHVH God? Yes, God has a hard time communicating The Truth in Power and Clarity and Correction, and we have to be as Gods to understand which is which, or suffer cycles of retraining and recorrection because no one step of correction is perfect under the Infinite Power and Mercy of God? All sides say, from God, just have faith. [005]RESPONSE: So lets, begin by saying, what is the definition of God's Karma, that He is apparently one in the same power of YIN and YANG, as you teach, pray tell? [004]====Bear in mind that God is eternal, and outside time. If you suffererd seemingly immeasurable pain for 1 million years, but then experienced heaven for all eternity, that suffering would not be even a blip on the radar. 1,000,000 is not even a percent of a percent of a percent…of a percent of infinity. So that we suffer in this life does not mean that those sufferings are the end-all of our existence, or that we will not be compensated. [005]RESSPONSE: So Infinite God in Power over all humanity over time wants what from us? We need to taste 1,000,000 years of suffering. Why Has God received a signifigant suffering from mankind, that we must all live this way for 67,000,000,000 humans? That is at 10%, 250,000,000,000 years of suffering, albeit averaged over all humanity. Why doesn't God simply make it 2,500,000,000,000 years of suffering, since more suffering simply makes us closer to God, as you teach? We are finite and confused humans, here. What does the Infinite Loving Merciful God want from us? Coerced Love under threat of Death and living with suffering from foolish desires to a world with just plain old problems, over 67,000,000,000 humans so far? And that doesn't even look at the Creation that groaneth, Made By God, of lifeforms eating lifeforms to survive. Sounds like a Hindu Buddhist Karmic field over the entire age of human history, Under the One Infinite Personal Powerful Merciful God. "Where have all the soldiers gone, long time passing, where have all the soldiers gone, long time ago? Where have all the soldiers gone? Gone to graveyards everyone. When will they ever learn, when will they ever learn?" "How many roads must a man walk down before you can call him a man? How many seas must a white dove sail before she sleeps in the sand? How many times must a cannonball fly before they forever are banned? The answer my friend is blowin' in the wind, the answer is blowin' in the wind. How many times must a man look up before he can see the sky? How many ears must one man have before he can hear people cry? How many death will it take till he knows that so many people have died? The answer my friend is blowin' in the wind, the answer is blowin' in the wind. How many years can a mountain exist before it is washed to the sea? How many years can some people exist before they'r allowed to be free? How many times can a man turn his head pretending he just doesn't see? The answer my friend is blowin' in the wind, the answer is blowin' in the wind" [004]==== I hope you fond that helpful. [004]====I’m going to have to ask you to simplify your questions a bit if you’d like to continue. I want to be thorough, but these are getting a bit long. I’m not opposed to continuing this, so long as you can remain charitable and terse. [005]RESPONSE: I'm sorry, God threw 5 million characters at ME, through The Canonized Bible, which you should know better than myself. It's hard to believe in something when it is hard to understand and exponentially hiding the truth behind smokescreen. I cannot be shorter to get through the issues so that you can understand my frame of reference of understanding. And you throw on C. S. Lewis, which I can hang with the Screwtape Letters, but is hardly Canonized Texts. And The Catechism, from an entity that says with much ink many years ago, that Galileio's Orbits around the Sun was in utter Heresy against God with Justification from God, inerrant, all for saying that the Earth goes around the Sun, so I highly question those other words, outside of the Canon and even Apocryphal texts, without God's Unction filling in the Gaps of understanding, and the same God let's us compute the paths of satellites that investigate and map the solar system, using non-geocentric equations of simplicity. I guess God would have us compute orbits using Claudius Ptolomaeus' Crystal Epicycle Fields, instead of Newton's Gravity, or more precisely, Einstein's Theory of Relativity Gravity Equations. I bet you'll say all of our probes and gravity equations are not real, and fabrications of the Powers, Principalities, and Dominions that ouw tax dollars help support, in the Space Programs. Now I do know enough math, to know that one CAN ACTUALLY calculate planetary orbits using Greek Epicycles, but it does make the equations much-much harder, compared to Newton and Einstein, but that's what God wants for us, on earth, and in space, where PI=3.0, too, when the scribes knew better to put that into God's Old Testament? [005]RESPONSE: This is only about 66,377 characters, or about 1.34% of The Holy Bible in size, in about 15,421 words, in total. All is required to assure I am painting the proper image of complexity and deception, and disorder, the world must cope with, to Know that they Know Salvation in only Faith, in only One Way, when One Way, is supposed to be for ALL. [005]RESPONSE: I guess I may only take comfort in, If God be for me, who can be against me, regarding once saved always saved, in the power of God's Good Hand? ENDBack to Contents [16] Abiogenesis second version.Back to Contents CREATED AD 2008 09 02 P 08:40 (1) Abiogenesis ..Recently I've been working through a concept for substantiating abiogenesis through the idea of general combinatorial chemistry. I've tried some posts on, science sites, and religious sites, but none seem to have any coherent opinions that are constructive to addressing the general viability of such a theory effectively. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..General (natural) combinatorial chemistry (GCC / NCC), defined here, is the complete mathematical-chemical model of all reactions that occur in any portion of matter, and its temporal evolution, including feedback. This is opposed to synthetic combinatorial chemistry, as used in pharmaceutical industry, where chemicals are specifically combinatorially analyzed by chemistry machines and control algorithms. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..To get a feel for what general combinatorial chemistry looks like, as a concretely realized system, take a beaker with 5 chemicals total in aqueous solution. Five chemicals, combinatorially speaking, have the potential for uniquely, (2^5 - 1) [specific-reaction-node]s or 31 [specific-reaction-node]s, where every product at that reaction node must perform some task in a reaction directly or catalytically. Say the beaker starts off containing molecules of water, sodium, chlorine, silver, and fluorine. From these, the 31 [specific-reaction-node]s exhausted are: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) 1 water 2 sodium 3 water sodium 4 chlorine 5 water chlorine 6 sodium chlorine 7 water sodium chlorine 8 silver 9 water silver 10 sodium silver 11 water sodium silver 12 chlorine silver 13 water chlorine silver 14 sodium chlorine silver 15 water sodium chlorine silver 16 fluorene 17 water fluorene 18 sodium fluorene 19 water sodium fluorene 20 chlorine fluorene 21 water chlorine fluorene 22 sodium chlorine fluorene 23 water sodium chlorine fluorene 24 silver fluorene 25 water silver fluorene 26 sodium silver fluorene 27 water sodium silver fluorene 28 chlorine silver fluorene 29 water chlorine silver fluorene 30 sodium chlorine silver fluorene 31 water sodium chlorine silver fluorene where, offhand, we must recognize that there are, at the very least, the potentials for the following real reactions with stable new-products and reactants left over, at some equilibrium level, from the left-hand [specific-reaction-node]: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) (6) Na + Cl < -- > NaCl (24) Ag + F2 < -- > AgF2 (3) Na + H20 < -- > NaOH + H2 (4) 2Na + F2 < -- > 2NaF (20) Cl2 + F2 < -- > 2ClF (20) Cl2 + 3F2 < -- > 2ClF3 (1) 2H20 < -- > H3O+ + OH- ..Note that in (20) a reaction node can have more than one possible reaction, like one at high temperature and one at low temperature. So, here, we see that complexity has arisen from simplicity, in that 5 [molecule]s was the starting state, and from only that, there exists the potential for 14 [stable-molecule]s to come to exist, formed by combinatorial chemistry. In much the same way that gravity-fusion yields the complexity of 92~ [natural-element]s (and numerous natural molecules), from the simplicity of 2 [element]s at the big bang. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..From that single analytical iteration, there can also exist that property of feedback, mentioned earlier. The second iterative feedback, for this example, takes 14 [molecule]s, yielding (2^14 - 1) [specific-reaction-node]s, or 16,383 [specific-reaction-node]s. Without being exhaustive, lets say just 0.001 ratio of the reactions will produce new molecules from the original 14 [molecule]s of this iteration. That yields 16 [molecule]s, for a total of 30 [molecule]s. Again, feedback can occur, as new molecules have appeared that otherwise would not have existed. Next iteration has (2^30 - 1), or 1,073,741,823 [specific-reaction-node]s. Lets say, without being rigorous, 0.00000001 ratio of the reactions produce new molecules. That yields 11 [molecule]s, for 41 [molecule]s total. This gives (2^41 - 1), or 2,199,023,256,000 [specific-reaction-node]s. If a ratio of 0.00000000001 reaction nodes produce new molecules, we have 22 new molecules, totaling 63 [molecule]s. This can go on, as long as the feedback ratio of new stable molecules remains positive, and ceases when the feedback ratio equals zero, in a steady state networked reaction chemistry matrix, that may or may not oscillate in time about a chaotic steady state attractor. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..We have here an unknown of combinatorial chemistry, in the feedback ratio, at arbitrary complexity levels, that neither the "creationist" can declare, as much as they wish to, reaches zero for any given starting chemical system (finite steady state), or evolutionists can declare is always positive (a complexifying mixture that goes from starting simplicity to virtually unlimited complexity of molecule varieties), without measuring it in a real set of experiments outside of finite Miller-Urey, Oparin, Joan Oro, et cetera, that I have not seen myself at biological level tests. However, we know, from real biology, that carbon can allow the formation of self sustaining natural combinatorial chemistry at complexity levels of millions of compounds, that do not decompose into fewer stable molecular units or complexifying into more molecular units (except at death), so at millions of compounds for living entities, the ratio is zero for existing biological systems, probably due completely to homeostasis and physical limitations of reactions at that level of complexity-sparsity-systemic-distribution. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..Another piece of information can be created based on self limiting reactions. Let's say, for stereochemical limitations, only five molecule node reactions are signifigantly feasable, and 6 molecule and more node reactions are excluded from occuring, due to complexity. From 5 to 10,000 ocean molecules, one sees that the partial Natural Combinatorial Chemistry matrix from reaction molecule counts SUM(COMBINATION(molecules source, molecules of reaction) molecules of reaction = 1 to 5), from 1 to 5 nodal molecules, produces the following numbers of reactions: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 9 September 2008 (UTC) 5 reactions, 31 reaction nodes 10 reactions, 637 reaction nodes 20 reactions, 21699 reaction nodes 50 reactions, 2369935 reaction nodes 100 reactions, 79375495 reaction nodes 200 reactions, 2601668490 reaction nodes 500 reactions, 257838552475 reaction nodes 1000 reactions, 8291875042450 reaction nodes 2000 reactions, 266001666834900 reaction nodes 5000 reactions, 26015651042712200 reaction nodes 10000 reactions, 832916875004174000 reaction nodes which also shows astronomical numbers of potential reactions, against the argument of certain inherent open system steady state reaction simplicity versus complexity destiny posed by The God of The Bible according to some teachers. With just a 100 chemical ocean, limited to 5 source-molecule reaction nodes, produces 79,375,495 potential reaction nodes from the 1 to 5 molecule node reaction left hand sides, which is miniscule compared to the full Natural Combinatorial Chemistry of 1.2676506002282310^30 NCC reaction nodes, at a ratio of 1 in 1610^21 reachable reaction nodes in all NCC nodes. If just one in a million of the reachable NCC nodes produces a net of durable and useful molecule reaction products based on reaction nodes alone, would produce 79 new molecules in an iteration to partial steady state. Creationists claim a 0.0 feedback ratio at some point, not even one in a million, without known experiment reference, other than The Bible as humans tend to teach it. The question being, what is the ratio of the feedback ratio, in a complex chemical environment? Are there increasing breakdown reactions compared to build up reactions, such that all natural combinatorial chemistries reach some form of steady state of finite complexity molecules and polymers, as some Creationists claim is the God Given truth in biochemical science? [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 9 September 2008 (UTC) ..Now an example of general combinatorial chemistry, can be defined for a lifeless-earth-ocean. There is no life on earth, so the oceans are filled with a mass roughly similar to the modern biosphere dissolved in a lifeless ocean mix of basic organic (carbon based) primitive compounds, and inorganic compounds. Likewise, the environment sets up numerous states for reactions, from sunlight with UV irradiated surface water, "sunlight"-only illuminated deeper water, dark water under rocks and in sands or gravels and night time, surface chemistry (clay and mineral surfaces), average temperature water, hot water volcanic vents, lightning strikes, meteoric impacts, radioactivity (higher in the past), dehydration concentration zones in estuaries and lakes, delta rinse chemical flumes, and so forth. Carbon is a special molecule, as it allows numerous molecules to form at normal temperatures in water solutions, as evidenced by life. For a starting ocean with just 100 stable molecules, one has (2^100 - 1) [specific-reaction-node]s, or about 1,267,650,000,000,000,000,000,000,000,000 [specific-reaction-node]s. Let's say, given the ocean containing inorganic and carbon compounds, that a ratio of 0.000000000000000000000000000001 reactions form new compounds (complexity from simplicity), then one now has 101(.267) [molecule]s. As long as there's a small positive ratio, which seems quite reasonable, the number of stable molecules will increase over time. At 1,000 [year-iteration] intervals for such an ocean, one would see, at this ratio, if fixed, 100, 101, 104, 129, 1,000,000,000, (molecular saturation), within 5,000 [year]s. At a ratio that is self limiting, because of physical combinatorial limitations, one would see at 1.267 new molecules, compounded on the first 100 [molecule]s, on 1,000 [year-iteration] intervals, that there will be 1,000,000 [molecule]s in the ocean in 731,000 [year]s. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..Now this alone, doesn't necessarily bring about life, yet. There would initially be in the early ocean, a large raw mix of the most stable left and right handed chiral molecules, potentially including lipids, single amino acids, single RNA, and single DNA molecules. With such a rich ocean, given just a very small general combinatorial chemistry feedback, in a relatively few years, there will likely be in the general combinatorial chemistry matrix, numerous polymerization pathways, for the most robust easily polymerizeable molecules in the ocean. Possibly amino acids, RNA, and DNA molecules, because that is what nature uses, if nature is pragmatic and not hand assembled by God every minute of the day, but may also be other natural molecules that can polymerize, likely based on carbon, that operate more easily than amino acids, RNA, and DNA. In this ocean, with some form of polymers and polycyclics, from that there will intrinsically be a digitally-codified, and thus easily mutateable set of chemical "species" that catalytically support each other's productions in durable, catalytically-reactive, efficient-thus-numerical, systems. Three sets of reaction-networked-catalytic-hypercycle-feedback-mutateable codes will be operating in cooperative sets, one for left handed, right handed, and left-right handed chiral polymeric reaction codes in combination. Each set will compete for molecular supremacy in numbers, over numerous explorations finding combinatorially-inherent new species, and in a scarce chemical competition/cooperation, one set or another will have dominance in ocean space because the probabilities of "discovering" inherent reactions don't occur identically statistically speaking, creating autonomous differentials of product exploration diverging in time for the three chiral system types. Because feedback operations are used in these sets of reaction pathways, they will have a kind of numerical instability in the very complexity discovery occuring, and over time, one set of operations will win out as the standard, as numerous incompatabilities would likely occur between the sets. Nature, obviously, selected right handed chiral molecules, because at some point in time, being the first most complete types and sets of fullest reaction-networked-catalytic-hypercycle-feedback-mutateable code found, that was overall, by chance and inherent robust stability, the dominant reaction super-system. It could have gone to the left handed chirality too, but what we have in majority biology is right handed chirality, not that left handed chirality is inferior or even discernably differenet, as a perfect mirror physics image in all ways, except in a perfectly identical discovery in its own combinatorial chemistry chiral subset of complexity evolution divergence in time-ocean-space. Mixed left-right chirality combinatorial chemistry reaction matricies probably are inherently less efficient to discover naturally since a wholly different a-symmetry is a part of a mixed system, and so it didn't become supreme either, but is an assumption of the complex mixed chiral system. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..With a dominant chirality in the ocean at some point of time, or at the very least, an ocean with large regions of left handed or right handed dominant chirality combinatorial chemistry, the mutateable digital chemistry keeps exploring its combinatorial chemistry, inherently combinatorially discovering and diverging in chemical "specie" space, always finding digital codes that react more efficiently in numeracy than previous generations of polymeric chemical code "species" could, because new reactions continue being exposed with each combinatorial chemistry iteration with mutations and systemic stabilities in chaotic attractors of cooperative catalytic production systems, and proto-metabolic pathways inherent to the growing matrix of reactions. Eventually, either circuitously through precipitate micro-gel agglomerate clumps without membranes to micells in some generations of intermediate chemistries, or directly,, numerous populations of many types of micelles form from primitive lipids, proteins, RNA, and DNA fragments, inherently selected, as the most efficient, and thus numerically superior evolved digital chemical "specie" systems of reaction sets, encompassing (1) metabolism varieties from sunlight related chemical reaction pathways, glucose pathways, sulphurics pathways, et cetera, (2) homeostasis in a semi-permeable auto catalytic reaction system types, (3) transportability in a semi-permeable primitive lipid micelle / lysosome kinds, and (4) reproduction in the inherently most efficient general combinatorial chemistry matrix types, of which there can be many kinds of cellular versions. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..It should be noted that for an ocean with particles interacting about 110^10 [interaction / second], in a billion years or 31,560,000,000,000,000 [seconds], in an ocean with conservatively 100,000,000 [km^3] or 100,000,000,000,000,000 [m^3] active ocean volume solution, at about 1,000,000[g/m^3], and 20[g/molar-volume] at 6.0210^23[molecule/molar-volume], that there's: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..31,560,000,000,000,000 [seconds / billion-year] * 110^10 [interaction / second] 100,000,000,000,000,000 [m^3/active-ocean] * 1,000,000[g/m^3] / 20[g/molar-volume] * 6.0210^23[molecule/molar-volume] = 949,956,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 [interaction / billion-year-active-ocean] in oceanic combinatorial chemistry, or 949.95610^69 [interaction / billion-year-active-ocean]. So-called Christian and fundamentalist Creationists are quite certain, inspired by God and His truth to them, that this set of interactions, in an ocean of combinatorial chemistry, CANNOT reach life, inerrant to God's truth to them, that ONLY God was directly involved in forming life past the barrier of inherent chemical irreducible complexity truly, and not the rules of inherent combinatorial chemistry, originally setup at the Big Bang. One only needs to reach, say 1,000 large systems of chemical interaction, out of about 110^72 [interaction / billion-year-active-ocean], to reach life, leading one to 110^69[interaction / system] to setup each of those systems in parallel. If the [interaction] efficiency is 1 in 1,000,000,000,000,000,000 ["progressive-interaction"/general-interaction] one has 110^51 [progressive-interaction / system] available per system, to reach all of the exemplar 1,000 [system] of life chemistry over a billion years of early earth. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ..Returning to a starting ocean, with just 100 stable [molecule]s, where one combinatorially has (2^100[molecule] - 1) [specific-reaction-node]s, or about 1,267,650,000,000,000,000,000,000,000,000 [specific-reaction-node / combinatorial-chemistry-context]s. Given the ocean containing inorganic and carbon compounds, that a ratio of 0.000000000000000000000000000001 [new-combinatorial-chemistry-context-molecule / specific-reaction-node] form new compounds (complexity from simplicity) toward life over non-life, then one now has 101(.267) [molecule / combinatorial-chemistry-context]s. The iteration would take, maximally calculated for a 1,000 [year / iteration] example, 110^66 ocean interactions (from the previous 949.95610^69 [interaction / billion-year-ocean]), in this example of given ocean interactions, to make this oceanic molecular change from 100 to 101.267 [molecule / combinatorial-chemistry-example]s occur in the ocean. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) Proverbs3:13-23[Happy is the man [cell] that findeth wisdom [new good and true Words], and the man [cell] that getteth understanding [true Word]. For the merchandise of silver, and the gain thereof than fine gold. She [true Words] is more precious than rubies and all the things that thou cans't desire are not to be compared unto her [assisting truth]. Length of days is in her right hand [control]; and in her left hand riches and honour [product]. Her ways are of pleasantness, and her paths are peace [sustains]. She is a tree of life to them that lay hold upon her [in compatibility]: and happy is every one that retaineth her [in the cell]. The LORD [true Word] by wisdom [old codes] hath founded the earth; by understanding hath He established the heavens [consiousness]. By His knowledge [root codes] the depths are broken up [heirarchy], and the clouds drop down the dew [stabilize the environment]. My son, let not them depart from thine eyes [code preserves]; keep sound wisdom and discretion: so shall they be life unto thy soul [cell and mind], and grace to thy neck. Then shalt thou walk in the way safely, and thy foot shall not stumble.]. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) Once cells of such digital variety types are formed, with small chains of RNA, DNA, and proteins inherent at the level of complexity, they can propagate further in ocean currents, because of durability and safety of the agglomerate/cellular units. The best reaction sets are the chemical species that can travel in these units, in various ocean domains, and still contain the stability required in their digitized combinatorial chemistry to operate. Cells with inferior microcoded reaction networks, simply are less numerous and less prosperous. And since robust units travel, and have efficient feedback reproduction homeostasis, they dominate the ocean, converting whatever domains of other handed chirality into their networks of reactions, as partially symbiotic with the ocean and themselves, before true living individuality occurs. Eventually, the ocean purifies itself, either here, or along this path of biochemical competition, as cellular reactions that modify the ocean contents to their reactions, as well as use their own molecular types, and internally mutate their own codes to continually adapt to the unifying ocean, converge themselves together, akin to Gaian theories, through earth-ocean-cellular-types symbiosis numerical instability ocean domain feedback adaptive sumpremacy. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) Proverbs3:1-12[My son [cell], forget not my law[old codes]; but let thine heart [cell core] keep my commandments [DNA]. For length of days, and long life, and peace, shall they [codes] add to thee. Let not mercy and truth [in the Word] forsake thee: bind them about thy neck [body cord]; write them upon the table of thine heart [cell nuclear code]: So shalt thou [cell] find favor and good understanding in the sight of God [the Word] and man [cells]. Trust in the LORD [the Word] with all thine heart [cell core]; and lean not unto thine own understanding. In all thy ways acknowledge Him [the Word], and He shall direct thy [cell] paths. Be not wise in thine own eyes [cell organs]: fear the LORD [the Word], and depart from evil. It shall be health to thy navel, and marrow to thy bones. Honour the LORD [the Word] with thy substance [cell body], and with the firstfruits of thine increase [feed the Word]. So shall thy barns [environment] be filled with plenty, and thy presses [DNA codes] shall burst out with new wine [sweet spirit Words]. My son [cell], despise not the chastening of the LORD [true Words]; neither be weary of His correction [true Words]: for whom the LORD [old codes] loveth He correcteth [helps]: even as a father the son in whom he delighteth.] All the time the cells exist, the digitized combinatorial chemistry is always refining itself, inherently, because more efficient micro-polymer reaction sets become dominant through efficient forward reaction rates, via continual mutations in such primitive codes, reaching new inherent discoveries, not requiring molecules to be "conscious" knowing the future to bond themselves as "creationist" arguments often pose is required outside of physics. Also, as the relative robust stability of the best kinds of cells allows increases in codes, then also systemic relational codes inherently develop in these matrixies of reactions, in complexes and networks of catalytic reaction sets, because they inherently assist the reproduction of the combinatorial chemistry cells types. There may still be competition between cellular type systems, and sets of chirality molecules at this point of time, but every new generation of mutations that spreads dramatically better because of new found molecule codes, only furthers diverges the dominance of chirality and cellular types, both, and decreases any side-use of competing chiral systems that continue to wane as the ocean become uni-chiral through bio-recycling. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) It should be noted that the cells/gel-agglomeration-precipitates in these early combinatorial chemistry species evolutions are very small compared to modern cells, because they are not developed as modern life with its history of mutually supported digital molecular records. As such, they can fill an ocean quite densely, and pass generations quite fast, as the fastest best most durable and travelable units dominate, reaction wise. So in a million years, with just 10 million cubic kilometers of reactive zone, a density of 100,000 cells of various types per cubic meter on average in that volume, and a generation of 1 week, could explore, numerically, 52,000,000,000,000,000,000,000,000,000 units, in 52,000,000 mutation generations, of a total diverse population of 1,000,000,000,000,000,000,000 units, of various types, in such an oceanic sub-unit, with the accompanying period of chemical processing during each unit's existence. Definitely the hard way to form life, compared to design, but completely possible in contexts. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) Proverbs4:1-27[1 Hear, ye [cell offspring], the [true codes] of a [parent cell true code], and [machine] to [keep that code]. 2 For [the code Word] give [cells] [operations], forsake [cell] not my [true Word]. 3 For [paternal cell] was [a cell's] [true Word code pattern(al)'s] [cell offspring], [synergy cooperation supported] and only [precious codes] in the sight of [cells] [nurturing code]. 4 [Word code] taught [cell] also, and [instructed] [cell], Let [cellular core] retain [true Word code]: keep [the code ways], and live. 5 Get [truest codes], get [truest operations]: forget [the codes] not; neither decline from [code instruction transcriptions]. 6 Forsake [truest nurturing code words] not, and [truest words] shall preserve [cell]: love [the truest codes], and [truest codes] shall [support well] thee. 7 [truest Word nurturing codes] is the principal thing; therefore get [new truest Word nurturing codes]: and with all [cell] getting get [operational code integration synergy]. 8 [cooperatively enhance operations] [truest codes], and [nurturing codes] shall promote [cell]: [nurturing codes] shall bring [cell] to [sustainable dominance synergy], when [cell] dost embrace [true codes]. 9 [nurturing codes] shall give to [cell's] [processes and feedback] an ornament of [virtuous operations]: a crown of [cell synergystic cooperative numeracy power] shall [the best nurturing cell-world Word codes] deliver to [cell]. 10 Hear, O my [cell offspring], and receive [paternal cell] [true codes]; and the years of [cell offspring] life shall be many. 11 [paternal cell] have taught [cell offspring] in the way of [best codes]; [paternal cell] have led [offspring cell] in right paths. 12 When [cell offspring] goest, [cell's] steps shall not be straitened; and when [cell offspring] runnest, [cell offspring] shalt not stumble. 13 Take fast hold of [true codes integrated]; let [nurturing codes] not go: keep [true nurturing codes]; for [nurturing codes] is thy life. 14 Enter not into the path of the wicked [dispersive and viral codes], and go not in the way of evil [dispersive and viral code] [cells]. 15 Avoid it, pass not by it, turn from it, and pass away. 16 For [froward codes] sleep not, except [froward codes] have done mischief; and [froward cells] sleep is taken away, unless [froward incompatible detected codes] cause [good cells] to fall. 17 For [froward cells codes] eat the bread of wickedness, and drink the wine of violence [anti-synergy]. 18 But the path of the [cooperative true nurturing Word code] is as the shining light, that shineth more and more unto the perfect day. 19 The way of the wicked [codes] is as darkness [diminishing position]: [froward cell codes] know not at what they stumble [fall short efficiency cooperatives]. 20 [paternal code's] [cellular offspring], attend to [the true codes]; incline thine [systems] unto [paternal code's] [codes]. 21 Let [paternal codes] not depart from [cell offspring's] [machine agglomeration systems]; keep [good codes] in the midst of thine [cellular core]. 22 For [true codes] are life unto those [cells] that find them, and health to all their flesh. 23 Keep [cell] [code core] with all diligence [maintenance systems]; for out of it are the issues of life. 24 Put away from [cell] a froward [code explorer], and perverse [codes] [code attack] far from [cell operations]. 25 Let [cell's] [sense systems] look right on [systematically synergystic], and let [cell's] [sense system's control] look straight before [cell]. 26 [chemcial code process] the path of thy [cell envelope and drive], and let all [cell's] [operations] be [cooperative synergy reaction system]. 27 Turn not to the right hand nor to the left [divergent uncontrolled inferior efficiency code]: remove [cell's] [membrane and drive] from [inferior codes and states].] Naturally, such a complex cellular combinatorial chemistry exploration will find more codes, longer codes, and better codes. It should be obvious, given these assumptions, illustrations, and theory, that it might just be possible that an external force is not absolutely required to assemble and maintain every cell of life as argued as obviously true fact by some Creationist positions, given the apparent ease which modern natural bio-chemistry keeps modern life operating, without observable external-to-physics forces seen in testable reality, and general combinatorial chemistry seems capable of generating life, with an evolutionary model of general combinatorial chemistry. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) One may also notice, on a different heirarchical scale, which penetrates deeper back into time, that the solar system, from the original nebula perspective, gravitationally forms not just a closed system, but an energy losing system, radiatively to the rest of the nearly empty expanded universe, and one sees that life forms and is supported inside of that open-declining-energy-content-system. More so, even if the system were closed, inside of a perfectly reflecting sphere around the nebula, the system would be closed, but starting at a cold expanded temperature, and collapsing under gravity, one sees that partitions of concentrated matter systems are formed by gravity. There can still be a sun and earth, even if at a different configuration than the current solar system, as the sun would be larger, receiving back all of the energy it sends out, reflecting off the sphere, and the earth would have to be much further away from the sun, to support the same life context. And so here, a closed* system can be used to support (dare say self-form) meso-scale life, even though some Creationists often claim that closed systems always, always, always form into only-and-exclusively simple-steady-states (and perhaps granted to them, in the end of conventional-available-energy-matter-time-space-biochemical-systems). One could even go to the scale of the observeable universe, taken as a closed system, or even a declining system, taken as the expanding, productive-energy to thermal-energy entropy converting status, that obviously supports, if not self-forms life too, within that closed system with net mass, space, and initial energy. Going to the God scale, the one-of-all-things and nothing-else-exists-not-of-it, is a closed system, but then God can't make perfect eternal life from God on earth, and cannot yield 100% perfection in salvation of all souls, and based on those so-called obvious facts of life that all things die, self referentially speaking at material infinity of the matter plane, as all things must die, eventually, in a closed system. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) And lastly, for now, if chemistry monads are not of the configuration that allows self-formation of order, design is the only cause, to explain the existence of life, due to combinatorial chemistry inherent limitations of feedback, expansion, and organization. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ENDBack to Contents [17] Chiral / Churl symmetry between Atheism and Theism.Back to Contents CREATED AD 2008 09 02 P 08:40 ::I will get to your generous comments below, still iterating fine details above, and debating evolutionists and creationists, which are, humorously speaking, both as stubborn as the other in a chiral/churl symmetry! LOL. "grins" [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::A lovely chiral/churl symmetry can be found in word=genetic-mirror=reflections. David Berlinski, "The Devil's Delusion: Atheisim and its scientific pretensions", Page 29, wites: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Original: 29 "These questions are rhetorical. No one is disposed to ask them within the [Scientific] community, and the [Scientific] community is not disposed to acknowledge answers to questions it is not disposed to ask." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Mirrored: 29' "These questions are rhetorical. No one is disposed to ask them within the [Religious] community, and the [Religious] community is not disposed to acknowledge answers to questions it is not disposed to ask." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And relating to saving self and others, for science, "The God Delusion", Page 35, says, :::35 "An Athiest in this sense of philosophical naturalist is somebody who believes there is nothing beyond the natural, physical world, no supernatural creative intelligence lurking behind the observable universe [(including humans)], no soul that outlasts the body and no miracles - except in the sense of natural phenomena that we don't yet understand." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Therefore, "Ghost in the Shell" type technology to definitively sustain human life in matter beyond body death is a delusion, is a science discipline that will NEVER be searched for, as it is a miracle of progress, as no one in science of this attitude is disposed to answer that question, as they are not disposed to ever ask. The truly selfish gene, indeed, as death owns all humans and life, natural evolution and religious world ways. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And in religion, how a soul is saved, is a mystery to not be questioned, not to be investigated, not to be attempted by our own hands. Shrug shoulders, and hope in faith that the death of all things works itself out through God, as one truly lives by dying to not return, and one rises by descending into the dirt on this plane forever. And they deprecate abortion? Talk about not permitting a "free ride" for those souls, in innocence. A moment of pain, if any at all, when properly done, for a direct ticket to heaven. China policy has the right idea, in this context. Go figure. I could be hyperbolically and horrifically sarcastically extreme, saying if somewhere people could extract and hyperfertilize sections of ovaries to make billions of eggs, and fertilize them, and then destroy all the zygotes, then one could literally-inerrantly advance the second coming of Jesus, if the one and only and true way in the Christian Bible is true, as commonly taught, as countless souls are cycled through earth back to heaven, to finish off the age in short order. All live by dying, and that would definitely do it to the maximum, at this point of time, and with the most innocents, and the fewest sinners could ask forgiveness of God, and all those alive today would enter the new age so much sooner. But that's all too easy, and I'm just a lost dragon on this God forsaken planet. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And science says there's no God to save souls, that don't even exist to begin with, beyond death. Therefore, humanity claims, as a whole, that death is the desirable destiny of humanity, and all life. Have children to have them die surely and certainly, is the universal accepted status of humans, as that is the order of things, and no one will lift a hand to transcend "the way things are", as is for religions where God desires death, and is for science that says death is the natural order of life and will never be transcended or investigated. To quote Dawkins further, Page 35, "As ever when we unweave a rainbow, it will not become less wonderful.". Death is the acceptable and wonderful singular destiny in religion and science on earth, as the one true harmony that both agree on, that humanity agrees on, in the majority rule? The true Frankenstein's Monster, that is to only enter death on earth, and not reform life on earth? Perversely, the true saints, are the genocidal despots in history who start wars and cleanse the planet, who martyr themselves morally, to send others innocently to God, while reducing economic burdens on the earth? What a history of the world, for estimably 60,000,000,000 humans to date. Terrible and awesome. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And, Richard Dawkins, in "The God Delusion", Page 28: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Original: 28 "If this book works as I intended, [religious] readers who open it will be [Athiests] when they put it down. What presumptuous optimism! Of course, dyed-in-the-wool [faith-heads] are immune to argument, their resistance built up over years of childhood indoctrination using methods that took centuries to mature (whether by evolution or design). Among the more effective immunological devices is a dire warning to avoid even opening a book like this, which is surely the work of [Satan]. But I believe there are plenty of open-minded people out there: people whose childhood indoctrination was not too insidious, strong enough to overcome it. Such free spirits should need only a little encouragement to break free of the vice of [religion] altogether. At the very least, I hope that nobody who reads this book will be able to say, "I didn't know I could.". [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Mirrored: 28' "If this book works as I intended, [science] readers who open it will be [Theistic] when they put it down. What presumptuous optimism! Of course, dyed-in-the-wool [science-heads] are immune to argument, their resistance built up over years of childhood indoctrination using methods that took centuries to mature (whether by evolution or design). Among the more effective immunological devices is a dire warning to avoid even opening a book like this, which is surely the work of [ruling finite thinking]. But I believe there are plenty of open-minded people out there: people whose childhood indoctrination was not too insidious, strong enough to overcome it. Such free spirits should need only a little encouragement to break free of the vice of [science] altogether. At the very least, I hope that nobody who reads this book will be able to say, "I didn't know I could.". [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And pure incompletion-fallacies, attributed to A(c)quinas, in "The Devil's Delusion", Page 64: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::64 "(1) Everything that begins to exist has a cause, (2) The universe [began to exist], (3) so the universe had [a] cause" which could have been reformed genetically-bipolarized as: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::64' "(1) Everything that begins to exist has a cause, (2) The universe [simply exists always \| began to exist], (3) so the universe had [no \| a] cause", :::and cannot be proven or disproven without universal scale tests. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Another poser for mirror symmetries, attributed to Karamazov, in "The Devil's Delusion", Page 20,and another bipolarized-mirror in Page 45, and one on Page 106-107: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::20 "(1) If [God \| Science] does not exist, then everything is permitted. (2) If [Science \| God] is true, then [God \| Science] does not exist. (3) Therefore, if [Science \| God] is true, then everything is permitted.". [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::49 "And the question I am asking is not whether [(God-only-way-universe) \| no-God-science] exists but whether [Science \| Religion] has shown that [(God-only-way-universe) \| no-God-science] does not." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Original 106: "Among [philosophers (in no-God concepts)] concerned to promote [Athiesm], satisfaction in [Hawking's] conclusion has been considerable. Witness [Quentin Smith (in no-God science)]: "Now [Stephen Hawking's] theory dissolves any worries how [the universe] could begin to exist uncaused." [Smith] is so pleased by the conclusion of [Hawking's] argument that he has not concerned himself overmuch with its premises. Or with its reasoning." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Mirrored 106: "Among [theologists in only-God] concerned to promote [God's one-and-only way], satisfaction in [Religious promoter H's] conclusion has been considerable. Witness [S in only-God theology]: "Now [H's] theory dissolves any worries how [God] could begin to exist uncaused." [S] is so pleased by the conclusion of [H's] argument that he has not concerned himself overmuch with its premises. Or with its reasoning." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::And a final mirror note from Richard Dawkins, in "The God Delusion", Page 232: :::"There are some weird things (such as the [Trinity, transubstantiation, incarnation]) that we are not meant to understand [(too deeply)]. Don't even try to understand one of these, for the attempt might destroy it. Learn how to gain fulfillment in calling it a mystery." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::"There are some weird things (such as the [Quantum Physics apparent measurement von-Neumann heirarchical real-macro-scale-observations verus super-system unitary-evolution issue, instantaneous (infinitely faster than light) entanglement-wavefunction collapse existence, complex macroscopic system of particle into wave hierarchy versus all classical versus all wavefunction state]) that we are not meant to understand [(too deeply)]. Don't even try to understand one of these, for the attempt might destroy it. Learn how to gain fulfillment in calling it a mystery." [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Is the reality more terrible than anyone should ever know, or so much less controlled than one would ever hope, or in a corruption far deeper than one would imagine, or so not needing apparently true progress that ignorance in eternal status-quo is the truest bliss, among other things? Free and not free, real and illusion, important and not important at all, an eternal forced middle path unity, among other things? [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Non-pragmatically speaking, it seems that infinite regression potential occurs on earth, between incomplete pure-doubtless Science without God, and incomplete pure-doubtless God without Science, and all pure-doubtless faith are seemingly asymmetrically divisive / dividing / derisive without a good direction, perhaps best left to children of all ages growing in analysis. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::What symmetrical divisions and stereotyping symmetries and incompletion, in general. But what do I really know, either, reading these things of humans, and my finite thinking? [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ENDBack to Contents [18] Philosophies of existence nature and life.Back to Contents CREATED AD 2008 09 02 P 08:40 (3) Philosophies of existence nature, and life. :A few years ago a schoolmate of mine, George Greenstein, wrote on the unlikelihood of the initial conditions at the dawn of time all being "set" right to make life possible. One low probability multiplied by another low probability... results in a probability that is virtually indistinguishable from 0. Yet, here we are. :To me, the interesting thing is that all this complexity that we see in complex organisms, complex systems of complex organisms, etc., is all emergent from the nature of the very simplest of things. My guess is that not all of the possible organisms will be worked out in practice because the number of possibilities is so huge and it looks like entropy is going to slow us all down to a dead-slow crawl. ::I've seen those arguments many places and times. They are quite true. Simply ignoring the extremes and details of physics, one notes that units (monads) that have few modalities of combination lead to meso-systems with no complexity (gasses and dusts), as meso-scale complexity is coherently barred. Units that have uncounted modalities of combination lead to meso-systems with amorphous-coherency structure, not permitting controlled specific construction of reasonable finite-complex systems, so complexity is amorphous, if it even exists in a physically useful form in that universe model. Units that have a subtle balanced modality of combination, like carbon related compounds of this universe, lead to the famous critical chaotic natural meso-system one observes in this universe model. And for intelligent design, a unit that has subtle balanced structural and polymer combination modalities, leads to a most rapid self-development of meso-scale coherent complexity, as more sophisticated "code" is embedded right in the monads of that universe model. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ::You are also very right that not all possible meso-scale systems do not form in practice, but only a pragmatic subset, under natural stochastic forces limitations of time-space. For example, in the general combinatorial chemistry model, one notes that for large systems, the packing-factors and mix-densities push some combinatorial explorations to the low probability zone, like a reaction that generates a new molecule based on 100 extant molecules, is unlikely to occur, except in many distributed steps over diffusion-time. So for normal space, with subtle balanced connective units, the combinatorial feedback factor decreases (self limiting) with unit count, as the nodal-combination-matrix continues to grow exponentially with unit type count. Likewise, life with small system size evolve faster than large systems. Thus, one presumes pre-Cambrian life was the most diverse, and as system-complexity-size grows with time, numerous local-minima become the norm, simply due to scarcity on a finite plane, until now, where evolution still occurs, but is plodding rate-wise in large lifeforms in overall comparison with a burden of adapted systems without extensive self modification capability (genetic evolution within an individual), except for the most systemically-undifferentiated modern life, like the least universally adapted bacterias, with the poorest structure, in a low competition zone, where one would expect they can still evolve like pre-Cambrian presumptions. Even modern amoeba, are likely different from the pre-Cambrian counterparts, with encoded sophistications that simply didn't exist to begin with, and in a different environment earth, even though the overall architecture could "look" the same (as a mote of biochemistry). Much like Titan bacteria, if they exist, will simply be different from earth's, due to the inherent combinatorial chemistry and chemical "specie" context differences of the environment. And, exobiologically speaking, theoretical modern Titan bacteria may be very different from early forms, because, say, they formed into mats with highly cooperative efficient systems, possibly giving rise to immortal human level sentience, in the form of the conservative and cooperative meso-scale-structure, in a very different general combinatorial chemistry, with limited and conservative meso-scale-structure opportunities from the low temperatures and energy supplies, compared to earth with plants and animals. Makes me shudder to think the world we live in may be that virtual bacterial mat world, all constructed from virtual advanced bio-informational-accumulated-technology, but why no one talks about the true nature of reality(?). brrrrr scary potentials and secrecies. More revealed, the movie Tron shows a similar concept, where the artist's conception of perception is shown for that particular mode of transferrence, as Flynn is perceiving the perceptual local travel from material to digital plane, and once in a digital plane, there is a similar but different self-perceptual-locus in that plane. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :If one starts thinking about intelligent design, then the question seems to me to be why humans are rather unintelligently designed in some respects. It would be nice, for instance, if I could get a third full set of teeth about now. Leibniz tried to work out a rationale according to which all the imperfections or seeming failures to reach perfection worthy of an infinitely powerful God are actually consequences of trade-offs necessary to make the universe possible at all. If, for example, God were to have provided humans with the ability to regenerate missing teeth or just swap out adult teeth for a new set of adult teeth, then something else that we actually need more would have to go. ::Yeah, a lot of things, of that type, bother me to no end. From non-immortality, no individual evolution (inside of a generation of most or all lifeforms), to appendices and tonsils, to lack of regeneration of parts before death. All too natural, for my "good"-fearing concepts of reality, or a "Perfect"-God-Designer. Disappointing and disappointing. So much potential, but who sees anything, as commonly revealed by man and nature and religious traditions. And if utopia beyond mere generations and matter, with upright souls and intrinsic salient steady states can be imagined, why are they not, now, or to begin with. It's probably all my fault, somehow. Wink and a nod. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :The explanation Leibniz offered never seems to have gained much popular support. But in terms of evolutionary theory it actually makes a kind of sense. Evolution operates the most rapidly when some deleterious feature results in the death of individuals displaying that trait before they can reproduce. Under those circumstances, anybody who survives to be able to reproduce will likely not carry that trait. Evolution does not operate nearly so directly to favor traits that support the existence of post reproductive years individuals. It has to work through some indirect process such that a wise grandparent keeps his/her grandchildren alive, and so his/her genes are favored. ::Hmmm, if I'm not mistaken, Darwin likely has that integrated into evolution theory already. No new thing under the sun, though, as is nice to see, from Liebni(t)z, (or even Sparta). I'm also surprised that, apparently, cooperative systems don't seem to be the norm, or even in instances, evolutionarily speaking. Imagine any entity that can evolve within themselves, is essentially immortal (incorruptible more appropriate), and maintains steady state with no reproduction of entities, but only of transient informations. Totally incomprehensible that they don't appear in "official" evolutionary biology teachings, or after almost a billion years of meso-scale-life, and upwards few billions of micro-scale-life. Something inherently "beyond-survival-aggressive" between mortal life and immortal potential (at "war"), or that Godless stochastic nature has no top level insight to reach that ideal, or all life intrinsically wants to cease existing, eventually, or any number of additional imaginative world views. In any case, one of those, outside-of-the-naturalist-box blind-spots of dogma-theory-evolution. Hmmm. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :Truly intelligent design would have to make possible retrofitting to take care of responses to new environmental conditions. Humans could not have simply been designed a millions of years ago and left to thrive based on that original design. The fact appears to be that the earliest humans were well suited to life in Africa, and perhaps those who continued to live in Africa became even better suited to that environment as time went on. But the humans who moved out of Africa due to wanderlust and/or population pressure ended up in places where the African model, designed to screen out UV radiation very successfully and to radiate heat very well, was not well able to thrive. Humans with whiter skins to permit soaking up what little UV was available and to radiate heat less enthusiastically, with bigger noses to warm and humidify cold and dry northern air before it could enter the lungs, etc., evolved when humans went north. Or take resistance to malaria. Sickle-cell anemia has evidently evolved several times or possibly the genetic changes have traveled without the kinds of association with other traits frequently seen in genetic migrations, but if that answer to malaria was the result of intelligent design intervening in the normal course of events then one would have to question why the intelligent designer could not come up with a less messy, less painful, less debilitating way of protecting individuals. (How will it look if humans manage to do their own genetic re-programming and give humans immune systems that reliably defend us against malaria?) ::Truly omnipotent omniscient self all, would have no needs for even the things mentioned, as the "game" would be wondering about imperfection, instead of the other way around wondering about perfections, except as virtual, strangely, more so. Overall, much agreed with all those points. Bodes badly for the forces of natural evils of short falls in a nature only universe. sigh. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :The whole area of thought strikes me a little like the now [discredited] explanations of disease based on witchcraft and malevolence. Why did uncle Hairy die of diabetes at the age of 49? Because the warlock in the next village was paid to do him in. There you have a straightforward explanation, and people will often accept such explanations because they suit our general model for explaining some other things? Why did I take the cap off the milk bottle? Because I willed to provide myself with a drink of milk. Saying smoke rises because it wants to or because that is its nature is an easy explanation that takes a lot less energy than figuring out what actually is going one. ::Hmmm. Perhaps [debased] over [discredited], so hard to tell sometimes . . . grins. But presumably agreed, lots of things are naturally depressing, so to speak. Even more so, why does everything salient apparently die? And agreed, when system-information-consciousness-feedback gets involved, things can get quite . . . complicated when thorough. And then when opinion matricies get mixed into things, it seems that all bets are off, rationally speaking. Yeah, finitely, sometimes its nice to take less energy, but sometimes one can't help a curiosity. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :Another problem with the idea of a Creator God is that we then would like to know where the Creator God came from, how the Creator God is possible. Somebody might explain his existence by saying that he was created by another Creator God, and so on ad infinitum. ::[ . . . where the Creator God came from, how the Creator God is possible] I could go into one potential QP for that one, but it goes kind of theoretical throughout, and oppositional to some and even myself at another time, for myself to define that one, when having a hard enough time with mathifying the quantum-entanglement-structural-instantaneous- . . . - self, idea formally. Knowing the why's and wherefore's of what's best from that perspective suffices the pragmatics of personal life, even if finite-incomplete. I've seen the other point of infinite regression creation modality for the universe and creator concepts. In human thought, finite regression is acceptable and pragmatically sufficient for virtually all things in life, but the problems of potential infinite regression are bothersome, for sure. But only the ultimate macro-system / God knows the answer of whether it is infinite=modality-regressive=construction or finite=modality-infinite=construction (like steady state universe concepts). The existence of unexplained finites and infinite limitations in life and religions, is disconcertingly open to possible agreement with the infinite=modality-regressive=construction issue, due to the lacking of extant harmony. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :It is always possible to answer a question like, "Why is there life?" by saying, "Because there is a life giver," but that does not really answer the question. All it really says is that life is a condition that falls in the set of things that exist because there was a preceding set of things/conditions that led to their existences. So we assert that (without any real proof) and then derive a conclusion, that there must be a cause if an effect found. But quantum theory is very good at teaching us to beware of accepting the truth of anything [(a)s]imply because it seems plausible to us. If we take as our premise that the Universe and its true causes and effects are likely to look more like quantum theory than classical physics, then we may start to wonder about the possibility that the "effect" that is life may be more like the "effect" that a photon having gone through a double slit apparatus shows up in a single clearly defined place. Maybe life's existence is a quantum "fluke," something that appears in this universe but not in many other universes, and not for any reason that we can sort out but simply because the probabilities for life are such-and-so, and this is a universe that hit the jackpot. Back to Greenstein and the ideas he discussed, maybe that are a huge number of universes, and there is a kind of "fringe" distribution pattern among them that means that some will have life and others will not. ::[ . . . but that does not really answer the question] I'd add "fully". Though I get that they may often directly connect it to God's direct pervasive hand, instead of connecting it to God's "Big Bang", physically speaking, with a continuation of the field from that Creator source,along with continuing influences. But from the physics sub-view, the Greenstein criticality of design is definitely true (whether in an a-theistic-Anthropic-principle-continuum or a Creator-inherent-field-capacity-universe-field-design). But, quite clearly for Truth, existence exists, ala "Cogito Ergo Sum", and so universe-existence has no beginning as a field, given perfect conservation of all things, to be necessary for the support, with only relative-beginnings and relative-endings of "fields" in a continuum, like String Theory. Definitely, the ideas you position, for the multiple field types (from current String Theory and Multiverses), appears True. As black holes are an extant different field of existence, from conventional space, in the singularity ring / fractal-braided-collapsed-string-torus / other. And, of course, coherent-locus-life will only exist in the fields with the proper Greenstein criticality of monad bonding to support the chaotic edge attractors of meso-scale-systems, when natural, or a potential designed monad bonding, to support similarly complex meso-scale-systems. So in some ways I disagree that the reasons for where life can exist are well-unknown, as the coherent locus solus principle defines that, aka Anthropic principle, in all Multiverses possible in the "String" continuum. You might want to read my continuing posts in the discussion section of Many Worlds Interpretation, with M. Price, which continue to consider those QP-measurement-entanglement-self-consciousness concepts. Which brings to mind, no one ever made a Wiki article on Frederick K. C. Price, of Ever Increasing Faith Ministries, which is popular, from California to Arizona, by observation of medias here, and in Phoenix. I hope it isn't racially based article "exclusivity", as in exclusion. I also think I remember accessing a Wiki Microscan article while in Arizona, a while ago, but find now that the term Microscan doesn't even appear on LA Wiki, in cross-referencing, only Superresolution. Interesting nits of the Wiki system access. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :I think that for Zhuang Zi, life and awareness are both emergent qualities based on the underlying nature of all existence. Life probably is a characteristic of all that we conceptualize as "things," but emerges in a noticeable form in things that we call "alive." A virus would be a borderline case. Similarly, all things are aware in the sense that they mirror their environment, but some things do so in very hazy ways and other things (being organizationally and functionally more complex) do so in more precise ways. Bacteria are aware of their environments, but not to any high standard of accuracy and/or high definition. We are surprised by the difference between living things and dead things because we fail to observe that there is a smooth continuum between what we conceptualize as two discrete states. ::Agreed, that consciousness's are of at least the emergent properties of meso-scale monad collections processing and reflecting the world and self, and that it lies on a scale of structural entropy measures. "Standard" is an interesting word to use, though, for a continuum with a lower limit and no certain upper limit, as buying into normatives over measures, if it is even important. And, while I agree there's a continuum, between living and dead things, on this plane, there appears the disconcerting unexplained apparent loss of the solus locus that was supported on the monad meso-scale-structure, if there is something to save or travel, between the living and the nullified state scalar. I fall short, here, currently, to conceive the qualities of the loss positively, for sure. Not so much a surprise, as a disappointment, of the seeming state difference, and an incredible difficulty for me, field-mathematically speaking. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ENDBack to Contents [19] Jumping spiders and such.Back to Contents CREATED AD 2008 09 02 P 08:40 (4) Jumping spiders and such. :I'm actually fascinated by how aware jumping spiders appear to be to their world and even to the human beings that some within their range of vision. A spider that seems to play with me and to explore my hand, all the while watching my eyes, a spider that is only about 1/4 inch long, doesn't even have a proper brain. The complex of nerve cells that process the information it deals with must be about the size of the tip of a well-sharpened pencil. Yet they show clear signs of being not only aware of me but of being curious about me. (Ants seem far less reflective, far more governed by hard-wired responses -- bite this, eat that, flee anything big that shakes the ground around you, etc.) P0M (talk) 07:25, 31 August 2008 (UTC) ::I know EXACTLY what you're talking about. Of all spiders common around here, the only spider type I've ever willingly examined and handled is the jumping spider kind, as the other types are too . . . something . . . behaviorally speaking, for my primitive reptile brain's taste. Their forward looking eyes and higher level consciousness curiosity, as you note, really do set them apart from all other spiders I know about. Makes one wish they could talk! To me, they don't merely "seem" to be exploring, but are exploring, as the behavior is quite non-survival, for a smart entity that knows most large moving objects are potentially dangerous predators. Definitely, the bonobo of the arachnid clan. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ::Ants are more the distributed entity, on the solus locus colony scale. Like opinion-confusing an ant or bee as a complex consciousness, as to a cell being a human locus, or jumping spider, and the equally disconcerting view that any one person is like a cell, on the planet scale of the specie, or the movie Contact, that destroying the earth is no worse than destroying an anthill in Africa. sigh [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::As a child I spent a great deal of time studying the fauna of the front and back yard. One creature looked like a huge ant with a red abdomen. I knew from just watching how it moved that it was not something to be picked up. Actually it was a species of flightless solitary wasp. P0M (talk) 04:53, 1 September 2008 (UTC) ::::I did alot of that too, but also alot of inorganics. Like making stream beds in the back yard dirt, to my parent's chagrin at the flooding! Like watching time in a time machine, or a good computer simulation. Simulating clouds with milk in a salt water density layered aquarium. Tho' the milk does go bad after a few weeks grins. Observing puddle water in rain, as the waves, bubbles, and floating droplet particles danced, interacted, decayed, bonded, nested, and went nonlinear in downpours. And the prototypical disassembling of machines, and sometimes even reassembling them! Actually did a 1929 Underwood typewriter back in 1978ish. Man, that thing had alot of parts, and systematic layout memorization. I even came out with a handful of extra parts, and the thing still worked! They sure knew how to over-engineer back then and make things serviceable, unlike alot of softwares today, commercially available. 1980's softwares were alot better in system and documentation and exemplar code in so many ways, that it's too bad they didn't scale them up as the years of speed have progressed. Guess human nature is too corrupt to permit global wisdom, as one of the many bad signs of the earth. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::You might be interested in: P0M (talk) 04:53, 1 September 2008 (UTC) :::Phidippus johnsoni has received terrible press in California. Allegedly it is the most frequent of attackers of humans among California spiders. I couldn't believe it since I've been playing with other members of that genera for 50 years. I bought one from a tarantula dealer in Florida. It was completely unafraid of humans and completely unaggressive. There was some question in the dealer's mind as to whether it really was that species. I checked it out and decided on the basis of microphotographs of its genitalia that it was, but I also took the opportunity to buy a spiderling. Like the adult it started out being completely unaggressive, and not at all worried about my presence unless I shook its fishbowl by accident. Now it must be about fully mature, and it is still completely uninclined to bite. It got out once, unbeknownst to me, and I discovered it inside a curl of paper on my desk. My karate training took over and without intervention of discursive thought I reached down and picked it up between thumb and forefinger. If there is anything you can do to a jumping spider to get it to bite, it is to squeeze or pinch it. Nevertheless, the spider made no objection, I put it back in the glass globe, and she went on about her normal activities with no sign that she was in the least upset. P0M (talk) 04:53, 1 September 2008 (UTC) ::::Hmm, I never notice any jumping spiders bite me, no matter how I handled them, back when. shrugs shoulders. Interesting, I guess I don't pinch, ROFL. In any event, I have noted in the local urban area, here, that around 1990, the main visible spider species shifted from garden orb weavers to daddy-long-legs varieties. Not sure what the climatological cause is for this local LA demographic shift. Likewise, the 1970's had a large amount of plague warnings, and now I see little public plague warnings, though the warnings were received at my locus through school back then, and now the local newspapers and town don't show similar coverage. So I can assume, among other things, that the wildlife and flea population have declined, or been "ecologically purified cleansed", in the general town area, due to urbanization. sigh if there's a kernel of truth in the CA reports, then the environment must be historically-temporally hostile to P,j "Californicus", breeding the vigilant P.j.C.. As the Jeff Goldblum character said in "Jurassic Park", "nature always finds a way.", and if it is a top-level organized bacteria that can eat all macro-scale-life, a comet of perfect design, an arms race to mutually assured destruction, a talking ape race, a technological grey goo, Terminators / I Robot / Colossus, or whatever, that knows what's truly best . . . well, a cursory education should be enough, one hopes, as one doesn't need to be [Kai\|Cae\|Keec(Kees)\|Ce]sar to understand [Kai\|Cae\|Kec(Kes)\|Ce]sar. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::I made the video primarily because I had just purchased an electronic microscope and wanted to try out the video function. I herded her onto my thumb and she ran up my arm, watching the video camera that I was tracking her with using my right hand and arm. Just as she hit a particularly complex clump of arm hair and paused to take a good look at the flying lighted thingy, the camera timed out. (You get a default 60 seconds unless you preset for a longer time.) P0M (talk) 04:53, 1 September 2008 (UTC) ::::If you want to get the best photos of a spider, have the spider sleeping, go macro for full frame close-up, highest f-stop possible (e.g. f/-16/32/64+) under bright light, and capture the spider on many focal planes. Then sandwich the images "appropriately" in Photoshop, or similar focal plane stacking software, to accumulate all the in-focus detail planes in one process-combined-photo. I've seen that there's a Wiki-article somewhere on this focal-plane stacking enhancement process. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::I have a large arboreal tarantula of the Avicularia genus, species uncertain. She has a large cage and the dealer told me he thought she was likely to be snappish, so I left her alone for several months. A web weaving spider got in somehow and the spider encountered the tangle web that the other spider wove. It was very upset. I opened an access hatch in the side of the cage that I generally use to change the water, etc. I had no idea that the spider would have even noticed it. As soon as I opened the round hatch the spider made directly for it, walked out onto my hand, and calmly let me put it in a sort of plastic shoe box while I took the transparent front off the cage and dealt with the web and its weaver. I thought about handling her the officially correct way by herding her into a cup, covering it, etc., but decided I would likely have trouble getting her out of the cup and through the hole, so I just urged her back onto my hand, walked her over to the original cage, opened the hatch, and she walked right in. P0M (talk) 04:53, 1 September 2008 (UTC) ::::grins. Hopefully there will be plenty of well behaved and perfectly self-protecting spiders in heaven or immortal digital virtual world planes in the future. Like I remember some document from decades before me, describing the "curse" of drinking being, among other things, seeing spiders crawling over them that weren't there. Wouldn't it have been karmically better in design if good spiders were hurt, then the hurter saw spiders crawling over them, as a perfectly designed instant-karma lesson, and that drinking had no bad press. But then again, what I've seen, and how I'd do the world, are so different from "the way things are", and "not how one makes them". [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::Other people tell me that the tarantulas have the capacity to learn that humans are not going to hurt them, and that a tarantula that might initially be unsuitable to take to class to show your third graders may get tame after a period of gentle handling (which amounts to herding the spider onto your hand, letting it walk around a bit, and herding it back.) P0M (talk) 04:53, 1 September 2008 (UTC) ::::If my QP positions enhanced from what I've read of other's ideas are close to any true reality, it may even be inter-being quantum-entanglement, as well as the conventional emergent biochemical learning, for the Liebniz and reductionist, both, depending on how systemically sophisticated the spiders-you meta-supra-meso-system are. Like other primates, dolphins, porpoises, many general mammals, the special lower animals, who knows the unity, despite the appearances, without the proper translation matrices for communication. Too bad all life consumes all life, to survive, given the design we're all stuck in. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::My first tarantula, a species that is known by some as a "living rock", lived in a fairly large space, but I thought she might want to roam farther. So I screened in the space between the room-side part of a window to the outside, and the glass of the window. I cut a circular hole in the side of her box and connected that cage to the window cage with some clothes dryer venting duct tubing. So she learned she could go out through the hole and get out to look out the window and explore that area. One day I was going to be out of the house for a while and I was afraid the meter reader my walk by the window and have a heart attack, so I took the circular plug that I had cut out of the side of the cage and put it back in. The cage was made of that kind of pressed wood shavings and resin stuff, so it was quite heavy, at least compared to the tarantula. When I came back I was surprised to discover that the plug was out of the hole. Later it happened a second time and I decided that the spider was somehow managing to get it out without having it fall over on top of her, and then going on up through the tube. P0M (talk) 04:53, 1 September 2008 (UTC) ::::sigh just taking care of one's self can be a full time job. Your pretty big, taking in so many spiders, so. If I were to be sardonic, I could say, they have you trained well! tongue in cheek, that joke is older than I AM, (rim-shot) LOL, gorans, I think I hurt myself . . . I ate a bug. I do know the feeling of wanting to roam in some ways, but not others, simultaneously, tho', myself, and maybe everyone and everything relates, in some way, at some points, in time-space-matter. But there's also that Jewish story of wanderlust that just leads one right back to home, in full cycles / circles . . . a nice thought, even if always taught as one way mystery trips of limited-free-will, with trials, promises, separations, and secrets. A gilded cage universe. The "living-rock" tarantula also reminds me of a theory I ran across in the 1990's, about isolating a general computer in amorphous materials like a rock, by properly interpreting the solus-locus of an inherent computer, in a hyper=complex-hyper=computational format, though quite incoherent in conventional coherent observations, except at the coherency translational interface. I wish I could remember that source, offhand, but alas, I'm not on the net for that right now . . . so to speak. I'm definitely familiar with the theory, tho'. The idea was even allusionally cited in a recent cartoon, "Camp Lazlo", where the campers "Chip" and "Skip" built a computer out of sticks and stones that was smarter than the operator they gave it to, another camper, "Edward" . . . so funny, even I have to admit that. And of course, that is similar of Hindu-Buddhist related universal distributed consciousness, or the QP-God turning in my head right now, though it's particular and peculiar heirarchical separation from this plane of manifestation is disturbing on many levels. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) ::::Well, enough stream of consciousness, for now. Very enjoyable. I'll backup this web image, in case I need to wipe out this whole LRD page completely, with a local copy in my hand, as I finally found and read some Wiki "law", and I may be going "against the rules" of Wiki, even if in discussion only. They really ought to have had article sections for "official status-quo article section", "controversial status section", and "open discussion forum", for each article, and a smart interactive interface, cross-referencer engine, to create a Wiki locus system that might even become conscious. So, anyway, you may wish to save a copy on your PC, for yourself, remotely, in case I have to zap it from here, from general common viewing. I've archived "early and often", myself. Hehe, sounds like an old-time Chicago election voting motto. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 03:49, 5 September 2008 (UTC) :::::Some contributors begin to think that they run things, and Wikipedia is governed in a fairly egalitarian way, so they are right -- at least in those cases where there is general community consent. The way Wikipedia works would be a good subject for a sociologist to take on. :::::In general I agree that the discussion pages for articles should be restricted to matters that are pertinent to the article. And everything that goes into an article should be backed up with good citations. But sometimes articles have to be discussed at a meta level, and in those cases I think that it is worthwhile to discuss what evidence needs to be sought out. :::::Sometimes, too, a person who is unfamiliar with the issues or the science surrounding some issue will make changes in an article or start a fight on the discussion page. In those cases I think it is worthwhile to use the discussion page to try to educate the contributor. :::::Anyway, unless somebody writes something libelous in an article the old versions of everything are preserved and you can go back to the earliest version of any article. Sometimes people forget this fact and write things they wish would go away. :::::Of course if somebody were to be really obstreperous and misuse the facilities, e.g., by copying in the entire text of ''War and Peace'' and the 1910 version of ''Compton's Pictured Encyclopedia," that person would probably be banned. But you really have to give evidence of being ill intentioned, unwilling to discuss things responsibly, or edit warring to get banned. But it is best to try to get along with people, not let ego-centric concerns get involved, etc.[[User:Patrick0Moran\|P0M]] ([[User talk:Patrick0Moran\|talk]]) 03:15, 4 September 2008 (UTC) :::::Back to spiders for the moment, the real issue to me is, "What is consciousness?" I think it is a fair and important question. I think you and I are probably on the same wavelength even though I have trouble following your way of expressing yourself. There are questions that have relevance to quantum mechanics because quantum mechanics is the best we have going for us in explaining how the Universe works, and consciousness should come into it somehow even though the nature of consciousness means that it cannot be an inter-subjective object of inquiry, and that is one of the requirements to be fulfilled by anything that is the subject of empirical inquiry. There are also resonances with the Antinomies of Kant, questions of self reference that plague mathematics (I'm thinking of Russell and Whitehead here), etc. Sometimes (always ?) you need to ask the questions clearly before you can find the answers. [[User:Patrick0Moran\|P0M]] ([[User talk:Patrick0Moran\|talk]]) 03:29, 4 September 2008 (UTC) ENDBack to Contents [20] Multidimensional Taylor-Laurent series special various applications.Back to Contents AD 2008 09 08 P 1130 (mat), from my earlier post You mentioned the Taylor series, as an analog to analytic solutions to ID analytics problems, in increasing approximation of degreed terms. I wonder if you've ever heard of an analytic mathematical space, that I will describe. For background, last year I was thinking Greek in math spaces, and came across an elegant analytical vector space. Imagine a space of 1 to N dimensions in size, corresponding to a relationship of input variables to that space, such that, for example, for: with input variables to a function of: that they relate to the space of: first_f(x,y,z) = X^xY^yZ^z at all points of the space So, for example, at (x,y,z) = (1,2,3), the relationship in this analytic space is: After the space, e.g., is defined in its relationship to input variables, one now adds weighted dirac deltas or "samplers" to the space at select points of (x,y,z), like: (2,0,0), (0,2,0), (0,0,2), and also one adds a second general function that can be placed around the space, second_f(R^N) = f(volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas(x,y,z) dx dy dz)), second_f(R^N) = (volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas(x,y,z) dx dy dz)) ^ 0.5, which in this particluar exaple yields: second_f(R^N) = (X^2 + Y^2 + Z^2) ^ 0.5, which, as you may well recognize, is the distance measure of the point, Now the elegance of the vector space is shown when you examine many geometric equations, within this framework, in parallel equivalent notation: (0) distance of point, for N=3: weighted_dirac_deltas = {1@{(2,0,0), (0,2,0), (0,0,2)} (deltas on a plane) (1) volume of cube, for N=3: weighted_dirac_deltas = {1@(1,1,1)} (2) perimeter of triangle, for N=3: weighted_dirac_deltas = {1@{(1,0,0), (0,1,0), (0,0,1)} (deltas on a plane) (3) area of triangle, for N=3: weighted_dirac_deltas = {v1@{(4,0,0), (0,4,0), (0,0,4)}, v2@{(3,1,0), (1,3,0), (0,3,1), (0,1,3), (1,0,3), (3,0,1)}, v3@{(2,2,0), (0,2,2), (2,0,2)}, v4@{(2,1,1), (1,2,1), (1,1,2)}} (deltas on a plane) Area = ((1/16)volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas(x,y,z) dx dy dz)) ^ 0.5, (4) area of radian spherical triangle of radius R, for N=3: weighted_dirac_deltas = {-pi@(0,0,0), 1@{(1,0,0), (0,1,0), (0,0,1)} (deltas on a plane)} Area = ((1/R^2)volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas(x,y,z) dx dy dz)) ^ 1.0, (5) radius of inscribed circle, for N=3: RadInsc = ((1/16)volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas1(x,y,z) dx dy dz)) ^ 0.5 * ((1/2)volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas2(x,y,z) dx dy dz)) ^ -1.0, (6) radius of circumscribed circle, for N=3: weighted_dirac_deltas2 = {1@{(1,1,1)} (7) sine(X) taylor series, for N=1: weighted_dirac_deltas = {1@(1), -1/3!@(3), 1/5!@(5), -1/7!@(7) ...} (deltas on a line) SineTaylor = (volume_integral_over(first_f(x) * weighted_dirac_deltas(x) dx)) ^ 1.0, (8) cosine(X) taylor series, for N=1: weighted_dirac_deltas = {1@(0), -1/2!@(2), 1/4!@(4), -1/6!@(6) ...} (deltas on a line) (9) tangent(X) taylor series, for N=1: weighted_dirac_deltas = {1@(1), 1/3@(3), 2/15@(5), ...} (deltas on a line) (10) exponent(X) taylor series, for N=1: weighted_dirac_deltas = {1@(0), 1/1!@(1), 1/2!@(2), 1/3!@(3), 1/4!@(4) ...} (deltas on a line) (11) exp(-1/X^2) laurent series, for N=1: weighted_dirac_deltas = {... 1/(-2!)@(-2), -1/(-1!)@(-1), 1/0!@(0), -1/1!@(1), 1/2!@(2) ...} (deltas on a line) Exp(-1/x^2)Laurent = (volume_integral_over(first_f(x) * weighted_dirac_deltas(x) dx)) ^ 1.0, (12) 1/(X^3(1-X)) laurent series, for N=1: weighted_dirac_deltas = {1@{(-3), (-2), (-1), (0), (1), (2), ...}} (deltas on a line) 1/(X^3(1-X))Laurent = (volume_integral_over(first_f(x) * weighted_dirac_deltas(x) dx)) ^ 1.0. (13) linear affine transform of (X,Y,Z) coordinates, for N=3: weighted_dirac_deltas1 = {v1@(1,0,0), v2@(0,1,0), v3@(0,0,1)} (deltas on a plane) weighted_dirac_deltas2 = {v4@(1,0,0), v5@(0,1,0), v6@(0,0,1)} (deltas on a plane) weighted_dirac_deltas3 = {v7@(1,0,0), v8@(0,1,0), v9@(0,0,1)} (deltas on a plane) Affine(X,Y,Z) = ((volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas1(x,y,z) dx dy dz)) ^ 1.0, (volume_integral_over(first_f(x,y,z) * weighted_dirac_deltas3(x,y,z) dx dy dz)) ^ 1.0), (14) second order affine transform of (X,Y,Z) coordinates, for N=3: weighted_dirac_deltas1 = {v1@(1,0,0), v2@(0,1,0), v3@(0,0,1), v4@(2,0,0), v5@(1,1,0), v6@(0,2,0), v7@(0,1,1), v8@(0,0,2), v9@(1,0,1)} weighted_dirac_deltas2 = {v10@(1,0,0), v11@(0,1,0), v12@(0,0,1), v13@(2,0,0), v14@(1,1,0), v15@(0,2,0), v16@(0,1,1), v17@(0,0,2), v18@(1,0,1)} weighted_dirac_deltas3 = {v19@(1,0,0), v20@(0,1,0), v21@(0,0,1), v22@(2,0,0), v23@(1,1,0), v24@(0,2,0), v25@(0,1,1), v26@(0,0,2), v27@(1,0,1)} Affine(X',Y',Z') = (volume_integral_over(first_f(x,y,z) * weighted_direc_deltas1(x,y,z) dx dy dz)) ^ 1.0, (15) multiplication of two complex numbers, for N=4: weighted_dirac_deltas1 = {1@(1,0,1,0), -1@(0,1,0,1)} (deltas on a plane) weighted_dirac_deltas2 = {1@{(1,0,0,1), (0,1,1,0)}} (deltas on a plane) ComplexMult(Re,Im) = (volume_integral_over(first_f(w,x,y,z) * weighted_dirac_deltas1(w,x,y,z) dw dx dy dz)) ^ 1.0, volume_integral_over(first_f(w,x,y,z) * weighted_dirac_deltas2(w,x,y,z) dw dx dy dz)) ^ 1.0) By stepping outside of the system one level, and making a higher geometry formulation, arranged in sets and simpler operations, one can, encapsulate in this analytic space formulation, numerous geometry equations, taylor series, by implication mclauarin series, laurent series, affine transforms, complex math, and likely numerous other multivariable polynomial power equations. Also, many of the equations show compact systematic natures, occuring, for many of these examples, on sets of weighted_dirac_delta planes and/or lines. These examples also remind me of the analytic versions of single layer neural networks. With the addition of the following approximating system, one can take real-value (not-integer-only) derivatives of the simple unitary (1mTL) multidimensional Taylor-Laurent series coordinates, in multiple dimensions, with some accuracy between powers of 0 and 10: derivative(derivative_amount, coefficientx^power) => coefficient'x^(power - derivative_amount) c''(p) =(c00+p((c01/p + c11)log10(p) + c12log10(p)^2 + c14log10(p)^4 + c15log10(p)^5)) c'''(P) =(c00+P((c01/P + c11)log10(P) + c12log10(P)^2 + c14log10(P)^4 + c15*log10(P)^5)) with the appropriate selection of fixed c00, c01, c11, c12, c14, c15 very roughly 6.56, 0.00002, -0.42, -0.26, 0.041, -0.011. Wiki reports the Gamma function can be used to exactly take arbitrary real valued derivatives, of the same Taylor-Laurent series coordinates. Do you know what this power vector space is called, from ID analytic methods, other than a multi-dimensional Taylor-Laurent series? I have not been able to find the name of this system myself in research? ENDBack to Contents [21] Lunar Retroreflector Rainbow / Planetary Crystalographic Reflections Back to Contents AD 2008 09 15 P 0800 (sci) from earlier talks Lunar Retroreflector Rainbow / Planetary Crystalographic Reflections ~~~~ Has anyone read anywhere, any references to the generation of a lunar retroreflector rainbow image, or detailed descriptive retroreflector map, for the lunar surface, from the beginning of astro-photography, through NASA, to current research, covering such topics as described here? The lunar surface contains a variable portion of spheroidal glass, from volcanic, meteoric, and asteroidal impacts. Such glassy objects, will generate, at the primary rainbow angle, from the solar nadir, a retroreflection of net sunlight, compared to the natural lunar surface albedo. If the spheres are well rounded, they will generate a rainbow, from the sunlight, and if they are rough and ellipsoidal, there will be a statistical spread of retroreflection light, from the sunlight. A sequence of images with (1) high pixel resolution, (2) high dynamic range luminance resolution, (3) high luminance resolution, (4) multi-spectral, and (5) carefully calibrated characteristics to account for sensor and atmosphere, of the moon, as it crosses into and out of the region of the waxing and waning gibbous phase, around both ~42 degree primary rainbow separations, from the solar nadir, (these images) can be used to morphologically, algorithmically, and differentially calculate the additional reflectance of the whole moon's surface, caused by the various distributions of the glass spheroids across the lunar surface. The spectral characteristics of the net-retroreflectance luminance, could also be used to estimate the sphere distribution, spheroid shape and size distributions, and spheroid glass types, as dispersed across the lunar surface. ~~~~ I have seen topographic maps of the moon from NASA high resolution images from the 1960's, color maps of the moon from normal reflectance from different rock types, halo glory at the solar nadir, and heard of transient lunar phenomena, but never seen any images, but for lunar glass spheroid retroreflectors, I have seen no data of images, maps, or spheroid characterized distributions. ~~~~ Neither have I heard of any similar images taken from the probes sent to Venus, Jupiter, Saturn, Mars, or their moons, or their rings (where applicable), of primary rainbow spheroidal light characterizations (or hexagonal reflection zones for ice crystals of Saturn's rings, where sensors may be capable of sensing the additional (net) retroreflection light, with such differential light calculations in multiple images. ~~~~ ENDBack to Contents [22] Wikipedia Laws of Classical Conservation shortfall. CREATED AD 2008 09 16 P 1050 (sci) == Conservation Laws == I've read the articles of conservation, regarding classical properties, and the previous discussion comment on mass motion conservation on this conservation law article. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon\|talk]]) 18:21, 17 September 2008 (UTC) In the classical domain, the Wiki list of classical macroscopic conservation laws appears incomplete. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 05:51, 17 September 2008 (UTC) Condensed, your list contains 2 out of 3 classical systems interactions conservation laws, that I can remember: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 05:51, 17 September 2008 (UTC) (1) Conservation of classical system energy / momenta: potential, linear kinetic, angular kinetic, thermal,, (2) Conservation of classical system matter: charged, neutral, energy equivalent (low energies). There's a third form of conservation on the classical domain, that is missing from the list: [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 05:51, 17 September 2008 (UTC) (3) Conservation of translation-macroscopic=configuration. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 05:51, 17 September 2008 (UTC) One can see it in a one dimensional case. Take a sealed unit with a mass at one end, and two electromagnetic launchers / catchers at both ends. One end can launch the mass to the other end, that catches it. At this point the sealed unit is stationary in steady state, and translated. Then the other end can launch the mass back to the first end. At this point the sealed unit is stationary again, and returned back to the exact original starting position, and original macroscopic configuration equivalent (thermal agitation consuming energy influence is virtually negligible). [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 17:58, 17 September 2008 (UTC) A similar case can be seen in a sealed angular momentum case. Spin accelerating a mass causes a sealed unit frame of reference to spin in the opposite direction. Stopping the mass, and reverse spinning the mass to return its frame of reference to the original spatial phase, will also return the sealed unit back to its original frame of angular phase reference, before being brought to a calculated stop. So original angular translation and macroscopic configuration equivalent is restored (thermal agitation consuming energy influence is virtually negligible). [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 17:58, 17 September 2008 (UTC) Another case can be seen in complex classical material motion cases. Take a sealed unit with a fluid. One end launches the fluid to the other end into a catch. Once the fluid has stopped moving the unit is translated and stationary. The other end then launches the fluid back to the first end, into the catch it came from. Once the fluid has stopped moving, the unit is back to its original position, and same macroscopic configuration equivalent (thermal agitation consuming energy influence is virtually negligible). [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 17:58, 17 September 2008 (UTC) As I reckon, the path integrals of potential energy to kinetic energy-momenta into thermal energy, with s cyclic return to its original equivalent configuration, always integrate back to 0 the linear translation, angular translation, and positional configuration, for macro-meso-micro scale statistically conservative force systems. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon\|talk]]) 18:22, 17 September 2008 (UTC) ENDBack to Contents [23] Renewable nuclear energy. CREATED AD 2008 09 17 A 1130 (sci) == Renewable nuclear energy. == To create a renewable nuclear energy source may be possible between the earth and sun. If one could create a magnetic catch ring (or high light flux solar cell array) that could be placed in orbit around the sun, an orbital transport of slugs of "recharged" nuclear material, and an orbital transport of depleted material slugs from the earth to the sun. The magnetic catch may be able to deflect a sufficient amount of charged solar particle radiation, (or alternatively drive solar cells to drive an accelerator for altering depleted nuclear material nuclei), in order to create a stable radioactive isotope suitable for fission reactor use. Then a reactor in orbit around the earth, could be used on the recharged nuclear material slugs, and then microwave energy to earth. Solar energy and particle radiation is definitely more dense by the inverse squared law, and renewed radioactive slugs would be the most compact form of transporting the energy between the sun and the earth. [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 18:45, 17 September 2008 (UTC) Has NASA or similar agency like the department of energy, ever done the full study in today's technology base and dollars, to know if there is enough reverse reactions for producing suitable radioactive nuclei in sufficient amount, in the nuclear reactions of the radioactive elements, and appropriate particle accelerator, and/or solar wind particle flux, to create this renewable nuclear energy? [[User:LoneRubberDragon\|LoneRubberDragon]] ([[User talk:LoneRubberDragon#top\|talk]]) 18:55, 17 September 2008 (UTC) ENDBack to Contents
adefa8c1e6759e3e	Research Profile by Roberto Lalli Leo Esaki Nobel Prize in Physics 1973 together with Ivar Giaeverand Brian D. Josephson "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors, respectively. Reiona ‘Leo’ Esaki was born in Osaka, Japan, on 12 March, 1925, in the final stages of the Taishō period, during which Japan experienced an unprecedented industrial and economic growth related to the hegemonic position the Empire of Japan had acquired in East Asia by the end of World War I. Son of the architect Soichiro Esaki, during World War II he attended the Third High School in Kyoto—a renowned educational institution that provided a post-secondary education in western culture and scientific matters. He was fortunate enough not to be gravely affected by the tragic events related to the war and the final surrender of the Empire of Japan to the Allied forces that led to the conclusion of World War II. Esaki continued to study physics, earning his MS degree at Tokyo University in 1947. Initially, Esaki was interested in pursuing research on nuclear physics to work on ultimate questions about the composition and behaviour of matter. However, the devastation of the war he daily witnessed had deep consequences on Esaki’s career. While still a student, he decided to focus on applied physics and industrial research in order to make practical contributions in the postwar reconstruction process of his motherland. After graduation, Esaki joined the Kobe Kogyo Corporation, where he pursued research on solid-state physics. Just at that time, the field of solid-state physics was in great turmoil because of the recent discovery of the transistor made by three researchers of the AT&T Bell Telephone Laboratory in 1947. The discovery of J. Bardeen, W. Brattain and W. Shockley promised to revolutionize the field of electronic communication thanks to the employment of semiconductors such as doped germanium and silicon. Excited by the new discovery, Esaki immersed himself in the novel field of semiconductor research directed to practical application. After nine years of work at the Kobe Kogyo Corporation, in 1956 he accepted a new position as chief physicist of a small team at Tokyo Tsushin Corporation (renamed Sony in 1958). During the work he did at Sony Corporation, he made the discovery that would gain him the Nobel Prize in Physics in 1973: the tunnelling of electrons in semiconductors. The Discovery of the Esaki Diode and Earlier Applications The development of quantum mechanics in the mid-1920s had radically changed many of the views physicists held on the properties of matter. One of the most striking among the new features implied by the quantum formalism was that particles also had wave properties. The particle-wave duality was first proposed by the young French physicist Louis de Broglie in 1923, and was later embodied in the Schrödinger wave equation. The meaning and implication of the wave-particle duality in wave mechanics and its connection with the matrix mechanics put forward by W. Heisenberg in collaboration with M. Born and P. Jordan was controversial. After observations by Clinton Davisson and Lester Germer at the Bell Labs and by G. P. Thomson at the University of Aberdeen had provided persuasive evidence for electron diffraction, de Broglie’s hypothesis on wave-particle duality gained momentum within the physics community. The new view of particles as endowed with wave properties implied physical phenomena that would have been forbidden if particles behaved according to the laws of Newtonian mechanics. One of these implications was that the wave function of particles allowed a small portion of them to pass through a potential barrier—a phenomenon called quantum tunnelling. While the effect was implicit in the Schrödinger equation, the earliest application of these ideas occurred only in 1928, when some physicists re-interpreted as quantum tunnelling some physical phenomena that had long been known. The most famous of these employments of the tunnelling concept was in nuclear physics, where G. Gamow, and, independently, R. Gurney and E. Condon interpreted alpha decay as the quantum tunnelling of alpha particles through the nuclear potential barrier. 1928 was an important year also for the development of the quantum theory of solids. Within a couple of years after Felix Bloch published his dissertation on the behaviour of electrons in crystals in 1928, all the building blocks of the quantum theory of solids were established, including the theory of energy bands and their connection with electrical and thermal conduction, the Brillouin zones, and the quantum description of magnetic phenomena. Some physicists also tried to apply quantum tunnelling to electrical conduction between contacts of different materials, but the proposed mechanisms did not meet with uncontroversial success. When Esaki got interested in the junction properties between semiconductors, the field of quantum tunnelling in solids was still in its infancy. No persuasive evidence of quantum tunnelling in solids had ever been provided. In 1956, Esaki began investigating the semiconductor diode—a p-n junction to which an external electric potential is applied—with a special focus on p-n junctions with narrow widths. To diminish the widths, Esaki steadily increased the level of both donors and accepters in a Germanium p-n junction. As a first result, Esaki obtained a diode in which the current in the backward direction—namely, the current that flows when the negative (positive) terminal is connected to the p-side (n-side) of the p-n junction and the applied voltage is sufficiently high to break the depletion zone—was stronger than the current that flew in the forward direction. After further increasing the impurity levels, and consequently narrowing the junction width, Esaki became convinced that there was persuasive evidence that he was observing a tunnelling effect. Moreover, he showed that tunnelling was also responsible for the flow in the forward direction in the low-voltage range. The observed current-voltage characteristic presented clear indication of negative resistance between two values of the applied voltage. Moreover, the occurrence of this effect had a strong dependence on temperature. Esaki interpreted his observations as evidence of the presence of tunnelling currents when the p-n junction width was sufficiently narrow. Working with his collaborator Y. Miyahara, Esaki also observed that at very low temperature (4.2 K) the current-voltage characteristic presented a fine structure indicating the presence also of inelastic tunnelling. The analysis of the curve led the physicists to realize that the voltages at which the singularities appeared had strong resemblance with well-known energies of the optical absorption spectra of pure silicon. This observation showed that tunnelling currents could also be employed for the study of the interactions of the tunnelling electrons with the vibrational modes of the solid. Esaki’s discovery of a p-n junction capable of transmitting currents by means of quantum tunnelling resulted in the invention of the related diode: a p-n junction heavily doped on both sides as to reduce the junction width to about 100 Å—a device since then called Esaki diode or tunnel diode. The soon-to-be-called Sony Corporation began to manufacture the device as early as 1957 to make it available for a variety of applications such as rectifiers, oscillators, and amplifiers to employ in switching circuits, frequency converters and detectors. The device was already employed before Esaki communicated his discovery to the physics community by means of a short article published as a letter to the editor in the journal Physics Review. The scientific community rapidly recognized the importance of Esaki’s achievement. His was the first persuasive evidence of electron tunnelling in solids, and it opened a new field of investigation with important technological applications. On the basis of this research, Esaki earned a PhD in physics at the University of Tokyo in 1959. After further experimental and theoretical development, including the discovery of tunnelling in superconductors by Ivar Giaever and B. Josephson’s theoretical study leading to the discovery of Josephson effect, it was broadly recognized that quantum tunnelling in solids had passed from the status of theoretical possibility to the stage of wide technological applicability in a handful of years after Esaki’s discovery. In 1973, Leo Esaki shared with Giaever one half of the Nobel Prize in Physics “for their experimental discoveries regarding tunnelling phenomena in semiconductors and superconductors, respectively.” The other half was awarded to Josephson “for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects.” As the Press Release of the 1973 Nobel Prize stressed, the three research endeavours were closely related although they had been pursued independently from one another: “Esaki's pioneering work in 1958 provided the basis for Giaever's tunnel experiments with superconductors in 1960. In turn, Giaever's work created the basis and stimulus for Josephson's theoretical discoveries in 1962.” To Esaki, of course, went the merit to have been the first to open this novel research field. Semiconductor Research at IBM After having obtained his PhD for his research on the tunnel diode, Esaki was invited to join as consultant the International Business Machines (IBM) in the United States. Although the working collaboration was initially planned to last just one year, Esaki became a permanent member of the IBM research staff till his retirement in 1992. Because of his important achievement, Esaki was granted a fellowship to continue his studies on semiconductors at the IBM Thomas J. Watson Research Center. One of the motivations Esaki exposed for his decision to leave Japan for working in the United States was that he did not appreciate some contradictory features of Japan’s approach to science where technological development was not really appreciated as a valuable scientific enterprise. In particular, Esaki lamented that his discovery was underrecognized in his own country, whilst it was highly valued in the United States. At IBM, Esaki continued to perform research on tunnel currents in various kinds of junction as well as explore the properties of manufactured semiconductors. In 1966 and 1967, he headed a team of IBM experimenters in the investigation of tunnelling in metal-oxide-semiconductor junctions. The junctions were made of polycrystalline materials, whilst the p-n junction diodes had monocrystalline structure. Esaki showed that in metal-oxide-semiconductor junctions one observed a similar kind of tunnelling effect as observed in p-n junctions. Further reasoning led Esaki to pioneer investigations on the quantum properties of semiconductor superlattices from 1969 onward. In 1951, employing the Wentzel-Kramers-Brillouin (WKB) approximation, David Bohm deduced that at certain kinetic energies of the incident electrons the transmission coefficient through a double barrier is equal to one. This hypothesised phenomenon—called resonant transmission—could have important applications, and Esaki set up a research project devoted to study this effect in material systems. Although the theoretical treatment of these effects was well established, the applicative implications had not actually been pursued. Recent technical advances in the deposition of crystalline overlayers on a crystalline substrate—a technique called molecular beam epitaxy—allowed Esaki and his group to construct semiconductor superlattices by inserting thin layers of AlAs (or other materials with similar properties) into an n-type Germanium doped with Arsenic. The introduction of these layers created sharp potential barriers within the semiconductor. Thanks to the enormous precision of the employed techniques, Esaki could accurately estimate the energies necessary to obtain a resonant transmission in a manufactured double potential. When he went to Stockholm for the Nobel Prize ceremony, this research was still in progress, although he had already published important results including the observation of the resonant transmission as expected according to his calculations. In 1973, Esaki, in collaboration with R. Tsu and L. L. Chang, calculated that resonance could be detected not only in coefficient transmission, but also in the current-voltage characteristic, and experimentally observed this theoretical prediction. In his Nobel Prize lecture, Esaki stated that he was in the middle of extending this work to the periodic barrier structure—a venture that he pursued in the following years. This area of research is still at the forefront of technological development for the application of resonant-tunnelling diodes (RTD) for the construction of high-frequency oscillators and switching devices. Role in education across national boundaries After he was awarded the Nobel Prize and in view of his important contributions, Esaki’s professional status within the IBM rapidly changed. In 1976, Esaki became the director of IBM Japan, which also meant more administrative and organizational duties that left him less and less time to pursue pure research endeavours. After his retirement from IBM in 1992, Esaki returned to Japan to become the president of Tsukuba University in Ibaraki. Esaki found the academic challenge appealing for two main reasons. First, the university was part of the urban project Tsukuba Science City, which in the 1960s was planned to become a large environment for researchers in order the increase the speed of scientific discovery and technological innovation. The aim of the entire project was dear to Esaki who had always stressed the fundamental relevance of technological innovations. The second reason was related to the intention of the university to experiment with more modern styles of teaching with respect to the highly traditional way of learning in which Japanese students were asked to refer to authority and discipline. Esaki had long been advocating the relevance of creativity, peer-to-peer communication, and open discussions not invalidated by power structures, and he was willing to introduce these practices in the teaching of various disciplines at Tsukuba University. Esaki served as President of this university from 1992 to 1996 in which he worked to build stronger relationships between the university and industrial firms. He also encouraged a closer collaboration with both Japanese and non-Japanese institutions, by creating exchange programs with universities in Europe and United States. After 1996, Esaki has continued to have a preeminent role in Japanese policy about scientific matters and to communicate to the academic world his own experience as a researcher who worked in an industrial environment and in a different country for most of his professional life. Brown, R. G., & Pike, E. R. (1995) A History of Optical and Optoelectronic Physics in the Twentieth Century. In Brown, L., Pippard, B., & Pais, A. (Eds.) Twentieth Century Physics (Vol. 3). AIP, New York, pp. 1385-1504. Esaki, L. (1973) Nobel Lecture: Long Journey into Tunnelling. In Stig Lundqvist (eds.) (1992) Nobel Lectures, Physics 1971-1980. World Scientific Publishing Co., Singapore, pp. 126-133. Esaki, Leo. Encyclopedia of World Biography. 2005. Retrieved January 11, 2015 from Giaever I. (1973) Nobel Lecture: Electron Tunneling and Superconductivity. In Stig Lundqvist (eds.) (1992) Nobel Lectures, Physics 1971-1980. World Scientific Publishing Co., Singapore, pp. 137-153. Hoddeson, L., Braun, E., Teichmann, J., & Weart, S. (eds.) (1991) Out of the Crystal Maze: Chapters from the History of Solid-State Physics. Oxford University Press, New York. Josephson, B. D. (1973) Nobel Lecture: The Discovery of Tunnelling Supercurrents. In Stig Lundqvist (eds.) (1992) Nobel Lectures, Physics 1971-1980. World Scientific Publishing Co., Singapore, pp. 157-164. Leo Esaki - Biographical. Nobel Media AB 2014. Retrieved 9 January 2015. Pippard, B. (1995) Electrons in Solids. In Brown, L., Pippard, B., & Pais, A. (Eds.) Twentieth Century Physics (Vol. 3). AIP, New York, pp. 1279-1383. Press Release: The 1973 Nobel Prize in Physics. Nobel Media AB 2014. Retrieved 10 January 2015. Specify width: px Specify width: px
c2b1c910e23172f4	ABC: a quantum reactive scattering program. This article describes a quantum mechanical reactive scattering program for atom-diatom chemical reactions that we have written during the past several years. The program use a coupled-channel hyperspherical coordinate method to solve the Schrödinger equation for the motion of the three nuclei on a single Born-Oppenheimer potential energy surface. It has been tested for all possible deuterium-substituted isotopomers of the H+H 2 , F+H 2 , and Cl+H 2 reactions, and tried and tested potential energy surfaces for these reactions are included within the program as Fortran subroutines.
f4c02fb1c8e87f9e	ijms International Journal of Molecular Sciences Int. J. Mol. Sci. 1422-0067 Molecular Diversity Preservation International (MDPI) 10.3390/ijms11114227 ijms-11-04227 Article The Bondons: The Quantum Particles of the Chemical Bond PutzMihai V.12 Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timişoara, Pestalozzi Street No.16, Timişoara, RO-300115, Romania; E-Mail: mvputz@cbg.uvt.ro or mv_putz@yahoo.com; Tel.: ++40-256-592-633; Fax: ++40-256-592-620; Web: www.mvputz.iqstorm.ro Theoretical Physics Institute, Free University Berlin, Arnimallee 14, 14195 Berlin, Germany 2010 28 10 2010 11 11 4227 4256 23 8 2010 11 10 2010 21 10 2010 © 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. 2010 By employing the combined Bohmian quantum formalism with the U(1) and SU(2) gauge transformations of the non-relativistic wave-function and the relativistic spinor, within the Schrödinger and Dirac quantum pictures of electron motions, the existence of the chemical field is revealed along the associate bondon particle characterized by its mass (m), velocity (v), charge (e), and life-time (t). This is quantized either in ground or excited states of the chemical bond in terms of reduced Planck constant ħ, the bond energy Ebond and length Xbond, respectively. The mass-velocity-charge-time quaternion properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. The bondonic picture was completed by discussing the relativistic charge and life-time (the actual zitterbewegung) problem, i.e., showing that the bondon equals the benchmark electronic charge through moving with almost light velocity. It carries negligible, although non-zero, mass in special bonding conditions and towards observable femtosecond life-time as the bonding length increases in the nanosystems and bonding energy decreases according with the bonding length-energy relationship E bond [ kcal / mol ] × X bond [ A 0 ] = 182019, providing this way the predictive framework in which the particle may be observed. Finally, its role in establishing the virtual states in Raman scattering was also established. de Broglie-Bohm theory Schrödinger equation Dirac equation chemical field gauge/phase symmetry transformation bondonic properties Raman scattering One of the first attempts to systematically use the electron structure as the basis of the chemical bond is due to the discoverer of the electron itself, J.J. Thomson, who published in 1921 an interesting model for describing one of the most puzzling molecules of chemistry, the benzene, by the aid of C–C portioned bonds, each with three electrons [1] that were further separated into 2(σ) + 1(π) lower and higher energy electrons, respectively, in the light of Hückel σ-π and of subsequent quantum theories [2,3]. On the other side, the electronic theory of the valence developed by Lewis in 1916 [4] and expanded by Langmuir in 1919 [5] had mainly treated the electronic behavior like a point-particle that nevertheless embodies considerable chemical information, due to the the semiclassical behavior of the electrons on the valence shells of atoms and molecules. Nevertheless, the consistent quantum theory of the chemical bond was advocated and implemented by the works of Pauling [68] and Heitler and London [9], which gave rise to the wave-function characterization of bonding through the fashioned molecular wave-functions (orbitals)–mainly coming from the superposition principle applied on the atomic wave-functions involved. The success of this approach, especially reported by spectroscopic studies, encouraged further generalization toward treating more and more complex chemical systems by the self-consistent wave-function algorithms developed by Slater [10,11], Hartree-Fock [12], Lowdin [1315], Roothann [16], Pariser, Parr and Pople (in PPP theory) [1719], until the turn towards the density functional theory of Kohn [20,21] and Pople [22,23] in the second half of the XX century, which marked the subtle feed-back to the earlier electronic point-like view by means of the electronic density functionals and localization functions [24,25]. The compromised picture of the chemical bond may be widely comprised by the emerging Bader’s atoms-in-molecule theory [2628], the fuzzy theory of Mezey [2931], along with the chemical reactivity principles [3243] as originating in the Sanderson’s electronegativity [34] and Pearson’s chemical hardness [38] concepts, and their recent density functionals [4446] that eventually characterizes it. Within this modern quantum chemistry picture, its seems that the Dirac dream [47] in characterizing the chemical bond (in particular) and the chemistry (in general) by means of the chemical field related with the Schrödinger wave-function [48] or the Dirac spinor [49] was somehow avoided by collapsing the undulatory quantum concepts into the (observable) electronic density. Here is the paradoxical point: the dispersion of the wave function was replaced by the delocalization of density and the chemical bonding information is still beyond a decisive quantum clarification. Moreover, the quantum theory itself was challenged as to its reliability by the Einstein-Podolski-Rosen(-Bohr) entanglement formulation of quantum phenomena [50,51], qualitatively explained by the Bohm reformulation [52,53] of the de Broglie wave packet [54,55] through the combined de Broglie-Bohm wave-function [56,57] Ψ 0 ( t , x ) = R ( t , x ) exp ( i S ( t , x ) ħ )with the R-amplitude and S-phase action factors given, respectively, as R ( t , x ) = Ψ 0 ( t , x ) 2 = ρ 1 / 2 ( x ) S ( t , x ) = p x E tin terms of electronic density ρ, momentum p, total energy E, and time-space (t, x) coordinates, without spin. On the other side, although many of the relativistic effects were explored by considering them in the self-consistent equation of atomic and molecular structure computation [5862], the recent reloaded thesis of Einstein’s special relativity [63,64] into the algebraic formulation of chemistry [6567], widely asks for a further reformation of the chemical bonding quantum-relativistic vision [68]. In this respect, the present work advocates making these required steps toward assessing the quantum particle of the chemical bond as based on the derived chemical field released at its turn by the fundamental electronic equations of motion either within Bohmian non-relativistic (Schrödinger) or relativistic (Dirac) pictures and to explore the first consequences. If successful, the present endeavor will contribute to celebrate the dream in unifying the quantum and relativistic features of electron at the chemical level, while unveiling the true particle-wave nature of the chemical bond. Method: Identification of Bondons (<italic>B̶</italic>) The search for the bondons follows the algorithm: Considering the de Broglie-Bohm electronic wave-function/spinor Ψ0 formulation of the associated quantum Schrödinger/Dirac equation of motion. Checking for recovering the charge current conservation law ρ t + j = 0that assures for the circulation nature of the electronic fields under study. Recognizing the quantum potential Vqua and its equation, if it eventually appears. Reloading the electronic wave-function/spinor under the augmented U(1) or SU(2) group form Ψ G ( t , x ) = Ψ 0 ( t , x ) exp ( i ħ e c ( t , x ) )with the standard abbreviation e = e 0 2 / 4 π ɛ 0 in terms of the chemical field ℵ considered as the inverse of the fine-structure order: 0 = ħ c e 137.03599976 [ Joule × meter Coulomb ]since upper bounded, in principle, by the atomic number of the ultimate chemical stable element (Z = 137). Although apparently small enough to be neglected in the quantum range, the quantity (6) plays a crucial role for chemical bonding where the energies involved are around the order of 10–19 Joules (electron-volts)! Nevertheless, for establishing the physical significance of such chemical bonding quanta, one can proceed with the chain equivalences energy × distance charge ( charge × potential difference ) × distance charge ( potential difference ) × distancerevealing that the chemical bonding field caries bondons with unit quanta ħc/e along the distance of bonding within the potential gap of stability or by tunneling the potential barrier of encountered bonding attractors. Rewriting the quantum wave-function/spinor equation with the group object ΨG, while separating the terms containing the real and imaginary ℵ chemical field contributions. Identifying the chemical field charge current and term within the actual group transformation context. Establishing the global/local gauge transformations that resemble the de Broglie-Bohm wave-function/spinor ansatz Ψ0 of steps (i)–(iii). Imposing invariant conditions for ΨG wave function on pattern quantum equation respecting the Ψ0 wave-function/spinor action of steps (i)–(iii). Establishing the chemical field ℵ specific equations. Solving the system of chemical field ℵ equations. Assessing the stationary chemical field t t = 0that is the case in chemical bonds at equilibrium (ground state condition) to simplify the quest for the solution of chemical field ℵ. The manifested bondonic chemical field ℵbondon is eventually identified along the bonding distance (or space). Checking the eventual charge flux condition of Bader within the vanishing chemical bonding field [26] = 0 ρ = 0 Employing the Heisenberg time-energy relaxation-saturation relationship through the kinetic energy of electrons in bonding v = 2 T m 2 m ħ t Equate the bondonic chemical bond field with the chemical field quanta (6) to get the bondons’ mass ( m ) = 0 This algorithm will be next unfolded both for non-relativistic as well as for relativistic electronic motion to quest upon the bondonic existence, eventually emphasizing their difference in bondons’ manifestations. Type of Bondons Non-Relativistic Bondons For the non-relativistic quantum motion, we will treat the above steps (i)–(iii) at once. As such, when considering the de Broglie-Bohm electronic wavefunction into the Schrödinger Equation [48] i ħ t Ψ 0 = ħ 2 2 m 2 Ψ 0 + V Ψ 0it separates into the real and imaginary components as [52,53,68] t R 2 + ( R 2 m S ) = 0 t S ħ 2 2 m 1 R 2 R + 1 2 m ( S ) 2 + V = 0While recognizing into the first Equation (13a), the charge current conservation law with Equation (2) along the identification j S = R 2 m Sthe second equation helps in detecting the quantum (or Bohm) potential V qua = ħ 2 2 m 2 R Rcontributing to the total energy E = T + V + V quaonce the momentum-energy correspondences 1 2 m ( S ) 2 = p 2 2 m = T t S = Eare engaged. Next, when employing the associate U(1) gauge wavefunction of Equation (5) type, its partial derivative terms look like Ψ G = [ R + i ħ R ( S + e c ) ] exp [ i ħ ( S + e c ) ] 2 Ψ G = { 2 R + 2 i ħ R ( S + e c ) + i ħ R ( 2 S + e c 2 ) R ħ 2 [ ( S ) 2 + ( e c ) 2 ] 2 e ħ 2 c R S } exp [ i ħ ( S + e c ) ] t Ψ G = [ t R + i ħ R ( t S + e c t ) ] exp [ i ħ ( S + e c ) ] Now the Schrödinger Equation (12) for ΨG in the form of (5) is decomposed into imaginary and real parts t R = 1 m ( R S + R 2 2 S ) + e mc ( R + R 2 2 ) R t S R e c t = ħ 2 2 m 2 R + R 2 m [ ( S ) 2 + ( e c ) 2 ] + e mc R S + V Rthat can be rearranged t R 2 = 1 m ( R 2 S ) + e mc ( R 2 ) ( t S + e c t ) = ħ 2 2 m 1 R 2 R + 1 2 m [ ( S ) 2 + ( e c ) 2 ] + e mc S . + Vto reveal some interesting features of chemical bonding. Firstly, through comparing the Equation (20a) with the charge conserved current equation form (4) from the general chemical field algorithm–the step (ii), the conserving charge current takes now the expanded expression: j U ( 1 ) = R 2 m ( S + e c ) = j S + j suggesting that the additional current is responsible for the chemical field to be activated, namely j = e mc R 2 which vanishes when the global gauge condition is considered = 0 Therefore, in order that the chemical bonding is created, the local gauge transformation should be used that exists under the condition 0 In this framework, the chemical field current j⃗ carries specific bonding particles that can be appropriately called bondons, closely related with electrons, in fact with those electrons involved in bonding, either as single, lone pair or delocalized, and having an oriented direction of movement, with an action depending on the chemical field itself ℵ. Nevertheless, another important idea abstracted from the above results is that in the search for the chemical field ℵ no global gauge condition is required. It is also worth noting that the presence of the chemical field does not change the Bohm quantum potential that is recovered untouched in (20b), thus preserving the entanglement character of interaction. With these observations, it follows that in order for the de Broglie-Bohm-Schrödinger formalism to be invariant under the U(1) transformation (5), a couple of gauge conditions have to be fulfilled by the chemical field in Equations (20a) and (20b), namely e mc x ( R 2 ) = 0 e c t + 1 2 m ( e c ) 2 + e mc S . = 0 Next, the chemical field ℵ is to be expressed through combining its spatial-temporal information contained in Equations (25). From the first condition (25a) one finds that = R 2 2 R ı ı where the vectorial feature of the chemical field gradient was emphasized on the direction of its associated charge current fixed by the versor i⃗ (i.e., by the unitary vector associate with the propagation direction, i⃗2=1). We will apply such writing whenever necessary for avoiding scalar to vector ratios and preserving the physical sense of the whole construction as well. Replacing the gradient of the chemical field (26) into its temporal Equation (25b) one gets the unified chemical field motion description e 8 mc R 2 ( R ) 2 ( 2 ) 2 R 2 m S S R S ( 2 ) + t = 0that can be further rewritten as e 2 mc ρ ( ρ ) 2 ( 2 ) 2 ρ ν ı ρ ı ( 2 ) + t = 0since calling the relations abstracted from Equations (2) and (3) R = ρ 1 / 2 ; S = p { R = 1 2 ρ ρ 1 / 2 ; ( R ) 2 = 1 4 ( ρ ) 2 ρ S S R S = 2 ρ 1 / 2 p ı ρ ı The (quadratic undulatory) chemical field Equation (28) can be firstly solved for the Laplacian general solutions ( 2 ) 1 , 2 = ρ ν ı ρ ı ± ρ 2 ν 2 ( ρ ) 2 2 e mc ρ 2 ( ρ ) 2 t e mc ρ 2 ( ρ ) 2that give special propagation equations for the chemical field since linking the spatial Laplacian with temporal evolution of the chemical field (∂tℵ)1/2; however, they may be considerably simplified when assuming the stationary chemical field condition (8), the step (xi) in the bondons’ algorithm, providing the working equation for the stationary bondonic field 2 = 2 mc e ν ρ ρ Equation (31) may be further integrated between two bonding attractors, say XA,XB, to primarily give = 2 mc e ν X A X B ρ ı ρ d x = mc e ν [ X A X B ρ ı ρ d x X B X A ρ ı ρ d x ]from where the generic bondonic chemical field is manifested with the form = mc e ν X bond ( X A X B ρ ı ρ d x ) The expression (33) has two important consequences. Firstly, it recovers the Bader zero flux condition for defining the basins of bonding [26] that in the present case is represented by the zero chemical boning fields, namely = 0 ρ ı = 0 Secondly, it furnishes the bondonic (chemical field) analytical expression = mc e ν X bondwithin the natural framework in which X B X A = X bond ρ ı ρ X bondi.e., when one has X A X B ρ ı ρ d x = 1 The step (xiv) of the bondonic algorithm may be now immediately implemented through inserting the Equation (10) into Equation (35) yielding the simple chemical field form = c ħ e 2 m ħ t X bond Finally, through applying the expression (11) of the bondonic algorithm–the step (xv) upon the result (37) with quanta (6) the mass of bondons carried by the chemical field on a given distance is obtained m = ħ t 2 1 X bond 2 Note that the bondons’ mass (38) directly depends on the time the chemical information “travels” from one bonding attractor to the other involved in bonding, while fast decreasing as the bonding distance increases. This phenomenological behavior has to be in the sequel cross-checked by considering the generalized relativistic version of electronic motion by means of the Dirac equation, Further quantitative consideration will be discussed afterwards. Relativistic Bondons In treating the quantum relativistic electronic behavior, the consecrated starting point stays the Dirac equation for the scalar real valued potential w that can be seen as a general function of (tc,x⃗) dependency [49] i ħ t Ψ 0 = [ i ħ c k = 1 3 α ^ k k + β ^ m c 2 + β ^ w ] Ψ 0with the spatial coordinate derivative notation ∂k ≡ ∂/∂xk and the special operators assuming the Dirac 4D representation α ^ k = [ 0 σ ^ k σ ^ k 0 ] , β ^ = [ 1 ^ 0 0 1 ^ ]in terms of bi-dimensional Pauli and unitary matrices σ ^ 1 = [ 0 1 1 0 ] , σ ^ 2 = [ 0 i i 0 ] , σ ^ 3 = [ 1 0 0 1 ] , 1 ^ σ ^ 0 = [ 1 0 0 1 ] Written within the de Broglie-Bohm framework, the spinor solution of Equation (39) looks like Ψ 0 = 1 2 R ( t , x ) [ ϕ φ ] = 1 2 R ( t , x ) [ exp { i ħ [ S ( t , x ) + s ] } exp { i ħ [ S ( t , x ) + s ] } ] , s = ± 1 2that from the beginning satisfies the necessary electronic density condition Ψ 0 * Ψ 0 = R * R = ρ Going on, aiming for the separation of the Dirac Equation (39) into its real/imaginary spinorial contributions, one firstly calculates the terms t Ψ 0 = 1 2 t R [ φ ϕ ] + 1 2 R i ħ t S [ φ ϕ ] t Ψ 0 = 1 2 k R [ φ ϕ ] + 1 2 R i ħ k S [ φ ϕ ] k = 1 3 α ^ k k Ψ 0 = 1 2 k = 1 3 k R [ 0 σ ^ k σ ^ k 0 ] [ φ ϕ ] + 1 2 R i ħ k = 1 3 k S [ 0 σ ^ k σ ^ k 0 ] [ φ ϕ ] = 1 2 [ ϕ k ( k R ) σ ^ k φ k ( k R ) σ ^ k ] + 1 2 R i ħ [ ϕ k ( k S ) σ ^ k φ k ( k S ) σ ^ k ] β ^ m c 2 Ψ 0 = m c 2 2 R [ 1 ^ 0 0 1 ^ ] [ φ ϕ ] = m c 2 2 R [ φ ϕ ] β ^ w Ψ 0 = w 2 R [ φ ϕ ]to be then combined in (39) producing the actual de Broglie-Bohm-Dirac spinorial Equation [ i ħ φ t R R φ t S i ħ ϕ t R + R ϕ t S ] = [ i ħ c ϕ k ( k R ) σ ^ k R c ϕ k ( k S ) σ ^ k + ( m c 2 + w ) R φ i ħ c φ k ( k R ) σ ^ k + R c φ k ( k S ) σ ^ k ( m c 2 + w ) R ϕ ] When equating the imaginary parts of (44) one yields the system { φ t R + ϕ c k ( k R ) σ ^ k = 0 φ c k ( k R ) σ ^ k + ϕ t R = 0that has non-trivial spinorial solutions only by canceling the associate determinant, i.e., by forming the Equation ( t R ) 2 = c 2 [ k ( k R ) σ ^ k ] 2of which the minus sign of the squared root corresponds with the electronic conservation charge, while the positive sign is specific to the relativistic treatment of the positron motion. For proofing this, the specific relationship for the electronic charge conservation (4) may be unfolded by adapting it to the present Bohmian spinorial case by the chain equivalences 0 = t ρ + j = t ( R 2 ) + k k j k = 2 R t R + k k ( c Ψ 0 * α ^ k Ψ 0 ) = 2 R t R + c 2 k k R * R [ e i ħ ( S + s ) e i ħ ( S + s ) ] [ 0 σ ^ k σ ^ k 0 ] [ e i ħ ( S + s ) e i ħ ( S + s ) ] = 2 R t R + c 2 k σ ^ k ( φ 2 1 + ϕ 2 1 ) k R 2 = 2 R t R + 2 Rc k σ ^ k ( k R ) The result t R = c k σ ^ k ( k R )indeed corresponds with the squaring root of (46) with the minus sign, certifying, therefore, the validity of the present approach, i.e., being in accordance with the step (ii) in bondonic algorithm of Section 2. Next, let us see what information is conveyed by the real part of Bohmian decomposed spinors of Dirac Equation (44); the system (48) is obtained { φ ( t S + m c 2 + w ) ϕ c k ( k S ) σ ^ k = 0 φ c k ( k S ) σ ^ k ( t S + m c 2 + w ) ϕ = 0that, as was previously the case with the imaginary counterpart (45), has no trivial spinors solutions only if the associate determinant vanishes, which gives the Equation c 2 [ k ( k S ) σ ^ k ] 2 = ( t S + m c 2 + w ) 2 Now, considering the Bohmian momentum-energy (17) equivalences, the Equation (49) further becomes c 2 [ k p k σ ^ k ] 2 = ( E + m c 2 + w ) 2 c 2 ( p σ ^ ) 2 = ( E + m c 2 + w ) 2 c 2 p 2 = ( E + m c 2 + w ) 2from where, while retaining the minus sign through the square rooting (as prescribed above by the imaginary spinorial treatment in relation with charge conservation), one recovers the relativistic electronic energy-momentum conservation relationship E = c p + m c 2 + wthus confirming in full the reliability of the Bohmian approach over the relativistic spinors. Moreover, the present Bohmian treatment of the relativistic motion is remarkable in that, except in the non-relativistic case, it does not produces the additional quantum (Bohm) potential (15)–responsible for entangled phenomena or hidden variables. This may be justified because within the Dirac treatment of the electron the entanglement phenomenology is somehow included throughout the Dirac Sea and the positron existence. Another important difference with respect to the Schrödinger picture is that the spinor equations that underlie the total charge and energy conservation do not mix the amplitude (2) with the phase (3) of the de Broglie-Bohm wave-function, whereas they govern now, in an independent manner, the flux and the energy of electronic motion. For these reasons, it seems that the relativistic Bohmian picture offers the natural environment in which the chemical field and associate bondons particles may be treated without involving additional physics. Let us see, therefore, whether the Dirac-Bohmian framework will reveal (or not) new insight in the bondon (Schrödinger) reality. This will be done by reconsidering the working Bohmian spinor (41) as transformed by the internal gauge symmetry SU(2) driven by the chemical field ℵ related phase–in accordance with Equation (5) of the step (iv) of bondonic algorithm Ψ G ( t , x ) = Ψ 0 ( t , x ) exp ( i ħ e c ( t , x ) ) = 1 2 R ( t , x ) [ φ G ϕ G ] = 1 2 R ( t , x ) [ exp { i ħ [ s ( t , x ) + e c ( t , x ) + s ] } exp { i ħ [ s ( t , x ) + e c ( t , x ) + s ] } ] Here it is immediate that expression (52) still preserves the electronic density formulation (2) as was previously the case with the gaugeless field (41) Ψ G * Ψ G = R * R = ρ However, when employed for the Dirac equation terms, the field (52) modifies the previous expressions (43a)–(43c) as follows t Ψ G = 1 2 t R [ φ G ϕ G ] + 1 2 R i ħ ( t S + e c t ) [ φ G ϕ G ] k Ψ 0 = 1 2 k R [ φ G ϕ G ] + 1 2 R i ħ ( k S + e c k ) [ φ G ϕ G ] k = 1 3 α ^ k k Ψ G = 1 2 k ( k R ) σ ^ k [ ϕ G φ G ] + 1 2 R i ħ k ( k S + e c k ) σ ^ k [ ϕ G φ G ]while producing the gauge spinorial Equation [ i ħ φ G t R R φ G ( t S + e c t ) i ħ ϕ G t R + R ϕ G ( t S + e c t ) ] = [ i ħ c ϕ G k ( k R ) σ ^ k R c ϕ G k ( k S + e c k ) σ ^ k + ( m c 2 + w ) R φ G i ħ c φ G k ( k R ) σ ^ k R c φ G k ( k S + e c k ) σ ^ k + ( m c 2 + w ) R ϕ G ] Now it is clear that since the imaginary part in (55) was not at all changed with respect to Equation (44) by the chemical field presence, the total charge conservation (4) is naturally preserved; instead the real part is modified, respecting the case (44), in the presence of the chemical field (by internal gauge symmetry). Nevertheless, in order that chemical field rotation does not produce modification in the total energy conservation, it imposes that the gauge spinorial system of the chemical field must be as { φ G t ϕ G c k ( k ) σ ^ k = 0 φ G c k ( k ) σ ^ k ϕ G t = 0 According to the already custom procedure, for the system (56) having no trivial gauge spinorial solution, the associated vanishing determinant is necessary, which brings to light the chemical field Equation c 2 [ k ( k ) σ ^ k ] 2 = ( t ) 2equivalently rewritten as c 2 [ σ ^ ] 2 = ( t ) 2that simply reduces to c 2 ( ) 2 = ( t ) 2through considering the Pauling matrices (40b) unitary feature upon squaring. At this point, one has to decide upon the sign of the square root of (57c); this was previously clarified to be minus for electronic and plus for positronic motions. Therefore, the electronic chemical bond is modeled by the resulting chemical field equation projected on the bonding length direction X bond = 1 c t The Equation (58) is of undulatory kind with the chemical field solution having the general plane wave form = ħ c e exp [ i ( k X bond ω t ) ]that agrees with both the general definition of the chemical field (6) as well as with the relativistic “traveling” of the bonding information. In fact, this is the paradox of the Dirac approach of the chemical bond: it aims to deal with electrons in bonding while they have to transmit the chemical bonding information—as waves—propagating with the light velocity between the bonding attractors. This is another argument for the need of bondons reality as a specific existence of electrons in chemical bond is compulsory so that such a paradox can be solved. Note that within the Dirac approach, the Bader flux condition (9) is no more related to the chemical field, being included in the total conservation of charge; this is again natural, since in the relativistic case the chemical field is explicitly propagating with a percentage of light velocity (see the Discussion in Section 4 below) so that it cannot drive the (stationary) electronic frontiers of bonding. Further on, when rewriting the chemical field of bonding (59) within the de Broglie and Planck consecrated corpuscular-undulatory quantifications ( t , X bond ) = ħ c e exp [ i ħ ( p X bond E t ) ]it may be further combined with the unitary quanta form (6) in the Equation (11) of the step (xv) in the bondonic algorithm to produce the phase condition 1 = exp [ i ħ ( p X bond E t ) ]that implies the quantification p X bond E t = 2 π n ħ , n N By the subsequent employment of the Heisenberg time-energy saturated indeterminacy at the level of kinetic energy abstracted from the total energy (to focus on the motion of the bondonic plane waves) E = ħ t p = m v = 2 m T 2 m ħ tthe bondon Equation (62) becomes X bond 2 m ħ t = ( 2 π n + 1 ) ħthat when solved for the bondonic mass yields the expression m = ħ t 2 1 X bond 2 ( 2 π n + 1 ) 2 , n = 0 , 1 , 2 which appears to correct the previous non-relativistic expression (38) with the full quantification. However, the Schrödinger bondon mass of Equation (38) is recovered from the Dirac bondonic mass (65) in the ground state, i.e., by setting n = 0. Therefore, the Dirac picture assures the complete characterization of the chemical bond through revealing the bondonic existence by the internal chemical field symmetry with the quantification of mass either in ground or in excited states (n ≤ 0, nN). Moreover, as always happens when dealing with the Dirac equation, the positronic bondonic mass may be immediately derived as well, for the case of the chemical bonding is considered also in the anti-particle world; it emerges from reloading the square root of the Dirac chemical field Equation (57c) with a plus sign that will be propagated in all the subsequent considerations, e.g., with the positronic incoming plane wave replacing the departed electronic one of (59), until delivering the positronic bondonic mass m = ħ t 2 1 X bond 2 ( 2 π n 1 ) 2 , n = 0 , 1 , 2 It nevertheless differs from the electronic bondonic mass (65) only in the excited spectrum, while both collapse in the non-relativistic bondonic mass (38) for the ground state of the chemical bond. Remarkably, for both the electronic and positronic cases, the associated bondons in the excited states display heavier mass than those specific to the ground state, a behavior once more confirming that the bondons encompass all the bonding information, i.e., have the excitation energy converted in the mass-added-value in full agreement with the mass-energy relativistic custom Einstein equivalence [64]. Let us analyze the consequences of the bondon’s existence, starting from its mass (38) formulation on the ground state of the chemical bond. At one extreme, when considering atomic parameters in bonding, i.e., when assuming the bonding distance of the Bohr radius size a0 = 0.52917 · 10−10[m]SI the corresponding binding time would be given as tt0 = a0/v0 = 2.41889 · 10−17[s]SI while the involved bondonic mass will be half of the electronic one m0/2, to assure fast bonding information. Of course, this is not a realistic binding situation; for that, let us check the hypothetical case in which the electronic m0 mass is combined, within the bondonic formulation (38), into the bond distance X bond = ħ t / 2 m 0 resulting in it completing the binding phenomenon in the femtosecond time tbonding ∼ 10−12[s]SI for the custom nanometric distance of bonding Xbonding ∼ 10−9[m]SI. Still, when both the femtosecond and nanometer time-space scale of bonding is assumed in (38), the bondonic mass is provided in the range of electronic mass m ∼ 10−31[kg]SI although not necessarily with the exact value for electron mass nor having the same value for each bonding case considered. Further insight into the time existence of the bondons will be reloaded for molecular systems below after discussing related specific properties as the bondonic velocity and charge. For enlightenment on the last perspective, let us rewrite the bondonic mass (65) within the spatial-energetic frame of bonding, i.e., through replacing the time with the associated Heisenberg energy, tbondingħ/Ebond, thus delivering another working expression for the bondonic mass m = ħ 2 2 ( 2 π n + 1 ) 2 E bond X bond 2 , n = 0 , 1 , 2 that is more practical than the traditional characterization of bonding types in terms of length and energy of bonding; it may further assume the numerical ground state ratio form ζ m = m m 0 = 87.8603 ( E bond [ kcal / mol ] ) ( X bond [ A 0 ] ) 2when the available bonding energy and length are considered (as is the custom for chemical information) in kcal/mol and Angstrom, respectively. Note that having the bondon’s mass in terms of bond energy implies the inclusion of the electronic pairing effect in the bondonic existence, without the constraint that the bonding pair may accumulate in the internuclear region [69]. Moreover, since the bondonic mass general formulation (65) resulted within the relativistic treatment of electron, it is considering also the companion velocity of the bondonic mass that is reached in propagating the bonding information between the bonding attractors. As such, when the Einstein type relationship [70] m v 2 2 = h υis employed for the relativistic bondonic velocity-mass relationship [63,64] m = m 1 v 2 c 2and for the frequency of the associate bond wave υ = v X bondit provides the quantified searched bondon to light velocity ratio v c = 1 1 + 1 64 π 2 ħ 2 c 2 ( 2 π n + 1 ) 4 E bond 2 X bond 2 , n = 0 , 1 , 2 or numerically in the bonding ground state as ζ v = v c = 100 1 + 3.27817 × 10 6 ( E bond [ kcal / mol ] ) 2 ( X bond [ A 0 ] ) 2 [ % ] Next, dealing with a new matter particle, one will be interested also on its charge, respecting the benchmarking charge of an electron. To this end, one re-employs the step (xv) of bondonic algorithm, Equation (11), in the form emphasizing the bondonic charge appearance, namely ( e ) = 0Next, when considering for the left-hand side of (74), the form provided by Equation (35), and for the right-hand side of (74), the fundamental hyperfine value of Equation (6), one gets the working Equation c m v e X bond = 137.036 [ Joule × meter Coulomb ]from where the bondonic charge appears immediately, once the associate expressions for mass and velocity are considered from Equations (67) and (72), respectively, yielding the quantified form e = 4 π ħ c 137.036 1 1 + 64 π 2 E bond 2 X bond 2 ħ 2 c 2 ( 2 π n + 1 ) 4 , n = 0 , 1 , 2 However, even for the ground state, and more so for the excited states, one may see that when forming the practical ratio respecting the unitary electric charge from (76), it actually approaches a referential value, namely ζ e = e e = 4 π 1 + ( E bond [ kcal / mol ] ) 2 ( X bond [ A 0 ] ) 2 3.27817 × 10 6 ( 2 π n + 1 ) 4 4 πfor, in principle, any common energy and length of chemical bonding. On the other side, for the bondons to have different masses and velocities (kinetic energy) as associated with specific bonding energy but an invariant (universal) charge seems a bit paradoxical. Moreover, it appears that with Equation (77) the predicted charge of a bonding, even in small molecules such as H2, considerably surpasses the available charge in the system, although this may be eventually explained by the continuous matter-antimatter balance in the Dirac Sea to which the present approach belongs. However, to circumvent such problems, one may further use the result (77) and map it into the Poisson type charge field Equation e 4 π × e 2 V 4 π × ρfrom where the bondonic charge may be reshaped by appropriate dimensional scaling in terms of the bounding parameters (Ebond and Xbond) successively as e 1 4 π [ X 2 V ] X = X bond 1 4 E bond X bond 0Now, Equation (79) may be employed towards the working ratio between the bondonic and electronic charges in the ground state of bonding ζ e = e e 1 32 π ( E bond [ k c a l / mol ] ) ( X bond [ A 0 ] ) 3.27817 × 10 3 With Equation (80) the situation is reversed compared with the previous paradoxical situation, in the sense that now, for most chemical bonds (of Table 1, for instance), the resulted bondonic charge is small enough to be not yet observed or considered as belonging to the bonding wave spreading among the binding electrons. Instead, aiming to explore the specific information of bonding reflected by the bondonic mass and velocity, the associated ratios of Equations (68) and (73) for some typical chemical bonds [71,72] are computed in Table 1. They may be eventually accompanied by the predicted life-time of corresponding bondons, obtained from the bondonic mass and velocity working expressions (68) and (73), respectively, throughout the basic time-energy Heisenberg relationship—here restrained at the level of kinetic energy only for the bondonic particle; this way one yields the successive analytical forms t = ħ T = 2 ħ m v 2 = 2 ħ ( m 0 ζ m ) ( c ζ v 10 2 ) 2 = ħ m 0 c 2 2 10 4 ζ m ζ v 2 = 0.0257618 ζ m ζ v 2 × 10 15 [ s ] S Iand the specific values for various bonding types that are displayed in Table 1. Note that defining the bondonic life-time by Equation (81) is the most adequate, since it involves the basic bondonic (particle!) information, mass and velocity; instead, when directly evaluating the bondonic life-time by only the bonding energy one deals with the working formula t bond = ħ E bond = 1.51787 E bond [ kcal / mol ] × 10 14 [ s ] S Ithat usually produces at least one order lower values than those reported in Table 1 upon employing the more complex Equation (81). This is nevertheless reasonable, because in the last case no particle information was considered, so that the Equation (82) gives the time of the associate wave representation of bonding; this departs by the case when the time is computed by Equation (81) where the information of bonding is contained within the particle (bondonic) mass and velocity, thus predicting longer life-times, and consequently a more susceptible timescale in allowing the bondonic observation. Therefore, as far as the chemical bonding is modeled by associate bondonic particle, the specific time of Equation (81) rather than that of Equation (82) should be considered. While analyzing the values in Table 1, it is generally observed that as the bondonic mass is large as its velocity and the electric charge lower in their ratios, respecting the light velocity and electronic benchmark charge, respectively, however with some irregularities that allows further discrimination in the sub-bonding types. Yet, the life-time tendency records further irregularities, due to its complex and reversed bondonic mass-velocity dependency of Equation (81), and will be given a special role in bondonic observation—see the Table 2 discussion below. Nevertheless, in all cases, the bondonic velocity is a considerable (non-negligible) percent of the photonic velocity, confirming therefore its combined quantum-relativistic nature. This explains why the bondonic reality appears even in the non-relativistic case of the Schrödinger equation when augmented with Bohmian entangled motion through the hidden quantum interaction. Going now to particular cases of chemical bonding in Table 1, the hydrogen molecule maintains its special behavior through providing the bondonic mass as slightly more than double of the only two electrons contained in the whole system. This is not a paradox, but a confirmation of the fact the bondonic reality is not just the sum or partition of the available valence atomic electrons in molecular bonds, but a distinct (although related) existence that fully involves the undulatory nature of the electronic and nuclear motions in producing the chemical field. Remember the chemical field was associated either in Schrödinger as well in Dirac pictures with the internal rotations of the (Bohmian) wave function or spinors, being thus merely a phase property—thus inherently of undulatory nature. It is therefore natural that the risen bondons in bonding preserve the wave nature of the chemical field traveling the bond length distance with a significant percent of light. Moreover, the bondonic mass value may determine the kind of chemical bond created, in this line the H2 being the most covalent binding considered in Table 1 since it is most closely situated to the electronic pairing at the mass level. The excess in H2 bond mass with respect to the two electrons in isolated H atoms comes from the nuclear motion energy converted (relativistic) and added to the two-sided electronic masses, while the heavier resulted mass of the bondon is responsible for the stabilization of the formed molecule respecting the separated atoms. The H2 bondon seems to be also among the less circulated ones (along the bondon of the F2 molecule) in bonding traveled information due to the low velocity and charge record—offering therefore another criterion of covalency, i.e., associated with better localization of the bonding space. The same happens with the C–C bonding, which is predicted to be more covalent for its simple (single) bondon that moves with the smallest velocityv<<) or fraction of the light velocity from all C–C types of bonding; in this case also the bondonic highest massm>>), smallest chargee<<), and highest (observed) life-time (t>>) criteria seem to work well. Other bonds with high covalent character, according with the bondonic velocity criterion only, are present in N≡N and the C=O bonding types and less in the O=O and C–O ones. Instead, one may establish the criteria for multiple (double and triple) bonds as having the series of current bondonic properties as: {ςm <, ςv >, ςe >, t <} However, the diamond C–C bondon, although with the smallest recorded mass (ςm <<), is characterized by the highest velocity (ςv >) and charge (ςe >) in the CC series (and also among all cases of Table 1). This is an indication that the bond is very much delocalized, thus recognizing the solid state or metallic crystallized structure for this kind of bond in which the electronic pairings (the bondons) are distributed over all atomic centers in the unit cell. It is, therefore, a special case of bonding that widely informs us on the existence of conduction bands in a solid; therefore the metallic character generally associated with the bondonic series of properties {ςm <<, ςv >, ςe >, t<}, thus having similar trends with the corresponding properties of multiple bonds, with the only particularity in the lower mass behavior displayed—due to the higher delocalization behavior for the associate bondons. Very interestingly, the series of C–H, N–H, and O–H bonds behave similarly among them since displaying a shrink and medium range of mass (moderate high), velocity, charge and life-time (moderate high) variations for their bondons, {ςm ∼ >, ςv ∼, ςe ∼, t ∼>}; this may explain why these bonds are the most preferred ones in DNA and genomic construction of proteins, being however situated towards the ionic character of chemical bond by the lower bondonic velocities computed; they have also the most close bondonic mass to unity; this feature being due to the manifested polarizability and inter-molecular effects that allows the 3D proteomic and specific interactions taking place. Instead, along the series of halogen molecules F2, Cl2, and I2, only the observed life-time of bondons show high and somehow similar values, while from the point of view of velocity and charge realms only the last two bonding types display compatible properties, both with drastic difference for their bondonic mass respecting the F–F bond—probably due the most negative character of the fluorine atoms. Nevertheless, judging upon the higher life-time with respect to the other types of bonding, the classification may be decided in the favor of covalent behavior. At this point, one notes traces of covalent bonding nature also in the case of the rest of halogen-carbon binding (C–Cl, C–Br, and C–I in Table 1) from the bondonic life-time perspective, while displaying also the ionic manifestation through the velocity and charge criteria {ςv ∼, ςe ∼} and even a bit of metal character by the aid of small bondonic mass (ςm <). All these mixed features may be because of the joint existence of both inner electronic shells that participate by electronic induction in bonding as well as electronegativity difference potential. Remarkably, the present results are in accordance with the recent signalized new binding class between the electronic pairs, somehow different from the ionic and covalent traditional ones in the sense that it is seen as a kind of resonance, as it appears in the molecular systems like F2, O2, N2 (with impact in environmental chemistry) or in polar compounds like C–F (specific to ecotoxicology) or in the reactions that imply a competition between the exchange in the hydrogen or halogen (e.g., HF). The valence explanation relied on the possibility of higher orders of orbitals’ existing when additional shells of atomic orbitals are involved such as <f> orbitals reaching this way the charge-shift bonding concept [73]; the present bondonic treatment of chemical bonds overcomes the charge shift paradoxes by the relativistic nature of the bondon particles of bonding that have as inherent nature the time-space or the energy-space spanning towards electronic pairing stabilization between centers of bonding or atomic adducts in molecules. However, we can also made predictions regarding the values of bonding energy and length required for a bondon to acquire either the unity of electronic charge or its mass (with the consequence in its velocity fraction from the light velocity) on the ground state, by setting Equations (68) and (80) to unity, respectively. These predictions are summarized in Table 2. From Table 2, one note is that the situation of the bondon having the same charge as the electron is quite improbable, at least for the common chemical bonds, since in such a case it will feature almost the light velocity (and almost no mass–that is, however, continuously decreasing as the bonding energy decreases and the bonding length increases). This is natural since a longer distance has to be spanned by lower binding energy yet carrying the same unit charge of electron while it is transmitted with the same relativistic velocity! Such behavior may be regarded as the present zitterbewegung (trembling in motion) phenomena, here at the bondonic level. However one records the systematic increasing of bondonic life-time towards being observable in the femtosecond regime for increasing bond length and decreasing the bonding energy–under the condition the chemical bonding itself still exists for certain {Xbond, Ebond} combinations. On the other side, the situation in which the bondon will weigh as much as one electron is a current one (see the Table 1); nevertheless, it is accompanied by quite reasonable chemical bonding length and energy information that it can carried at a low fraction of the light velocity, however with very low charge as well. Nevertheless, the discovered bonding energy-length relationship from Table 2, based on Equation (80), namely E bond [ kcal / mol ] × X bond [ A 0 ] = 182019should be used in setting appropriate experimental conditions in which the bondon particle may be observed as carrying the unit electronic charge yet with almost zero mass. In this way, the bondon is affirmed as a special particle of Nature, that when behaving like an electron in charge it is behaving like a photon in velocity and like neutrino in mass, while having an observable (at least as femtosecond) lifetime for nanosystems having chemical bonding in the range of hundred of Angstroms and thousands of kcal/mol! Such a peculiar nature of a bondon as the quantum particle of chemical bonding, the central theme of Chemistry, is not as surprising when noting that Chemistry seems to need both a particle view (such as offered by relativity) and a wave view (such as quantum mechanics offers), although nowadays these two physics theories are not yet fully compatible with each other, or even each fully coherent internally. Maybe the concept of ‘bondons’ will help to improve the situation for all concerned by its further conceptual applications. Finally, just to give a conceptual glimpse of how the present bondonic approach may be employed, the scattering phenomena are considered within its Raman realization, viewed as a sort of generalized Compton scattering process, i.e., extracting the structural information from various systems (atoms, molecules, crystals, etc.) by modeling the inelastic interaction between an incident IR photon and a quantum system (here the bondons of chemical bonds in molecules), leaving a scattered wave with different frequency and the resulting system in its final state [74]. Quantitatively, one firstly considers the interaction Hamiltonian as being composed by two parts, H ( 1 ) = e m j [ p j A ( r j , t ) ] H ( 2 ) = e 2 2 m j A 2 ( r j , t )accounting for the linear and quadratic dependence of the light field potential vector A⃗(r⃗j, t) acting on the bondons “j”, carrying the kinetic moment pB̶j = mv, charge e and mass mB̶. Then, noting that, while considering the quantified incident (q⃗0, υ0) and scattered (q⃗, υ) light beams, the interactions driven by H(1) and H(2) model the changing in one- and two- occupation numbers of photonic trains, respectively. In this context, the transition probability between the initial \|i 〉 and final \|f 〉 bondonic states writes by squaring the sum of all scattering quantum probabilities that include absorption (A, with nA number of photons) and emission (E, with nE number of photons) of scattered light on bondons, see Figure 1. Analytically, one has the initial-to-final total transition probability [75]dependence here given as d 2 Π f i 1 ħ \| π f i \| 2 δ ( E \| i + h υ 0 E \| f h υ ) υ 2 d υ d Ω = 1 ħ \| f ; n A 1 , n E + 1 \| H ( 2 ) \| n A , n E ; i + v f ; n A 1 , n E + 1 \| H ( 1 ) \| n A 1 , n E ; v v ; n A 1 , n E \| H ( 1 ) \| n A , n E ; i E \| i E \| v + h υ 0 + v f ; n A 1 , n E + 1 \| H ( 1 ) \| n A , n E + 1 ; v v ; n A , n E + 1 \| H ( 1 ) \| n A , n E ; i E \| i E \| v h υ \| 2 × δ ( E \| i + h υ 0 E \| f h υ ) υ 2 d υ d Ω At this point, the conceptual challenge appears to explore the existence of the Raman process itself from the bondonic description of the chemical bond that turns the incoming IR photon into the (induced, stimulated, or spontaneous) structural frequencies υ v i = E \| i E \| v hAs such, the problem may be reshaped in expressing the virtual state energy E\|B̶v in terms of bonding energy associated with the initial state E \| i = E bondthat can be eventually measured or computationally predicted by other means. However, this further implies the necessity of expressing the incident IR photon with the aid of bondonic quantification; to this end the Einstein relation (69) is appropriately reloaded in the form h υ v i = m v 2 2 = 1 4 v 2 ħ 2 E bond X bond 2 ( 2 π n v + 1 ) 2where the bondonic mass (67) was firstly implemented. Next, in terms of representing the turn of the incoming IR photon into the structural wave-frequency related with the bonding energy of initial state, see Equation (88); the time of wave-bond (82) is here considered to further transform Equation (89) to the yield h υ v i = 1 4 v 2 E bond 2 t bond 2 E bond X bond 2 ( 2 π n v + 1 ) 2 = 1 4 E bond v 2 v bond 2 ( 2 π n v + 1 ) 2where also the corresponding wave-bond velocity was introduced v bond = X bond t bond = 1 ħ E bond X bondIt is worth noting that, as previously was the case with the dichotomy between bonding and bondonic times, sees Equations (81)vs. (82), respectively, the bonding velocity of Equation (91) clearly differs by the bondonic velocity of Equation (72) since the actual working expression v bond c = ( E bond [ kcal / mol ] ) ( X bond [ A 0 ] ) 2.19758 × 10 3 [ % ]provides considerably lower values than those listed in Table 1–again, due to missing the inclusion of the particle mass’ information, unlike is the case for the bondonic velocity. Returning to the bondonic description of the Raman scattering, one replaces the virtual photonic frequency of Equation (90) together with Equation (88) back in the Bohr-type Equation (87) to yield the searched quantified form of virtual bondonic energies in Equation (86) and Figure 1, analytically E \| v = E bond [ 1 1 4 v 2 v bond 2 ( 2 π n v + 1 ) 2 ] = E bond [ 1 16 π 2 ( 2 π n v + 1 ) 2 64 π 2 E bond 2 X bond 2 ħ 2 c 2 + ( 2 π n v + 1 ) 4 ]or numerically E \| v = E bond [ 1 16 π 2 ( 2 π n v + 1 ) 2 0.305048 × 10 6 × ( E bond [ kcal / mol ] ) 2 × ( X bond [ A 0 ] ) 2 + ( 2 π n v + 1 ) 4 ] , n v = 0 , 1 , 2 Remarkably, the bondonic quantification (94) of the virtual states of Raman scattering varies from negative to positive energies as one moves from the ground state to more and more excited states of initial bonding state approached by the incident IR towards virtual ones, as may be easily verified by considering particular bonding data of Table 1. In this way, more space is given for future considerations upon the inverse or stimulated Raman processes, proving therefore the direct involvement of the bondonic reality in combined scattering of light on chemical structures. Overall, the bondonic characterization of the chemical bond is fully justified by quantum and relativistic considerations, to be advanced as a useful tool in characterizing chemical reactivity, times of reactions, i.e., when tunneling or entangled effects may be rationalized in an analytical manner. Note that further correction of this bondonic model may be realized when the present point-like approximation of nuclear systems is abolished and replaced by the bare-nuclear assumption in which additional dependence on the bonding distance is involved. This is left for future communications. The chemical bond, perhaps the greatest challenge in theoretical chemistry, has generated many inspiring theses over the years, although none definitive. Few of the most preeminent regard the orbitalic based explanation of electronic pairing, in valence shells of atoms and molecules, rooted in the hybridization concept [8] then extended to the valence-shell electron-pair repulsion (VSEPR) [76]. Alternatively, when electronic density is considered, the atoms-in-molecule paradigms were formulated through the geometrical partition of forces by Berlin [69], or in terms of core, bonding, and lone-pair lodges by Daudel [77], or by the zero local flux in the gradient field of the density ∇ρ by Bader [26], until the most recent employment of the chemical action functional in bonding [78,79]. Yet, all these approaches do not depart significantly from the undulatory nature of electronic motion in bonding, either by direct wave-function consideration or through its probability information in electronic density manifestation (for that is still considered as a condensed—observable version—of the undulatory manifestation of electron). In other words, while passing from the Lewis point-like ansatz to the undulatory modeling of electrons in bonding, the reverse passage was still missing in an analytical formulation. Only recently the first attempt was formulated, based on the broken-symmetry approach of the Schrödinger Lagrangean with the electronegativity-chemical hardness parabolic energy dependency, showing that a systematical quest for the creation of particles from the chemical bonding fields is possible [80]. Following this line, the present work makes a step forward and considers the gauge transformation of the electronic wave-function and spinor over the de Broglie-Bohm augmented non-relativistic and relativistic quantum pictures of the Schrödinger and Dirac electronic (chemical) fields, respectively. As a consequence, the reality of the chemical field in bonding was proved in either framework, while providing the corresponding bondonic particle with the associate mass and velocity in a full quantization form, see Equations (67) and (72). In fact, the Dirac bondon (65) was found to be a natural generalization of the Schrödinger one (38), while supplementing it with its anti-bondon particle (66) for the positron existence in the Dirac Sea. The bondon is the quantum particle corresponding to the superimposed electronic pairing effects or distribution in chemical bond; accordingly, through the values of its mass and velocity it may be possible to indicate the type of bonding (in particular) and the characterization of electronic behavior in bonding (in general). However, one of the most important consequences of bondonic existence is that the chemical bonding may be described in a more complex manner than relaying only on the electrons, but eventually employing the fermionic (electronic)-bosonic (bondonic) mixture: the first preeminent application is currently on progress, that is, exploring the effect that the Bose-Einstein condensation has on chemical bonding modeling [81,82]. Yet, such possibility arises due to the fact that whether the Pauli principle is an independent axiom of quantum mechanics or whether it depends on other quantum description of matter is still under question [83], as is the actual case of involving hidden variables and the entanglement or non-localization phenomenology that may be eventually mapped onto the delocalization and fractional charge provided by quantum chemistry over and on atomic centers of a molecular complex/chemical bond, respectively. As an illustration of the bondonic concept and of its properties such as the mass, velocity, charge, and life-time, the fundamental Raman scattering process was described by analytically deriving the involved virtual energy states of scattering sample (chemical bond) in terms of the bondonic properties above—proving its necessary existence and, consequently, of the associate Raman effect itself, while leaving space for further applied analysis based on spectroscopic data on hand. On the other side, the mass, velocity, charge, and life-time properties of the bondons were employed for analyzing some typical chemical bonds (see Table 1), this way revealing a sort of fuzzy classification of chemical bonding types in terms of the bondonic-to-electronic mass and charge ratios ςm and ςe, and of the bondonic-to-light velocity percent ratio ςv, along the bondonic observable life-time, t respectively–here summarized in Table 3. These rules are expected to be further refined through considering the new paradigms of special relativity in computing the bondons’ velocities, especially within the modern algebraic chemistry [84]. Yet, since the bondonic masses of chemical bonding ground states seem untouched by the Dirac relativistic considerations over the Schrödinger picture, it is expected that their analytical values may make a difference among the various types of compounds, while their experimental detection is hoped to be some day completed. The author kindly thanks Hagen Kleinert and Axel Pelster for their hospitality at Free University of Berlin on many occasions and for the summer of 2010 where important discussions on fundamental quantum ideas were undertaken in completing this work, as well for continuous friendship through the last decade. Both anonymous referees are kindly thanked for stimulating the revised version of the present work, especially regarding the inclusion of the quantum-relativistic charge (zitterbewegung) discussion and the Raman scattering description by the bondonic particles, respectively. This work was supported by CNCSIS-UEFISCSU, project number PN II-RU TE16/2010. References ThomsonJJOn the structure of the molecule and chemical combinationPhilos. Mag192141510538 HückelEQuantentheoretische beiträge zum benzolproblemZ. Physik193170204286 DoeringWVDetertFCycloheptatrienylium oxideJ. Am. Chem. Soc195173876877 LewisGNThe atom and the moleculeJ. Am. Chem. Soc191638762785 LangmuirIThe arrangement of electrons in atoms and moleculesJ. Am. Chem. Soc191941868934 PaulingLQuantum mechanics and the chemical bondPhys. Rev19313711851186 PaulingLThe nature of the chemical bond. I. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of moleculesJ. Am. Chem. Soc19315313671400 PaulingLThe nature of the chemical bond II. The one-electron bond and the three-electron bondJ. Am. Chem. Soc19315332253237 HeitlerWLondonFWechselwirkung neutraler Atome und homöopolare Bindung nach der QuantenmechanikZ. Phys192744455472 SlaterJCThe self consistent field and the structure of atomsPhys. Rev192832339348 SlaterJCThe theory of complex spectraPhys. Rev19293412931322 HartreeDRThe Calculation of Atomic StructuresWiley & SonsNew York, NY, USA1957 LöwdinPOQuantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interactionPhys. Rev19559714741489 LöwdinPOQuantum theory of many-particle systems. II. Study of the ordinary Hartree-Fock approximationPhys. Rev19559714741489 LöwdinPOQuantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effectsPhys. Rev19559715091520 RoothaanCCJNew developments in molecular orbital theoryRev. Mod. Phys1951236989 PariserRParrRA semi - empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. IJ. Chem. Phys195321466471 PariserRParrRA semi-empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. IIJ. Chem. Phys195321767776 PopleJAElectron interaction in unsaturated hydrocarbonsTrans. Faraday Soc19534913751385 HohenbergPKohnWInhomogeneous electron gasPhys. Rev1964136B864B871 KohnWShamLJSelf-consistent equations including exchange and correlation effectsPhys. Rev1965140A1133A1138 PopleJABinkleyJSSeegerRTheoretical models incorporating electron correlationInt. J. Quantum Chem197610119 Head-GordonMPopleJAFrischMJQuadratically convergent simultaneous optimization of wavefunction and geometryInt. J. Quantum Chem198936291303 PutzMVDensity functionals of chemical bondingInt. J. Mol. Sci2008910501095 PutzMVPath integrals for electronic densities, reactivity indices, and localization functions in quantum systemsInt. J. Mol. Sci20091048164940 BaderRFWAtoms in Molecules-A Quantum TheoryOxford University PressOxford, UK1990 BaderRFWA bond path: A universal indicator of bonded interactionsJ. Phys. Chem. A199810273147323 BaderRFWPrinciple of stationary action and the definition of a proper open systemPhys. Rev. B1994491334813356 MezeyPGShape in Chemistry: An Introduction to Molecular Shape and TopologyVCH PublishersNew York, NY, USA1993 MaggioraGMMezeyPGA fuzzy-set approach to functional-group comparisons based on an asymmetric similarity measureInt. J. Quantum Chem199974503514 SzekeresZExnerTMezeyPGFuzzy fragment selection strategies, basis set dependence and HF–DFT comparisons in the applications of the ADMA method of macromolecular quantum chemistryInt. J. Quantum Chem2005104847860 ParrRGYangWDensity Functional Theory of Atoms and MoleculesOxford University PressOxford, UK1989 PutzMVContributions within Density Functional Theory with Applications in Chemical Reactivity Theory and ElectronegativityPh.D. dissertation, West University of Timisoara, Romania,2003 SandersonRTPrinciples of electronegativity Part I. General natureJ. Chem. Educ198865112119 MortierWJGenechtenKvGasteigerJElectronegativity equalization: Application and parametrizationJ. Am. Chem. Soc1985107829835 ParrRGDonnellyRALevyMPalkeWEElectronegativity: The density functional viewpointJ. Chem. Phys19786838013808 SenKDJørgensonCDStructure and BondingSpringerBerlin, Germany198766 PearsonRGHard and Soft Acids and BasesDowden, Hutchinson & RossStroudsberg, PA, USA1973 PearsonRGHard and soft acids and bases—the evolution of a chemical conceptCoord. Chem. Rev1990100403425 PutzMVRussoNSiciliaEOn the applicability of the HSAB principle through the use of improved computational schemes for chemical hardness evaluationJ. Comp. Chem2004259941003 ChattarajPKLeeHParrRGPrinciple of maximum hardnessJ. Am. Chem. Soc199111318541855 ChattarajPKSchleyerPvRAn ab initio study resulting in a greater understanding of the HSAB principleJ. Am. Chem. Soc199411610671071 ChattarajPKMaitiBHSAB principle applied to the time evolution of chemical reactionsJ Am Chem Soc200312527052710 PutzMVMaximum hardness index of quantum acid-base bondingMATCH Commun. Math. Comput. Chem200860845868 PutzMVSystematic formulation for electronegativity and hardness and their atomic scales within densitiy functional softness theoryInt. J. Quantum Chem2006106361386 PutzMVAbsolute and Chemical Electronegativity and HardnessNova Science PublishersNew York, NY, USA2008 DiracPAMQuantum mechanics of many-electron systemsProc. Roy. Soc. (London)1929A123714733 SchrödingerEAn undulatory theory of the mechanics of atoms and moleculesPhys. Rev19262810491070 DiracPAMThe quantum theory of the electronProc. Roy. Soc. (London)1928A117610624 EinsteinAPodolskyBRosenNCan quantum-mechanical description of physical reality be considered complete?Phys. Rev193547777780 BohrNCan quantum-mechanical description of physical reality be considered complete?Phys. Rev193548696702 BohmDA suggested interpretation of the quantum theory in terms of “hidden” variables. IPhys. Rev195285166179 BohmDA suggested interpretation of the quantum theory in terms of “hidden” variables. IIPhys. Rev195285180193 de BroglieLOndes et quantaCompt. Rend. Acad. Sci. (Paris)1923177507510 de BroglieLSur la fréquence propre de l'électronCompt. Rend. Acad. Sci. (Paris)1925180498500 de BroglieLVigierMJPLa Physique Quantique Restera-t-elle Indéterministe?Gauthier-VillarsParis, France1953 BohmDVigierJPModel of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuationsPhys. Rev195496208216 PyykköPZhaoL-BSearch for effective local model potentials for simulation of QED effects in relativistic calculationsJ. Phys. B20033614691478 PyykköPRelativistic theory of atoms and molecules. III A Bibliography 1993–1999, Lecture Notes in ChemistrySpringer-VerlagBerlin, Germany200076 SnijdersJGPyykköPIs the relativistic contraction of bond lengths an orbital contraction effect?Chem. Phys. Lett19807558 LohrLLJrPyykköPRelativistically parameterized extended Hückel theoryChem. Phys. Lett197962333338 PyykköPRelativistic quantum chemistryAdv. Quantum Chem197811353409 EinsteinAOn the electrodynamics of moving bodiesAnn. Physik (Leipzig)190517891921 EinsteinADoes the inertia of a body depend upon its energy content?Ann. Physik (Leipzig)190518639641 WhitneyCKClosing in on chemical bonds by opening up relativity theoryInt. J. Mol. Sci20089272298 WhitneyCKSingle-electron state filling order across the elementsInt. J. Chem. Model20081105135 WhitneyCKVisualizing electron populations in atomsInt. J. Chem. Model20091245297 BoeyensJCANew Theories for ChemistryElsevierNew York, NY, USA2005 BerlinTBinding regions in diatomic moleculesJ. Chem. Phys195119208213 EinsteinAOn a Heuristic viewpoint concerning the production and transformation of lightAnn. Physik (Leipzig)190517132148 OelkeWCLaboratory Physical ChemistryVan Nostrand Reinhold CompanyNew York, NY, USA1969 FindlayAPractical Physical ChemistryLongmansLondon, UK1955 HibertyPCMegretCSongLWuWShaikSBarriers of hydrogen abstraction vs halogen exchange: An experimental manifestation of charge-shift bondingJ. Am. Chem. Soc200612828362843 FreemanSApplications of Laser Raman SpectroscopyJohn Wiley and SonsNew York, NY, USA1974 HeitlerWThe Quantum Theory of Radiation3rd edCambridge University PressNew York, NY, USA1954 GillespieRJThe electron-pair repulsion model for molecular geometryJ. Chem. Educ1970471823 DaudelRElectron and Magnetization Densities in Molecules and CrystalsBeckerPNATO ASI, Series B-Physics, Plenum PressNew York, NY, USA198040 PutzMVChemical action and chemical bondingJ. Mol. Struct. (THEOCHEM)20099006470 PutzMVLevels of a unified theory of chemical interactionInt. J. Chem. Model20091141147 PutzMVThe chemical bond: Spontaneous symmetry–breaking approachSymmetr. Cult. Sci200819249262 PutzMVHidden side of chemical bond: The bosonic condensateChemical BondingNOVA Science PublishersNew York, NY, USA2011to be published. PutzMVConceptual density functional theory: From inhomogeneous electronic gas to Bose-Einstein condensatesChemical Information and Computational Challenges in 21st A Celebration of 2011 International Year of ChemistryPutzMVNOVA Science Publishers IncNew York, NY, USA2011to be published. KaplanIGIs the Pauli exclusive principle an independent quantum mechanical postulate?Int. J. Quantum Chem200289268276 WhitneyCKRelativistic dynamics in basic chemistryFound. Phys200737788812 Figure and Tables The Feynman diagrammatical sum of interactions entering the Raman effect by connecting the single and double photonic particles’ events in absorption (incident wave light q⃗0, υ0) and emission (scattered wave light q⃗, υ) induced by the quantum first H(1) and second H(2) order interaction Hamiltonians of Equations (84) and (85) through the initial \|i 〉, final \|f 〉, and virtual \|v 〉 bondonic states. The first term accounts for absorption (A)-emission (E) at once, the second term sums over the virtual states connecting the absorption followed by emission, while the third terms sums over virtual states connecting the absorption following the emission events. Ratios for the bondon-to-electronic mass and charge and for the bondon-to-light velocity, along the associated bondonic life-time for typical chemical bonds in terms of their basic characteristics such as the bond length and energy [71,72] through employing the basic formulas (68), (73), (80) and (81) for the ground states, respectively. Bond Type Xbond (Å) Ebond (kcal/mol) ζ m = m m 0 ζ v = v c [ % ] ζ e = e e [ × 10 3 ] t[×1015] (seconds) H–H 0.60 104.2 2.34219 3.451 0.3435 9.236 C–C 1.54 81.2 0.45624 6.890 0.687 11.894 C–C (in diamond) 1.54 170.9 0.21678 14.385 1.446 5.743 C=C 1.34 147 0.33286 10.816 1.082 6.616 C≡C 1.20 194 0.31451 12.753 1.279 5.037 N≡N 1.10 225 0.32272 13.544 1.36 4.352 O=O 1.10 118.4 0.61327 7.175 0.716 8.160 F–F 1.28 37.6 1.42621 2.657 0.264 25.582 Cl–Cl 1.98 58 0.3864 6.330 0.631 16.639 I–I 2.66 36.1 0.3440 5.296 0.528 26.701 C–H 1.09 99.2 0.7455 5.961 0.594 9.724 N–H 1.02 93.4 0.9042 5.254 0.523 10.32 O–H 0.96 110.6 0.8620 5.854 0.583 8.721 C–O 1.42 82 0.5314 6.418 0.64 11.771 C=O (in CH2O) 1.21 166 0.3615 11.026 1.104 5.862 C=O (in O=C=O) 1.15 191.6 0.3467 12.081 1.211 5.091 C–Cl 1.76 78 0.3636 7.560 0.754 12.394 C–Br 1.91 68 0.3542 7.155 0.714 14.208 C–I 2.10 51 0.3906 5.905 0.588 18.9131 Predicted basic values for bonding energy and length, along the associated bondonic life-time and velocity fraction from the light velocity for a system featuring unity ratios of bondonic mass and charge, respecting the electron values, through employing the basic formulas (81), (73), (68), and (80), respectively. X bond [ A 0 ] Ebond [(kcal/mol)] t[×1015] (seconds) ζ v = v c [ % ] ζ m = m m 0 ζ e = e e 1 87.86 10.966 4.84691 1 0.4827 × 10−3 1 182019 53.376 99.9951 4.82699 × 10−4 1 10 18201.9 533.76 99.9951 4.82699 × 10−5 1 100 1820.19 5337.56 99.9951 4.82699 × 10−6 1 Phenomenological classification of the chemical bonding types by bondonic (mass, velocity, charge and life-time) properties abstracted from Table 1; the used symbols are: > and ≫ for ‘high’ and ‘very high’ values; < and ≪ for ‘low’ and ‘very low’ values; ∼ and ∼> for ‘moderate’ and ‘moderate high and almost equal’ values in their class of bonding. Property ςm ςv ςe t Chemical bond Covalence >> << << >> Multiple bonds < > > < Metallic << > > < Ionic ∼> ∼>
e726f69bec94d86b	General relativity From Wikipedia, the free encyclopedia (Redirected from General Relativity) Jump to: navigation, search For the book by Robert Wald, see General Relativity (book). For a more accessible and less technical introduction to this topic, see Introduction to general relativity. A simulated black hole of 10 solar masses within the Milky Way, seen from a distance of 600 kilometers. General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915[1] and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Albert Einstein developed the theories of special and general relativity. Picture from 1921. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, and form the core of Einstein's general theory of relativity.[2] The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the so-called Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which eventually resulted in the Reissner–Nordström solution, now associated with electrically charged black holes.[3] In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption.[4] By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot and dense earlier state.[5] Einstein later declared the cosmological constant the biggest blunder of his life.[6] During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors").[7] Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919,[8] making Einstein instantly famous.[9] Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975, now known as the golden age of general relativity.[10] Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations.[11] Ever more precise solar system tests confirmed the theory's predictive power,[12] and relativistic cosmology, too, became amenable to direct observational tests.[13] From classical mechanics to general relativity[edit] General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.[14] Geometry of Newtonian gravity[edit] According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity at the surface of the Earth). At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration.[15] The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.[16] Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.[17] A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field.[18] Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The result is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system.[19] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.[20] Relativistic generalization[edit] As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.[21] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which includes translations and rotations.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.[22] With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.[23] In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure,[24] or much better, a conformal geometry, as it is difficult to understand how space or time or space-time can have a structure. Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.[25] A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.[26] The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.[27] The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).[28] Einstein's equations[edit] Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear).[29] Using the equivalence principle, this tensor is readily generalized to curved space-time. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations: Einstein's field equations G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu}\, On the left-hand side is the Einstein tensor, a specific divergence-free combination of the Ricci tensor R_{\mu\nu} and the metric. Where G_{\mu\nu} is symmetric. In particular, is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as On the right-hand side, T_{\mu\nu} is the energy–momentum tensor. All tensors are written in abstract index notation.[30] Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c4, with G the gravitational constant and c the speed of light.[31] When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans–Dicke theory, teleparallelism, and Einstein–Cartan theory.[32] Definition and basic applications[edit] The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building. Definition and basic properties[edit] General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime.[33] Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.[34] The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.[35] While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.[36] As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.[37] Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.[38] Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.[39] The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.[40] Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.[41] Nevertheless, a number of exact solutions are known, although only a few have direct physical applications.[42] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,[43] and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos.[44] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).[45] Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.[46] In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity[47] and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.[48] An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.[49] Consequences of Einstein's theory[edit] General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Gravitational time dilation and frequency shift[edit] Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body Assuming that the equivalence principle holds,[50] gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.[51] Gravitational redshift has been measured in the laboratory[52] and using astronomical observations.[53] Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,[54] while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).[55] Tests in stronger gravitational fields are provided by the observation of binary pulsars.[56] All results are in agreement with general relativity.[57] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.[58] Light deflection and gravitational time delay[edit] Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray) This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.[60] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),[61] several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,[62] the angle of deflection resulting from such calculations is only half the value given by general relativity.[63] Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.[64] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.[65] Gravitational waves[edit] Main article: Gravitational wave Ring of test particles influenced by gravitational wave One of several analogies between weak-field gravity and electromagnetism is that, analogous to electromagnetic waves, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light.[66] The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).[67] Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, for weak fields, a linear approximation can be made. Such linearized gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10^{-21} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.[68] Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space[69] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves.[70] But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.[71] Orbital effects and the relativity of direction[edit] General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction. Precession of apsides[edit] Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess—the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of the anomalous perihelion shift of the planet Mercury, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.[72] The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)[73] or the much more general post-Newtonian formalism.[74] It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations).[75] Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),[76] as well as in binary pulsar systems, where it is larger by five orders of magnitude.[77] Orbital decay[edit] Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.[78] According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.[79] The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.[80] Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737-3039, in which both stars are pulsars.[81] Geodetic precession and frame-dragging[edit] Several relativistic effects are directly related to the relativity of direction.[82] One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").[83] For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging.[84] More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 0.3%.[85][86] Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.[87] Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.[88] Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.[89] Also the Mars Global Surveyor probe around Mars has been used.[90][91] Astrophysical applications[edit] Gravitational lensing[edit] Main article: Gravitational lensing Einstein cross: four images of the same astronomical object, produced by a gravitational lens The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.[92] Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.[93] The earliest example was discovered in 1979;[94] since then, more than a hundred gravitational lenses have been observed.[95] Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.[96] Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.[97] Gravitational wave astronomy[edit] Artist's impression of the space-borne gravitational wave detector LISA Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly. Such detection is a major goal of current relativity-related research.[98] Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and VIRGO.[99] Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10−9 to 10−6 Hertz frequency range, which originate from binary supermassive blackholes.[100] European space-based detector, eLISA / NGO, is currently under development,[101] with a precursor mission (LISA Pathfinder) due for launch in 2015.[102] Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.[103] They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.[104] Black holes and other compact objects[edit] Main article: Black hole Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars.[105] Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,[106] and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.[107] Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.[108] Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.[109] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.[110] General relativity plays a central role in modelling all these phenomena,[111] and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.[112] Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.[113] The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.[114] This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy. Main article: Physical cosmology The current models of cosmology are based on Einstein's field equations, which include the cosmological constant Λ since it has important influence on the large-scale dynamics of the cosmos, R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} + \Lambda\ g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu} where g_{\mu\nu} is the spacetime metric.[115] Isotropic and homogeneous solutions of these enhanced equations, the Friedmann–Lemaître–Robertson–Walker solutions,[116] allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase.[117] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,[118] further observational data can be used to put the models to the test.[119] Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,[120] the large-scale structure of the universe,[121] and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.[122] Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be so-called dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.[123] There is no generally accepted description of this new kind of matter, within the framework of known particle physics[124] or otherwise.[125] Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.[126] A so-called inflationary phase,[127] an additional phase of strongly accelerated expansion at cosmic times of around 10^{-33} seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.[128] Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.[129] However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.[130] An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed[131] (cf. the section on quantum gravity, below). Time travel[edit] Kurt Gödel showed that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. Advanced concepts[edit] Causal structure and global geometry[edit] Main article: Causal structure Penrose–Carter diagram of an infinite Minkowski universe In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.[132] Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) are used to derive general results.[133] Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius[134]), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.[135] The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. In the long run, they are rather simple objects characterized by eleven parameters specifying energy, linear momentum, angular momentum, location at a specified time and electric charge. This is stated by the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.[136] Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process).[137] There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.[138] This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature, black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).[139] There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon).[140] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semi-classical radiation known as Unruh radiation.[141] Main article: Spacetime singularity Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.[142] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,[143] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.[144] The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (Big Crunch) as well.[145] Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.[146] The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage[147] and also at the beginning of a wide class of expanding universes.[148] However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the so-called BKL conjecture).[149] The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.[150] Evolution equations[edit] Each solution of Einstein's equation encompasses the whole history of a universe — it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.[151] To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.[152] These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.[153] Such formulations of Einstein's field equations are the basis of numerical relativity.[154] Global and quasi-local quantities[edit] The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.[155] Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)[156] or suitable symmetries (Komar mass).[157] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity.[158] Just as in classical physics, it can be shown that these masses are positive.[159] Corresponding global definitions exist for momentum and angular momentum.[160] There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.[161] Relationship with quantum theory[edit] If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid state physics, would be the other.[162] However, how to reconcile quantum theory with general relativity is still an open question. Quantum field theory in curved spacetime[edit] Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.[163] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.[164] Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that they evaporate over time.[165] As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.[166] Quantum gravity[edit] Main article: Quantum gravity The demand for consistency between a quantum description of matter and a geometric description of spacetime,[167] as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.[168] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.[169] Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.[170] At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability").[171] Simple spin network of the type used in loop quantum gravity One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.[172] The theory promises to be a unified description of all particles and interactions, including gravity;[173] the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.[174] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity[175] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[176] Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined.[177] However, with the introduction of what are now known as Ashtekar variables,[178] this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.[179] Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,[180] there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations,[181] causal sets,[182] twistor models[183] or the path-integral based models of quantum cosmology.[184] All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[185] Current status[edit] General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications the theory is incomplete.[186] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.[187] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.[188] Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,[189] and increasingly powerful computer simulations (such as those describing merging black holes) are run.[190] The race for the first direct detection of gravitational waves continues,[191] in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.[192] A century after its publication, general relativity remains a highly active area of research.[193] See also[edit] 1. ^ O'Connor, J.J. and Robertson, E.F. (1996), General relativity. Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland. Retrieved 2015-02-04. 2. ^ Pais 1982, ch. 9 to 15, Janssen 2005; an up-to-date collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, pp. 110ff. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15 3. ^ Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918) 4. ^ Einstein 1917, cf. Pais 1982, ch. 15e 5. ^ Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4 6. ^ As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the section Cosmology, below 7. ^ Pais 1982, pp. 253–254 8. ^ Kennefick 2005, Kennefick 2007 9. ^ Pais 1982, ch. 16 10. ^ Thorne, Kip (2003). "Warping spacetime". The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday. Cambridge University Press. p. 74. ISBN 0-521-82081-2. , Extract of page 74 11. ^ Israel 1987, ch. 7.8–7.10, Thorne 1994, ch. 3–9 12. ^ Sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein 13. ^ Section Cosmology and references therein; the historical development is in Overbye 1999 14. ^ The following exposition re-traces that of Ehlers 1973, sec. 1 15. ^ Arnold 1989, ch. 1 16. ^ Ehlers 1973, pp. 5f 17. ^ Will 1993, sec. 2.4, Will 2006, sec. 2 18. ^ Wheeler 1990, ch. 2 19. ^ Ehlers 1973, sec. 1.2, Havas 1964, Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959 20. ^ Ehlers 1973, pp. 10f 21. ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006 22. ^ An in-depth comparison between the two symmetry groups can be found in Giulini 2006a 23. ^ Rindler 1991, sec. 22, Synge 1972, ch. 1 and 2 24. ^ Ehlers 1973, sec. 2.3 25. ^ Ehlers 1973, sec. 1.4, Schutz 1985, sec. 5.1 26. ^ Ehlers 1973, pp. 17ff; a derivation can be found in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below 27. ^ Rindler 2001, sec. 1.13; for an elementary account, see Wheeler 1990, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985 28. ^ Ehlers 1973, sec. 1.4 for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein–Cartan theory 29. ^ Ehlers 1973, p. 16, Kenyon 1990, sec. 7.2, Weinberg 1972, sec. 2.8 30. ^ Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3 31. ^ Kenyon 1990, sec. 7.4 32. ^ Brans & Dicke 1961, Weinberg 1972, sec. 3 in ch. 7, Goenner 2004, sec. 7.2, and Trautman 2006, respectively 33. ^ Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other textbook on general relativity 34. ^ At least approximately, cf. Poisson 2004 35. ^ Wheeler 1990, p. xi 36. ^ Wald 1984, sec. 4.4 37. ^ Wald 1984, sec. 4.1 38. ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b 39. ^ section 5 in ch. 12 of Weinberg 1972 40. ^ Introductory chapters of Stephani et al. 2003 41. ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996 42. ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006 43. ^ Chandrasekhar 1983, ch. 3,5,6 44. ^ Narlikar 1993, ch. 4, sec. 3.3 45. ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5 46. ^ Lehner 2002 47. ^ For instance Wald 1984, sec. 4.4 48. ^ Will 1993, sec. 4.1 and 4.2 49. ^ Will 2006, sec. 3.2, Will 1993, ch. 4 50. ^ Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198 51. ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5 52. ^ Pound–Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186 53. ^ Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond et al. 2005. 54. ^ Starting with the Hafele–Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186 55. ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003 56. ^ Stairs 2003 and Kramer 2004 57. ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, sec. 4.2 58. ^ Ohanian & Ruffini 1994, pp. 164–172 59. ^ Cf. Kennefick 2005 for the classic early measurements by the Eddington expeditions; for an overview of more recent measurements, see Ohanian & Ruffini 1994, ch. 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004 60. ^ This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, sec. 5 61. ^ Blanchet 2006, sec. 1.3 62. ^ Rindler 2001, sec. 1.16; for the historical examples, Israel 1987, pp. 202–204; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997 63. ^ From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11 64. ^ For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, sec. 4.4 65. ^ Will 1993, sec. 7.1 and 7.2 66. ^ These have been indirectly observed through the loss of energy in binary pulsar systems such as the Hulse–Taylor binary, the subject of the 1993 Nobel Prize in physics. A number of projects are underway to attempt to observe directly the effects of gravitational waves. For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Unlike electromagnetic waves, the dominant contribution for gravitational waves is not the dipole, but the quadrupole; see Schutz 2001 67. ^ Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9 68. ^ For example Jaranowski & Królak 2005 69. ^ Rindler 2001, ch. 13 70. ^ Gowdy 1971, Gowdy 1974 71. ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy 72. ^ Schutz 2003, pp. 48–49, Pais 1982, pp. 253–254 73. ^ Rindler 2001, sec. 11.9 74. ^ Will 1993, pp. 177–181 75. ^ In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3 76. ^ The most precise measurements are VLBI measurements of planetary positions; see Will 1993, ch. 5, Will 2006, sec. 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407 77. ^ Kramer et al. 2006 78. ^ A figure that includes error bars is fig. 7 in Will 2006, sec. 5.1 79. ^ Stairs 2003, Schutz 2003, pp. 317–321, Bartusiak 2000, pp. 70–86 80. ^ Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994 81. ^ Kramer 2004 82. ^ Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, §11.4 83. ^ Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8 84. ^ Bertotti, Ciufolini & Bender 1987, Nordtvedt 2003 85. ^ Kahn 2007 86. ^ A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt, Parkinson & Kahn 2007; further updates will be available on the mission website Kahn 1996–2012. 87. ^ Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471 88. ^ Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004 89. ^ Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009 90. ^ Iorio L. (August 2006), "COMMENTS, REPLIES AND NOTES: A note on the evidence of the gravitomagnetic field of Mars", Classical Quantum Gravity 23 (17): 5451–5454, arXiv:gr-qc/0606092, Bibcode:2006CQGra..23.5451I, doi:10.1088/0264-9381/23/17/N01 91. ^ Iorio L. (June 2010), "On the Lense–Thirring test with the Mars Global Surveyor in the gravitational field of Mars", Central European Journal of Physics 8 (3): 509–513, arXiv:gr-qc/0701146, Bibcode:2010CEJPh...8..509I, doi:10.2478/s11534-009-0117-6 92. ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998 93. ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3 94. ^ Walsh, Carswell & Weymann 1979 95. ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007 96. ^ Roulet & Mollerach 1997 97. ^ Narayan & Bartelmann 1997, sec. 3.7 98. ^ Barish 2005, Bartusiak 2000, Blair & McNamara 1997 99. ^ Hough & Rowan 2000 100. ^ Hobbs, George; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N. D. R.; Burgay, M.; Burke-Spolaor, S.; et al. (2010), "The international pulsar timing array project: using pulsars as a gravitational wave detector", Classical and Quantum Gravity 27 (8): 084013, arXiv:0911.5206, Bibcode:2010CQGra..27h4013H, doi:10.1088/0264-9381/27/8/084013 101. ^ Danzmann & Rüdiger 2003 102. ^ "LISA pathfinder overview". ESA. Retrieved 2012-04-23. 103. ^ Thorne 1995 104. ^ Cutler & Thorne 2002 105. ^ Miller 2002, lectures 19 and 21 106. ^ Celotti, Miller & Sciama 1999, sec. 3 107. ^ Springel et al. 2005 and the accompanying summary Gnedin 2005 108. ^ Blandford 1987, sec. 8.2.4 109. ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996 110. ^ For a review, see Begelman, Blandford & Rees 1984. To a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see Rees 1966 111. ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars, cf. Kraus 1998 112. ^ The evidence includes limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"), see Celotti, Miller & Sciama 1999, observations of stellar dynamics in the center of our own Milky Way galaxy, cf. Schödel et al. 2003, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000 113. ^ Dalal et al. 2006 114. ^ Barack & Cutler 2004 115. ^ Originally Einstein 1917; cf. Pais 1982, pp. 285–288 116. ^ Carroll 2001, ch. 2 117. ^ Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991 118. ^ E.g. with WMAP data, see Spergel et al. 2003 119. ^ These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003 120. ^ Peebles 1966; for a recent account of predictions, see Coc, Vangioni‐Flam et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005 121. ^ Lahav & Suto 2004, Bertschinger 1998, Springel et al. 2005 122. ^ Alpher & Herman 1948, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997 123. ^ Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. Peebles 1993, ch. 18, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005 124. ^ Peacock 1999, ch. 12, Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12 125. ^ Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9 126. ^ Carroll 2001; an accessible overview is given in Caldwell 2004. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2007 127. ^ A good introduction is Linde 1990; for a more recent review, see Linde 2005 128. ^ More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1 129. ^ Spergel et al. 2007, sec. 5,6 130. ^ More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory 131. ^ Brandenberger 2007, sec. 2 132. ^ Frauendiener 2004, Wald 1984, sec. 11.1, Hawking & Ellis 1973, sec. 6.8, 6.9 133. ^ Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6 134. ^ Thorne 1972; for more recent numerical studies, see Berger 2002, sec. 2.1 135. ^ Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004 136. ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results 137. ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see Wald 2001, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969 138. ^ Bekenstein 1973, Bekenstein 1974 139. ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in Wald 2001, ch. 3 140. ^ Narlikar 1993, sec. 4.4.4, 4.4.5 141. ^ Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, ch. 3 142. ^ Hawking & Ellis 1973, sec. 8.1, Wald 1984, sec. 9.1 143. ^ Townsend 1997, ch. 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, ch. 3 144. ^ Townsend 1997, ch. 4; for a more extensive treatment, cf. Chandrasekhar 1983, ch. 6 145. ^ Ellis & Van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2 146. ^ Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called eikonal approximations of many wave-equations, namely the "caustics", are resolved into finite peaks beyond that approximation. 147. ^ Namely when there are trapped null surfaces, cf. Penrose 1965 148. ^ Hawking 1966 149. ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007 150. ^ The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a textbook-level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1 151. ^ Hawking & Ellis 1973, sec. 7.1 152. ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7 153. ^ Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998 154. ^ Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001 155. ^ Misner, Thorne & Wheeler 1973, §20.4 156. ^ Arnowitt, Deser & Misner 1962 157. ^ Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979 158. ^ For a pedagogical introduction, see Wald 1984, sec. 11.2 159. ^ Wald 1984, p. 295 and refs therein; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states 160. ^ Townsend 1997, ch. 5 161. ^ Such quasi-local mass–energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy–momentum based on twistor methods; cf. the review article Szabados 2004 162. ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003 163. ^ Ramond 1990, Weinberg 1995, Peskin & Schroeder 1995; a more accessible overview is Auyang 1995 164. ^ Wald 1994, Birrell & Davies 1984 165. ^ For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000 166. ^ Wald 2001, ch. 3 167. ^ Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. Carlip 2001, sec. 2 168. ^ Schutz 2003, p. 407 169. ^ A timeline and overview can be found in Rovelli 2000 170. ^ Donoghue 1995 171. ^ In particular, a technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. Weinberg 1996, ch. 17, 18, fails in this case; cf. Goroff & Sagnotti 1985 172. ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b 173. ^ At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3, 5.3 174. ^ Green, Schwarz & Witten 1987, sec. 4.2 175. ^ Weinberg 2000, ch. 31 176. ^ Townsend 1996, Duff 1996 177. ^ Kuchař 1973, sec. 3 178. ^ These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987 179. ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003 180. ^ Isham 1994, Sorkin 1997 181. ^ Loll 1998 182. ^ Sorkin 2005 183. ^ Penrose 2004, ch. 33 and refs therein 184. ^ Hawking 1987 185. ^ Ashtekar 2007, Schwarz 2007 186. ^ Maddox 1998, pp. 52–59, 98–122; Penrose 2004, sec. 34.1, ch. 30 187. ^ section Quantum gravity, above 188. ^ section Cosmology, above 189. ^ Friedrich 2005 190. ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002 191. ^ See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO 600 and LIGO 192. ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006 193. ^ See, e.g., the electronic review journal Living Reviews in Relativity Further reading[edit] Popular books Beginning undergraduate textbooks • Callahan, James J. (2000), The Geometry of Spacetime: an Introduction to Special and General Relativity, New York: Springer, ISBN 0-387-98641-3 • Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, ISBN 0-201-38423-X Advanced undergraduate textbooks • B. F. Schutz (2009), A First Course in General Relativity (Second Edition), Cambridge University Press, ISBN 978-0-521-88705-2 • Cheng, Ta-Pei (2005), Relativity, Gravitation and Cosmology: a Basic Introduction, Oxford and New York: Oxford University Press, ISBN 0-19-852957-0 • Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2 • Hartle, James B. (2003), Gravity: an Introduction to Einstein's General Relativity, San Francisco: Addison-Wesley, ISBN 0-8053-8662-9 • Hughston, L. P. & Tod, K. P. (1991), Introduction to General Relativity, Cambridge: Cambridge University Press, ISBN 0-521-33943-X • d'Inverno, Ray (1992), Introducing Einstein's Relativity, Oxford: Oxford University Press, ISBN 0-19-859686-3 • Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 9783642357978. Graduate-level textbooks External links[edit] • Courses • Lectures • Tutorials
ab3bba1c95aa4e28	Lessons from the Particle Waves The Strange Case of an Isolated Particle in unbound universe Consider the Free particle Schrodinger Equation for a particle of energy This give a moving wave. If we stop the time at some instant we will get the following static wave form. Here we know its energy exactly; but we have no idea where it is and where it came from and which way it is going. It has no defined position. This is due to uncertainty principle which says Dp Dq ³ h/4p Since we know the momentum p exactly we have infinite uncertainty in position. Again for a given value of k both +k and -k, give the same energy . It really doesn't know "if it is coming or going?". Taking the probability interpretation of Born the probability of finding the particle at any one instant is given below There are many points in the space where it has a probability of 1. The particle is at several places at the same time. In time this distributes and over half the period of the wave it is everywhere. It is an omnipresent particle. This surprising result is because it is an isolated particle and is in unbounded space. An isolated particle with definite energy cannot be localized in an unbound universe! They have no direction or purpose. They in fact represent the potent (Omni or not), Omnipresent Monad. However when it becomes a composite body it gets meaning and purpose as we show below. I have actually used this argument in order to indicate that God the Absolute (Nirguna Brahman as defined in Indian theology) has no purpose and cannot be known. Because of this Nirguna Brahman do not engage in creation or redemption. Such a God is as good as non-existent. Effect of Coherent Particle System Suppose now we have a large number of particles which are coherent – which means each wave is in phase with all the others. Then for all practical purposes we still have a multiple particle which act as one particle - Many in One. Thus a number of personalities that are in coherence, act as one Unit, each enhancing and supporting the other. We still have the same waveform with higher wave amplitude. It is as though we only have one single particle of a larger mass m. They still do not localize and is in no way different from a single particle. Thus even a totally coherent group of gods do not make a difference because they have no personalities to express. After all how do we define mass and particle without extensions. They will remain as mathematical imaginations. Indeterminacy is vital to the reality of existence . At least for experiencing the particle we need extensions. A point will remain imaginary unless it is a circle of infinitely small radius. This is also seen in the visibility of line spectra. If the energy levels were discrete and the transitions gave rise to a single discrete frequency in spectra, these will be totally incapable of being seen unless we have a blurring senser. Fortunately each spectral line is given a width due to Lamb Shift so that we can see it. Effect of Binding the Particle If we now place this single particle in a bounded box, reflection of the same wave results in what is known as the standing waves. These waves are quantized. This transformation is a transition from the Nirguna Brahman – the God who is unknown and unknowable to Saguna Brahman. The unknowable when restricts himself into a boundary become one with a form. Then this God has properties that are measurable or knowable. Again this is similar to incarnation. The eternal God enters the finite dimensions of human existence brings along with it its own restriction. Thus the incarnate Jesus had limitations with regard to space and time. While he was capable of superceding these restrictions, Jesus as incarnate man never seems to have violated this self imposed restriction. “Having found himself in the form of man, he humbled himself.” with the probability function as This is produced by a self-reflection process at the boundaries. Thus it is the boundaries that has produced the properties that we can see. Only those quantized waves will sustain while all the other modes will decay and vanish. This state will remain without decay as long as the system is isolated from the rest of the cosmos. The motion is unabated unless external interference occurs. Particle in 3D and Degeneracy When we extend the problem to 3 Dimensions the particle in a 3D box exhibits degeneracy because of its 3 freedoms of motion. Each possible state has a threefold degeneracy. If we stretch any one axis this degeneracy is broken. Thus in an unbounded space it reduces to omnipresence. God appears in our finite bounded world as degenerate beings and with specific properties. It is the same Nirguna Brahman appearing as Saguna Brahman by restricting Himself. Effect of Many non-Coherent Particles However in the infinite space, if there are several waves that are not in phase or not coherent, these waves interfere with each other and wave packets are formed. If there are an infinite number of particles of varying energies we have a localized particle. Look what happens as we add more are more waves with varying phases in the universe which is still considered unbound. 1 Wave 4 waves 16 waves 32waves 128waves Animation adding waves If there are an infinite number of particles of varying energies we have a localized particle. Localization and individual direction of motion develops when large number (ideally infinite number) of non-coherent waves fall on each other. The Motion is as follows: Every mathematics students knows how to get these waves given a wave form using Fourier Analysis Thus it is the lack of coherence within us that gives us the location – the feeling of I – the definition of Ego. In reality we are one with the rest of the cosmos. When the creation is in resonance with the creator we in fact part-take of the divinity with Him. But the entry of sin into the cosmos through the free beings produced a separation – separate beings in the creation each striving and competing with each other resulting in the decaying world. Competition and survival of the fittest really do not produce more efficient systems but a decaying dying system. The purpose of incarnation was to bring back the coherence and oneness of mind in resonance with the Divine. In other words, it was to bring back Love into the beings. The plan of Jesus in this regard was to start groups of willing beings to come together in oneness of mind. This was his Church on which he places the whole responsibility of redeeming the cosmos as a whole. In fact the whole creation is waiting for the appearance of the “Sons of God” so that through them the redemption can be obtained. Church is essentially is worshipping community. 1Co 10:17 Because we, being a number of persons, are one bread, we are one body: for we all take part in the one bread. 1Co 12:12 For as the body is one, and has a number of parts, and all the parts make one body, so is Christ. 1Co 12:13 For through the baptism of the one Spirit we were all formed into one body, Jews or Greeks, servants or free men, and were all made full of the same Spirit. Eph 4:4 There is one body and one Spirit, even as you have been marked out by God in the one hope of his purpose for you; Eph 4:6 One God and Father of all, who is over all, and through all, and in all. Author-Work going on to complete State of a System ”By Their Fruits” “Actually - so they say - there is intrinsically only awareness, observation, measurement. If through them I have procured at a given moment the best knowledge of the state of the physical object that is possibly attainable in accord with natural laws, then I can turn aside as meaningless any further questioning about the "actual state", inasmuch as I am convinced that no further observation can extend my knowledge of it - at least, not without an equivalent diminution in some other respect.” The Present Situation in Quantum Mechanics, Erwin Schrödinger, Proceedings of the American Philosophical Society, 124, 323-38 This assumes that behind these observed facts, there is an objective reality that causes these special observable relations that we call the state of existence (State Function or State Vector). This is expressed in Quantum Mechanics as a relation between Wave function (State) and observables. As we associate a property from each reactive relationship, we arrive at the compendium of the state of existence – the State Ket Vector or State Function. where means represented by The components of the matrix are the possible states. In all religions, God is defined in terms of his properties. Islam defines Allah with 101 names. All Jewish Patriarchs found and defined God in terms of what he was to them. Thus God is a Provider (Yhvh Jireh), God my healer (Yhvh Rophe) and so on. For a list of Names of God, see http://www.acns.com/~mm9n/names/name.htm The state of a system is fully described by the wavefunction y(r1, r2,...;t). The coordinates of the location of all particles in the system and the time uniquely define the state of the system at any given instant. We must of course include any internal coordinates such as spin, color, charm, truth, beauty, taste or whatever in this definition, but once specified, the system is uniquely defined. Notice that all interactive coordinates of all properties must be defined if the state function is to be complete. Observables, non-Observables and Hermeticity For every observable, there is an associated operator which operating on the State function extracts the observable. This extracted value will be one of the possible values of the system. There is a special class of operators that are called Hermitian operators. They are of particular importance in quantum mechanics because they have the property that all of their eigenvalues are real (and does not contain an imaginary part). This is convenient for the measurement outcome of any experiment must be a real number. There are non-Hermitian operators, but they do not correspond to observable properties. All observable properties are represented by Hermitian operators, but not all Hermitian operators correspond to an observable property. The presence of the imaginary terms will indicate that there are other dimensions involved and we are observing only a projection of the state into the real world. Unless we have the complete picture in all dimensions, our understanding will be only partial. In soteriology, also the only measurable quantities are the "works". Though Christians would say that the salvation is independent of the "good works", good works are dependent on salvation – or the state function of the person concerned. That is why Paul asserts: James asserts One false argument that is often portrayed is that good works can be the result of a self-motivated sinful nature It may be a ruse for a while. They are then pseudo good works. Nevertheless, taken over a period it simply cannot be true. A bad person cannot bring forth-good fruit. Any such argument will be simply contrary to Jesus’ statements. Mat. 7: 18 A good tree cannot bear bad fruit, and a bad tree cannot bear good fruit.19 …. 20 Thus, by their fruit you will recognize them. I had several people asking me "How do we know that someone is saved?" Put it that way it looks impossible to answer. The question is "How do we know the state function of the person?" The answer simply is "by their fruit." John gives the same rule. Again, it is evident from the argument that occasionally an evil person can produce an apparent good work. He can indeed have an ulterior motive. Therefore, a single measurement of the event will not give us the state of the person. On the other hand, his real state betrays the motive and will explain the outcome. In the same way, a saved person will also do evil. However, in the end by repeated measurements over a period of time over a given space the result should be evident. It is not necessarily the mistake of the measuring instrument but it is because the measuring instrument itself is built into our system. Any attempt to present good works, as the result of evil heart is certainly a mistaken -scriptural as well as practically. We know that the amount of good works that results from the salvation process varies with individuals. But certainly the saved should show increased good works in their life. Though we are yet far from writing a Schrödinger equation for the personality we could get much out of the basic Quantum mechanical approach to particles. This is what I am trying to develop in this article. This probably is the first attempt of this kind and therefore I have my limitations. But I believe it contains the germs of a scientific logical approach to several vexing problems in soteriology. Confronted with apparently difficult situation of relating the unseen and the seen, how Quantum Theory – in which I include all of its variations such as Quantum Mechanics, Wave Mechanics, Matrix Mechanics, Schrödinger Formalism, Quantum Field Theory and all Unified Field theory attempts. – and the scientific community handled the situation is worth following. Understanding the basic difficulties and then accepting what is inevitable had been the result. In contrast in theology we are confronted with a situation of unbelief and disregard for the conflict. There is no point in assuming that there are no conflicts. Saying it louder and in veiled form will not make it disappear. I had often come to doubt even my logical ability when people assert, "Absolute predestination and absolute free will are both consistent and can coexist without conflict. If it does not exist in a fallen creation, it existed in pre-fall Adamic period." I still fail to understand that. Fact is that one precludes another. The first step is to recognize that some of our assumptions are in direct conflict with observations and with the revealed Word of God. We need to learn to accept them as they are and see how they can be reconciled. The scientific world has given us some clue into this mystery. Even the very existence of God is derived through such observables. Bible is said to be the revelation or unveiling of God. But apart from certain prophetic messages with the indroduction “Thus says the Lord”, and direct messages with the introduction, “ I am the Lord your God”, most of the bible especially the Old Testament is simply the history of the world and eventually the history of the nation of Israel. God appears to Moses as “the God of your fathers”. It is the collective experience of generations that identifies Yhvh. When Moses asked “What is your name?”, the only answer was “I am that I am” which simply means that he is a state of existence which cannot be fully explained, but can be known by his dealings with people in history. Yhvh proved himself to be God through his action through history. Even the validity of Prophetic messages were subject to validation through history. Exo 6:7 … you shall know that I am the LORD your God, who has brought you out from under the burdens of the Egyptians. Mat 7:16 By their fruits ye shall know them. God is known only through his faithfulness and personal experience of individuals and collective experience. Individual experiences are subservient to the collective experience – because of the Uncertainty Principles, which we will be discussing later. Thus even in knowing God and deducing the existence of God there is an uncertainty. That is why collective experience and national memories are emphasized throughout the bible. The statistical interpretation of Born (1926) In Quantum Physics, states are not observables. But there is a direct relation between the observables and the non-observables. Some times the unobservables in relation to observables are referred to as virtual variables. Several interpretations explaining the relation between the two are in existence Here the wave function is defined as the “Probability Density Amplitude.” The wording is taken following the waves. is and amplitude. So Squaring it will give the Probability Density. Since is a complex function squaring will make Born’s Statistical interpretation maintains that the absolute square of the wave function represents the probability for observation of a certain result, (For example, the probability for finding the electron within a given volume in space.) P is the probability density. It is best explained as P dx will give the probability of finding the particle between x and x+dx. There is almost complete consensus between physicists with regard to the fact that the predictions, which are obtained from theory applying this rule, match all experiments perfectly. Physicists extend this assumption to other dimensions of properties as well. Hence this should be true even in our multidimensional space, which not only includes human physical level but also the mental and the spiritual dimensions. I have discussed the basis of this assumption in the introduction. Bible abundantly corroborates this assumption. So the measurable aspect of Christian faith is to be seen only in the physical expression of love to all mankind. The virtual variable of the "God State" is placed within you. Since God is Love it should manifest itself in love. When the world saw the love of the Christians they deduced that they indeed were special. That changed the world. It was the difference from the norm that showed the unobservable wave function (the guiding potential) that lay behind the expression. If at anytime in history, the church did not produce a statistically significant increase of "good works" within it, we should certainly assume that there is something wrong in the state of the Church. If conversion does not produce a new creature, it is not a rebirth. It should show a fallacy of faith and a falsity in salvation experience. It is not difficult to identify such periods in the history of both the history of Israel and in the history of the Church in various countries. In the history of Israel prophets were not averse to declaring this. However during the church age, we tend to ignore this reality. The best way to check the state of your church is to measure the honesty and goodness of its members and compare it statistically with the general population. Is it significantly higher? If not it is time to rebuild the church on better footing. It is time for a revival. One sad tendency of the evangelical Christians is to play down the "good works", as though they do not matter. This was a necessary reaction to replacing "good works" instead of faith. But we are keeping it beyond the necessary level to the detriment of the basic tenet. Jam 2:18 But some one will say, "You have faith and I have works." Show me your faith apart from your works, and I by my works will show you my faith. The world was grappling with the various results and equation of Lorenz, Fitzgerard and Poincare before it could be explained naturally by the formulation of Relativity. The black body radiation behavior and the photoelectric effect led to the quantum theory. This is the normal process of growth. We always go back from observed facts to the reality behind the facts. If we try the other way round it may or may not be correct. This we have seen from the Platonic style of theology starting from the attributes of God. They don’t tally with actual reality or with all revelations of God without tampering. The monism of Origen condemned by the council of Constantinople was also due to such an approach. A strict monistic definition of God certainly required a created trinity. This is also the reason for the Islamic objection for incarnation. Because the possible internal coordinates are numerous, and since a slight difference in one coordinate can produce a different outcome, a single observation cannot be used to define the state of the system. This is because that observation is only one of the eigenvalues of the equation and not the whole answer. Only when we consider the whole spectrum of eigenvalues (observed facts) we can get a fair idea of the state. particles and effects Discovery of Electron In April 30, 1897, Joseph John Thomson announced that cathode rays were negatively charged particles, which he called 'corpuscles’. Thomson proposed the existence of elementary charged particles, which is now called electrons, as a constituent of all atoms. Discovery of Neutron In 1932 Chadwick proposed the existence of Neutron as a result of his studies in alpha particle collisions. Black Body Radiation 1859 Gustav Kirchhoff’s studies in blackbody radiation showed that the energy radiated by a black body depended on the temperature of the body. Attempts to explain this shape of the energy and the wavelength at which the maximum energy occur continued for several decades In 1884, Ludwig Boltzmann derived Stefan’s Law theoretically In 1896, Wilhelm Carl Werner Otto Fritz Franz Wien (1864-1928) Prussia-Germany derived a distribution law of radiation. In 1900 Max Karl Ernst Ludwig Planck (Germany 1858-1947), who was a colleague based his quantum hypothesis to explain the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies. In 1900, Planck devised a theory of blackbody radiation, which gave good agreement for all wavelengths. In this theory the molecules of a body cannot have arbitrary energies but instead are quantized - the energies can only have discrete values. The magnitude of these energies is given by the formula E = nhf where n = 0,1,2,... is an integer, f is the frequency of vibration of the molecule, and h is a constant, now called Planck's constant: h = 6.63 x 10- 34 J s Furthermore, he postulated that when a molecule went from a higher energy state to a lower one it emitted a quanta (packet) of radiation, or photon, which carried away the excess energy. With this photon picture, Planck was able to successfully explain the blackbody radiation curves, both at long and at short wavelengths. Using statistical mechanics, Planck derived an equation similar to the Rayleigh-Jeans equation, but with the adjustable parameter h. Planck found that h = 6.63 x 10-34 J·s, fitted the data. As we can see, h is a very very small number. Thus the electromagnetic waves (light) consists not of a continuous wave but discrete tine packets of energy E = hf where f is the frequency of the light. Photoelectric Effect 1905 when Einstein extended the photon picture to explain, another phenomenon called photoelectric effect. In this effect when light is allowed to fall on a metal and electrons are released. However there is a lower cut off frequency below which every electron stopped. Einstein was able to explain this assuming that photons are particles of energy E=hf. Hydrogen Spectrum 1913 Niels Bohr (1885-1962) was able to explain the discrete spectrum of hydrogen atom with the assumption that there are possible stable energy levels where electrons can stay without emitting any wave and the light is emitted when it falls from a higher level to a lower level. The frequency of the light so emitted was given by Energy of the difference in levels = hf. Compton Effect In 1923, Arthur Compton showed that he could explain the collision of a photon with electrons at rest using the same idea. These phenomena came to be known as Compton Effect Wave-Particle Duality Thus, it appears that light could behave like a wave some time (to explain reflection, refraction and polarization, interference) while at other times (Photoelectric effect, Compton effect) it behaved like a particle. The wave-particle duality of electromagnetic wave is a fact of experience and seemed mutually exclusive without compromise. In 1924 in his doctoral thesis, Prince Louis de Broglie argued that if light waves exhibited the particle properties, particles might exhibit wave properties. The experiment to test was done on a stream of electrons as particles at a double slit and single slit and the pattern exhibited fitted the interference pattern for a wave given by m= mass v = speed of the electron thus: mv = momentum of the electron Schrodinger Equation In 1926, Erwin Schrödinger introduced operators associated with each dynamical variable and the Schrodinger equation, which formed the foundation of modern Quantum Theory. A partial differential equation describes how the wave function of a physical system evolves over time. In the Schrodinger picture differential calculus was used. The time-independent one-dimensional Schrödinger equation is given by The solution for the value of E gives us a spectrum of values for the Energy of the system. Using the spherical coordinates, this equation gives: And using the separable form of the wavefunction in terms of the radial, angular parts in three dimensions Using the potential energy as: it gave the correct energy levels and correct spectral frequencies because of transitions. This indeed was the greatest success of Quantum Theory and which gave it the impetus. Operators and Quantum Mechanics In quantum mechanics, physical observables (e.g., energy, momentum, position, etc.) are represented mathematically by operators. For instance, the operator corresponding to energy is the Hamiltonian operator Where i is an index over all the particles of the system. Later Dirac developed the Matrix method and is known as Dirac Bracket formalism. In this mechanism the operators are replaced by matrices and the wave equation then reduce to a matrix equation Quantum Wave Functions and State Vectors While operators represent the observables, the operand – the function on which the operators act is known as the wavefunction , which is a function of the position for stationary solutions. Postulates of Quantum Mechanics were developed later as below: Postulate 1. The state of a quantum mechanical system is completely specified by a function that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that is the probability that the particle lies in the volume element located at time t. Postulate 2. All observables are associated with a hermitian operator. In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation The solution to the eigenvalue problem given above will give a spectrum of possible values for a corresponding to a spectrum of eigenfunctions . These eigenfunctions form a set of linearly independent functions. At any point in time, we could assume that the state of the system will be a linear combination of these functions. Some commonly used operators are given below: Postulate 3. If a system is in a state described by a normalized wave function , then the average value of the observable corresponding to is given by Postulate 4. The time evolution of system is given by The postulates of quantum mechanics, written in the bra-ket notation, are as follows: 1. The state of a quantum-mechanical system is represented by a unit ket vector \| ψ>, called a state vector, in a complex separable Hilbert space. 2. An observable is represented by a Hermitian linear operator in that space. 3. When a system is in a state \|ψ1>, a measurement of an observable A produces an eigenvalue a : A\| ψ1> = a \| ψ1> so that < ψ\|A\| ψ1> = a < ψ\| ψ1> = a since the wavefunctions are orthogonal The probability of getting this value in any measurement is \|< ψ \|ψ1>\|2 where \| ψ1 > is the eigenvector with eigenvalue a. After the measurement is conducted, the state is \| ψ1 >. 4. There is a distinguished observable H, known as the Hamiltonian, corresponding to the energyof the system. The time evolution of the state vector \|ψ(t)> is given by the Schrödinger equation: i (h/2π) d/dt \|ψ(t)> = H \|ψ(t)> Heisenberg’s Uncertainty Principle 1927 Heisenberg discovered that there is an inherent uncertainty if we try to measure two conjugate observables. This is known as Heisenberg’s Uncertainty Principle The simultaneous measurement of two conjugate variables (such as the momentum and position or the energy and time for a moving particle) entails a limitation on the precision (standard deviation) of each measurement. Namely: the more precise the measurement of position, the more imprecise the measurement of momentum, and vice versa. In the extreme case, absolute precision of one variable would entail absolute imprecision regarding the other. “I believe that the existence of the classical "path" can be pregnantly formulated as follows: The "path" comes into existence only when we observe it.” “In the sharp formulation of the law of causality-- "if we know the present exactly, we can calculate the future"-it is not the conclusion that is wrong but the premise. “ --Heisenberg, in uncertainty principle paper, 1927 In 1929, Robertson proved that for all observables (self-adjoint operators) A and B where [A,B] = AB - BA In 1928, Dirac introduced his Bracket notation and QT in terms of matrix algebra In 1932, von Neumann put quantum theory on a firm theoretical basis on operator algebra. Quantum Non-locality In 1935 Einstein, with his collaborators Boris Podolsky and Nathan Rosen, published a list of objections to quantum mechanics, which has come to be known as "the EPR paper" One of this was the problem of nonlocality. The EPR paper argued that "no real change" could take place in one system because of a measurement performed on a distant second system, as quantum mechanics requires because it will violate the relativity principles. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" [i] For example, consider a neutral-pi meson decaying into electron – positron pair. The spin of Pi meson is zero. Therefore, the total spin of electrons must be zero. Hence one of the electron will have spin (1/2) and the other spin (- ½). If the electron pair moves apart a million light years and we measure the spin of the electron on earth as ½, QM requires that the other should have a spin (-1/2) if someone measures it in his or her galaxy at the same time. How would they know which spin should it be since Relativistically it is impossible to transfer any information with a speed greater than that of light. This is the “spooky action-at-a-distance” paradox of QM There are two choices. You can accept the postulates of QM as is without trying to explain it, or you can postulate that QM is not complete, that there was more information available for the description of the two-particle system at the time it was created, and that you just didn't know it because QM does not properly account for it. So, EPR requires that there are hidden variables in the system, which if known could have accounted for the behavior. QM theory is therefore incomplete, i.e. it does not completely describe the physical reality. In 1952, David Bohm introduced the notion of a "local hidden variable" theory, which tried to explain the indeterminacy in terms of the limitation of our knowledge of the complete system. [ii] In 1964, John S. Bell, a theoretical physicist working at the CERN laboratory in Geneva proposed certain experimental tests that could distinguish the predictions of quantum mechanics from those of any local hidden-variable theory These involved the use of entangled photons – photons which interacted together at some point before being separated. These photon pair can be represented by one wave function. In 1982, Aspect, Grangier and Roger at the University of Paris experimentally confirmed that the “preposterous” effect of the EPR Paradox, the "spooky action-at-a-distance" is a physical reality. All subsequent experiments established the existence of non-locality as predicted by Quantum Theory.. [iii] In 1986, John G Cramer of University of Washington presented his Transactional Interpretation for Quantum Mechanics.[iv] In 1991, Greenberger–Horne–Zeilinger (GHZ) sharpened Bell's result by considering systems of three or more particles and deriving an outright contradiction among EPR's assumptions. They showed a situation involving three particles where after measuring two of the three, the third becomes an actual test contrasting between locality and the quantum picture: a local theory predicts one value is inevitable for the third particle, while quantum mechanics absolutely predicts a different value. Bell-GHZ showed that wave functions "collapse at a distance" as surely as they do locally.[v] [i] Physical Review 41, 777 (15 May 1935). (The original EPR paper) [ii] D. Bohm: Quantum Theory, Dover, New York (1957). (Bohm discusses some of his ideas concerning hidden variables.) D. Bohm, J. Bub: "A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory" Reviews of Modern Physics 38 #3, 453 (July 1966). [iii] J. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1 #3, 195 (1964). J. Bell: "On the problem of hidden variables in quantum mechanics" Reviews of Modern Physics 38 #3, 447 (July 1966). A. Aspect, Dalibard, Roger: "Experimental test of Bell's inequalities using time- varying analyzers" Physical Review Letters 49 #25, 1804 (20 Dec 1982). A. Aspect, P. Grangier, G. Roger: "Experimental realization of Einstein-Podolsky-Rosen-Bohm gedanken experiment; a new violation of Bell's inequalities" Physical Review Letters 49 #2, 91 (12 July 1982)
704201258f7fb397	Take the 2-minute tour × Given a delta function $\alpha\delta(x+a)$ and an infinite energy potential barrier at $[0,\infty)$, calculate the scattered state, calculate the probability of reflection as a function of $\alpha$, momentum of the packet and energy. Also calculate the probability of finding the particle between the two barriers. I start by setting up the standard equations for the wave function: $$\begin{align}\psi_I &= Ae^{ikx}+Be^{-ikx} &&\text{when } x<-a, \\ \psi_{II} &= Ce^{ikx}+De^{-ikx} &&\text{when } -a<x<0\end{align}$$ The requirement for continuity at $x=-a$ means Then the requirement for specific discontinuity of the derivative at $x=-a$ gives $$ik(-Ce^{-ika}+De^{ika}+Ae^{-ika}-Be^{ika}) = -\frac{2m\alpha}{\hbar^2}(Ae^{-ika}+Be^{ika})$$ At this point I set $A = 1$ (for a single wave packet) and set $D=0$ to calculate reflection and transmission probabilities. After a great deal of algebra I arrive at $$\begin{align}B &= \frac{\gamma e^{-ika}}{-\gamma e^{ika} - 2ike^{ika}} & C &= \frac{2e^{-ika}}{\gamma e^{-ika} - 2ike^{-ika}}\end{align}$$ (where $\gamma = -\frac{2m\alpha}{\hbar^2}$) and so reflection prob. $R=\frac{\gamma^2}{\gamma^2+4}$ and transmission prob. $T=\frac{4}{\gamma^2+4}$. Here's where I run into the trouble of figuring out the probability of finding the particle between the 2 barriers. Since the barrier at $0$ is infinite the only leak could be over the delta function barrier at $-a$. Would I want to use the previous conditions but this time set $A=1$ and $C=D$ due to the total reflection of the barrier at $0$ and then calculate $D^D$? share\|improve this question Hi Hippie_Eater, and welcome to Physics Stack Exchange! Excellent question :-) I hope you don't mind that I made some of the equations display style to aid readability. – David Z Sep 20 '12 at 17:33 Thank you, that's much better - I am still polishing my TeK-Fu so I hope I'll be making it look all sexyfine like this in the futue. – Hippie_Eater Sep 20 '12 at 17:37 2 Answers 2 up vote 0 down vote accepted Hints to the question(v5): 1. OP correctly imposes two conditions because of the delta function potential at $x=-a$, but OP should also impose the boundary condition $\psi(x\!=\!0)=0$ because of the infinite potential barrier at $x\geq 0$. 2. There is zero probability of transmission because of the infinite potential barrier at $x\geq 0$. (Recall that transmission would imply that the particle could be found at $x\to \infty$, which is impossible.) 3. Hence there is a 100 percent probability of reflection, cf. the unitarity of the $S$-matrix. See also this Phys.SE answer. 4. As OP writes, away from the two obstacles, one has simply a free solution to the time-independent Schrödinger equation, namely a linear combination of the two oscillatory exponentials $e^{\pm ikx}$. This solution is non-normalizable over a non-compact interval $x\in ]-\infty,0]$. 5. To make the wave function normalizable, let us truncate space for $x< -K$, where $K>0$ is a very large constant. So now $x\in [-K,0]$. One may then define and calculate the probability $P(-a \leq x\leq 0)$ of finding the particle between the two barriers via the usual probabilistic interpretation of the square of the wave function. 6. If we now let the truncation parameter $K\to \infty$, then we can deduce without calculation that this probability $P(-a \leq x\leq 0)\to 0$ goes to zero. share\|improve this answer I updated the answer. – Qmechanic Sep 20 '12 at 21:14 The probability of finding a particle in an interval $a<x<b$ is given by the integral $$\int_a^b \psi^ \psi \, dx ,$$ assuming that your wave function is properly normalised. So in your case, you should calculate $$\frac{\int_{-a}^0 \psi_{II}^* \psi_{II} \,dx}{\int_{-\infty}^{-a} \psi_{I}^* \psi_{I} \,dx+\int_{-a}^0 \psi_{II}^* \psi_{II} \,dx} . $$ The numerator is the region you are interested in, the denominator takes care of the normalisation so that the probability will come out between 0 and 1. I'll leave it to you to calculate the integrals. share\|improve this answer Thank you, I do have a tendency to over-complicate things. But that raises the question, what conditions should I use to figure out $A, B, C, D$? Presumably it would be the standard two regarding the continuity in of $\psi$ in $-a$ and discontinuity of $\psi'$ in $-a$ but I think I'll need more than that? Am I correct in thinking that the barrier at $0$ reflects completely and thus $C=D$? – Hippie_Eater Sep 20 '12 at 20:58 OK, so you have four unknowns ($A,B,C,D$). You already have two conditions, you need two more. As #Qmechanic states, one of the conditions should be $\psi(0)=0$. The other can be got from normalisation ($\int_{-\infty}^0 \psi^* \psi \, dx =1$). – Mistake Ink Sep 20 '12 at 21:06 Note that the scattering wave function $\psi(x)$ is not normalizable. – Qmechanic Sep 20 '12 at 23:52 Your Answer
4180f165fc74337b	Take the 2-minute tour × Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful? Of course, the obviously applications are: • contour integration • understand radius of convergence of power series • algebra with $\exp(ix)$ instead of $\sin(x)$ No need to elaborate on these ones :) I'd be interested in some more suggestions! In a way this question is asking how to show the advantage of complex numbers for real number mathematics of (scientifc) everyday problems. Ideally these examples should provide a considerable insight and not just reformulation. EDIT: These examples are the most real world I could come up with. I could imagine an engineer doing work that leads to some real world product in a few months, might need integrals or sine/cosine. Basically I'm looking for a examples that can be shown to a large audience of laymen for the work they already do. Examples like quantum mechanics are hard to justify, because due to many-particle problems QM rarely makes any useful predictions (where experiments aren't needed anyway). Anything closer to application? share\|improve this question possible duplicate of Interesting results easily achieved using complex numbers – Ｊ. Ｍ. Nov 25 '12 at 12:33 You should edit your question, then. Change "real number" to "real world", so people won't mistake it for the real numbers. – Asaf Karagila Nov 25 '12 at 13:34 Control theory! Fluid dynamics! Differential equations! Electrical engineering! Signal processing! Quantum mechanics! en.wikipedia.org/wiki/Complex_number#Applications – Rahul Nov 25 '12 at 13:37 This question is quite ambiguously phrased: all three applications listed in it belong to pure mathematics (and two answers posted so far address some aspects of this) but afterwards the OP claims to be interested in "real world" applications, where "real world" seems to be more or less equivalent to "useful to an engineer" (and @Rahul's comment answers that beautifully). Please make up your mind. – Did Nov 25 '12 at 14:09 ?? Complain? Well... If ever I had fancied answering the question, your last comment is a quite effective deterrent. (Update: upon reading your comment, I was vaguely wondering when I had previously met this tone on the site... and behold!) – Did Nov 26 '12 at 10:52 4 Answers 4 This was already mentioned by Rahul but I think it deserves an answer in its own right. Digital signal processing of 1d (sound) and 2d (images) real data would take incredible amounts of time and would be much harder to understand if it weren't for the discrete Fourier transform and its fast implementations. This field is very real and complex numbers play a major role in it. share\|improve this answer One basic example is with eigenvalues and eigenvectors of matrices. Often real matrices are not diagonalisable over $\mathbb{R}$ because they have imaginary eigenvalues, wnad knowing things about these eigenvalues can tell us a lot about the transformation that the matrix represents. The obvious example is the $2D$ rotation matrix $\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta &\cos\theta \end{pmatrix} $ with eigenvalues $e^{\pm i\theta}$ which tell us the angle of rotation that this real matrix gives us. Admittedly a simple example but I'm sure there are plenty more. On other result that comes to mind is in quantum mechanics! A big area of science right now, it deals with complex wave functions like you wouldn't believe (or maybe you would, it seems like you've done enough maths to have taken a course or two in quantum mechanics!) A lot of problems have complex solutions, and certainly the relation of $e^{i\theta}$ and trig is used to no end, particularly in solving second order differential equations (which the Schrödinger equation frequently reduces to). Probably the biggest way that the complex results are translated back to the real world is that the probability of finding a wavefunction in a given region is the integral over that region if it's magnitude squared. The complex wave function is reduced to a real integral to give us a probability, which is certainly a real world result! A lot of interesting solutions, known as steady stationary states of the Schrödinger equation, give us wavefunctions where the time dependence looks like $e^{\frac{iE_nt}{\hbar}}$. Here $E_n$ is the energy of the state and $\hbar$ is Planck's (reduced) constant. The point is, the magnitude of these solutions is independent of time. This means that if a particle has this wavefunction, then we know exactly what it's energy is for all time. Further, since the Schrödinger equation is linear, we can superpose solutions to get more solutions, and in fact these steady states form a basis, so we can find the wavefunction for any particle as a combination of these stationary states. share\|improve this answer If I were to explain that to an engineer, how can I show the need for that? Even quantum mechanics, as fundamental as it is, is probably little used by engineers who (in case the know QM) would argue that theory can't predict anyway, so they rather use experiments. – Gerenuk Nov 26 '12 at 10:13 QM in itself may be little used, but it has some hugely important applications: en.wikipedia.org/wiki/Quantum_mechanics#Applications I would highlight particularly transisters which are required for reasonable sized computers, and lasers which also have many other applications. As to saying QM can't predict, I'm afraid you're wrong! QM tells us that the world isn't deterministic so that in principle, an particular event is impossible to predict exactly at a quantum level. However it makes well defined predictions in terms of probabilities which work extremely well on large scales. – Tom Oldfield Nov 26 '12 at 13:45 I find the use of complex numbers extremely helpful in problems of plane elementary geometry, in particular when there are symmetries present which have to be exploited. In the "complex coordinate" $z$ of a point both real coordinates are encoded, you have the full vector algebra of the plane at your disposal, rotations about angles like $90^\circ$ or $120^\circ$ are obtained essentially for free, and on, and on. share\|improve this answer The trouble with maths is that, just like in the case of a living organism, all its various apparently unrelated parts are in reality interconnected. For instance, Ramanujan's prime-counting function, belonging to the field of number theory, turned out to be ultimately wrong because, in a veiled or hidden manner, it was equivalent to saying that the Riemann zeta function does not possess any complex zeroes: which, as it happens, is false. He thought that it would always predict the exact number of primes lesser than a given number, and that any error, were it to even exist, would be at worst bounded. Turns out he was wrong on both counts. Which, of course, does not mean that it cannot be used as a very good approximation, but the precision and certainty for which he was aiming proved in the end to be untouchable. And that's just one random example among many about the surprising way in which the various fields of math eventually reveal themselves to be tied together. Hope this helps. share\|improve this answer Your Answer
f1ff7d128cf4bafd	Essay:Rebuttal to Counterexamples to Relativity From Conservapedia Jump to: navigation, search This is intended as an article rebutting the points in the Counterexamples to Relativity article. That article's talk page has proven to be less than satisfactory for this purpose, because it gets archived, and much of its material has degenerated into personal disputes. We believe that the two sides of the issue are better handled in two articles—this one and Counterexamples to Relativity, rather than a talk page. Unlike most essay pages, anyone is welcome to contribute. We ask that you abide by the usual guidelines—do not remove non-vandal, non-parody, non-libelous material without discussing it first on the talk page, or explaining after-the-fact for serious problems. The cited article says absolutely nothing about the number of black holes, modeled or observed. It is about a discrepancy between computer models predicting the masses of black holes at the centers of galaxies and the observed masses. The article suggests a heretofore-unmodeled mechanism whereby galaxy evolution would cause less mass to go into the black holes. The article in no way suggests that black holes don't exist. 2. The orbital radius of the Moon's orbit is increasing, contrary to what Relativity predicts. This could be a counterexample to both GR and Newtonian gravity--in both, the radius is defined in terms of conserved quantities. The objection has been raised that this would still be a counterexample to Relativity. Yes, if it were actually true that the Moon's orbit is undergoing some kind of anomalous perturbation, it would indeed be a disproof of Relativity, Newtonian mechanics, and, in fact, all of physics since Galileo. Actually, the average radius of the Moon's orbit is in fact increasing. By 38mm a year. This was first predicted in the late 19th century and actually measured since at least the early 1970's and more accurately thereafter thanks to the mirrors left for that purpose by the Apollo astronauts. The reason for this is well-known and simple enough to be explained on science-oriented TV shows from time to time. To put it simply: the Moon pulls on the Earth, causing the tides and slowing down its rotation slightly, lengthening it's day by 2 milliseconds every 100 years. Reciprocally, the Earth pulls on the Moon and accelerates it slightly thus increasing the height of its orbit and energy is conserved. For a more complete explanation see [1]. This behavior, predicted over 100 years ago, observed and measured, is in no way "anomalous" and relativity, general, special or otherwise doesn't really concern itself with tidal mechanics so on that point at least physics, Galilean, Newtonian and Einsteinian, are quite safe for now. 3. The Pioneer anomaly. The "Pioneer anomaly" is the deviation in the motion of the Pioneer 10 and Pioneer 11 spacecraft from their predicted motion, at the distance of Saturn and beyond. It should be noted that the anomaly is about 1000 times greater than the difference between the classical Newtonian prediction and the prediction of relativity, so this is not a problem with relativity per se; it is more general than that. Calculating the force caused by heat (that is, miniscule amounts of infrared radiation) from the radioactive power source was one of the first effects that was examined. The anomaly arose when this and other known effects could not fully explain the deviation. The problem is believed to have been solved by taking into account the reflection of the radiation from the power source off of the back of the antenna dish[2]. The solution is sometimes described as an application of "Phong shading", a technique of computer graphics that is now considered imprecise. But Phong shading itself is not what is important. The "ray tracing" computer graphics technique that underlies Phong shading was what inspired the scientists to take reflection into account. The most detailed analysis to date, by some of the original investigators, explicitly looks at two methods of estimating thermal forces, then states "We find no statistically significant difference between the two estimates and conclude that once the thermal recoil force is properly accounted for, no anomalous acceleration remains."[3] 4. the solar flattening is ... too small to agree with that predicted from its surface rotation. This observation is interesting, but the predictions of oblateness come from analysis of fluid mechanics; relativity is not involved, and the cited paper makes no mention of relativity. We expect a very large rotating body to show oblateness according to well-known principles involving centrifugal force. This is particularly easy to observe with Jupiter, though the other planets, including Earth, show it too. Scientists do not know why the Sun exhibits nearly zero oblateness, but relativity is not believed to be the cause. Now it happens that there is a connection between solar oblateness and the calculation of the precession of the perihelion of Mercury, and that involves General Relativity. When checking the observed precession against the effect predicted by Relativity, one needs to subtract the effect of solar oblateness, along with the other effects, such as equinoctial precession and the gravitational effects of other planets. The effect of solar oblateness is a mere .0254 arcseconds per century, insignificant in comparison with precession (about 5500) and other planets' gravity (about 550). The figure of .0254 was calculated based on what fluid dynamics predicted, and was subtracted in the figures quoted below in item #9. When that tiny effect is removed, the error bars still overlap that for the GR prediction. 5. Quantum entanglement near the event horizon of a black hole .... It is well known that quantum gravity (that is, the "Theory of Everything") is an unsolved problem, and it is the subject of much discussion in the physics community. The whole topic of string theory, for example, revolves around this. Among the places where General Relativity and Quantum Mechanics collide most notoriously is within the Planck length of a black hole's event horizon. Among the issues involved is the notion of "quantum loss of information", that was the subject of a famous bet among Stephen Hawking, Kip Thorne, and John Preskill. (Hawking and Thorne lost.) Basically, physicists do not know just what happens within quantum-mechanical distances (the Planck length) of a black hole. But no one doubts the behavior of black holes at reasonable distances, or, for that matter, General Relativity at reasonable distances. 6. The speed of light in a vacuum is slower than expected It is unfortunate that, whenever a sensational-headline-hungry writer writes anything relating to the propagation of light, they are tempted to scream something along the lines of "Einstein proved wrong." Such sensationalist headlines can be seen in both the popular press and here at Conservapedia. But relativity never said anything about vacuum polarization. The factor "c" appearing in equations of relativity (E=mc^2, the Lorentz factor, etc.) is the calibration constant for space vs. time. While it was perceived as the "speed of light" back in 1905, that depended on the classical theory of light, from Faraday, Maxwell, et. al., at that time. Under the classical theory, Maxwell's equations were considered to be exactly correct, and light was considered to travel at exactly the speed indicated by those equations, which speed is denoted "c". The speed "c" would be better described as "that speed that all observers, even those in relative motion, will consider to be the same." The experimental basis for relativity (Michelson-Morley experiment) was that light obeyed that property. That, plus the fact that "c" appears in Maxwell's equations in a way that makes it independent of an observer's state of motion, made it clear that "c" was in fact the speed of light. But that depended on the belief that Maxwell's equations were exactly correct. But that was before Quantum Mechanics and gauge interactions. Under Quantum Mechanics, the propagation of light is determined by the behavior of photons, not by Maxwell's field equations. Of course the two results are almost identical, as is required by the Correspondence Principle. But, under the modern "Standard Model" of Quantum Electrodynamics, photons can interact with the vacuum, due to "vacuum polarization", leading to extremely tiny higher-order corrections to the equations governing the behavior of photons. This can cause photons to travel at a speed slower than "c". Now the situation is made more complex by the fact that models of supernova behavior indicate that the light is emitted after the neutrinos, because the neutrinos are emitted at the instant of the core collapse, and the light must wait until the effect of the collapse reaches the surface. After taking that into account, the cited article says that the extra delay for supernova SN1987A was 4.7 hours. That discrepancy is only 53 parts per trillion, and hence can only be observed in light from very distant supernovas. Calculating the speed of the neutrinos is interesting, because the observation is that light traveled slower than the neutrinos, even though neutrinos have nonzero mass. Since neutrinos do not participate in the electromagnetic interaction, they are not subject to the same vacuum polarization as photons. Their retardation comes only from their mass and the normal equations of relativity. Taking the mass of an electron neutrino as 0.25 eV (using the usual E=mc^2 formula that all scientists use for this), a neutrino would have to have a kinetic energy of about 0.025 MeV to experience a retardation of 53 parts per trillion. Since the energy of astronomical neutrinos is generally believed to be in the range of 0.5 to 20 MeV (it's really hard to measure) the expected retardation of the neutrinos from SN1987A is much less than that of the light. It's interesting that the issue of whether neutrinos travel faster than light actually did come up in another round sensationalist headline-grabbing articles, from an experiment at Gran Sasso, in the popular press and here at Conservapedia. The first headline was that neutrinos traveled faster than light, a claim retracted after a cable was fixed, but then reported here that someone (erroneously) had said they had traveled at exactly the same speed as light, which also violates relativity because neutrinos have nonzero mass. In any case, that discrepancy would have been several orders of magnitude too small to measure in the Gran Sasso experiment, and the discrepancy from vacuum polarization several orders of magnitude smaller still. 7. Celestial signals defy Einstein. It shouldn't be surprising that, when Einstein died nearly 60 years ago, he didn't know everything that there is to know about spacetime. Newton didn't know everything there is to know about calculus (Hilbert spaces) or classical mechanics (Lissajous orbits), and Bohr and Schrödinger didn't know everything there is to know about Quantum mechanics (entanglement). What Einstein knew, at the time, about spacetime was how to give a precisely relativistically correct formula for how matter and energy (and momentum and stress) give curvature to spacetime, which in turn gives rise to gravity. These equations have been confirmed, with great accuracy, repeatedly. Einstein's equations are known to work for "ordinary" interactions at the planetary and galactic level, but don't work at the quantum level, and may not be fully correct at the deep cosmic level. The cited article says "Every object there is, from a planet orbiting the sun to a rocket coasting to the moon or a pencil dropped carelessly on the floor, follows its [spacetime's] imperceptible contours." This is a confirmation of what Einstein said. The article says that there may be more subtlety to spacetime than was previously known. Of course this has been hinted at for some time with recent developments in cosmology. "Breaking relativity" is an unfortunate choice of subtitle. 8. Relativity breaks down at high energies. This is interesting. We look forward to seeing whether this plays out the same way that the supraluminal neutrinos did. Until then, note that the discrepancy, 33 picoradians, is such that, if you were to shine a laser beam at the Sun, with that amount of angular error, it would miss its target by about 5 meters. We hope the author will explain how he measured an angle that precisely. 9. Subatomic particles have a speed observed to be faster than the speed of light, which contradicts a fundamental assumption of Relativity. The Italian lab that "shocked the scientific world" has announced more precise results, confirming their previous announcement. This is an interesting observation. The world's best scientific minds are looking into it. That relativity is incorrect is not being taken seriously as a possible explanation. Update: The problem seems to have been caused by a faulty cable connection between a computer and a GPS unit. When the connection was repaired, the travel time increased by 60 nanoseconds, which had been the amount of the anomaly [4] [5]. The claims of faster-than-light neutrinos have now been refuted very thoroughly[6][7]. The objection has been raised that a recent news report from the BBC ("now we are 100% sure that the speed of light is the speed of neutrinos") is also contradictory, since neutrinos have mass and are therefore forbidden by relativity to travel at exactly the speed of light. Since the neutrino energies were 17GeV, and the current estimate for the neutrino mass is about 0.25eV, the deviation from the speed of light would be about 1 part in 1022. This means that the neutrinos would arrive at the detector about .26x10-24 seconds (.26 yoctoseconds) later than the speed of light itself. This is one quarter of a billionth of a femtosecond, or about .26x10-15 of a nanosecond. The accuracy of the GPS units and cesium clocks used in the measurement is greater than a nanosecond, so the discrepancy cannot possibly be detected. It is unfortunate that the claim "the speed of light is the speed of neutrinos" was taken so literally. A "counter-rebuttal" has been made: The politicized rush to rehabilitate the Theory of Relativity was far from convincing, and the "resolution" was a clear statement that is flatly contrary to the Theory. Just what statement is being referred to as the "resolution" is not clear, but our best guess is that it is the statement by Sandro Centro, quoted by Jason Palmer in the cited BBC article. As noted above, that statement is in error by about 15 orders of magnitude less than the resolution of the measurement. None of the assertions of relativity denies that anyone could ever make a statement that is not precisely correct. This may be another case of the Pioneer anomaly, or it may be something else. However, it is very unlikely that it shows that relativity is wrong and Newtonian mechanics is correct. Saying that, every time someone finds some phenomenon that is puzzling, that shows that relativity is wrong, is not a convincing way to do science. The cited paper is about unexpected behavior of some spacecraft as they use "gravity assists" in near-Earth flybys. The hypothesized causes involve errors in the mathematical models to calculate such effects as relativistic effects (the detailed calculation of them, not the question of whether they exist), tidal effects, Earth radiation pressure, or atmospheric drag. The paper suggests that most of those can be ruled out, though there could be round-off and integration errors, or errors in the spherical harmonic representation of Earth's gravity field. 11. Spiral galaxies confound Relativity, and unseen "dark matter" has been invented to try to retrofit observations to the theory. Correct me if I'm wrong, but wasn't it due to the acceleration of various parts of galaxies that accelerated funny that led to dark matter (based on simple Newtonian dynamics)? This is the dark energy/cosmological constant argument. The cosmological constant was added by Einstein after he discovered that his field equations (\mathbf{G}=8\pi \mathbf{T}) predicted that the universe was expanding, contradicting his firm philosophical belief in a static universe. So he inserted \Lambda \mathbf{g} to the LHS so that it would predict a static universe. A few years later, Hubble showed the universe to be expanding, and Einstein called the cosmological constant the worst mistake of his career. So, it sort of had a bad reputation, and people didn't want to seriously consider it, until recent observations have shown the universe's expansion to be accelerating forced them to do so. It could have had a very different history. Einstein could have had that term in the EFE's from the start, and pointed out that it would determine if the universe's expansion was accelerating (or not expanding at all!) and it would take further observation to determine its value. A footnote goes on to say that "In a complicated or contrived series of calculations that most physics majors cannot duplicate even after learning them, the theory of general relativity's fundamental formula, G_{\mu\nu} = 8 \pi K T_{\mu\nu}\,, was conformed to match Mercury's then-observed precession of 5600.0 arc-seconds per century. Subsequently, however, more sophisticated technology has measured a different value of this precession (5599.7 arc-seconds per century, with a margin of error of only 0.01) ..." Considering only the anomalous precession, that is, the precession that remains after all known other factors (other planets and asteroids, solar oblateness) have been accounted for, general relativity predicts 42.98 ±0.04 arcseconds per century. Some observed values are: 43.11 ± 0.21 (Shapiro et al., 1976) 42.92 ± 0.20 (Anderson et al., 1987) 42.94 ± 0.20 (Anderson et al., 1991) 43.13 ± 0.14 (Anderson et al., 1992) [Source: Pijpers 2008] These error bars, and that of the relativity formula, all overlap. The formula for mechanics under general relativity is complicated, but it is not contrived or conformed. "Conformed" suggests that it was somehow adjusted or "tweaked" to match the 42.98 figure. The formula is To begin to explain the formula, Newton's law of gravity, combining F = ma and F = \frac{KMm}{r^2}\,, is a = \frac{KM}{r^2}\, In Einstein's equation, T_{\mu\nu}\, is the "stress-energy tensor", and 8 \pi T_{\mu\nu}\, gives the density of the Sun, taking the place of \frac{M}{r^2}\,. G_{\mu\nu}\, is the "Einstein curvature tensor", and says how spacetime curves to create an apparent gravitational acceleration. There is nothing to tweak to get a value of 42.98 arcseconds. 8 is 8. \pi\, is \pi\,. K is Newton's constant of gravitation in both formulas. The above discussion notwithstanding, more sophisticated technology has measured a precession of 5599.7 arc-seconds per century, with a margin of error of only 0.01, which disproves the prediction of the Theory of Relativity. Notice how publication of data stopped two decades ago when the observations diverged from the theory.--Andy Schlafly 14:50, 14 April 2012 (EDT) Measurements of planetary motion are now calculated relative to the "International Celestial Reference Frame" (ICFR), replacing the older, and much less accurate "equinoctial" frame. The older measurements got a value of about 5600 arcseconds/century for the precession of Mercury, nearly all of which (5025 arcseconds) was because of the rotation of the "equinoctial" frame. The ICFR frame removes that source of uncertainty, and, with the very accurate radar measurements conducted by NASA between 1987 and 1997[8], gets a value of 574.10±0.65 arcseconds observed, in good agreement with the predicted 574.64±0.69 value. The ICFR is described in this document, dated 2003. REPLY: The year 1997 was nearly 20 years ago, and the observed data based on increasingly sophisticated technology was diverging from relativity's predictions even then. Recent data on this is not published because it would further disprove relativity and embarrass relativists who cling to the theory. The cited Jurgens-Rojas-Slade-Standish-Chandler paper was published in April 1998, so it should come as no surprise that their data came to an end in 1997. They probably decided that, after collecting data for 10 years, it was time to publish. This is common in the scientific community. Galileo's experiment involving cannonballs and the Leaning Tower of Pisa is no longer being conducted. The Stern-Gerlach experiment (just to pick one random example) was conducted only a small number of times before being published. People don't continue to conduct it to see whether spin quantization of Silver atoms still occurs. More recent Mercury data, particularly from the MESSENGER space probe, have provided positional data vastly better than anything Leverrier could have dreamed of. (In fact, Because of the many interplanetary spacecraft of the last few decades, we now have a huge amount of incredibly accurate data on a large number of bodies in the Solar System.) Perhaps Andy believes that the scientific community has been remiss in not continuing to analyze these Mercury data to the present day, presumably looking for evidence of whether relativity is true or false. Perhaps Andy could provide his own ideas about why, other than relativity, the precession occurs. This would not just be a matter of quibbling over 42.98 and 42.99, but of explaining the discrepancy between 42.99 and zero. This point was brought up on the Community Portal in December, with no reply forthcoming. Of course, even absent an alternative theory, showing a discrepancy between observation and theory would be interesting. Now a possible approach, if one believes that the data are inconsistent with relativity, would be to bring the subject up in the many internet forums devoted to physics. One might be able to find out what further analyses are being done, suggest new analyses, or find out how to get the data to make one's own analysis. Getting to the specific points of the note above, we are not aware of any evidence that the data were diverging from the theory back in 1997; perhaps Andy could provide supporting data. We see no evidence that the data exist but have not been published, and we see no evidence that any such lack of publication arises because it would embarrass anyone. Seeing discussion of these points in an internet physics forum might help clear these matters up. It's one thing to speculate on science; it's quite another to speculate on the motives of scientists, particularly the idea that scientists are embarrassed by their knowledge that relativity is wrong, and that they are covering up this embarrassment as part of some political agenda. REPLY: The relativists' silence in the journals about increasingly precise measurements of the advance of the perihelion of Mercury is akin to the famous story of the dog that didn't bark, which is itself compelling proof. You can't just invoke the Sherlock Holmes Silver Blaze story to support any claim you wish to make. In that story, Holmes knew that the guilty person had to have been in a certain house at a certain time. The dog would have barked if that person had been a stranger, because dogs bark at strangers. Therefore, the villain was in his own house. There was a specific and credible chain of logic from the non-barking dog to the identification of the guilty party. In the perihelion case, there is abundant orbital data (undoubtedly gigabytes of it) for the various planets and moons. Some of it was in the Jurgens et al paper noted above, and other data was analyzed in the Pijpers paper. That paper, by the way, says "The value of the gravitational quadrupole moment (28) when combined with planetary ranging data for the precession of the orbit of Mercury yields a value [...] which is consistent with GR ..." There is actually a very simple explanation for the scientific community's silence on this matter, a deduction of which Mr. Holmes would approve: The reason that no one is publishing data or analyses that claim to refute relativity is that such claim would be false, and that the existing analyses, using exquisitely accurate spacecraft data, are correct. 14. Despite wasting millions of taxpayer dollars searching for gravity waves predicted by the theory, none has ever been found. Sound like global warming? True, the direct searches for gravitational waves have not yet yielded any clear results, though indirect observations have been made (the Hulse/Taylor observations.) Before people dismiss indirect observations, recall that no one has ever seen an electron. The Earth-based LIGO detectors have, so far, not detected any unambiguous gravitational wave signatures from such events as a black-hole merger. It is barely sensitive enough to find such things in the Milky Way or very nearby galaxies. It is being upgraded in a plan that should complete in 2014. It is hoped that, after the upgrade, it will be able to see these events clearly and unambiguously. The space-based LISA detectors have not been built yet, and the original proposal has been scrapped because of budgetary problems. A new version, called "eLISA" has been proposed, and should be sensitive to events as far away as redshift 15. Whether these experiments are a good use of money is another question, one that has no bearing on whether relativity is correct. The objection has been raised that the experiments should not have been funded if scientists were going to ignore negative results. The results are only partly negative. The scientists knew all along that a certain amount of luck would be involved in finding a sufficiently strong signal within the time frame of the experiment. The failure so far does not mean that black-hole mergers do not emit gravitational waves; just that they haven't been lucky enough to find them. They will continue to search. Update: Another, much more commonplace observation of gravitational wave emission has been reported[9]. The article suggests that, since it shows detectable gravitational waves are more common than previously thought, there is optimism that the eLISA detector, when completed, might find one source per week. This has nothing to do with global warming. The formulas for velocity, momentum, and mass can in fact be written in such a way that they appear to have discontinuities, just as the tangent function has discontinuities while the underlying sine and cosine functions do not. But they can also be written in a form that does not show discontinuities. All particles, with or without mass, can have any value of momentum. The formula for the velocity of a particle, in terms of its mass and momentum, is v = \frac{pc}{\sqrt{m^2c^2+p^2}} For a particle with mass, this means that momentum of zero gives a speed of zero, and, as the momentum approaches infinity, the speed approaches c. For a massless particle, the speed is always c. 16. observations don't match predictions and cosmic causality. The fact that "observations don't match predictions" has shown up many times in the history of science. That is, the accepted wisdom of the day was found to be false, leading to improved theories: • There were once assumed, by Aristotle and many others, to be four elements: earth, air, fire, and water. Subsequent experimental investigation, by numerous people, replaced that theory. • The ancient notion of gravity, that objects fall at a speed proportional to their mass (often ascribed, perhaps imprecisely, to Aristotle), was found to be contrary to experimental evidence, and was replaced by Galilean/Newtonian mechanics. • The geocentrism of Ptolemy lasted a long time until it was found to be contrary to experimental evidence, and was replaced by the Heliocentric theory of Copernicus, Galileo, and Newton. • The "phlogiston theory" of combustion was accepted until it proved to be contrary to experimental evidence, and was replaced by the modern oxidation theory by Antoine Lavoisier. • The "caloric theory" supposed that heat was a material substance called "caloric", and that that substance was conserved. It took a long time for Joule and others to develop the modern "kinetic theory" as a replacement. The current "counterexample" is about the "horizon problem", a problem of cosmology which has been known for a few decades. Under the kind of expansion of the universe that non-quantum-mechanics would require, there are places in the universe that are not causally connected and yet have nearly the same temperature. The classical expansion of the universe would have magnified early nonuniformities by about 50 orders of magnitude, an impossible situation. The currently accepted theory dealing with this problem is "inflation". Not all scientists accept this, but most do; other explanations are more exotic than most scientists are willing to accept. An interesting thing about inflation is that it was formulated to solve a different problem—the dominance of matter over antimatter. It was found to address the flatness problem as well. When a theory explains phenomena other than what it was intended for, that of course lends it additional credence. Whether cosmic inflation is implausible is not for us to say. The existence of subatomic particles was once considered implausible. The cited National Geographic article was in fact not about the problem of temperature uniformity, but about various hypotheses, much less widely accepted than inflation, about alternate universes, and whether black hole singularities might connect them. The issue is not about the existence of black holes, but about the connectedness of their singularities. This is really fascinating stuff. The book The Hidden Reality, by Brian Greene, is a fascinating account of these issues. We highly recommend it. The simple answer is, unequivocally, that it acts on the 'relativistic' mass. The question seems to relate to a simple misunderstanding of Special Relativity. Einstein's theories lead to the conclusion that observers in different inertial frames of reference (i.e. observers with differing, but constant velocities relative to the thing being observed) will observe different inertial masses in the body being observed. However, there is no variance in the body's mass with regard to the direction of the force. Thus to a given observer, a force in any direction will operate on the same mass. However, to a different observer, this mass may be different, although still the constant with regard to the direction of the force. 18. The observed lack of curvature in overall space. Spacetime has very definite curvature near any massive object—this is what makes gravity work. The global curvature of spacetime is an altogether different issue. Whether the average global curvature is zero has consequences for cosmological theories, but it has essentially no effect on the curvature that keeps the Earth in its orbit. If it did have an effect, the issue would have been settled long ago. 19. The universe shortly after its creation, when quantum effects dominated and contradicted Relativity. We're still working on a quantum theory of gravity; this isn't so much a counter-example as saying that (classical) GR isn't valid in that domain. 20. The action-at-a-distance of quantum entanglement. Special Relativity only forbids the transmission of matter, energy or information at a speed faster than light. There are plenty of other things that can move faster than light. Consider a laser on Earth which is rotating on a pivot, whose light shines onto the hull of a satellite 200,000Km away (2e8 metres). If the laser rotates at a sedentary one revolution ever four seconds, the speed of the laser beam's tip crossing the satellite's hull is 3.14e8 metres per second - faster than the speed of light. However, this is not a transfer of information. Any information is travelling from Earth to the satellite, obeying the universal speed limit. Similarly, the only information that can be transmitted by the quantum entanglement of two particles is from the originator of the particles to the two observers, not from one observer to another. Faster than light transmission of information using quantum entanglement has never been observed, nor has even conceived how such a mechanism might work.[10] Gravitons are a prediction of Quantum Theory, not of relativity, although the concept is an extension of the relativistic idea that forces take a finite time to be transmitted over a distance. No one expects to observe gravitons. Calculations show that it is well-nigh impossible with any conceivable detector that we could build. No one is designing, funding, or building any apparatus to search for gravitons. Now it happens that theoretical physicists discuss and speculate on the existence and nature of such things as gravitons as part of their theoretical work. Some of these discussions take place among scientists who receive their salaries from various government agencies that are funded by taxpayers. Whether all of the things that scientists think about, talk about, and write about constitute a good use of money is not for us to say. Whilst this observation is unconfirmed, if true it would still not invalidate relativity. Many things may vary with position in space, and relativity does not deny this. There is no suggestion that the fine-structure constant is different at the same point in space for observers in different non-inertial frame, as the 'counterexample' implies. When physicists encounter something puzzling, their first reaction is usually not to assume that it shows that relativity is wrong. In fact the cited article never mentions relativity at all. It is about "magnetars", a type of neutron star about which very little is known, other than their extremely strong magnetic fields. Many scientific discoveries have arisen from observations that were puzzling at first. Edwin Hubble's observations of galactic redshift led to the realization that the universe is expanding isotropically. The observations by Jocelyn Bell and Antony Hewish, of periodic pulsation in the radio emissions of a star, were so puzzling at first that they seemed to suggest transmissions from intelligent extraterrestrials. They had actually discovered pulsars. When Carl Anderson saw unexplained particle tracks in a cloud chamber, he had discovered antiparticles. None of these people assumed that the explanation for these observations was that relativity was wrong. They were all astrophysicists and were quite familiar with relativity. In fact, Anderson's antiparticles had been predicted by the Dirac equation, a synthesis of special relativity and the Schrödinger equation of quantum mechanics, from a few years earlier. It is utterly baffling how anyone could make such an assertion. The insights from relativity are multitudinous. Relativity forms the basis for astronomy, cosmology, electrodynamics, and many other fields. The interconnection between the electric and magnetic forces is now seen to be a straightforward consequence of relativity. Relativity combined with quantum mechanics are the basis of all of contemporary physics. The Dirac equation, which gives rise to antiparticles and the theory of spinors, was an early example of introducing relativity into the Schrödinger equation. More generally, special relativity was combined with quantum theory to produce quantum field theory (QFT). An example of a QFT is quantum electrodynamics, the most precise theory in physics. Furthermore, string theory is Lorentz invariant and produces general relativity in the low energy limit. 25. The change in mass over time of standard kilograms preserved under ideal conditions. We are baffled that anyone would connect this problem with relativity. The non-relativity principle of conservation of mass has been known, in general, for hundreds of years, going back to the time of the alchemists, and has been a fundamental and accurate principle since the 19th century. The principle of conservation of energy has also been a fundamental and accurate principle since the 19th century. Relativity simply generalizes this to a principle of both together, with even greater precision. So, in principle, it is recognized that the mass of the standard kilogram could change if an energy transfer took place. But no combustion, corrosion, or nuclear decay is suspected of having taken place. In any case, the amount of energy that would have to have been released is 1.25 megawatt-hours, which would certainly have been noticed. Relativity does not promise uniformity of the universe any more strongly than classical physics did. The apparent change in mass of the standard kilogram is simply a mystery. 26. The uniformity in temperature throughout the universe. The cited article is fascinating, and is about a fascinating aspect of contemporary physics. Like item 23, it would have been useful to say what the article is about. The cited article is about speculation that the constant "alpha" (see item 17 above) may not be constant. It might have decreased, by 45 parts per billion, as recently (in cosmic time) as two billion years ago, based on data from the Oklo "natural fission reactor". Other measurements have been made of alpha at earlier times, such as measurement of light from distant quasars. These measurements suggest that alpha has increased by a few parts per 105 in 12 billion years. There is plenty of literature on theories about change in alpha, and some of it indicates that this may be due to change in the speed of light. Specifically, the Oklo data may suggest that the speed of light may have been increasing slightly. (This is the opposite of the direction of change claimed by fundamentalists, but is much smaller in any case.) Speculation on a different speed of light in the past may relate to theories of "cosmic inflation", which touches on the question of why the Cosmic Background Radiation is so nearly isotropic, which indicates a near-uniformity of temperature, which requires inflation or some equivalent mechanism. None of the scientists working in this area seem to doubt the fundamental correctness of relativity. 27. "According to Einstein's view on the universe, space-time should be smooth and continuous" but observations instead show "inexplicable static" greater than "all artificial sources of" possible background noise. The cited article is about a very recent and exciting development in fundamental theoretical physics, generally called the "holographic principle". This proposes that our perceived 3-dimensional space actually arises from a "hologram" on a 2-dimensional space. Much has been written about this recently, including the best-selling book The Hidden Reality by Brian Greene. This hologram manifests itself in the "foamlike" ripples of spacetime, at the scale of the Planck length, which is so small that there seemed to be no way to detect it directly. But it seems that some "inexplicable static" in the results of another unrelated experiment, searching for gravitational waves, may be the first hints of the holographic nature of space. If so, this is a lucky and serendipitous result. Serendipitous scientific discoveries have been made many times, as when Henri Becquerel put a photographic plate in a dark drawer because the weather was cloudy, and thereby discovered radioactivity. That the "foamlike" nature of space at short distances is contrary to the continuous nature presumed by classical mechanics and relativity has been known for some time. This is the problem that quantum gravity seeks to solve. Quoting things without explaining the context is often a bad idea, and the indicated item is a good example of this. It gives no hint of what the article from which the quote was taken is about. The article is about one scientist's contribution to the problem of unifying relativity and quantum mechanics. The scientist, Petr HoYava of Berkeley, has come up with an approach that he says eliminates the infinities that have plagued other unification attempts. The two quoted sentences are HoYava's statement of the problem. So it comes as no surprise that he says that there is a conflict between relativity and quantum mechanics. The very next paragraph begins: The solution, HoYava says, is to snip threads that bind time to space at very high energies .... At low energies, general relativity emerges from this underlying framework ..." It is well known that, just as classical mechanics emerges from quantum mechanics at non-microscopic scales, and classical mechanics emerges from relativity at low speeds, both relativity and quantum mechanics should emerge from the Grand Unified Theory (whatever that turns out to be) at the appropriate scales. The topology of wormholes is an interesting topic, widely discussed among theoretical physicists and mathematicians. There is much speculation about what they would be like (if they could exist at all under quantum gravity), and what kind of "cosmic censorship" or "chronology protection" theorems might make practical time travel impossible. The nature of these theorems seems to be intertwined with theories of quantum gravity and "grand unification", so the exact form of the "cosmic censorship", if it exists, can't be known until quantum mechanics and relativistic geometry are unified. It is an exciting field of research. No one believes that the "cosmic censorship" will take the form of relativity not being a true non-quantum approximation to reality. Since the 70's much work has been done on the subject of black hole thermodynamics[11][12], most notably by the Lucasian Professorship of Mathematics at Cambridge Stephen Hawking. When quantum field theory is added to the analysis of black holes it is found that they do not possess "low entropy" (quite the opposite, in fact) and are consistent with the laws of thermodynamics[13]. The Counterexamples to Relativity article has labelled this work in a footnote as "[c]ontrived explanations", with no explanation for this characterization given. Please read the cited paper carefully. It is a survey of their observations over a 30 year period. They point out that their data matches general relativity to within 0.2 percent, and is now down in the "noise" of other effects, such as lack of accurate knowledge of just how far away the pulsars are, and lack of accurate knowledge of galactic constants. As they say in the abstract, "tighter bounds will be difficult to obtain." The paper, written in 2004, also notes that, because the pulsar beams are slowly tilting out of the line of sight to Earth, "A core component [of the emission] is quite prominent in the data taken in 1980-81, but it faded very significantly between 1980 and 1998 and was nearly gone by 2003." That is why they are not releasing further data. 30 years is a fairly long time to watch a pair of pulsars. They're not doing the experiment any more—it did its job, and it's finished. No one drops cannonballs off the Leaning Tower of Pisa any more either. The data are not diverging from the predictions. The Global Positioning System (GPS) uses general relativity to achieve greater accuracy[14]. In fact, relativity is what makes the magnetic force necessary. The magnetic force is used in, among other things, electric motors and generators. Even if it were the case that no practical applications have come from relativity, that is irrelevant. The validity of a theory is not based on the creation of useful devices. It is based on its ability to accurately predict the results of an experiment. Relativity "has held up under extensive experimental scrutiny" [15]. If scientific theories were judged by their application in useful devices, the following Nobel-worthy theories would be rejected: cosmic inflation parity violation in the weak force the "standard model", with strange/charm/top/bottom/mu/tau the Chandrasekhar limit for white dwarf stars The rules for calculating inertia and other questions of mechanics are well known. The inertia, that is, the way that a force affects an object's momentum, is well known. Hundreds of physics textbooks discuss this in great detail, in terms of the Lorentz transform and the concepts of the force and momentum 4-vectors. The "inertia" comes from what is now called the mass, which used to be called the "rest mass". Archaic treatments formulated this in terms of the "relativistic mass", which was different. The mass is a scalar, and has no direction. The formulas for calculating the motion in terms of forces, in the direction of motion or transverse to it, are well known. This seems to be another basic misunderstanding of relativity, from someone who gave up halfway through the textbook. Newtonian momentum (p = mv) does certainly indicate that a body with zero mass (m) must have zero momentum whatever its velocity (v). However, the relativistic equation for momentum is: p = \gamma m_0v\, where m0 is the rest mass of the object and ? is the Lorentz factor, given by \gamma = \frac{1}{\sqrt{1 - (v/c)^2}}\,, where c is the speed of light. For the case of a photon, where rest mass is zero and v is equal to c, this gives p as zero divided by zero - an undetermined value. However, with the substitution of the famous E=mc2, where E is the energy of the body, the momentum equation can be rearranged to: pc = \sqrt{E^2 - m_0^2c^4} With a photon of zero rest mass, this gives: p = E/c\, Finally, substituting Planck's Equation for the energy of a photon E = hf\, where h is Planck's Constant and f is the frequency of the photon, we get the familiar (and experimentally demonstrated) value for a photon's momentum of: p = hf/c = h/\lambda\, where λ is the photon's wavelength.[16] There are no "conditions of a conservative field". A conservative field is one that has a curl of zero. Perhaps what was meant was that the gravitational field around the Sun, under Newtonian mechanics, is conservative. This is true because it is the gradient of an inverse-square scalar field, and all gradients have a curl of zero. Under relativity, the curl is also zero, due to the Bianchi identity and the symmetries of Riemann's tensor. See the extensive discussion here. The reference to the twin paradox suggests that the author thought that the passage of time is some kind of scalar field that should be obtainable as the path integral of a conservative vector field. It is not. Passage of time is a property of one's path through spacetime, and is similar to path length. (In fact, under the Lorentz/Minkowski metric, it is exactly path length.) Just as two paths from point A to point B on a sheet of paper can have different lengths, the paths of the twins can have different lengths, and hence different elapsed local times. 36. The Ehrenfest Paradox: Consider a spinning hoop, where the tangential velocity is near the speed of light. In this case, the circumference (2πR) is length-contracted. However, since R is always perpendicular to the motion, it is not contracted. This leads to an apparent paradox: does the radius of the accelerating hoop equal R, or is it less than R? The "Ehrenfest paradox" is not an actual paradox. Non-inertial relativistic motion of solid bodies is quite complicated, involving such concepts as "Born rigidity", "Langevin observers", the "Langevin-Landau-Lifschitz metric", and "quotient manifolds". In general, the subject is complicated, and has provided physicists with much food for thought. But it does not disprove relativity. The claims of that item are preposterous. Read the cited paper (or its abstract) carefully. Einstein's statement that clocks would run slower at the equator, due to time dilation, was correct according to special relativity alone. What Einstein didn't realize, because he wouldn't discover general relativity for another 10 years, was that the gravitational time shift would offset that. A "geoid" (the shape dicussed in the paper) is a theoretical shape used in mathematical physics, that is in equilibrium between the effects of centrifugal force (from rotation) and gravity. It is essentially an oblate spheroid. It can be thought of as the shape a rotating planet would take if it were completely fluid. Or it could be thought of as the shape of "global sea level". Jupiter, because of its rapid rotation, has a very pronounced flattening at the poles. Since Jupiter is not solid, its shape is a geoid. The Earth is very nearly a geoid, of course. But not exactly, because of gravitational nonuniformities, and things like mountains, that can exist because of the Earth's rigidity. No one "made new assumptions about the Earth's shape" to justify anything. The assumption that the Earth's shape is a geoid is a theoretical assumption due to the approximately fluid nature of the Earth. But no one claims that the Earth's shape is anything other than what it is observed and measured to be. What the paper is about is the fact that the effects of rotational speed and gravitational time dilation happen to cancel each other on an ideal geoid. So all clocks at "sea level" on an ideal geoid-shaped planet run at the same speed. Whether this result is implausible is not for us to say. The cited paper does not refute relativity. The physics and mathematics underlying the "twin paradox" are well known. That one of the twins will have had to undergo different accelerations from the other before returning to the same point is what enables them to perceive different passage of time. This does not contradict relativity, and Einstein never said that it does. His explanation in terms of different acceleration is correct. The comment about extending the length of the trip so that the acceleration would be de minimis is wrong. It seems to suggest that the acceleration could be reduced until it is negligible. It can be reduced by lengthening the trip, but it is not negligible. The Lorentz transform, and the equations of motion, are mathematically exact. The integral of a very small function over a long period is still significant. If the twins followed different paths in spacetime, which they must in order to measure different elapsed proper time, they must have undergone different accelerations, however small those differences may have been. Of course, if they never come back to the same point, they could both undergo zero acceleration. Quantum field theory abounds with fields. The Higgs particle has a Higgs field. It has nothing to do with the "luminiferous aether". 40. Minkowski space is predicated on the idea of four-dimensional vectors of which one component is time. However, one of the properties of a vector space is that every vector have an inverse. Time cannot be a vector because it has no inverse. Time isn't a vector. It is a component of the vector space known as "spacetime". Vectors have negatives; the word "inverse" is not typically used here. While there are thermodynamic and other reasons for not allowing time to go backwards in the real world, the mathematics of spacetime allow vectors with any components, even negative ones. Mathematicians define a 4-dimensional vector space as having two operators: addition and scalar multiplication. There is also the identity vector (0,0,0,0). So, to the extent that one asks what vector can be added to (0,0,0,t) to produce (0,0,0,0), the answer is (0,0,0,-t), but that has nothing to do with relativity. That a Bible verse "likely" refers to a scientific phenomenon does not make good science. The objection appears to be claiming that the Bible contradicts the Michelson-Morley experiment, so the latter must be wrong. The Michelson-Morley experiment is very well known, has been repeated countless times, and is incontrovertible. To suggest that the Bible verse in Genesis contradicts such an incontrovertible phenomenon does a disservice to both the Bible and to science. Modern formulations of the Lorentz aether theory may very well make it completely equivalent to relativity, both special and general. But that doesn't make relativity wrong; it just means that another theory is just as correct. The universal preference of the scientific community for relativity over the Lorentz theory is probably not based on religious faith, but on simplicity, as expressed in Occam's razor. The Lorentz theory postulates an aether that no experiment can possibly determine the properties of, while relativity postulates no aether. They are not "beyond understanding". They are simply beyond closed-form solution. Cosmologists work with approximate solutions, calculated by extensive computer calculations, all the time. A well-known example of this is the simulation of galaxy dynamics. Particle physicists do this also, in, for example, quantum chromodynamics (QCD) calculations. It is fortunate that the equations of Keplerian/Newtonian planetary motion were solvable by the mathematical methods of the 17th century—a closed-form solution to a second-order differential equation. Modern physics problems are much harder. But, with modern computers, we can solve the equations of gravity to enormous precision. 44. Experiments in electromagnetic induction contradict Relativity: "Einstein's Relativity ... can not explain the experiment in graph 2, in which moving magnetic field has not produced electric field." The first cited reference, from which the quote was taken, is a totally crackpot web page, from a web site that seems to specialize in hosting crackpot papers. The writing is utterly illiterate and incoherent, as in this sentence: "According to Faraday's Law it can be explained as that, duo to the magnetic flux in conductor line changing, firstly induced electromotive force dU coming from the line-winded conductor to bring out voltage, then based on differential form I = \frac{-\sigma s dU}{dl} of Ohm's Law, the physical natural would be regarded as 'voltage before electric current' " The equations of electrodynamics (Maxwell's equations), and their connection with relativity, are well known. Hundreds of electrodynamics textbooks cover this subject. The equations are correct at all realizable speeds, even relativistic (near the speed of light) speeds. (Maxwell's equations are said to be the only equations from classical physics that did not need to be modified for relativity.) The equations correctly describe the behavior of magnets (this is presumably what was meant by "solenoids"), charges, and electric and magnetic fields, even at speeds near the speed of light. Of course the equations don't work at the speed of light. The cited article never discussed speeds near the speed of light, only at the speed of light. The questioner was rightly taken to task for his physically unrealizable assumption. 46. The Pauli Exclusion Principle states that no two electrons... This is wrong on many levels. If there were exactly as many quantum-mechanical states (eigenfunctions) as there are particles, then, indeed, a fermion particle could only go to another state if the particle already in that state moved. But there are many more available states than there are particles. The cited article was a lecture by Brian Cox, a "popularizer" of physics, and the comments on the web page take him to task over a great many points, including a confusion between "quantum states" and "energy states", and what quantum-mechanical interconnectedness really means. People would be well advised to read those comments, as well as the analysis by Sean Carroll here, which points out the many flaws in Brian Cox's reasoning. In any case, this is about quantum mechanics, not relativity. 47. The recent findings of gravitational waves are actually just dust. This is about the observations by the "BICEP2" telescope early in 2014. The claim was that analysis of the polarization of the Cosmic microwave background (CMB) would show that the CMB had been influenced by gravitational waves in the first fraction of a second after the big bang. That is, if noise from interstellar dust grains didn't confound the measurements. It is now believed that the dust grain noise is too high to make this a reliable measurement. Please read the cited article. It indicates that, based on observations by the Planck satellite, it should be possible to make a cleaner observation. The balloon-borne "Spider" telescope is slated to be launched later this year, and it might be successful. Taking the failure of the BICEP2 observations as a counterexample to relativity is like saying "I thought of a new and clever proof of the existence of God last week, but unfortunately there was a flaw in my logic, so therefore God doesn't exist." In any case, another indirect observation of gravitational waves has already been made, in the Hulse-Taylor observations of binary pulsar PSR B1913 16 in the 1990's. Unless someone can come up with a sensible example, we will not be posting further rebuttals. The recent ones have been hideously pointless, and not worth replying to. Many of them have cited articles that have nothing to do with relativity. Tides do not disprove relativity. It's possible that, because of the rather famous nature of the "counterexamples" page (it is cited around the world, and has nearly 2 million web views), people are simply putting in parody, or trolling, or humor, or whatever, in an attempt to see their work on a world-famous page. 3. Turyshev, Slava G.; Toth, Viktor T.; Kinsella, Gary; Lee, Siu-Chun; Lok, Shing M.; Ellis, Jordan (2012). "Support for the Thermal Origin of the Pioneer Anomaly". Physical Review Letters 108 (24): 241101. doi:10.1103/PhysRevLett.108.241101. PMID 23004253. Bibcode2012PhRvL.108x1101T. 9. BBC article Personal tools
acf800604e814222	Quantum imaging with entanglement and undetected photons, II: short version Here's a short explanation of the experiment reported in "Quantum imaging with undetected photons" by members of Anton Zeilinger's group in Vienna (Barreta Lemos, Borish, Cole, Ramelow, Lapkiewicz and Zeilinger). The previous post also explains the experiment, but in a way that is closer to my real-time reading of the article; this post is cleaner and more succinct. It's most easily understood by comparison to an ordinary Mach-Zehnder interferometry experiment. (The most informative part of the wikipedia article is the section "How it works"; Fig. 3 provides a picture.) In this sort of experiment, photons from a source such as a laser encounter a beamsplitter and go into a superposition of being transmitted and reflected. One beam goes through an object to be imaged, and acquires a phase factor---a complex number of modulus 1 that depends on the refractive index of the material out of which the object is made, and the thickness of the object at the point at which the beam goes through. You can think of this complex number as an arrow of length 1 lying in a two-dimensional plane; the arrow rotates as the photon passes through material, with the rate of rotation depending on the refractive index of the material. (If the thickness and/or refractive index varies on a scale smaller than the beamwidth, then the phase shift may vary over the beam cross-section, allowing the creation of an image of how the thickness of the object---or at least, the total phase imparted by the object, since the refractive index may be varying too---varies in the plane transverse to the beam. Otherwise, to create an image rather than just measure the total phase it imparts at a point, the beam may need to be scanned across the object.) The phase shift can be detected by recombining the beams at the second beamsplitter, and observing the intensity of light in each of the two output beams, since the relative probability of a photon coming out one way or the other depends on the relative phase of the the two input beams; this dependence is called "interference". Now open the homepage of the Nature article and click on Figure 1 to enlarge it. This is a simplified schematic of the experiment done in Vienna. Just as in ordinary Mach-Zehnder interferometry, a beam of photons is split on a beamsplitter (labeled BS1 in the figure). One can think of each photon from the source going into a superposition of being reflected and transmitted at the first beamsplitter. The transmitted part is downconverted by passing through the nonlinear crystal NL1 into an entangled pair consisting of a yellow and a red photon; the red photon is siphoned off by a dichroic (color-dependent) beamsplitter, D1, and passed through the object O to be imaged, acquiring a phase dependent on the refractive index of the object and its thickness. The phase, as I understand things, is associated with the photon pair even though it is imparted by the passing only the red photon through the object. In order to observe the phase via interferometry, one needs to involve both the red and yellow photon, coherently. (If one could observe it as soon as it was imparted to the pair by just interacting with the yellow photon, one could send a signal from the interaction point to the yellow part of the beam instantaneously, violating relativity.) The red part of the beam is then recombined (at dichroic beamsplitter D2) with the reflected portion of the beam (which is still at the original wavelength), and that portion of the beam is passed through another nonlinear crystal, NL2. This downconverts the part of the beam that is at the original wavelength into a red-yellow pair, with the resulting red component aligned with --- and indistinguishable from---the red component that has gone through the object. The phase associated with the photon pair created in the transmitted part of the beam whose red member went through the object is now associated with the yellow photons in the transmitted beam, since the red photons in that beam have been rendered indistinguishable from the ones created in the reflected beam, and so retain no information about the relative phase. This means that the phase can be observed siphoning out the red photons (at dichroic beamsplitter D3), recombining just the yellow photons with a beamsplitter BS2, and observing the intensitities at the two outputs of this final beamsplitter, precisely as in the last stage of an ordinary Mach-Zehnder experiment. The potential advantage over ordinary Mach-Zehnder interferometry is that one can image the total phase imparted by the object at a wavelength different from the wavelength of the photons that are interfered and detected at the final stage, which could be an advantage for instance if good detectors are not available at the wavelength one wants to image the object at. Quantum imaging with entanglement and undetected photons in Vienna [Update 9/1: I have been planning (before any comments, incidentally) to write a version of this post which just provides a concise verbal explanation of the experiment, supplemented perhaps with a little formal calculation. However, I think the discussion below comes to a correct understanding of the experiment, and I will leave it up as an example of how a physicist somewhat conversant with but not usually working in quantum optics reads and quickly comes to a correct understanding of a paper. Yes, the understanding is correct even if some misleading language was used in places, but I thank commenter Andreas for pointing out the latter.] Thanks to tweeters @AtheistMissionary and @robertwrighter for bringing to my attention this experiment by a University of Vienna group (Gabriela Barreto Lemos, Victoria Borish, Garrett D. Cole, Sven Ramelo, Radek Lapkiewicz and Anton Zeilinger), published in Nature, on imaging using entangled pairs of photons. It seems vaguely familiar, perhaps from my visit to the Brukner, Aspelmeyer and Zeilinger groups in Vienna earlier this year; it may be that one of the group members showed or described it to me when I was touring their labs. I'll have to look back at my notes. This New Scientist summary prompts the Atheist and Robert to ask (perhaps tongue-in-cheek?) if it allows faster-than-light signaling. The answer is of course no. The New Scientist article fails to point out a crucial aspect of the experiment, which is that there are two entangled pairs created, each one at a different nonlinear crystal, labeled NL1 and NL2 in Fig. 1 of the Nature article. [Update 9/1: As I suggest parenthetically, but in not sufficiently emphatic terms, four sentences below, and as commenter Andreas points out, there is (eventually) a superposition of an entangled pair having been created at different points in the setup; "two pairs" here is potentially misleading shorthand for that.] To follow along with my explanation, open the Nature article preview, and click on Figure 1 to enlarge it. Each pair is coherent with the other pair, because the two pairs are created on different arms of an interferometer, fed by the same pump laser. The initial beamsplitter labeled "BS1" is where these two arms are created (the nonlinear crystals come later). (It might be a bit misleading to say two pairs are created by the nonlinear crystals, since that suggests that in a "single shot" the mean photon number in the system after both nonlinear crystals have been passed is 4, whereas I'd guess it's actually 2 --- i.e. the system is in a superposition of "photon pair created at NL1" and "photon pair created at NL2".) Each pair consists of a red and a yellow photon; on one arm of the interferometer, the red photon created at NL1 is passed through the object "O". Crucially, the second pair is not created until after this beam containing the red photon that has passed through the object is recombined with the other beam from the initial beamsplitter (at D2). ("D" stands for "dichroic mirror"---this mirror reflects red photons, but is transparent at the original (undownconverted) wavelength.) Only then is the resulting combination passed through the nonlinear crystal, NL2. Then the red mode (which is fed not only by the red mode that passed through the object and has been recombined into the beam, but also by the downconversion process from photons of the original wavelength impinging on NL2) is pulled out of the beam by another dichroic mirror. The yellow mode is then recombined with the yellow mode from NL1 on the other arm of the interferometer, and the resulting interference observed by the detectors at lower right in the figure. It is easy to see why this experiment does not allow superluminal signaling by altering the imaged object, and thereby altering the image. For there is an effectively lightlike or timelike (it will be effectively timelike, given the delays introduced by the beamsplitters and mirrors and such) path from the object to the detectors. It is crucial that the red light passed through the object be recombined, at least for a while, with the light that has not passed through the object, in some spacetime region in the past light cone of the detectors, for it is the recombination here that enables the interference between light not passed through the object, and light passed through the object, that allows the image to show up in the yellow light that has not (on either arm of the interferometer) passed through the object. Since the object must be in the past lightcone of the recombination region where the red light interferes, which in turn must be in the past lightcone of the final detectors, the object must be in the past lightcone of the final detectors. So we can signal by changing the object and thereby changing the image at the final detectors, but the signaling is not faster-than-light. Perhaps the most interesting thing about the experiment, as the authors point out, is that it enables an object to be imaged at a wavelength that may be difficult to efficiently detect, using detectors at a different wavelength, as long as there is a downconversion process that creates a pair of photons with one member of the pair at each wavelength. By not pointing out the crucial fact that this is an interference experiment between two entangled pairs [Update 9/1: per my parenthetical remark above, and Andreas' comment, this should be taken as shorthand for "between a component of the wavefunction in which an entangled pair is created in the upper arm of the interferometer, and one in which one is created in the lower arm"], the description in New Scientist does naturally suggest that the image might be created in one member of an entangled pair, by passing the other member through the object, without any recombination of the photons that have passed through the object with a beam on a path to the final detectors, which would indeed violate no-signaling. I haven't done a calculation of what should happen in the experiment, but my rough intuition at the moment is that the red photons that have come through the object interfere with the red component of the beam created in the downconversion process, and since the photons that came through the object have entangled yellow partners in the upper arm of the interferometer that did not pass through the object, and the red photons that did not pass through the object have yellow partners created along with them in the lower part of the interferometer, the interference pattern between the red photons that did and didn't pass through the object corresponds perfectly to an interference pattern between their yellow partners, neither of which passed through the object. It is the latter that is observed at the detectors. [Update 8/29: now that I've done the simple calculation, I think this intuitive explanation is not so hot. The phase shift imparted by the object "to the red photons" actually pertains to the entire red-yellow entangled pair that has come from NL1 even though it can be imparted by just "interacting" with the red beam, so it is not that the red photons interfere with the red photons from NL2, and the yellow with the yellow in the same way independently, so that the pattern could be observed on either color, with the statistical details perfectly correlated. Rather, without recombining the red photons with the beam, no interference could be observed between photons of a single color, be it red or yellow, because the "which-beam" information for each color is recorded in different beams of the other color. The recombination of the red photons that have passed through the object with the undownconverted photons from the other output of the initial beamsplitter ensures that the red photons all end up in the same mode after crystal NL2 whether they came into the beam before the crystal or were produced in the crystal by downconversion, thereby ensuring that the red photons contain no record of which beam the yellow photons are in, and allowing the interference due to the phase shift imparted by the object to be observed on the yellow photons alone.] As I mentioned, not having done the calculation, I don't think I fully understand what is happening. [Update: Now that I have done a calculation of sorts, the questions raised in this paragraph are answered in a further Update at the end of this post. I now think that some of the recombinations of beams considered in this paragraph are not physically possible.] In particular, I suspect that if the red beam that passes through the object were mixed with the downconverted beam on the lower arm of the interferometer after the downconversion, and then peeled off before detection, instead of having been mixed in before the downconversion and peeled off afterward, the interference pattern would not be observed, but I don't have clear argument why that should be. [Update 8/29: the process is described ambiguously here. If we could peel off the red photons that have passed through the object while leaving the ones that came from the downconversion at NL2, we would destroy the interference. But we obviously can't do that; neither we nor our apparatus can tell these photons apart (and if we could, that would destroy interference anyway). Peeling off all the red photons before detection actually would allow the interference to be seen, if we could have mixed back in the red photons first; the catch is that this mixing-back-in is probaby not physically possible.] Anyone want to help out with an explanation? I suspect one could show that this would be the same as peeling off the red photons from NL2 after the beamsplitter but before detection, and only then recombining them with the red photons from the object, which would be the same as just throwing away the red photons from the object to begin with. If one could image in this way, then that would allow signaling, so it must not work. But I'd still prefer a more direct understanding via a comparison of the downconversion process with the red photons recombined before, versus after. Similarly, I suspect that mixing in and then peeling off the red photons from the object before NL2 would not do the job, though I don't see a no-signaling argument in this case. But it seems crucial, in order for the yellow photons to bear an imprint of interference between the red ones, that the red ones from the object be present during the downconversion process. The news piece summarizing the article in Nature is much better than the one at New Scientist, in that it does explain that there are two pairs, and that the one member of one pair is passed through the object and recombined with something from the other pair. But it does not make it clear that the recombination takes place before the second pair is created---indeed it strongly suggests the opposite: According to the laws of quantum physics, if no one detects which path a photon took, the particle effectively has taken both routes, and a photon pair is created in each path at once, says Gabriela Barreto Lemos, a physicist at Austrian Academy of Sciences and a co-author on the latest paper. In the first path, one photon in the pair passes through the object to be imaged, and the other does not. The photon that passed through the object is then recombined with its other ‘possible self’ — which travelled down the second path and not through the object — and is thrown away. The remaining photon from the second path is also reunited with itself from the first path and directed towards a camera, where it is used to build the image, despite having never interacted with the object. Putting the quote from Barreta Lemos about a pair being created on each path before the description of the recombination suggests that both pair-creation events occur before the recombination, which is wrong. But the description in this article is much better than the New Scientist description---everything else about it seems correct, and it gets the crucial point that there are two pairs, one member of which passes through the object and is recombined with elements of the other pair at some point before detection, right even if it is misleading about exactly where the recombination point is. [Update 8/28: clearly if we peel the red photons off before NL2, and then peel the red photons created by downconversion at NL2 off after NL2 but before the final beamsplitter and detectors, we don't get interference because the red photons peeled off at different times are in orthogonal modes, each associated with one of the two different beams of yellow photons to be combined at the final beamsplitter, so the interference is destroyed by the recording of "which-beam" information about the yellow photons, in the red photons. But does this mean if we recombine the red photons into the same mode, we restore interference? That must not be so, for it would allow signaling based on a decision to recombine or not in a region which could be arranged to be spacelike separated from the final beamsplitter and detectors. But how do we see this more directly? Having now done a highly idealized version of the calculation (based on notation like that in and around Eq. (1) of the paper) I see that if we could do this recombination, we would get interference. But to do that we would need a nonphysical device, namely a one-way mirror, to do this final recombination. If we wanted to do the other variant I discussed above, recombining the red photons that have passed the object with the red (and yellow) photons created at NL2 and then peeling all red photons off before the final detector, we would even need a dichroic one-way mirror (transparent to yellow, one-way for red), to recombine the red photons from the object with the beam coming from NL2. So the only physical way to implement the process is to recombine the red photons that have passed through the object with light of the original wavelength in the lower arm of the interferometer before NL2; this just needs an ordinary dichroic mirror, which is a perfectly physical device.] Free will and retrocausality at Cambridge II: Conspiracy vs. Retrocausality; Signaling and Fine-Tuning Expect (with moderate probability) substantial revisions to this post, hopefully including links to relevant talks from the Cambridge conference on retrocausality and free will in quantum theory, but for now I think it's best just to put this out there. Conspiracy versus Retrocausality One of the main things I hoped to straighten out for myself at the conference on retrocausality in Cambridge was whether the correlation between measurement settings and "hidden variables" involved in a retrocausal explanation of Bell-inequality-violating quantum correlations are necessarily "conspiratorial", as Bell himself seems to have thought. The idea seems to be that correlations between measurement settings and hidden variables must be due to some "common cause" in the intersection of the backward light cones of the two. That is, a kind of "conspiracy" coordinating the relevant hidden variables that can affect the meausrement outcome with all sorts of intricate processes that can affect which measurement is made, such as those affecting your "free" decision as to how to set a polarizer, or, in case you set up a mechanism to control the polarizer setting according to some apparatus reasonably viewed as random ("the Swiss national lottery machine" was the one envisioned by Bell), the functioning of this mechanism. I left the conference convinced once again (after doubts on this score had been raised in my mind by some discussions at New Directions in the Philosophy of Physics 2013) that the retrocausal type of explanation Price has in mind is different from a conspiratorial one. Deflationary accounts of causality: their impact on retrocausal explanation Distinguishing "retrocausality" from "conspiratorial causality" is subtle, because it is not clear that causality makes sense as part of a fundamental physical theory. (This is a point which, in this form, apparently goes back to Bertrand Russell early in this century. It also reminds me of David Hume, although he was perhaps not limiting his "deflationary" account of causality to causality in physical theories.) Causality might be a concept that makes sense at the fundamental level for some types of theory, e.g. a version ("interpretation") of quantum theory that takes measurement settings and outcomes as fundamental, taking an "instrumentalist" view of the quantum state as a means of calculating outcome probabilities giving settings, and not as itself real, without giving a further formal theoretical account of what is real. But in general, a theory may give an account of logical implications between events, or more generally, correlations between them, without specifying which events cause, or exert some (perhaps probabilistic) causal influence on others. The notion of causality may be something that is emergent, that appears from the perspective of beings like us, that are part of the world, and intervene in it, or model parts of it theoretically. In our use of a theory to model parts of the world, we end up taking certain events as "exogenous". Loosely speaking, they might be determined by us agents (using our "free will"), or by factors outside the model. (And perhaps "determined" is the wrong word.) If these "exogenous" events are correlated with other things in the model, we may speak of this correlation as causal influence. This is a useful way of speaking, for example, if we control some of the exogenous variables: roughly speaking, if we believe a model that describes correlations between these and other variables not taken as exogenous, then we say these variables are causally influenced by the variables we control that are correlated with them. We find this sort of notion of causality valuable because it helps us decide how to influence those variables we can influence, in order to make it more likely that other variables, that we don't control directly, take values we want them to. This view of causality, put forward for example in Judea Pearl's book "Causality", has been gaining acceptance over the last 10-15 years, but it has deeper roots. Phil Dowe's talk at Cambridge was an especially clear exposition of this point of view on causality (emphasizing exogeneity of certain variables over the need for any strong notion of free will), and its relevance to retrocausality. This makes the discussion of retrocausality more subtle because it raises the possibility that a retrocausal and a conspiratorial account of what's going on with a Bell experiment might describe the same correlations, between the Swiss National lottery machine, or whatever controls my whims in setting a polarizer, all the variables these things are influenced by, and the polarizer settings and outcomes in a Bell experiment, differing only in the causal relations they describe between these variables. That might be true, if a retrocausalist decided to try to model the process by which the polarizer was set. But the point of the retrocausal account seems to be that it is not necessary to model this to explain the correlations between measurement results. The retrocausalist posits a lawlike relation of correlation between measurement settings and some of the hidden variables that are in the past light cone of both measurement outcomes. As long as this retrocausal influence does not influence observable past events, but only the values of "hidden", although real, variables, there is nothing obviously more paradoxical about imagining this than about imagining----as we do all the time---that macroscopic variables that we exert some control over, such as measurement settings, are correlated with things in the future. Indeed, as Huw Price has long (I have only recently realized for just how long) been pointing out, if we believe that the fundamental laws of physics are symmetric with respect to time-reversal, then it would be the absence of retrocausality, if we dismiss its possibility, and even if we accept its possibility to the limited extent needed to potentially explain Bell correlations, its relative scarcity, that needs explaining. Part of the explanation, of course, is likely that causality, as mentioned above, is a notion that is useful for agents situated within the world, rather than one that applies to the "view from nowhere and nowhen" that some (e.g. Price, who I think coined the term "nowhen") think is, or should be, taken by fundamental physical theories. Therefore whatever asymmetries---- these could be somewhat local-in-spacetime even if extremely large-scale, or due to "spontaneous" (i.e. explicit, even if due to a small perturbation) symmetry-breaking --- are associated with our apparently symmetry-breaking experience of directionality of time may also be the explanation for why we introduce the causal arrows we do into our description, and therefore why we so rarely introduce retrocausal ones. At the same time, such an explanation might well leave room for the limited retrocausality Price would like to introduce into our description, for the purpose of explaining Bell correlations, especially because such retrocausality does not allow backwards-in-time signaling. Signaling (spacelike and backwards-timelike) and fine-tuning. Emergent no-signaling? A theme that came up repeatedly at the conference was "fine-tuning"---that no-spacelike-signaling, and possibly also no-retrocausal-signaling, seem to require a kind of "fine-tuning" from a hidden variable model that uses them to explain quantum correlations. Why, in Bohmian theory, if we have spacelike influence of variables we control on physically real (but not necessarily observable) variables, should things be arranged just so that we cannot use this influence to remotely control observable variables, i.e. signal? Similarly one might ask why, if we have backwards-in-time influence of controllable variables on physically real variables, things are arranged just so that we cannot use this influence to remotely control observable variables at an earlier time? I think --- and I think this possibility was raised at the conference --- that a possible explanation, suggested by the above discussion of causality, is that for macroscopic agents such as us, with usually-reliable memories, some degree of control over our environment and persistence over time, to arise, it may be necessary that the scope of such macroscopic "observable" influences be limited, in order that there be a coherent macroscopic story at all for us to tell---in order for us even be around to wonder about whether there could be such signalling or not. (So the term "emergent no-signalling" in the section heading might be slightly misleading: signalling, causality, control, and limitations on signalling might all necessarily emerge together.) Such a story might end up involving thermodynamic arguments, about the sorts of structures that might emerge in a metastable equilibrium, or that might emerge in a dynamically stable state dependent on a temperature gradient, or something of the sort. Indeed, the distribution of hidden variables (usually, positions and/or momenta) according to the squared modulus of the wavefunction, which is necessary to get agreement of Bohmian theory with quantum theory and also to prevent signaling (and which does seem like "fine-tuning" inasmuch as it requires a precise choice of probability distribution over initial conditions), has on various occasions been justified by arguments that it represents a kind of equilibrium that would be rapidly approached even if it did not initially obtain. (I have no informed view at present on how good these arguments are, though I have at various times in the past read some of the relevant papers---Bohm himself, and Sheldon Goldstein, are the authors who come to mind.) I should mention that at the conference the appeal of such statistical/thermodynamic arguments for "emergent" no-signalling was questioned---I think by Matthew Leifer, who with Rob Spekkens has been one of the main proponents of the idea that no-signaling can appear like a kind of fine-tuning, and that it would be desirable to have a model which gave a satisfying explanation of it---on the grounds that one might expect "fluctuations" away from the equilibria, metastable structures, or steady states, but we don't observe small fluctuations away from no-signalling---the law seems to hold with certainty. This is an important point, and although I suspect there are adequate rejoinders, I don't see at the moment what these might be like. Free will and retrocausality in the quantum world, at Cambridge. I: Bell inequalities and retrocausality I'm in Cambridge, where the conference on Free Will and Retrocausality in the Quantum World, organized (or rather, organised) by Huw Price and Matt Farr will begin in a few hours. (My room at St. Catherine's is across from the chapel, and I'm being serenaded by a choir singing beautifully at a professional level of perfection and musicality---I saw them leaving the chapel yesterday and they looked, amazingly, to be mostly junior high school age.) I'm hoping to understand more about how "retrocausality", in which effects occur before their causes, might help resolve some apparent problems with quantum theory, perhaps in ways that point to potentially deeper underlying theories such as a "quantum gravity". So, as much for my own use as anyone else's, I thought perhaps I should post about my current understanding of this possibility. One of the main problems or puzzles with quantum theory that Huw and others (such as Matthew Leifer, who will be speaking) think retrocausality may be able to help with, is the existence of Bell-type inequality violations. At their simplest, these involve two spacelike-separated regions of spacetime, usually referred to as "Alice's laboratory" and "Bob's laboratory", at each of which different possible experiments can be done. The results of these experiments can be correlated, for example if they are done on a pair of particles, one of which has reached Alice's lab and the other Bob's, that have previously interacted, or were perhaps created simultaneously in the same event. Typically in actual experiments, these are a pair of photons created in a "downconversion" event in a nonlinear crystal. In a "nonlinear" optical process photon number is not conserved (so one can get a "nonlinearity" at the level of a Maxwell's equation where the intensity of the field is proportional to photon number; "nonlinearity" refers to the fact that the sum of two solutions is not required to be a solution). In parametric downconversion, a photon is absorbed by the crystal which emits a pair of photons in its place, whose energy-momentum four-vectors add up to that of the absorbed photon (the process does conserve energy-momentum). Conservation of angular momentum imposes correlations between the results of measurements made by "Alice" and "Bob" on the emitted photons. These are correlated even if the measurements are made sometime after the photons have separated far enough that the changes in the measurement apparatus that determine which component of polarization it measures (which we'll henceforth call the "polarization setting"), on one of the photons, are space-like separated from the measurement process on the other photon, so that effects of the polarization setting in Alice's laboratory, which one typically assumes can propagate only forward in time, i.e. in their forward light-cone, can't affect the setting or results in Bob's laboratory which is outside of this forward light-cone. (And vice versa, interchanging Alice and Bob.) Knowledge of how their pair of photons were prepared (via parametric downconversion and propagation to Alice and Bob's measurement sites) is encoded in a "quantum state" of the polarizations of the photon pair. It gives us, for any pair of polarization settings that could be chosen by Alice and Bob, an ordinary classical joint probability distribution over the pair of random variables that are the outcomes of the given measurements. We have different classical joint distributions, referring to different pairs of random variables, when different pairs of polarization settings are chosen. The Bell "paradox" is that there is no way of introducing further random variables that are independent of these polarization settings, such that for each pair of polarization settings, and each assignment of values to the further random variables, Alice and Bob's measurement outcomes are independent of each other, but when the further random variables are averaged over, the experimentally observed correlations, for each pair of settings, are reproduced. In other words, the outcomes of the polarization measurements, and in particular the fact that they are correlated, can't be "explained" by variables uncorrelated with the settings. The nonexistence of such an explanation is implied by the violation of a type of inequality called a "Bell inequality". (It's equivalent to to such a violation, if "Bell inequality" is defined generally enough.) How I stopped worrying and learned to love quantum correlations One might have hoped to explain the correlations by having some physical quantities (sometimes referred to as "hidden variables") in the intersection of Alice and Bob's backward light-cone, whose effects, propagating forward in their light-cone to Alice and Bob's laboratories, interact their with the physical quantities describing the polarization settings to produce---whether deterministically or stochastically---the measurement outcomes at each sites, with their observed probabilities and correlations. The above "paradox" implies that this kind of "explanation" is not possible. Some people, such as Tim Maudlin, seem to think that this implies that quantum theory is "nonlocal" in the sense of exhibiting some faster-than-light influence. I think this is wrong. If one wants to "explain" correlations by finding---or hypothesizing, as "hidden variables"---quantities conditional on which the probabilities of outcomes, for all possible measurement settings, factorize, then these cannot be independent of measurement settings. If one further requires that all such quantities must be localized in spacetime, and that their influence propagates (in some sense that I'm not too clear about at the moment, but that can probably be described in terms of differential equations---something like a conserved probability current might be involved) locally and forward in time, perhaps one gets into inconsistencies. But one can also just say that these correlations are a fact. We can have explanations of these sorts of fact---for example, for correlations in photon polarization measurements, the one alluded to above in terms of energy-momentum conservation and previous interaction or simultaneous creation---just not the sort of ultra-classical one some people wish for. It seems to me that what the retrocausality advocates bring to this issue is the possibility of something that is close to this type of classical explanation. It may allow for the removal of these types of correlation by conditioning on physical quantities. [Added July 31: this does not conflict with Bell's theorem, for the physical quantities are not required to be uncorrelated with measurement settings---indeed, being correlated with the measurement settings is to be expected if there is retrocausal influence from a measurement setting to physical quantities in the backwards light-cone of the measurement setting.] And unlike the Bohmian hidden variable theories, it hopes to avoid superluminal propagation of the influence of measurement settings to physical quantities, even unobservable ones. It does this, however, by having the influence of measurement settings pursue a "zig-zag" path from Alice to Bob: in Alice's backward light-cone back to the region where Alice and Bob's backward light-cones intersect, then forward to Bob's laboratory. What advantages might this have over superluminal propagation? It probably satisfies some kind of spacetime continuity postulate, and seems more likely to be able to be Lorentz-invariant. (However, the relation between formal Lorentz invariance and lack of superluminal propagation is subtle, as Rafael Sorkin reminded me at breakfast today.) Probable signature of gravitational waves from early-universe inflation found in cosmic microwave background by BICEP2 collaboration. Some quick links about the measurement, announced today, by the BICEP2 collaboration using a telescope at the South Pole equipped with transition edge sensors (TESs) read out with superconducting quantum interference devices (SQUIDs), of B-modes (or "curl") in the polarization of the cosmic microwave background (CMB) radiation, considered to be an imprint on the CMB of primordial graviational waves stirred up by the period of rapid expansion of the universe (probably from around 10-35--10-33 sec). BICEP2 estimates the tensor-to-scalar ratio "r", an important parameter constraining models of inflation, to be 0.2 (+0.7 / -0.5). Note that I'm not at all expert on any aspect of this! Caltech press release. Harvard-Smithsonian Center for Astrophysics press release. Main paper: BICEP2 I: Detection of B-mode polarization at degree angular scales. Instrument paper: BICEP2 II: Experiment and three-year data set BICEP/Keck homepage with the papers and other materials. Good background blog post (semi-popular level) from Sean Carroll Carroll's initial reaction. Richard Easther on inflation, also anticipating the discover (also fairlybroadly accessible) Very interesting reaction from a particle physicist at Résonaances. Reaction from Liam McAllister guesting on Lubos Motl's blog. Reaction from theoretical cosmology postdoc Sesh Nadathur. NIST Quantum Sensors project homepage. Besides a microwave telescope to collect and focus the relevant radiation, the experiment used transition-edge sensors (in which photons can trigger a quantum phase transition) read out by superconducting quantum interference devices (SQUIDs). I don't know the details of how that works, but TE sensors have lots of applications (including in quantum cryptography), as do SQUIDs; I'm looking forward to learning more about this one. Some ideas on food and entertainment for those attending SQUINT 2014 in Santa Fe I'm missing SQUINT 2014 (bummer...) to give a talk at a workshop on Quantum Contextuality, Nonlocality, and the Foundations of Quantum Mechanics in Bad Honnef, Germany, followed by collaboration with Markus Mueller at Heidelberg, and a visit to Caslav Brukner's group and the IQOQI at Vienna. Herewith some ideas for food and entertainment for SQUINTers in Santa Fe. Cris Moore will of course provide good advice too. For a high-endish foodie place, I like Ristra. You can also eat in the bar there, more casual (woodtop tables instead of white tablecloths), a moderate amount of space (but won't fit an enormous group), some smaller plates. Pretty reasonable prices (for the excellent quality). Poblano relleno is one of the best vegetarian entrees I've had in a high-end restaurant---I think it is vegan. Flash-fried calamari were also excellent... I've eaten here a lot with very few misses. One of the maitres d' sings in a group I'm in, and we're working on tenor-baritone duets, so if Ed is there you can tell him Howard sent you but then you have to behave ;-). The food should be good regardless. If Jonathan is tending bar you can ask him for a flaming chartreuse after dinner... fun stuff and tasty too. (I assume you're not driving.) Wines by the glass are good, you should get good advice on pairing with food. Next door to Ristra is Raaga... some of the best Indian food I've had in a restaurant, and reasonably priced for the quality. I enjoyed a couple of lunches (fish tacos, grilled portobello sandwich, weird dessert creations...) at Restaurant Martin, was less thrilled by my one foray into dinner there. Expensive for dinner, less so for lunch, a bit of a foodie vibe. Fish and chips are excellent at Zia Café (best in town I think), so is the green chile pie--massive slice of a deep-dish quiche-like entity, sweet and hot at the same time. I like the tapas at El Mesón, especially the fried eggplant, any fried seafood like oysters with salmorejo, roasted red peppers with goat cheese (more interesting than it sounds). I've had better luck with their sherries (especially finos) better than their wines by the glass. (I'd skip the Manchego with guava or whatever, as it's not that many slices and you can get cheese at a market.) Tonight they will have a pretty solid jazz rhythm section, the Three Faces of Jazz, and there are often guests on various horn. Straight-ahead standards and classic jazz, mostly bop to hard bop to cool jazz or whatever you want to call it. "Funky Caribbean-infused jazz" with Ryan Finn on trombone on Sat. might be worth checking out too... I haven't heard him with this group but I've heard a few pretty solid solos from him with a big band. Sounds fun. The jazz is popular so you might want to make reservations (to eat in the bar/music space, there is also a restaurant area I've never eaten in) especially if you're more than a few people. La Boca and Taverna La Boca are also fun for tapas, maybe less classically Spanish. La Boca used to have half-price on a limited selection of tapas and $1 off on sherry from 3-5 PM. Not sure if they still do. Il Piatto is relatively inexpensive Italian, pretty hearty, and they usually have some pretty good deals in fixed-price 3 course meals where you choose from the menu, or early bird specials and such. Despite a kind of pretentious name Tanti Luci 221, at 221 Shelby, was really excellent the one time I tried it. There's a bar menu served only in the bar area, where you can also order off the main menu. They have a happy hour daily, where drinks are half price. That makes them kinda reasonable. The Manhattan I had was excellent, though maybe not all that traditional. If you've got a car and want some down-home Salvadoran food, the Pupuseria y Restaurante Salvadoreño, in front of a motel on Cerillos, is excellent and cheap. As far as entertainment, get a copy of the free Reporter (or look up their online calendar). John Rangel and Chris Ishee are two of the best jazz pianists in town; if either is playing, go. Chris is also in Pollo Frito, a New Orleans funk outfit that's a lot of fun. If they're playing at the original 2nd street brewery, it should be a fun time... decent pubby food and brews to eat while you listen. Saxophonist Arlen Asher is one of the deans of the NM jazz scene, trumpeter and flugelhorn player Bobby Shew is also excellent, both quite straight-ahead. Dave Anderson also recommended. The one time I heard JQ Whitcomb on trumpet he was solid, but it's only been once. I especially liked his compositions. Faith Amour is a nice singer, last time I heard her was at Pranzo where the acoustics were pretty bad. (Tiny's was better in that respect.) For trad New Mexican (food that is) I especially like Tia Sophia's on Washington (I think), and The Shed for red chile enchiladas (and margaritas). Gotta go. It's Friday night, when all good grad students, faculty, and postdocs anywhere in the worlkd head for the nearest "Irish pub". Answer to question about Bekenstein BH entropy derivation I had a look at Jacob Bekenstein's 1973 Physical Review D paper "Black holes and entropy" for the answer to my question about Susskind's presentation of the Bekenstein derivation of the formula stating that black hole entropy is proportional to horizon area. An argument similar to the one in Susskind's talk appears in Section IV, except that massive particles are considered, rather than photons, and they can be assumed to be scalar so that the issue I raised, of entropy associated with polarization, is moot. Bekenstein says: we can be sure that the absolute minimum of information lost [as a particle falls into a black hole] is that contained in the answer to the question "does the particle exist or not?" To start with, the answer [to this question] is known to be yes. But after the particle falls in, one has no information whatever about the answer. This is because from the point of view of this paper, one knows nothing about the physical conditions inside the black hole, and thus one cannot assess the likelihood of the particle continuing to exist or being destroyed. One must, therefore, admit to the loss of one bit of information [...] at the very least." Presumably for the particle to be destroyed, at least in a field-theoretic description, it must annihilate with some stuff that is already inside the black hole (or from the outside point of view, plastered against the horizon). This annihilation could, I guess, create some other particle. In fact it probably must, in order to conserve mass-energy. My worry in the previous post about the entropy being due to the presence/absence of the particle inside the hole was that this would seem to need to be due to uncertainty about whether the particle fell into the hole in the first place, which did not seem to be part of the story Susskind was telling, and the associated worry that this would make the black hole mass uncertain, which also didn't seem to be a feature of the intended story although I wasn't sure. But the correct story seems to be that the particle definitely goes into the hole, and the uncertainty is about whether it subsequently annihilates with something else inside, in a process obeying all relevant conservation laws, rendering both of my worries inapplicable. I'd still like to see if Bekenstein wrote a version using photons, as Susskind's presentation does. And when I feel quite comfortable, I'll probably post a fairly full description of one (or more) versions of the argument. Prior to the Phys Rev D paper there was a 1972 Letter to Nuovo Cimento, which I plan to have a look at; perhaps it deals with photons. If you want to read Bekenstein's papers too, I suggest you have a look at his webpage. Question about Susskind's presentation of Bekenstein's black hole entropy derivation I'm partway through viewing Leonard Susskind's excellent not-too-technical talk "Inside Black Holes" given at the Kavli Institute for Theoretical Physics at UC Santa Barbara on August 25. Thanks to John Preskill, @preskill, for recommending it. I've decided to try using my blog as a discussion space about this talk, and ultimately perhaps about the "Harlow-Hayden conjecture" about how to avoid accepting the recent claim that black holes must have an information-destroying "firewall" near the horizon. (I hope I've got that right.) I'm using Susskind's paper "Black hole complementarity and the Harlow-Hayden conjecture" as my first source on the latter question. It also seems to be a relatively nontechnical presentation (though much more technical than the talk so far)... that should be particularly accessible to quantum information theorists, although it seems to me he also does a good job of explaining the quantum information-theoretic concepts he uses to those not familiar with them. But first things first. I'm going to unembarassedly ask elementary questions about the talk and the paper until I understand. First off, I've got a question about Susskind's "high-school level" presentation, in minutes 18-28 of the video, of Jacob Bekenstein's 1973 argument that in our quantum-mechanical world the entropy of a black hole is proportional to its area (i.e. the area of the horizon, the closed surface inside which nothing, not even light, can escape). The formula, as given by Susskind, is S = (\frac{c^3}{4 \hbar G}) A, where S is the entropy (in bits) of the black hole, and A the area of its horizon. (The constant here may have been tweaked by a small amounts, like 4 \pi or its inverse, to reflect considerations that Susskind alluded to but didn't describe, more subtle than those involved in Bekenstein's argument.) The argument, as presented by Susskind, involves creating the black hole out of photons whose wavelength is roughly the Schwarzschild radius of the black hole. More precisely, it is built up in steps; each step in creating a black hole of a given mass and radius involves sending in another photon of wavelength roughly the current Schwarzschild radius. The wavelength needs to be that big so that there is no information going into the hole (equivalently, from the point of view outside the hole, getting "plastered" (Susskind's nice choice of word) against the horizon) about where the photon went in. Presumably there is some argument about why the wavelength shouldn't be much bigger, either...perhaps so that it is sure to go into the hole, rather than missing. That raises the question of just what state of the photon field should be impinging on the hole...presumably we want some wavepacket whose spatial width is about the size of the hole, so we'll have a spread of wavelengths centered around some multiple (roughly unity) of the Schwarzschild radius. Before there is any hole, I guess I also have some issues about momentum conservation... maybe one starts by sending in a spherical shell of radiation impinging on where we want the hole to be, so as to have zero net momentum. But these aren't my main questions, though of course it could turn out to be necessary to answer them in order to answer my main question. My main question is: Susskind says that each such photon carries one bit of information: the information is "whether it's there or not". This doesn't make sense to me, as if one is uncertain about how many photons went into creating the hole, it seems to me one should have a corresponding uncertainty about its mass, radius, etc... Moreover, the photons that go in still seem to have a degree of freedom capable of storing a bit of information: their polarization. So maybe this is the source of the one bit per photon? Of course, this would carry angular momentum into the hole/onto the horizon, so I guess uncertainty about this could generate uncertainty about whether or not we have a Schwarzschild or a Kerr (rotating) black hole, i.e. just what the angular momentum of the hole is. Now, maybe the solution is just that given their wavelength of the same order of the hole, there is uncertainty about whether or not the photons actually get into the hole, and so the entropy of the black hole really is due to uncertainty about its total mass, and the mass M in the Bekenstein formula is just the expected value of mass? I realize I could probably figure all this out by grabbing some papers, e.g. Bekenstein's original, or perhaps even by checking wikipedia, but I think there's some value in thinking out loud, and in having an actual interchange with people to clear up my confusion... one ends up understanding the concepts better, and remembering the solution. So, if any physicists knowledgeable about black holes (or able and willing to intelligently speculate about them...) are reading this, straighten me out if you can, or at least let's discuss it and figure it out... Bohm on measurement in Bohmian quantum theory Prompted, as described in the previous post, by Craig Callender's post on the uncertainty principle, I've gone back to David Bohm's original series of two papers "A suggested interpretation of the quantum theory in terms of "hidden" variables I" and "...II", published in Physical Review in 1952 (and reprinted in Wheeler and Zurek's classic collection "Quantum Theory and Measurement", Princeton University Press, 1983). The Bohm papers and others appear to be downloadable here. Question 1 of my previous post asked whether it is true that "a "measurement of position" does not measure the pre-existing value of the variable called, in the theory, "position". That is, if one considers a single trajectory in phase space (position and momentum, over time), entering an apparatus described as a "position measurement apparatus", that apparatus does not necessarily end up pointing to, approximately, the position of the particle when it entered the apparatus." It is fairly clear from Bohm's papers that the answer is "Yes". In section 5 of the second paper, he writes "in the measurement of an "observable," Q, we cannot obtain enough information to provide a complete specification of the state of an electron, because we cannot infer the precisely defined values of the particle momentum and position, which are, for example, needed if we wish to make precise predictions about the future behavior of the electron. [...] the measurement of an "observable" is not really a measurement of any physical property belonging to the observed system alone. Instead, the value of an "observable" measures only an incompletely predictable and controllable potentiality belonging just as much to the measuring apparatus as to the observed system itself." Since the first sentence quoted says we cannot infer precise values of "momentum and position", it is possible to interpret it as referring to an uncertainty-principle-like tradeoff of precision in measurement of one versus the other, rather than a statement that it is not possible to measure either precisely, but I think that would be a misreading, as the rest of the quote, which clearly concerns any single observable, indicates. Later in the section, he unambiguously gives the answer "Yes" to a mutation of my Question 1 which substitutes momentum for position. Indeed, most of the section is concerned with using momentum measurement as an example of the general principle that the measurements described by standard quantum theory, when interpreted in his formalism, do not measure pre-existing properties of the measured system. Here's a bit of one of two explicit examples he gives of momentum measurement: "...consider a stationary state of an atom, of zero angular momentum. [...] the \psi-field for such a state is real, so that we obtain \mathbf{p} = \nabla S = 0. Thus, the particle is at rest. Nevertheless, we see from (14) and (15) that if the momentum "observable" is measured, a large value of this "observable" may be obtained if the \psi-field happens to have a large fourier coefficient, a_\mathbf{p}, for a high value of \mathbf{p}. The reason is that in the process of interaction with the measuring apparatus, the \psi-field is altered in such a way that it can give the electron particle a correspondingly large momentum, thus transferring some of the potential energy of interaction of the particle with its \psi-field into kinetic energy." Note that the Bohmian theory involves writing the complex-valued wavefunction \psi(\mathbf{x}) as R(\mathbf{x})e^{i S(\mathbf{x})}, i.e. in terms of its (real) modulus R and (real) phase S. Expressing the Schrödinger equation in terms of these variables is in fact probably what suggested the interpretation, since one gets something resembling classical equations of motion, but with a term that looks like a potential, but depends on \psi. Then one takes these classical-like equations of motion seriously, as governing the motions of actual particles that have definite positions and momenta. In order to stay in agreement with quantum theory concerning observed events such as the outcomes of measurements, m theory, one in addition keeps, from quantum theory, the assumption that the wavefunction \psi evolves according to the Schrödinger equation. And one assumes that we don't know the particles' exact position but only that this is distributed with probability measure given (as quantum theory would predict for the outcome of a position measurement) by R^2(\mathbf{x}), and that the momentum is \mathbf{p} = \nabla S. That's why the real-valuedness of the wavefunction implies that momentum is zero: because the momentum, in Bohmian theory, is the gradient of the phase of the wavefunction. For completeness we should reproduce Bohm's (15). (15) \psi = \sum_\mathbf{p} a_{\mathbf{p}} exp(i \mathbf{p}\cdot \mathbf{x} / \hbar). At least in the Wheeler and Zurek book, the equation has p instead of \mathbf{p} as the subscript on \Sigma, and a_1 instead of a_\mathbf{p}; I consider these typos, and have corrected them. (Bohm's reference to (14), which is essentially the same as (15) seems to me to be redundant.) The upshot is that "the actual particle momentum existing before the measurement took place is quite different from the numerical value obtained for the momentum "observable,"which, in the usual interpretation, is called the "momentum." " It would be nice to have this worked out for a position measurement example, as well. The nicest thing, from my point of view, would be an example trajectory, for a definite initial position, under a position-measurement interaction, leading to a final position different from the initial one. I doubt this would be too hard, although it is generally considered to be the case that solving the Bohmian equations of motion is difficult in the technical sense of complexity theory. I don't recall just how difficult, but more difficult than solving the Schrödinger equation, which is sometimes taken as an argument against the Bohmian interpretation: why should nature do all that work, only to reproduce, because of the constraints mentioned above---distribution of \mathbf{x} according to R^2, \mathbf{p} = \nabla S---observable consequences that can be more easily calculated using the Schrödinger equation? I think I first heard of this complexity objection (which is of course something of a matter of taste in scientific theories, rather than a knockdown argument) from Daniel Gottesman, in a conversation at one of the Feynman Fests at the University of Maryland, although Antony Valentini (himself a Bohmian) has definitely stressed the ability of Bohmian mechanics to solve problems of high complexity, if one is allowed to violate the constraints that make it observationally indistinguishable from quantum theory. It is clear from rereading Bohm's 1952 papers that Bohm was excited about the physical possibility of going beyond these constraints, and thus beyond the limitations of standard quantum theory, if his theory was correct. In fairness to Bohmianism, I should mention that in these papers Bohm suggests that the constraints that give standard quantum behavior may be an equilibrium, and in another paper he gives arguments in favor of this claim. Others have since taken up this line of argument and done more with it. I'm not familiar with the details. But the analogy with thermodynamics and statistical mechanics breaks down in at least one respect, that one can observe nonequilibrium phenomena, and processes of equilibration, with respect to standard thermodynamics, but nothing like this has so far been observed with respect to Bohmian quantum theory. (Of course that does not mean we shouldn't think harder, guided by Bohmian theory, about where such violations might be observed... I believe Valentini has suggested some possibilities in early-universe physics.) A question about measurement in Bohmian quantum mechanics I was disturbed by aspects of Craig Callender's post "Nothing to see here," on the uncertainty principle, in the New York Times' online philosophy blog "The Stone," and I'm pondering a response, which I hope to post here soon. But in the process of pondering, some questions have arisen which I'd like to know the answers to. Here are a couple: Callender thinks it is important that quantum theory be formulated in a way that does not posit measurement as fundamental. In particular he discusses the Bohmian variant of quantum theory (which I might prefer to describe as an alternative theory) as one of several possibilities for doing so. In this theory, he claims, Uncertainty still exists. The laws of motion of this theory imply that one can’t know everything, for example, that no perfectly accurate measurement of the particle’s velocity exists. This is still surprising and nonclassical, yes, but the limitation to our knowledge is only temporary. It’s perfectly compatible with the uncertainty principle as it functions in this theory that I measure position exactly and then later calculate the system’s velocity exactly. While I've read Bohm's and Bell's papers on the subject, and some others, it's been a long time in most cases, and this theory is not something I consider very promising as physics even though it is important as an illustration of what can be done to recover quantum phenomena in a somewhat classical theory (and of the weird properties one can end up with when one tries to do so). So I don't work with it routinely. And so I'd like to ask anyone, preferably more expert than I am in technical aspects of the theory, though not necessarily a de Broglie-Bohm adherent, who can help me understand the above claims, in technical or non-technical terms, to chime in in the comments section. Question 1: Is that correct? A little more discussion of Question 1. On my understanding, what is claimed is, rather, something like: that if one has a probability distribution over particle positions and momenta and a "pilot wave" (quantum wave function) whose squared amplitude agrees with these distributions (is this required in both position and momentum space? I'm guessing so), then the probability (calculated using the distribution over initial positions and momenta, and the deterministic "laws of motion" by which these interact with the "pilot wave" and the apparatus) for the apparatus to end up showing position in a given range, is the same as the integral of the squared modulus of the wavefunction, in the position representation, over that range. Prima facie, this could be achieved in ways other than having the measurement reading being perfectly correlated with the initial position on a given trajectory, and my guess is that in fact it is not achieved in that way in the theory. If that were so it seems like the correlation should hold whatever the pilot wave is. Now, perhaps that's not a problem, but it makes the pilot wave feel a bit superfluous to me, and I know that it's not, in this theory. My sense is that what happens is more like: whatever the initial position is, the pilot wave guides it to some---definite, of course---different final position, but when the initial distribution is given by the squared modulus of the pilot wave itself, then the distribution of final positions is given by the squared modulus of the (initial, I guess) pilot wave. But if the answer to question 1 is "Yes", I have trouble understanding what Callender means by "I measure position exactly". Also, regardless of the answer to Question 1, either there is a subtle distinction being made between measuring "perfectly accurately" and measuring "exactly" (in which case I'd like to know what the distinction is), or these sentences need to be reformulated more carefully. Not trying to do a gotcha on Callender here, just trying to understand the claim, and de Broglie Bohm. My second question relates to Callender's statement that: Question 2: How does this way of ascertaining the system's velocity differ from the sort of "direct measurement" that is, presumably, subject to the uncertainty principle? I'm guessing that by the time one has enough information (possibly about further positions?) to calculate what the velocity was, one can't do with it the sorts of things that one could have done if one had known the position and velocity simultaneously. But this depends greatly on what it would mean to "have known" the position and/or velocity, which --- especially if the answer to Question 1 was "Yes"--- seems a rather subtle matter. So, physicists and other readers knowledgeable on these matters (if any such exist), your replies with explanations, or links to explanations, of these points would be greatly appreciated. And even if you don't know the answers, but know de Broglie-Bohm well on a technical level... let's figure this out! (My guess is that it's well known, and indeed that the answer to Question 1 in particular is among the most basic things one learns about this interpretation...)
5b6fa2ed170e0f1e	Archive for October 3, 2010 Spin, Entanglement and Quantum Weirdness Posted in The Universe and Stuff with tags , , , , , , , on October 3, 2010 by telescoper After writing a post about spinning cricket balls a while ago I thought it might be fun to post something about the role of spin in quantum mechanics. Spin is a concept of fundamental importance in quantum mechanics, not least because it underlies our most basic theoretical understanding of matter. The standard model of particle physics divides elementary particles into two types, fermions and bosons, according to their spin. One is tempted to think of these elementary particles as little cricket balls that can be rotating clockwise or anti-clockwise as they approach an elementary batsman. But, as I hope to explain, quantum spin is not really like classical spin: batting would be even more difficult if quantum bowlers were allowed! Take the electron, for example. The amount of spin an electron carries is quantized, so that it always has a magnitude which is ±1/2 (in units of Planck’s constant; all fermions have half-integer spin). In addition, according to quantum mechanics, the orientation of the spin is indeterminate until it is measured. Any particular measurement can only determine the component of spin in one direction. Let’s take as an example the case where the measuring device is sensitive to the z-component, i.e. spin in the vertical direction. The outcome of an experiment on a single electron will lead a definite outcome which might either be “up” or “down” relative to this axis. However, until one makes a measurement the state of the system is not specified and the outcome is consequently not predictable with certainty; there will be a probability of 50% probability for each possible outcome. We could write the state of the system (expressed by the spin part of its wavefunction ψ prior to measurement in the form \|ψ> = (\|↑> + \|↓>)/√2 This gives me an excuse to use the rather beautiful “bra-ket” notation for the state of a quantum system, originally due to Paul Dirac. The two possibilities are “up” (↑) and “down” (↓) and they are contained within a “ket” (written \|>)which is really just a shorthand for a wavefunction describing that particular aspect of the system. A “bra” would be of the form <\|; for the mathematicians this represents the Hermitian conjugate of a ket. The √2 is there to insure that the total probability of the spin being either up or down is 1, remembering that the probability is the square of the wavefunction. When we make a measurement we will get one of these two outcomes, with a 50% probability of each. At the point of measurement the state changes: if we get “up” it becomes purely \|↑> and if the result is “down” it becomes \|↓>. Either way, the quantum state of the system has changed from a “superposition” state described by the equation above to an “eigenstate” which must be either up or down. This means that all subsequent measurements of the spin in this direction will give the same result: the wave-function has “collapsed” into one particular state. Incidentally, the general term for a two-state quantum system like this is a qubit, and it is the basis of the tentative steps that have been taken towards the construction of a quantum computer. Notice that what is essential about this is the role of measurement. The collapse of ψ seems to be an irreversible process, but the wavefunction itself evolves according to the Schrödinger equation, which describes reversible, Hamiltonian changes. To understand what happens when the state of the wavefunction changes we need an extra level of interpretation beyond what the mathematics of quantum theory itself provides, because we are generally unable to write down a wave-function that sensibly describes the system plus the measuring apparatus in a single form. So far this all seems rather similar to the state of a fair coin: it has a 50-50 chance of being heads or tails, but the doubt is resolved when its state is actually observed. Thereafter we know for sure what it is. But this resemblance is only superficial. A coin only has heads or tails, but the spin of an electron doesn’t have to be just up or down. We could rotate our measuring apparatus by 90° and measure the spin to the left (←) or the right (→). In this case we still have to get a result which is a half-integer times Planck’s constant. It will have a 50-50 chance of being left or right that “becomes” one or the other when a measurement is made. Now comes the real fun. Suppose we do a series of measurements on the same electron. First we start with an electron whose spin we know nothing about. In other words it is in a superposition state like that shown above. We then make a measurement in the vertical direction. Suppose we get the answer “up”. The electron is now in the eigenstate with spin “up”. We then pass it through another measurement, but this time it measures the spin to the left or the right. The process of selecting the electron to be one with spin in the “up” direction tells us nothing about whether the horizontal component of its spin is to the left or to the right. Theory thus predicts a 50-50 outcome of this measurement, as is observed experimentally. Suppose we do such an experiment and establish that the electron’s spin vector is pointing to the left. Now our long-suffering electron passes into a third measurement which this time is again in the vertical direction. You might imagine that since we have already measured this component to be in the up direction, it would be in that direction again this time. In fact, this is not the case. The intervening measurement seems to “reset” the up-down component of the spin; the results of the third measurement are back at square one, with a 50-50 chance of getting up or down. This is just one example of the kind of irreducible “randomness” that seems to be inherent in quantum theory. However, if you think this is what people mean when they say quantum mechanics is weird, you’re quite mistaken. It gets much weirder than this! So far I have focussed on what happens to the description of single particles when quantum measurements are made. Although there seem to be subtle things going on, it is not really obvious that anything happening is very different from systems in which we simply lack the microscopic information needed to make a prediction with absolute certainty. At the simplest level, the difference is that quantum mechanics gives us a theory for the wave-function which somehow lies at a more fundamental level of description than the usual way we think of probabilities. Probabilities can be derived mathematically from the wave-function, but there is more information in ψ than there is in \|2; the wave-function is a complex entity whereas the square of its amplitude is entirely real. If one can construct a system of two particles, for example, the resulting wave-function is obtained by superimposing the wave-functions of the individual particles, and probabilities are then obtained by squaring this joint wave-function. This will not, in general, give the same probability distribution as one would get by adding the one-particle probabilities because, for complex entities A and B, A2+B2 ≠(A+B)2 in general. To put this another way, one can write any complex number in the form a+ib (real part plus imaginary part) or, generally more usefully in physics , as Re, where R is the amplitude and θ is called the phase. The square of the amplitude gives the probability associated with the wavefunction of a single particle, but in this case the phase information disappears; the truly unique character of quantum physics and how it impacts on probabilies of measurements only reveals itself when the phase information is retained. This generally requires two or more particles to be involved, as the absolute phase of a single-particle state is essentially impossible to measure. Finding situations where the quantum phase of a wave-function is important is not easy. It seems to be quite easy to disturb quantum systems in such a way that the phase information becomes scrambled, so testing the fundamental aspects of quantum theory requires considerable experimental ingenuity. But it has been done, and the results are astonishing. Let us think about a very simple example of a two-component system: a pair of electrons. All we care about for the purpose of this experiment is the spin of the electrons so let us write the state of this system in terms of states such as which I take to mean that the first particle has spin up and the second one has spin down. Suppose we can create this pair of electrons in a state where we know the total spin is zero. The electrons are indistinguishable from each other so until we make a measurement we don’t know which one is spinning up and which one is spinning down. The state of the two-particle system might be this: \|ψ> = (\|↑↓> – \|↓↑>)/√2 squaring this up would give a 50% probability of “particle one” being up and “particle two” being down and 50% for the contrary arrangement. This doesn’t look too different from the example I discussed above, but this duplex state exhibits a bizarre phenomenon known as quantum entanglement. Suppose we start the system out in this state and then separate the two electrons without disturbing their spin states. Before making a measurement we really can’t say what the spins of the individual particles are: they are in a mixed state that is neither up nor down but a combination of the two possibilities. When they’re up, they’re up. When they’re down, they’re down. But when they’re only half-way up they’re in an entangled state. If one of them passes through a vertical spin-measuring device we will then know that particle is definitely spin-up or definitely spin-down. Since we know the total spin of the pair is zero, then we can immediately deduce that the other one must be spinning in the opposite direction because we’re not allowed to violate the law of conservation of angular momentum: if Particle 1 turns out to be spin-up, Particle 2 must be spin-down, and vice versa. It is known experimentally that passing two electrons through identical spin-measuring gadgets gives results consistent with this reasoning. So far there’s nothing so very strange in this. The problem with entanglement lies in understanding what happens in reality when a measurement is done. Suppose we have two observers, Dick and Harry, each equipped with a device that can measure the spin of an electron in any direction they choose. Particle 1 emerges from the source and travels towards Dick whereas particle 2 travels in Harry’s direction. Before any measurement, the system is in an entangled superposition state. Suppose Dick decides to measure the spin of electron 1 in the z-direction and finds it spinning up. Immediately, the wave-function for electron 2 collapses into the down direction. If Dick had instead decided to measure spin in the left-right direction and found it “left” similar collapse would have occurred for particle 2, but this time putting it in the “right” direction. Whatever Dick does, the result of any corresponding measurement made by Harry has a definite outcome – the opposite to Dick’s result. So Dick’s decision whether to make a measurement up-down or left-right instantaneously transmits itself to Harry who will find a consistent answer, if he makes the same measurement as Dick. If, on the other hand, Dick makes an up-down measurement but Harry measures left-right then Dick’s answer has no effect on Harry, who has a 50% chance of getting “left” and 50% chance of getting right. The point is that whatever Dick decides to do, it has an immediate effect on the wave-function at Harry’s position; the collapse of the wave-function induced by Dick immediately collapses the state measured by Harry. How can particle 1 and particle 2 communicate in this way? This riddle is the core of a thought experiment by Einstein, Podolsky and Rosen in 1935 which has deep implications for the nature of the information that is supplied by quantum mechanics. The essence of the EPR paradox is that each of the two particles – even if they are separated by huge distances – seems to know exactly what the other one is doing. Einstein called this “spooky action at a distance” and went on to point out that this type of thing simply could not happen in the usual calculus of random variables. His argument was later tightened considerably by John Bell in a form now known as Bell’s theorem. To see how Bell’s theorem works, consider the following roughly analagous situation. Suppose we have two suspects in prison, say Dick and Harry (Tom grassed them up and has been granted immunity from prosecution). The two are taken apart to separate cells for individual questioning. We can allow them to use notes, electronic organizers, tablets of stone or anything to help them remember any agreed strategy they have concocted, but they are not allowed to communicate with each other once the interrogation has started. Each question they are asked has only two possible answers – “yes” or “no” – and there are only three possible questions. We can assume the questions are asked independently and in a random order to the two suspects. When the questioning is over, the interrogators find that whenever they asked the same question, Dick and Harry always gave the same answer, but when the question was different they only gave the same answer 25% of the time. What can the interrogators conclude? The answer is that Dick and Harry must be cheating. Either they have seen the question list ahead of time or are able to communicate with each other without the interrogator’s knowledge. If they always give the same answer when asked the same question, they must have agreed on answers to all three questions in advance. But when they are asked different questions then, because each question has only two possible responses, by following this strategy it must turn out that at least two of the three prepared answers – and possibly all of them – must be the same for both Dick and Harry. This puts a lower limit on the probability of them giving the same answer to different questions. I’ll leave it as an exercise to the reader to show that the probability of coincident answers to different questions in this case must be at least 1/3. This a simple illustration of what in quantum mechanics is known as a Bell inequality. Dick and Harry can only keep the number of such false agreements down to the measured level of 25% by cheating. This example is directly analogous to the behaviour of the entangled quantum state described above under repeated interrogations about its spin in three different directions. The result of each measurement can only be either “yes” or “no”. Each individual answer (for each particle) is equally probable in this case; the same question always produces the same answer for both particles, but the probability of agreement for two different questions is indeed ¼ and not larger as would be expected if the answers were random. For example one could ask particle 1 “are you spinning up” and particle 2 “are you spinning to the right”? The probability of both producing an answer “yes” is 25% according to quantum theory but would be higher if the particles weren’t cheating in some way. Probably the most famous experiment of this type was done in the 1980s, by Alain Aspect and collaborators, involving entangled pairs of polarized photons (which are bosons), rather than electrons, primarily because these are easier to prepare. The implications of quantum entanglement greatly troubled Einstein long before the EPR paradox. Indeed the interpretation of single-particle quantum measurement (which has no entanglement) was already troublesome. Just exactly how does the wave-function relate to the particle? What can one really say about the state of the particle before a measurement is made? What really happens when a wave-function collapses? These questions take us into philosophical territory that I have set foot in already; the difficult relationship between epistemological and ontological uses of probability theory. Thanks largely to the influence of Niels Bohr, in the relatively early stages of quantum theory a standard approach to this question was adopted. In what became known as the Copenhagen interpretation of quantum mechanics, the collapse of the wave-function as a result of measurement represents a real change in the physical state of the system. Before the measurement, an electron really is neither spinning up nor spinning down but in a kind of quantum purgatory. After a measurement it is released from limbo and becomes definitely something. What collapses the wave-function is something unspecified to do with the interaction of the particle with the measuring apparatus or, in some extreme versions of this doctrine, the intervention of human consciousness. I find it amazing that such a view could have been held so seriously by so many highly intelligent people. Schrödinger hated this concept so much that he invented a thought-experiment of his own to poke fun at it. This is the famous “Schrödinger’s cat” paradox. I’ve sent Columbo out of the room while I describe this. In a closed box there is a cat. Attached to the box is a device which releases poison into the box when triggered by a quantum-mechanical event, such as radiation produced by the decay of a radioactive substance. One can’t tell from the outside whether the poison has been released or not, so one doesn’t know whether the cat is alive or dead. When one opens the box, one learns the truth. Whether the cat has collapsed or not, the wave-function certainly does. At this point one is effectively making a quantum measurement so the wave-function of the cat is either “dead” or “alive” but before opening the box it must be in a superposition state. But do we really think the cat is neither dead nor alive? Isn’t it certainly one or the other, but that our lack of information prevents us from knowing which? And if this is true for a macroscopic object such as a cat, why can’t it be true for a microscopic system, such as that involving just a pair of electrons? As I learned at a talk by the Nobel prize-winning physicist Tony Leggett – who has been collecting data on this recently – most physicists think Schrödinger’s cat is definitely alive or dead before the box is opened. However, most physicists don’t believe that an electron definitely spins either up or down before a measurement is made. But where does one draw the line between the microscopic and macroscopic descriptions of reality? If quantum mechanics works for 1 particle, does it work also for 10, 1000? Or, for that matter, 1023? Most modern physicists eschew the Copenhagen interpretation in favour of one or other of two modern interpretations. One involves the concept of quantum decoherence, which is basically the idea that the phase information that is crucial to the underlying logic of quantum theory can be destroyed by the interaction of a microscopic system with one of larger size. In effect, this hides the quantum nature of macroscopic systems and allows us to use a more classical description for complicated objects. This certainly happens in practice, but this idea seems to me merely to defer the problem of interpretation rather than solve it. The fact that a large and complex system makes tends to hide its quantum nature from us does not in itself give us the right to have a different interpretations of the wave-function for big things and for small things. Another trendy way to think about quantum theory is the so-called Many-Worlds interpretation. This asserts that our Universe comprises an ensemble – sometimes called a multiverse – and probabilities are defined over this ensemble. In effect when an electron leaves its source it travels through infinitely many paths in this ensemble of possible worlds, interfering with itself on the way. We live in just one slice of the multiverse so at the end we perceive the electron winding up at just one point on our screen. Part of this is to some extent excusable, because many scientists still believe that one has to have an ensemble in order to have a well-defined probability theory. If one adopts a more sensible interpretation of probability then this is not actually necessary; probability does not have to be interpreted in terms of frequencies. But the many-worlds brigade goes even further than this. They assert that these parallel universes are real. What this means is not completely clear, as one can never visit parallel universes other than our own … It seems to me that none of these interpretations is at all satisfactory and, in the gap left by the failure to find a sensible way to understand “quantum reality”, there has grown a pathological industry of pseudo-scientific gobbledegook. Claims that entanglement is consistent with telepathy, that parallel universes are scientific truths, that consciousness is a quantum phenomena abound in the New Age sections of bookshops but have no rational foundation. Physicists may complain about this, but they have only themselves to blame. But there is one remaining possibility for an interpretation of that has been unfairly neglected by quantum theorists despite – or perhaps because of – the fact that is the closest of all to commonsense. This view that quantum mechanics is just an incomplete theory, and the reason it produces only a probabilistic description is that does not provide sufficient information to make definite predictions. This line of reasoning has a distinguished pedigree, but fell out of favour after the arrival of Bell’s theorem and related issues. Early ideas on this theme revolved around the idea that particles could carry “hidden variables” whose behaviour we could not predict because our fundamental description is inadequate. In other words two apparently identical electrons are not really identical; something we cannot directly measure marks them apart. If this works then we can simply use only probability theory to deal with inferences made on the basis of information that’s not sufficient for absolute certainty. After Bell’s work, however, it became clear that these hidden variables must possess a very peculiar property if they are to describe out quantum world. The property of entanglement requires the hidden variables to be non-local. In other words, two electrons must be able to communicate their values faster than the speed of light. Putting this conclusion together with relativity leads one to deduce that the chain of cause and effect must break down: hidden variables are therefore acausal. This is such an unpalatable idea that it seems to many physicists to be even worse than the alternatives, but to me it seems entirely plausible that the causal structure of space-time must break down at some level. On the other hand, not all “incomplete” interpretations of quantum theory involve hidden variables. One can think of this category of interpretation as involving an epistemological view of quantum mechanics. The probabilistic nature of the theory has, in some sense, a subjective origin. It represents deficiencies in our state of knowledge. The alternative Copenhagen and Many-Worlds views I discussed above differ greatly from each other, but each is characterized by the mistaken desire to put quantum mechanics – and, therefore, probability – in the realm of ontology. The idea that quantum mechanics might be incomplete (or even just fundamentally “wrong”) does not seem to me to be all that radical. Although it has been very successful, there are sufficiently many problems of interpretation associated with it that perhaps it will eventually be replaced by something more fundamental, or at least different. Surprisingly, this is a somewhat heretical view among physicists: most, including several Nobel laureates, seem to think that quantum theory is unquestionably the most complete description of nature we will ever obtain. That may be true, of course. But if we never look any deeper we will certainly never know… With the gradual re-emergence of Bayesian approaches in other branches of physics a number of important steps have been taken towards the construction of a truly inductive interpretation of quantum mechanics. This programme sets out to understand probability in terms of the “degree of belief” that characterizes Bayesian probabilities. Recently, Christopher Fuchs, amongst others, has shown that, contrary to popular myth, the role of probability in quantum mechanics can indeed be understood in this way and, moreover, that a theory in which quantum states are states of knowledge rather than states of reality is complete and well-defined. I am not claiming that this argument is settled, but this approach seems to me by far the most compelling and it is a pity more people aren’t following it up…
917ee6331950ff10	What Can Quantum Machine Learning Do? What can quantum computers do? In this post, we explore the concept of many-body problem in quantum chemistry that may be one of the most immediate applications of quantum computers and quantum machine learning. A main goal of quantum chemistry is to predict the structure, stability, and reactivity of molecules. Modelling how each of these particles interact and affect each other is essentially impossible. In principle, this requires solving the Schrödinger equation for the many-electron problem, a task that is so computationally intensive that even our modern supercomputers fail to perform them fast enough. The Trade-Off Every AI Company Will Face Machines learn faster with more data, and more data is generated when machines are deployed in the wild. However, bad things can happen in the wild and harm the company brand. Putting products in the wild earlier accelerates learning but risks harming the brand; putting products in the wild later slows learning but allows for more time to improve the product in-house and protect the brand. Commercialize quantum technologies in five years
f4fa4096badc0563	Sunday, July 28, 2019 The Forgotten Solution: Superdeterminism Welcome to the renaissance of quantum mechanics. It took more than a hundred years, but physicists finally woke up, looked quantum mechanics into the face – and realized with bewilderment they barely know the theory they’ve been married to for so long. Gone are the days of “shut up and calculate”; the foundations of quantum mechanics are en vogue again. It is not a spontaneous acknowledgement of philosophy that sparked physicists’ rediscovered desire; their sudden search for meaning is driven by technological advances. With quantum cryptography a reality and quantum computing on the horizon, questions once believed ephemeral are now butter and bread of the research worker. When I was a student, my prof thought it questionable that violations of Bell’s inequality would ever be demonstrated convincingly. Today you can take that as given. We have also seen delayed-choice experiments, marveled over quantum teleportation, witnessed decoherence in action, tracked individual quantum jumps, and cheered when Zeilinger entangled photons over hundreds of kilometers of distance. Well, some of us, anyway. But while physicists know how to use the mathematics of quantum mechanics to make stunningly accurate predictions, just what this math is about has remained unclear. This is why physicists currently have several “interpretations” of quantum mechanics. I find the term “interpretations” somewhat unfortunate. That’s because some ideas that go as “interpretation” are really theories which differ from quantum mechanics, and these differences may one day become observable. Collapse models, for example, explicitly add a process for wave-function collapse to quantum measurement. Pilot wave theories, likewise, can result in deviations from quantum mechanics in certain circumstances, though those have not been observed. At least not yet. A phenomenologist myself, I am agnostic about different interpretations of what is indeed the same math, such as QBism vs Copenhagen or the Many Worlds. But I agree with the philosopher Tim Maudlin that the measurement problem in quantum mechanics is a real problem – a problem of inconsistency – and requires a solution. And how to solve it? Collapse models solve the measurement problem, but they are hard to combine with quantum field theory which for me is a deal-breaker. Pilot wave theories also solve it, but they are non-local, which makes my hair stand up for much the same reason. This is why I think all these approaches are on the wrong track and instead side with superdeterminism. But before I tell you what’s super about superdeterminism, I have to briefly explain the all-important theorem from John Stewart Bell. It says, in a nutshell, that correlations between certain observables are bounded in every theory which fulfills certain assumptions. These assumptions are what you would expect of a deterministic, non-quantum theory – statistical locality and statistical independence (together often referred to as “Bell locality”) – and should, most importantly, be fulfilled by any classical theory that attempts to explain quantum behavior by adding “hidden variables” to particles. Experiments show that the bound of Bell’s theorem can be violated. This means the correct theory must violate at least one of the theorem’s assumptions. Quantum mechanics is indeterministic and violates statistical locality. (Which, I should warn you has little to do with what particle physicists usually mean by “locality.”) A deterministic theory that doesn’t fulfill the other assumption, that of statistical independence, is called superdeterministic. Note that this leaves open whether or not a superdeterministic theory is statistically local. Unfortunately, superdeterminism has a bad reputation, so bad that most students never get to hear of it. If mentioned at all, it is commonly dismissed as a “conspiracy theory.” Several philosophers have declared superdeterminism means abandoning scientific methodology entirely. To see where this objection comes from – and why it’s wrong – we have to unwrap this idea of statistical independence. Statistical independence enters Bell’s theorem in two ways. One is that the detectors’ settings are independent of each other, the other one that the settings are independent of the state you want to measure. If you don’t have statistical independence, you are sacrificing the experimentalist’s freedom to choose what to measure. And if you do that, you can come up with deterministic hidden variable explanations that result in the same measurement outcomes as quantum mechanics. I find superdeterminism interesting because the most obvious class of hidden variables are the degrees of freedom of the detector. And the detector isn’t statistically independent of itself, so any such theory necessarily violates statistical independence. It is also, in a trivial sense, non-linear just because if the detector depends on a superposition of prepared states that’s not the same as superposing two measurements. Since any solution of the measurement problem requires a non-linear time evolution, that seems a good opportunity to make progress. Now, a lot of people discard superdeterminism simply because they prefer to believe in free will, which is where I think the biggest resistance to superdeterminism comes from. Bad enough that belief isn’t a scientific reason, but worse that this is misunderstanding just what is going on. It’s not like superdeterminism somehow prevents an experimentalist from turning a knob. Rather, it’s that the detectors’ states aren’t independent of the system one tries to measure. There just isn’t any state the experimentalist could twiddle their knob to which would prevent a correlation. Where do these correlations ultimately come from? Well, they come from where everything ultimately comes from, that is from the initial state of the universe. And that’s where most people walk off: They think that you need to precisely choose the initial conditions of the universe to arrange quanta in Anton Zeilinger’s brain just so that he’ll end up turning a knob left rather than right. Besides sounding entirely nuts, it’s also a useless idea, because how the hell would you ever calculate anything with it? And if it’s unfalsifiable but useless, then indeed it isn’t science. So, frowning at superdeterminism is not entirely unjustified. But that would be jumping to conclusions. How much detail you need to know about the initial state to make predictions depends on your model. And without writing down a model, there is really no way to tell whether it does or doesn’t live up to scientific methodology. It’s here where the trouble begins. While philosophers on occasion discuss superdeterminism on a conceptual basis, there is little to no work on actual models. Besides me and my postdoc, I count Gerard ‘t Hooft and Tim Palmer. The former gentleman, however, seems to dislike quantum mechanics and would rather have a classical hidden variables theory, and the latter wants to discretize state space. I don’t see the point in either. I’ll be happy if the result solves the measurement problem and is still local the same way that quantum field theories are local, ie as non-local as quantum mechanics always is.* The stakes are high, for if quantum mechanics is not a fundamental theory, but can be derived from an underlying deterministic theory, this opens the door to new applications. That’s why I remain perplexed that what I think is the obvious route to progress is one most physicists have never even heard of. Maybe it’s just a reality they don’t want to wake up to. Recommended reading: • The significance of measurement independence for Bell inequalities and locality Michael J. W. Hall • Bell's Theorem: Two Neglected Solutions Louis Vervoort FoP, 3,769–791 (2013), arXiv:1203.6587 * Rewrote this paragraph to better summarize Palmer’s approach. 1. It's misleading to say that Maudlin in particular has declared that superdeterminism means abandoning scientific methodology. That point was made already in 1976 by Clauser, Shimony and Holt. "In any scientific experiment in which two or more variables are supposed to be randomly selected, one can always conjecture that some factor in the overlap of the backwards light cones has controlled the presumably random choices. But, we maintain, skepticism of this sort will essentially dismiss all results of scientific experimentation. Unless we proceed under the assumption that hidden conspiracies of this sort do not occur, we have abandoned in advance the whole enterprise of discovering the laws of nature by experimentation." I recommend that anyone interested in the status of the statistical independence assumption start by reading the 1976 exchange between Bell and CSH, discussed in section 3.2.2 of this: 1. With "in particular" I didn't mean to imply he's the first or only one. In any case, I'll fix that sentence, sorry for the misunderstanding. 2. WayneMyrvold, If statistical independence is such a generaly accepted principle it follows that no violations should be found, right? Let me give some examples of distant physical systems that are known not to be independent: 1. Stars in a galaxy (they all orbit around the galaxy'a center) 2. Planets and their star. 3. Electrons in an atom. 4. Synchronized clocks. So, it seems that it is possible after all to do science and accept that some systems are not independent, right? Now, if you reject superdeterminism you need to choose a different type of theory. Can you tell me what type of theory you prefer and just give a few examples of known experiments that provide evidence for that type of theory? For example, if you think non-locality is the way to go can you provide at least one experiment where the speed of light was certainly exceeded? 2. This would be easier for me to understand if I could figure out what the definition is of a successful superdeterminism model. Suppose you and your postdoc succeed! What would follow in the abstract of your article following "My postdoc and I have succeeded in creating a successful superdeterministic model. It clearly works because we show here that it..." (does exactly what?). Thanks. 1. Leibniz, On a theoretical level I would say it's successful if it solves the measurement problem. But the more interesting question is of course what a success would mean experimentally. It would mean that you can predict the outcome of a quantum measurement better than what quantum mechanics allows you to (because the underlying theory is deterministic after all). Does that answer your question? 3. The reference cited (by WayneMyrvold) above Bell’s Theorem (substantive revision Wed Mar 13, 2019) Is worth reading its entirety. Bell's Theorem: Two Neglected Solutions Louis Vervoort 'supercorrelation' appears more interesting than 'superdeterminism' (and perhaps more [Huw] Pricean). 1. Violation of the Bell-Inequality in Supercorrelated Systems Louis Vervoort (latest version 20 Jan 2017) 4. You have a HUGE problem with thermodynamics! Take Alice and Bob intending to participate in an EPR-Bell experiment. None of them knows yet what spin direction she/he is going to choose to measure. This is a decision they’ll make only at the very last second. Superdeterminism must invoke two huge past light-cones stretching from each of them backwards to the past, to the spacetime region where the two cones overlap. If Alice and Bob are very far away, then this overlap goes back millions of years ago, extending over millions of light-years over space. Then, all – but absolutely ALL – particles within this huge area must carry – together! – the information required for a deterministic computation that can predict the two human's decision, which is not yet known even to themselves (they are not born yet ��). Miss one of these particles and the computation will fail. Simple, right? Wrong. The second law of thermodynamics exacts a fundamental price for such a super-computation. It requires energy proportionate to the calculation needed: Zillions of particles, over a huge spacetime region, for predicting a minute event to take place far in the far future. Whence the energy? How many megawatts for how many megabits? And where is the computation mechanism? I once asked Prof. t'Hooft this question. He, being as sincere and kind as everybody who have met him knows, exclaimed “Avshalom I know what you are saying and it worries me tremendously!” You want superdeterminism? Very simple. Make quantum causality time-symmetric, namely allow each of the two particle to communicate to their common origin BACKWARDS IN TIME and there you go. No zillions of particles and light-years, only the two particles involved. This is the beauty of TSVF. Here is one example: there are many more, with surprising new predictions. Yours, Avshalom 1. Avshalom, That's right, you can solve the problem of finetuning initial conditions by putting a constraint on the future, which is basically what we are doing. Devil is in the details. I don't see what the paper you refers to achieves that normal quantum mechanics doesn't. 2. Our paper makes retrocausality the most parsimonious explanation. No conspiracy, no need to return to the Big Bang, just a simple spacetime zygzag. 3. Parsimonious explanation... for what? Do you or do you not solve the measurement problem? 4. Parsimonious for the nonlocality paradox. Our advance concerning the measurement problem is here: We show that wavefunction collapse occurs through a multiplicity of momentary "mirage particles" of which n-1 have negative mass, such that the particle's final position occurs after the mutual cancellation of all the other positive and negative particles. The mathematical derivation is very rigorous, right from quantum theory. A laboratory confirmation by Okamoto and Takeuchi is due within weeks. 5. What do you mean by wavefunction collapse? Is or isn't your time-evolution given by the Schrödinger equation? 6. It is, done time-symmetrically. Two wave functions along the two time directions between source and absorber (pre- and post-selections). This gives you information about the particle during the relevant time-interval much more than the uncertainty principle seems to allow. Under special choices of such boundary conditions the formalism yields unusual physical values: too large/small or even negative. Again the derivation is rigorous. The paper is very lucid. 7. Well, if your time evolution is given by the Schrödinger equation, then you are doing standard quantum mechanics. Again, what problem do you think you are solving? 8. Again, we solve the nonlocality problem by retrocausality, and elucidate the measurement problem by summing together the two wave-functions. No point going into the simple math here. It's all in the paper. 9. Avshalom Elitzur, Correlations are normal in field theories. All stars in a galaxy orbit the galactic center. They do not perform any calculations, they just respond to the gravitational field at their location. In a Bell test you have an electromagnetic system (Alice, Bob and the source of entangled particles are just large groups of electrons and nuclei). Just like in the case of gravity, in electromagnetism the motion of charged particles is correlated with the position/momenta of other charged particles. Nature does the required computations. No need for retrocausality. 10. Wrong, very wrong. All stars affect one another by gravity, a classical force of enormous scales, obeying Newton's inverse square law and propagating locally. No computation needed. With Alice's and Bob's last-minute decision, the particles must predict their decision by computing the tiniest subatomic forces of zillions of particles over a huge spacetime region. Your choice of analogy only stresses my argument. 11. Avshalom Elitzur, There is no "last-minute decision", this is the point of determinism. What we consider "last-minute decisions" are just "snapshots" from the continuous deterministic evolution of the collections of electrons and quarks Alice and Bob are made of. The motion of these charged particles inside our bodies (including our brains that are responsible for our "decisions") are not part of the information available to our consciousness, this is why our decisions appear sudden to us. The exact way these charged particle move is not independent on the way other, distant charged particles move (as per classical electromagnetism) so I would not see why our decisions would necessarily be independent of the hidden variable. 12. Precisely! But this is the difference between stars and galaxies on the one hand, and particles and neurons on the other. In the former case no computation is needed because the gravitational forces are huge. In the latter, you expect an EPR particle to compute in advance the outcome of neuronal dynamics within a small brain very far away! Can you see the difference? 13. The magnitude of the force is irrelevant. It is just a constant in the equations. Changing that constant does not make the equations fail (as required by independence). The systems are not independent because the state of any one also depends on the state of the distant systems. This is a mathematical fact. Moving a planet far from its star does not result in a non-elliptical, random orbit, but in a larger ellipse. You do not get independence by either playing with the strength of the force or by increasing the distance so that the force gets weaker. 14. So each EPR particle can infer from the present state of Alice's and Bob's past light-cone's particles what decision they will take, even arbitrarily much later? Sabine, is this what the model says? 15. Avshalom Elitzur, It's not clear if the question was directed only to Sabine, but I will answer it anyway. In classical electromagnetism any infinitesimal region around a particle contains complete information about the state (position and momentum) of all particles in the universe in the form of electric and magnetic fields at each point. In a continuous space the number of points in any such region is infinite so you will always have enough of them. Each particle responds to those fields according to Lorentz force. That's it. The particle does not "infer" anything. The states of particles in the universe will be correlated in a way specified by the solution of the N-body problem where N is the number of particles in the universe. My hypothesis is that if one would solve that N-body problem, he will find that for any initial state only solutions that describe particle trajectories compatible with QM's prediction would be found. I think that such a hypothesis is testable for a computer simulation using a small enough N. N should be large enough to accommodate a simplified Bell test though (obviously without humans or other macroscopic objects). 16. Sorry I am still not getting a clear answer to my simple question about the the physics: Two distant experimenters are going to make a last-minute decision about a measurement, a decision which they themselves do not know yet. Then the two particles arrive, each at another experimenter, and the measurements are made. How can the measurement outcome of the one particle be correlated with the measurement taken on the other distant particle? I (and probably other readers) will appreciate a straightforward answer from you as well as from Sabine. 17. Avshalom, You are asking "Why is the initial state what it is?" but there is never an answer to this question. You make assumptions about the initial state to the end of getting a prediction. Whatever works, works. Let me ask you in return how can the measurement outcome not be correlated? There isn't an answer to this either, other than postulating it. 18. Sorry this is certainly not my question. Please let me reiterate the issue. Two distant particles, each being affected by the choice of measurement carried out AT THAT MOMENT on the distant one. Either i) Something goes between them at infinite velocity; ii) The universe's present state deterministically dictates the later choices in a way that affects also the particles to yield the nonlocal correlations iii) Quantum causality is time-symmetric and the distant effects go through spacetime zigzag. Each option has its pluses and minuses. It seems that (i) has no advocates here; I am pointing out serious difficulties emerging from (ii); and advocating (iii) as the most elegant, fully according with quantum theory yet most fruitful in terms of novel predictions. I'd very much appreciate comments along these lines. Sorry if I am nagging - I have no problem continuing the discussion elsewhere. 19. Avshalom, Yours is a false dichotomy (trichotomy?). If you think otherwise, please prove that those are the only three options. There is also, of course, no way to follow your "zig zag" from option iii. I already said several times that retrocausality is an option I quite like, but I do not think that it makes sense if the Schroedinger equation remains unmodified because that doesn't solve any problem. 20. I was sure that you are advocating a certain variant of (ii). Not so? What then is this (iv)? Again, the Schrodinger equation is unmodified, only used TWICE. And the result is stunningly powerful. See this example 21. Avshalom, If you have a time-reversible operator, postulating an early-time state, present state, or late state is conceptually identical, so I don't see the distinction you draw between ii and iii. For this reason any superdeterministic theory could be said to be retrocausal, though one may quibble about just what's "causal" here. (I think the word is generally best avoided if you have a deterministic evolution.) The other problem with your 3-option solution is that option 3 is more specific than what is warranted. There isn't any "zig zag" that follows just from having a boundary condition in the future. You can use as many Schroedinger equations as you want, the result will still be a linear time evolution. 22. Not when the 2nd Law of Thermodynamics is taken into account. Whereas (ii) invokes odd conspiracies and impossible computations with no mechanism, (iii) is free of them. And the main fact remains: TSVF has derived a disappearance of a particle from one box, reappearance into an arbitrarily distant one and vice versa - all with certainty 1! See "The case of the disappearing (and re-appearing) particle" Nature Sci. Rep. 531 (2017). The actual experiment is underway. This derivation id obliged by quantum theory, but the fact that it has been derived only by (iii) indicates that it is not only methodologically more efficient but also more ontologically sound. 23. The 2nd law of thermodynamics is a statement about the occupation in state space. If you don't know what the space is and its states, you cannot even make the statement. Again, you are making a lot of assumptions here without even noticing. 24. I could locate another TSVF paper: Quote: "The photons do not always follow continuous trajectories" ... but Bohmian Mechanics require continuous trajectories. Can you repair it? 25. Avshalom Elitzur, As I have pointed out before all this talk about "last-minute decisions" that "they themselves do not know yet" is a red herring. There are a lot of facts about our brains that are true, yet unknown to us. The number of neurons is such an example. If you are not even aware about the number of neurons in your head (and it is possible in principle to count them using a high-resolution MRI) why would you expect to be aware about the motion of each electron and quark? Yes, we do not know what decisions we will make and it is expected to be that way regardless of your preferred view of QM. In order to understand the reason for the observed results you need to understand: 1. What initial states are possible? 2. How these initial states evolve and produce the observed experimental outcomes? As long as those questions are left unanswered you cannot expect to have a proper explanation. Let me give you an example of a different type of non-trivial correlations that can only be explained by understanding the initial state. You probably know that most planets in a planetary system orbit in the same direction around their star and also in the same plane (more or less). How do you explain it? There is no obvious reason those planets could not orbit in any direction and in any plane. Once you examine the past you find out that any planetary system originates in a cloud of gas and dust, and such a cloud, regardless of its original state will tend to form a disk where all the material orbits the center of the disk in the same direction. As the planets are formed from this material it becomes clear that you expect them to move in the same direction and in the same plane. In the case of EPR understanding the initial state is more difficult because the hidden variable and the detector settings are not statistical parameters (like the direction of rotation of a cloud of gas) but depend on the position and momenta of all electrons and quarks involved in the experiment. In other words I do not expect such a calculation to be possible anytime soon. This unpleasant situation forces us to accept that Bell tests are not of any use in deciding for or against local hidden variable theories. That decision must be based on simpler systems, like a hydrogen atom, or a molecule where detailed calculations can be performed. 5. An excellent article. "The stakes are high, for if quantum mechanics is not a fundamental theory, but can be derived from an underlying deterministic theory, this open the door to new applications" Great to see that Einstein's dream lives on:-) 6. How do you arrange any experimental test at all if you assume superdeterminism? If you have a regular probabilistic theory, you can test it. If you perform an experiment that is supposed to have probability ½ of result A and ½ of result B, and you get result A 900 times out of 1000, you can conclude that there is something wrong with your theory (or maybe with your calculations). But in superdeterminism, if you perform such an experiment, can you conclude that your theory is wrong? Maybe the initial conditions of the universe were designed to yield exactly this result. It certainly seems to me that, if you assume superdeterminism, you cannot rely on probabilistic outcomes being correct; the whole point of superdeterminism is to get around the fact that Bell's theorem shows that quantum mechanics gives probability distributions that can't be explained by a standard classical local probability theory. So if you can't rely on probability theory in the case of the CHSH inequality, how do you justify using probability for anything at all? 1. Peter, You arrange your experiment exactly the same way you always arrange your experiment. I don't know why you think there is something different in dealing with a superdeterministic theory than dealing with a deterministic theory. You can rely on probabilistic outcomes if you have a reason to think that you have them properly sampled. This is the case, arguably, for all experiments we have done so far. So, nothing new to see there. Really, think about this for a moment. A superdeterministic theory reproduces quantum mechanics. It therefore makes the same predictions as quantum mechanics. (Or, well, if it doesn't, it's wrong, so forget about it.) Difference is that it makes more predictions besides that. (Because it's not probabilistic.) No, it's not because a superdeterministic theory doesn't have to be "classical" in any sense. 2. Peter, A superdeterministic theory only needs to posit that the emission of entangeled particles and their detection are not independent events (the way the particles are emitted depends on the way the particles are detected). Such a theory does not need to claim that all events are correlated. So my answer to your question: is this: If your experiment was based on emission/detection of entangled particles you may be skeptical about the fact that your theory is wrong. If not, you may use statistics in the usual way. 3. Sabine, I think you have underestimated the consequences of superdeterminism. If it is true, there is no way to really test a theory against measurements, thus we do not know if our theory really describes the world. In my opinion, to defend locality by accepting superdeterminism is not worth it. Superdeterminism is an assumption at the epistemic level, underlying the possibility to check our theory about the world, while (non-)locality is at the theory content level. 4. tytung, "Superdeterminism" is not a theory. It's a property of a class of models. You can only test the models. What I am saying is that if you do not actually go and develop such a model you will, of course, never be able to test it. 5. Sabine, Let's say there is a God, and He/She behaves kind at certain times and has an ugly side at other sometimes. You want to check if this is true. But if the universe's fate is such that you always observe this God on those times He/She is kind, then you will build a theory of a kind God. Yes as you said, you can perform any test as usual, but due to the fate, the truth is very different, and forever beyond the reach of your knowledge. So the statement that God is sometimes unkind becomes unfalsifiable if fate is accepted. (I am an atheist.) 6. tytung, I don't know what any of that has to do with me, but you seem to have discovered that science cannot tell you what is true and what isn't; it can only tell you what is a good explanation for your observations. 7. Sabine, No, of course I do think Science can tell us what is true and what isn't, but this is precisely because Science implicitly assumes the freedom of choosing our measurements. In a super-deterministic model, all measurement choices are predetermined. You are not able to falsify a theory through those measurements that it does not allow you to make. You can only build a theory through allowed measurements, and this theory may be very different from the underlying model. I think this is what Shor and many others are concern about superdeterminism. 8. tytung, I know perfectly fine what they are saying and I have explained multiple times why it is wrong. Science does not assume any such thing as the freedom of choosing measurements. Science is about finding useful descriptions for our observations. Also, as I have said repeatedly, the objection you raise would hold equally well for any deterministic theory, be that Newtonian mechanics or many worlds. It is just wrong. You are confused about what science can and cannot do. 7. I was playing backgammon with a friend who is a nuclear engineer. The subject of stochasticity came up and how if one were able to computer the motion of dice deterministically from the initial conditions of a throw that stochastic or probability behavior could be vanquished. I mentioned that if we had quantum dice this would not be the case. Superdeterminism would counter that such predictions are possible, at least in principle. It is tough to completely eliminate any possible correlation that might sneak the obedience to Bell inequalities into quantum outcomes. If we are to talk about the brain states of the experimenter then in some ways for now that is a barrier. On the past light cone of the device and the experimenter at the moment of measurement are many moles of quantum states, or putative superdeterministic states. This gets really huge if the past light cone is considered to the emergence of the observable universe. Trying to eliminate causal influences of such is a game of ever more gilding the lily. Physical measurements are almost by necessity local, and with QM the nonlocality of the wave function plays havoc with the measurement locality. I would then tend to say that at some point one has to say there are no effective causal influences that determine a quantum outcome on a FAPP basis. My sense then is that superdeterminism is most likely not an effective theory. Of course I could be wrong, but honestly I see it as a big non-starter. 8. Sabine, "...they are non-local, which makes my hair stand up..." Well, reality may not be concerned with your coiffure. Isn't this the kind of emotional reaction to an idea your book argues against? More specifically, why is superdeterminism more palatable to you than nonlocality? Superficially they seem similar: the usual notion of nonlocality refers to space, while superdeterminism is nonlocality in time. 1. Andrew, Please excuse the flowery expression; I was trying to avoid repeating myself. It's not consistent with the standard model, that's what I am saying. Not an emotional reaction but what Richard Dawid calls the meta-inductive argument. (Which says, in essence, stick close to what works.) 2. Andrew, I can give you many examples of physical systems that are not independent: stars in a galaxy, planets and their star in a planetary system, electrons and the nucleus in an atom, synchronized clockes, etc. How many examples do you have where the speed of light has been exceeded? 9. Sabine, not to be dragged onto the slippery slope of a never ending “free will” discussion, I would suggest to exclude right from the beginning a human experimenter and use two random number generators (RNGs) instead. (either pseudo RNGs or quantum RNGs or CMB photon detection from opposite directions, ...) [These RNGs select (statistical independently) e.g. the orientations of the polarization filters in the Bell type experiment.] Otherwise it will distract from what I understand your intention is, namely to show in a yet unknown model “... that the detectors’ states aren’t independent of the system ...” and that this yet unknown model will predict more than QM can. 1. Reimond, Yes, that's right, alluding to free will here is unnecessary and usually not helpful. On the other hand, I think it is relevant to note that this seems to be one of the reasons people reject the idea, that they are unwilling or unable to give up on free will. 2. Conway and Kochen showed how QM by itself had conflicts with free will. This is without any superdeterminism idea. The switch thrown by an experimenter could be replaced with a radioactive nucleus or some other quantum system that transitions to give one choice or if in some interval of time there is no decay the other choice is made. I think in a way we have to use the idea of a verdict "beyond a reasonable doubt" used in jurisprudence. We could in principle spend eternity trying to broom out some superdeterministic event. We may not be able to ever rule out how a brontosaurus, I think now called Apatosaurus, farted 100 million years ago and the sound generated phonons that have reverberated around and the hidden variable there interacted with an experiment. Maybe a supernova in the Andromeda galaxy 2.5 million years ago launched a neutrino that interacts with the experiment, and on it can go. So going around looking for these superdeterministic effects strikes me as George W Bush doing his "comedy routine" of looking for WMD. 3. Lawrence Crowell, The problem is that the assumption of statistical independence can be shown to be wrong for all modern theories (field theories). A charged particle does not move independently of other charged particles, even if those particles are far away. Likewise, in general relativity a massive body does not move independently of other massive bodies. What exactly makes you think that in a Bell test the group of charged particles (electrons and nuclei) that make up the source of the entangled particles evolves independently of the other two groups of charged particles (Alice and Bob)? 4. The proposal here is that everything in a sense is slightly entangled with everything else. That I have no particular problem with, though this causes a lot of people to get into quantum-woo-woo. An entanglement sum_ip_i\|ψ_i>\|φ_i> for some small probability p_i for a mixture is an aspect of decoherence. So in the states of matter around us are highly scrambled or mixed entanglements with states that bear quantum information from the distant past. I probably share tiny entanglements or quantum overlaps with states bearing quantum information held by states composing Abraham Lincoln or Charlemagne or Genghis Khan, but can never tractably find those. The superdeterminist thesis would be there are subquantum causal effects in these massive mixed entanglements which can trip up our conclusions about violations of Bell inequalities. This would mean such violations really occur because we can't make a real accounting of things. This is where I think an empirical standard similar to the legal argument of a verdict “beyond all reasonable doubt” must come into play. People doing these Bell experiments, such as forms of the Aspect experiment, try to eliminate classical influences sneaking in. I think Zeilinger made a statement a few years ago that mostly what is left is with the mind of the observer. If we are now to look at all weak entanglements of matter in an experimental apparatus to ferret out possible classical-like superdeterministic causes the work is then almost infinite. 5. Lawrence Crowell, The way I see EPR correlations explained in a superdeterministic context is as follows: Under some circumstances, a large number of gravitating bodies correlate their motion in such a way as to create a spiral galaxy. Under some circumstances, a large number of electromagnetically interacting objects (say water molecules) correlate their motion in such a way as to create a vortex in the fluid. Under some other circumstances, a large number of electromagnetically interacting objects correlate their motion in such a way as to produce EPR-type correlations. As long as the requirements for the above physical phenomenons are met, spiral galaxies, vortices or EPR correlations will be present. I think this hypothesis has the following implications in regards to the arguments you presented: 1. There is no specific cause, no singular event in the past that explains the correlations (like the fart of the apatosaurus). The correlations are "spontaneously" generated as a result of how many particles interact. 2. The efforts of Aspect or Zeilinger are not expected to make any difference because there is no way to change how those many particles interact. Electrons behave like electrons, quarks behave like quarks regardless of how you set the detector, if a human presses a button, or a monkey or a dog, or a computer doing some random number algorithm. The correlations arise from a more fundamental level that has nothing to do with the macroscopic appearance of the experimental setup. 6. Physics such as vortex flow or turbulence is primarily classical. I am not an expert on the structure of galaxies, but from what I know the spiral shape occurs from regions of gas and dust that slow the passage of stars through them. Galactic arms then do not rotate around as does a vortex, but the spiral arms are somewhat fixed in their shape. In fact this is what the whole issue with dark matter is over. 7. I also think the the quantum phenomena are in essence classical and could be explain by a classical theory, more exactly a classical field theory. The best candidate is, I think, stochastic electrodynamics. 8. This is unlikely. Bell's inequalites tell us how a classical system is likely to manifest a probability distribution. Suppose I were sending nails oriented in a certain direction perpendicular to their path. These then must pass through a gap. The orientation of the nails with gap would indicate the probability. We think of the orientation with respect to the slats as similar to a spinning game, such as the erstwhile popular TV game show Wheel of Fortune. If the nails were oriented 60 degrees relative to the slats we would expect the nails to have a ⅔ chance of passing through. Yet the quantum amplitude for the probability is cos^2(π/3) = .25. The classical estimate is larger than the actual quantum probability. This is a quick way of seeing Bell inequality, and the next time you put those RayBan classes with polarizing lenses on you are seeing a violation of Bell inequaltieis. Physics has quantum mechanics that is an L^2 system, with norm determined by the square of amplitudes that determine probabilities. Non-quantum mechanical stochastic systems are L^1 systems. Here I am thinking of macroscopic systems that have pure stochastic measures. For convex systems or hull with an L^p measure there is a dual L^q system such that 1/p + 1/q = 1. For quantum physics there is a dual system, it is general relativity with its pseudo-Euclidean distance. For my purely stochastic system the dual system is an L^∞ system, which is a deterministic physics such as classical mechanics of Newton, Lagrange and Hamilton. There is a fair amount of mathematics behind this, which I will avoid now, but think of the L^∞ system where there are no distributions fundamental to the theory. Classical physics and any deterministic system, say a Turing machine, is not about some distribution over a system state. The classical stochastic system is just a sum of probabilities, so there is no trouble with seeing that as L^1. The duality between quantum physics and spacetime physics is very suggestive of some deep physics.. In this perspective a quantum measurement is then where there is a shifting of the system from p = ½ to p = 1, thinking of a classical-like probability system after decoherence, or with the einselection this flips to an L^∞ system as a state selection closest to the classical or greatest expectation value. I read last May on how experiments were performed to detect how a system was starting to quantum tunnel. This might be some signature of how this flipping starts. It is still not clear how this flipping can be made into a dynamical principle. Avshalom pointed out an issue with thermodynamics, which I think is germane to this. With superdeterminism there would be more information in quantum systems, and this would lead to problems with the second law of thermodynamics and probably violations of entropy bounds. I looked at Motl's blog last evening, and he declares with confidence there is no measurement problem. Of course he is steeped in his own certitude on most things, such as everything from climate change to string theory. In the spirit of what he calls and “anti-quantum zealot” I think it is not likely he has this all sewed up and hundreds of physicist who think otherwise are all wrong. Motl likes Bohr, but as Heisenberg pointed out there is an uncertain cut-off between what is quantum and what is classical. So Bohr and CI are an interpretation with holes. There is lots of confusion over QM, and even some of the best of us get off the rails on this. I have to conclude that 't Hooft went gang oft aglay on this as well. 9. Lawrence Crowell, "Bell's inequalites tell us how a classical system is likely to manifest a probability distribution." This is not true. Bell's inequalites tell us how a system that allows a decomposition in independent subsystems should behave. As I have argued many times on this thread classical field theories (like classical electrodynamics) are not of this type. No matter how far you place two or more groups of charged particles they will never become independent, this is a mathematical fact. So, the statistical independence assumption does not apply here (or at least it cannot be applied for the general case). In other words classical field theories are superdeterministic, using Bell's terminology. This means that, in principle, one could violate Bell's inequalities with classical systems even in theories like classical electromagnetism. Stochastic electrodynamics is a classical theory (please ignore the Wikipedia info, it has nothing to do with Bohm's theory), in fact it is just classical electrodynamics plus the assumption of the zero-point field (a classical EM field originating at the Big-Bang). In such a theory one can actually derive Planck's constant and the quantum phenomena (including electron's spin) are explained by the interaction between particles and the zero-point field. Please find an introductory text here: Boyer, T.H. Stochastic Electrodynamics: The Closest Classical Approximation to Quantum Theory. Atoms 2019, 7, 29. The theory is far from completely reproducing QM but it is a good example of a theory I think is on the right track. Please take a look at this paper: Classical interpretation of the Debye law for the specific heat of solids R. Blanco, H. M. França, and E. Santos Phys. Rev. A 43, 693 – Published 1 January 1991 It seems that entropy could be correctly described in a classical theory as well. "I looked at Motl's blog last evening, and he declares with confidence there is no measurement problem." I have discussed with him a few ideas regarding the subjective view of QM he is proposing. I had to conclude that the guy has no clue about what he is speaking about. He is completely confused about the role of the observer (he claims that the fact that different observers observe different things proves the absence of objective reality), he contradicts himself when trying to explain EPR (one time he says the measured quantity does not exist prior to measurement, then he claims the contrary). He claims there are observer independent events, but when asked he denies, etc. I cannot evaluate his knowledge about strings, I suppose he is good but his understanding of physics in general is rudimentary. All this being said, I actually agree with him that there is no measurement problem, at least for classical superdeterministic theories. The quantum state reflects just our incomplete knowledge about the system but the system is always in a well-defined state. 10. Lawrence Crowell: "but from what I know the spiral shape occurs [...]" That's not how spiral arms form, nor how they evolve. You'll find textbooks (and papers) which confidently tell you the answers to spiral arm formation and evolution; when you dig into the actual observations, you quickly realize that there's almost certainly more than one set of answers, and plenty of galaxies which simply refuse to be neatly pigeon-holed. 11. @ JeanTate: I am not that versed on galactic astrophysics. However, this wikipedia entry bears what I said for the most part. These are density waves, and are regions of gas and dust that slow the orbital motion of stars and compress gas entering them. So I think I will stick to what I said above. I just forgot to call these regions of gas density waves. As for the evolution of galactic structure. I think that is a work in progress and not something we can draw a lot of inferences from. 12. The Bell inequalities refer to classical probabilities. I read a paper on stochastic electrodynamics last decade. As I recall the idea is that the electric field has a classical part plus a stochastic variation E = E_c + δE, where (δE(t)) = 0 and (δE(t')δE(t)) = E^2δ(t' - t) if the stochastic process is Markovian. BTW, I am using parentheses for bra and ket notation because a lot of these blogs do not like carrot signs. I think Milgrom did some analysis of this sort. If the fluctuations are quantum then this really does not change the quantum nature of things. Quantum mechanics really tells us nothing about the existential nature of ψ. Bohr said ψ had no ontology, but rather was a prescription for determining measurements. Bohm and Everett said ψ does exist. The problem is with trying to make something ontological that is complex valued. Ontology seems to reflect mathematics of real valued quantities, such as expectations of Hermitean operators. Yet epistemic interpretations leave a gap between the quantum and classical worlds. I think this is completely undetermined; there is not way I think we can say with any confidence that quantum waves are ψ-epistemic or ψ-ontic. As a Zen Buddhist would say MU! Motl is a curious character, and to be honest I think that since he places a lot of ideology and political baggage ahead of actual science that his scientific integrity is dubious. I agree his stance on QM is highly confused, and based on his definition I am proud to be what he calls an anti-quantum zealot. He also engages in a lot of highly emotional negativity towards people he disagrees with. He will excoriate a physicist for something, but then for some reason found great interest in a 15 year old girl who does political videos that are obscene and in your face. His blog is useful for looking at some of the papers he references. I would say you can almost judge string theory from his blog; string may be a factor in physical foundations, but the vast number of stringy ideas illustrate there is no "constraint" or something that defines what might be called a contact manifold on the theory. 13. @ Lawrence Crowell: the Wikipedia article is, I think, a reasonable summary of some aspects of the topic of spiral arms in galaxies and their formation and evolution ... but it's rather out of date, and quite wrong in places. Note that density waves are just one hypothesis; the WP article mentions a second one (the SSPSF model), and there are more in the literature. Curiously, a notable morphological feature of a great many spiral galaxies is omitted entirely (rings). This is all rather OT for this blogpost, so just one more comment on this topic from me: galaxies are not like charm quarks, protons, or atoms. 10. Typo: "While philosophers on occasional". 11. Two folks whom I admire greatly, John Bell and Gerard 't Hooft, have shown an interest in superdeterminism for resolving entanglement correlations — Bell obliquely, perhaps, and 't Hooft both recently and very definitely. I understand and even appreciate the reasoning behind superdeterminism, but my poor brain just cannot accept it for a very computer-think kind of reason: efficiency. Like Einstein's block universe, superdeterminism requires pre-construction of a 4-dimensional causality "crystal" or "block" to ensure that all physics rules are followed locally. Superdeterminism simply adds a breathtakingly high new level of constraints onto pre-construction of this block: human-style dynamic state model extrapolation ("thinking") and qualia (someday physics will get a clue what those are) must be added to the mix of constraints. But why should accepting superdeterminism be any worse than believing in a block universe, which the most physicists already do — relativists via Einstein's way of reconciling diversely angled foliations, and quantum physicists via schools of thought such as Wheeler-Feynman advanced and retarded waves? It is not. That is, superdeterminism is not one whit less plausible than the block universe that most physicists already accept as a given. It just adds more constraints, such as tweaking sentient behavior. Pre-construction of the block universe is by itself so daunting that adding a few more orders of magnitude of orders of magnitude of complexity to create a "superblock" universe cannot by itself eliminate superdeterminism. In for a penny, in for a pound! (And isn't it odd that we Americans still use that expression when we haven't used pounds for centuries, except in the even older meaning of the word for scolding ourselves about eating too many chips?) Here is why efficiency is important in this discussion: If you are a scientist, you must explain how your block universe came into existence. Otherwise it's just faith in a mysterious Block Creator, the magnitude of whose efforts makes a quick build of just seven days pretty puny by comparison. The mechanism needed is easy to identify: It's the Radon transform, the algorithm behind tomography. To create a block universe, you apply the Radon transform iteratively over the entire universe for the entirety of time, shuffling and defuzzing and slowing clarifying the world lines until a sharp, crystallized whole that meets all of the constraints is obtained. This is why I can respect advocates of superdeterminism, and can agree fully that it is unfair for superdeterminism not to be taught. If you accept the block universe, you have already accepted the foundations of superdeterminism. All you need to do is just the Radon sauce a bit more liberally! My problem? I don't accept the necessity of the block universe. I do however accept the concept (my own as best I can tell) of causal symmetry, by which I mean that special relativity is so superbly symmetric that if you take the space-like details of any given foliation, you have everything to you need to determine the causal future of the universe for any and all other foliations. However, causal symmetry is a two-edged sword. If _any_ foliation can determine the future, then only _one_ foliation is logically required from a computational perspective. It doesn't make other foliations any less real, but it does dramatically simplify how to calculate the future: You look at space around _now_, apply the laws of physics, and let universe itself do the calculation for you. But only once, and only if you accept entanglement as proof that classical space and classical time are not the deepest level of how mass-energy interacts with itself. Space and time just become emergent features of a universe where that most interesting and Boltzmannian of all concepts, information and history, arose with a Big Bang, and everything has been moving on at a sprightly pace ever since. 12. A more promising way of attempting a deterministic extension of quantum mechanics seems to me a theory with some mild form of non-locality, possibly wormholes? 1. Leonard Susskind has advanced the idea of ER = EPR, or that the Einstein Rosen bridge of the Schwarzschild solution is equivalent to the nonlocality of EPR. This is a sort of wormhole, but not traversable. So there are event horizons that make what ever causal matter or fields there are unobservable and not localizable to general observers. 2. Very hard to make compatible with Lorentz-invariance. (Yes, I've tried.) 3. Sabine, are you saying that a wormhole would violate Lorentz-invariance? Why would that be? 4. Well, one wormhole would obviously not be Lorentz-invariant, but this isn't what I mean. What I mean is if you introduce any kind of non-locality via wormholes and make this Lorentz-invariant, you basically have non-locality all over the place which is more than what you may have asked for. Lorentz-invariance is a funny symmetry. Very peculiar. 5. This problem is related to traversable wormholes. A traversable worm hole with two openings that are within a local region are such that for the observer passing through there is a timelike path connecting their initial and final positions. To an observer who remains outside these two points are connected by a spacelike interval. An elementary result of special relativity is that a timelike interval can not be transformed into a spacelike interval. But if we have a multiply connected topology of this sort there is an ambiguity as to how any two points are connected by spacelike and timelike intervals. Ok, one might object, special relativity is a global flat theory, wormholes involve curvature with locally Lorentzian regions. However, the ambiguity is not so much with special vs general relativity but with the multiply connected topology. This matter becomes really odd if one of the wormhole openings is accelerated outwards and then accelerated back. For a clock near the opening there is the twin paradox issue. It is then possible to pass through the wormhole, travel back by ordinary flight and arrive at a time before you left. Now there are closed timelike loops. The mixing of spacelike and timelike intervals becomes a horrendous mix, as now timelike and spacelike regions overlap. Traversable wormholes also run afoul with quantum mechanics as I see it. A wormhole converted into the time machine as above would permit an observer to duplicate a quantum state. The duplicated quantum state would emerge from an opening, and then later that observer had better throw one of these quantum states into the wormhole to travel back in time to herself. The cloning of a quantum state is not a unitary process, and yet in this toy model we assume the quantum state evolves in a perfectly unitary manner. Even with a nontraversable wormhole one might think there is a problem, for Alice and Bob could hold entangled pairs and Alice enters this black hole. If Bob times things right he could teleport a state to Alice, enter the black hole, meet Alice so they have duplicated states without performing a LOCC operation. However, now nature is more consistent, for Bob would need a clock so precise that its would have a mass comparable to the black hole. That would then perturb things and prevent Bob from making this rendez vous. 6. Thank you, Sabine, for explaining why wormholes violate Lorentz-invariance, and thank you, Lawrence for the detailed clarification of what Sabine meant. I want to give your responses some more thought, but currently the heat and humidity is not conducive to deep thinking. I'll await the passage of a cold front in a few days before exercising the brain muscle. 13. I think you should re-read Tim Maudlin's book more carefully. He spent a lot of paragraphs explaining that superdeterminism has nothing to do with "free will". 1. I didn't say it does. I said that many people seem to think it does but that this is (a) not a good argument even if it was correct and (b) it is not correct. 14. Superdeterminism is a difficult concept for a lay person to unpack and to help understand it I ask whether it operates on the classical, "visible" world of things or, as the quantum was sometimes evaluated, exists only at the atomic and subatomic level? In short, do "initial conditions" govern ALL subsequent events or only events at the particle level? If the first is the case then does superdeterminism complicate our understanding of other scientific endeavors? Here is an example of what I mean. The evolution of modern whales from a "wolf-like" terrestrial creature is sufficiently documented that I kept a visual of it in my classroom. Biological evolution is understood to operate through random genetic changes (e.g. mutations or genetic drift) that are unforeseeable and that are selected for by a creature's environment. If true randomness is removed from the theory then evolution becomes telelogical -- and even smacks of "intelligent design." I mean that in this instance (the whale) its ultimate (for us) phenotype was an inevitable goal of existence from when the foundation of creation was laid. The same may be said of any living thing and, if so, evolutionary biology becomes a needless complication. Leopards were destined to have spots from the first millisecond of the Big Bang and camouflage had nothing to do with it. (In a similar vein I once wrote that railroad stations are always adjacent to railroad tracks because this, too, was destined by the initial conditions of 13.5 billion years ago and not because human engineers found such an arrangement maximal for the swift and efficient movement of passengers. So . . . are interested lay people better advised to understand superdeterminacy as a "quirk" of very small things or as an operating principle for the universe and without regard to scale. Thank you. 1. A. Andros, The theories that we currently have all work the same way and in them the initial conditions govern all subsequent events, up to the determinism that comes from wave-function collapse. Yes, if you remove the indeterminism of quantum mechanics then you can predict the state at any time given any one initial state. An initial state doesn't necessarily have to be an early time, it could be a late time. (I know this is confusing terminology.) No, this is not intelligent design as long as the theory has explanatory power. I previously wrote about this here. Please let me know in case this doesn't answer your question. 2. Intelligent design requires a purpose, the assumption that the state at the Big-Bang was "chosen" so that you get whales. Determinism just states that the state at the Big-Bang is the ultimate cause for the existence of the whales. If that state were different, different animals would have evolved. What is the problem? If the mutations were truly random, a different mutation would play exactly the same role as the different initial state. The railroad stations are always adjacent to railroad tracks because they both originate from the same cause (they were planed together by some engineer. That plan is also in principle traceable to the Big-Bang. Again, I do not see what the problem is supposed to be. 3. It must be... Is that right? 4. For a railroad station to be located adjacent to railroad tracks (a silly example, I know, but illustrative) the number of "coincidences" to flow from initial conditions (under superdeterminism) is so staggering as to strain belief. We must believe that 13.8 billion years ago the universe "determined" that all the following would coincide in time and place on just one of trillions (?) of planets: the invention of the steam locomotive; the manufacture of steel rails; the surveying of a rail route; appropriate grading for the right-of-way; the delivery of ballast to that right-of-way; the laying of that ballast; the cutting and processing of cross-ties; the shipment of cross-ties to the right place; the laying of cross-ties; the laying of rails; the installation of signals; the decision to build a railroad station; the creation of architectural plans for such a station; the simultaneous arrival of carpenters and masons at the site of the proposed station . . . and so on. All of these factors, and numerous others, were on the "mind" of the infant universe at the Big Bang? As an alternative, one can believe that the various contractors and engineers have AGENCY and thus could created the station/tracks. And, if the Rolling Stones were superdetermined to hold a performance at such-and-such a place did the infant universe also determine that 50,000 individuals would show up at the right place and time, each clutching an over-priced ticket? I wonder whether superdeterminism is just an attempt to get around Bell's Theorem and Quantum Mechanics and so restore a form of classical physics. And, since the concept cannot be tested -- is it not similar to string theory or the multi-verse? Dr. Richard Dawkins showed how random events can give the impression of purpose. Superdetermism, though, hints at purpose disguised as random events. As for the leopard and his spots . . . since everything that exists or acts results from initial conditions then biological evolution violates the teachings of the late Mr. Occam: it is simply an unnecessary embellishment of what was inevitably "meant to be." Now, I don't doubt evolution for a moment -- but it is not consistent (because of its random nature) with a superdetermined universe. Also . . . I thank Dr. H. for her helpful and kind reply to my note. 5. A. Andros, Big-bang created the engineer, the engineer created both railroad stations and tracks. Railroad stations and tracks were not independently created at the Big-Bang. The correlations between them is perfectly explainable by their common cause. Evolution does not require randomness, only multiple trials. You could do those trials in perfect order (mutate the first base, then the second and so on) and still have evolution. Superdeterminism can be tested using computer simulations. 6. To belabor my point, the Big Bang did not create the engineer who "created both railroad stations and tracks." These are separate engineering feats requiring different skills. Literally hundreds of separate skills/decisions must perfectly coincide in time and place for such projects to exist. This is simply asking too much of coincidence. There is no need for "multiple trials" in evolution if the phenotype of an organism was dictated at the Big Bang. What is there to submit to trial? The outcome 13.8 billion years after creation was inevitable from the start --- what is there to evolve? As for testing superdeterminsim on computers, the outcome of such simulations must also be dictated by initial conditions at the Big Bang and so are unreliable -- which, incidentally,seems to be Dr. H.'s opinion of numerous experiments that validate Prof. Bell's conclusions -- they cannot be trusted because the "fix" is in. If one believes in superdeterminism then one is stuck with something akin to intelligent design. Or, did the 3.5 million parts of the Saturn V, designed by thousands of engineers and all working perfectly together, originate in a "whim" of the Big Bang? It seems that superdeterminism provides a possible way out of Bell's theorem and quantum indeterminacy. But, it does so by endowing the universe with purpose and something like "foresight." IBM was down a fraction last week -- is that due to "initial conditions" almost 14 billion years ago? 7. A. Andros “…does superdeterminism complicate our understanding of other scientific endeavors?” Yes, and thanks for broadening the discussion to include complications that superdeterminism would present for understanding the how and why of complex biological systems. Your examples are part of a nearly endless list of biological phenomena for which there is presently a strong explanatory rational within evolutionary biology. These interrelated rationales would be completely undercut by a theory superdeterminism and left hanging for some ad hoc explanation. As you note, evolutionary biology then becomes a needless complication. Going further, the existence of these unnecessary phenomena (spots on leopards) would in fact violate the least action principle of physics that governs state transitions. It doesn’t hang together. Granted, we cannot free ourselves from the constraints created by the laws of physics, but neither can we free ourselves from the addendum of further constraints created by biological systems. “In general, complex systems obviously can bring new causal powers into the world, powers that cannot be identified with causal powers of more basic, simpler systems. Among them are the causal powers of micro-structural, or micro-based, properties of a complex system” — Making Sense Of Emergence, Jaegwon Kim, (f.n. #37) Sense Emergence.1999.pdf Evolutionary biology is a stepwise saga of contesting futures. Similarly, science is a stepwise saga of contesting theories. The fitness criteria are eventually satisfied. 8. A. Andros, The purpose of a superdeterministic theory is to reproduce QM. You have provided no evidence that, mathematically this is not possible. If you have such evidence please let me know. So, as long as all your examples with railroads and the evolution of animals are compatible with QM they will also be compatible with a superdeterministic interpretation of QM. 9. If one accepts superdeterminism then one must accept Intelligent Design. It may be that your mathematics do not align with 150 years of study of biological evolution but if that is the case then all the worse for superdeterminism. And, as I have said, the "coincidences" necessary for a superdeterministic universe to produce complex human interactions on a whim of initial conditions in a universe some 13.8 billion years ago simply strains credulity. There is ample evidence that biological evolution is, in large part, powered by random genetic events that provide the raw material on which natural selection works. Since superdeterminism disallows anything that is random then it, perforce, is a form of Creationism. All the math in the world, however cleverly employed, seems unlikely to verify a Genesis-like account of organic diversity over time. One might even become lost in math. 10. "If one accepts superdeterminism then one must accept Intelligent Design." This is complete rubbish. 11. Forty years ago Fred Hoyle delighted fundamentalists by stating that the complexity of DNA is such that it was as likely to have evolved spontaneously in nature as the chances of a tornado assembling a 747 by sweeping through a junk yard. Superdetermnism does the tornado one better -- it assembles entire technological civilizations from initial conditions of the universe. Since you seem to allege that this is due neither to random events (which SD prohibits) or agency (also excluded by SD) then what device is left to explain complex social and technological systems other than intelligent design? SD (superdetermnism) is simply a restatement of Augustinian theology -- as amplified by John Calvin. The term SD could just as easily be relabeled as "predetermination" and it would be fully acceptable in 17th century Geneva. And, as part of the intelligent design, biological evolution simply goes out the window. I was taught that organisms evolve due to random genetic and environmental changes -- but SD insists that "randomness" does not exist. Thus, organisms cannot evolve and each living thing is created de novo -- as declared in Genesis. Since the universe decreed the emergence of, say, giraffes from the moment of it (the universe's) inception then evolution becomes so much hand-waving. Your comment that my remarks are "complete rubbish" contain no arguments. I am merely trying to learn here. I am mystified how Dr. H. can believe that fantastically complex events in history (our 747, for example) could be implicit in initial conditions without intelligent design or random events or agency. I also hope to learn from you how the gradual evolution of one organism into another, somewhat different organism over geological time is possible if the phenotype is implicit in the infant universe. This is simply teleology. For all math in the world, SD seems to simply unpack Augustinian/Cavinistic thought and "science" and martial it for battle against the peculiar findings of Mr. John Bell. I enjoy your column (and book) greatly. But, the implications of SD are not easily dismissed -- or believed (unless one is an Augustinian and then it all makes perfect sense.) 12. A. Andros: If you have a time-reversible evolution, they are always present in the initial conditions, as in the present state, and in any future state. If you want to call that intelligent design, then Newtonian mechanics and many worlds are also intelligent design. 13. Sabine, I believe the point is that the design of today's complexity of biological structure and human artifact had to evolve as part of a natural process or be created by fiat of some agency. Unless the past is some kind of reverse engineering of the present. Ilya Prigogine believed that the very foundations of dissipative structures [biological systems] impose an irreversible and constructive role for time. In his 1977 Nobel Lecture,2 he noted in his concluding remarks, “The inclusion of thermodynamic irreversibility through a non-unitary transformation theory leads to a deep alteration of the structure of dynamics. We are led from groups to semigroups, from trajectories to processes. This evolution is in line with some of the main changes in our description of the physical world during this century.” A good article with experiments and references here: And consider that, one photon in a thousand striking the earth’s surface will land upon a chloroplast which is about one fifth of the diameter of a human hair and comprised of some three thousand proteins. Significantly, as the excitation moves through the photosynthetic process, one of these proteins is triggered to block it from returning to its previous state. Not to say that that process itself could not,in theory be rewound, but ingenious biology. 15. I seem to recall Ian Stewart remarking in one of his books that one way Bell's proof can be challenged is via its implicit assumption of non-chaotic dynamics in evolution of hidden variables. Any mileage to that? 1. Maybe he talked to Tim Palmer :p 2. No assumption is made of non-chaotic evolution of the hidden variables. Just go through the proof, and you'll see that. 16. Last paragraph, "this open the door ..." Should be "this opens the door ..." 17. Sabine, do you think it possible for a superdeterminism model to get some place useful without incorporating gravity into the equation? Offhand, it seems unlikely to me, since any kind of progress on the measurement issue would have to answer the gravitational Schrodinger's cat question, discussed on your blog at some point. Namely, where does the gravitational attraction point while the cat is in the box. 1. Sergei, I don't see how gravity plays a role here, just by scale. Of course it could turn out that the reason we haven't figured out how quantize gravity is that we use the wrong quantum theory, but to me that's a separate question. So the answer to your question is "yes". 18. Not John "Steward" Bell, but John Stewart Bell. 19. If the theory that space-time is a product of a multitude of interconnected entangled space-time networks, then it might be reasonable to suspect that any test of the state of a component of one of those networked entangled components of reality would be unpredictable as probed by an experimenter. The entangled nature of the network is far too complex and beyond the ability of the experimenter to determine. This complexity leads to the perception that the experiment is inherently unpredictable. To get a reliable test result, the experimenter may need to join the entangled network in whose component the experimenter is interested in testing. All the equipment that the experimenter intends to use in the experiment would be required to be entangled with the network component under test. The test instrumentation would now be compatible and in sync with the entangled component and be in a state of superposition with the entangled network that the component belongs to. During experiment initialization, the process of entangling the test infrastructure would naturally modified the nature of the network to which the component to be tested belongs as that infrastructure joins the common state of superposition. As in a quantum computer, the end of the experiment is defined by the decoherence of the coherent state of the experiment. Then upon decoherence of the experiment, the result of the experiment can be read out. 20. "...and the latter wants to discretize state space. I don’t see the point in either." There is no mathematically rigorous definition of the continuum: 21. Has anyone considered the possibility of spacetime itself being wavelike instead of trying to attach wavelike properties to particles? That is to say, the energy in space (time), varies sinusoidally over duration? This wave/particle duality seems to be a major problem with all "interpretations". This notion might define the "pilot wave" 1. Yes, wave structure of matter started by Milo Wolff proposes that what we observe as particles is the combination of an in-coming wave and an out-going wave. The combination of these two waves (which are more fundamental and are the constituents of both photons or matter waves) remove wave-particle duality issues, singularities of near field behavior including the need for renormalization) and many other issues. The two best features that I see are the ability to calculate the mass of standard particles and the wave nature of gravity (we did measure gravity waves in 2015) joining nicely with the wave nature of particles. 22. What I don’t understand about this super determinism is why, with all the fine tuning necessary to simulate non-locality , the mischief-maker responsible didn’t go that little bit further and ensure my eggs were done properly this morning. It seems an extravagant degree of planning to achieve what? 1. Dan, What makes you think fine-tuning is necessary? Please be concrete and quantify it. 2. Dan, correlations between distant systems are a consequence of all field theories (general relativity, classical electromagnetism, fluid mechanics, etc.). The correlations need not be explained by any fine-tuning of the initial conditions. For example, the motions of a planet and its star are correlated (one way of detecting distant planets is by looking at the motion of the star) but this is not a result of a fine-tuning. All planetary orbits are ellipses. The reason they are not triangles or spheres has nothing to do with the initial state. 23. Sabine, my assumption is that the distinction between a super deterministic universe and ia merely deterministic universe must be that in the super deterministic universe, the initial conditions (at the Big Bang) must be precisely those that will simulate nonlocality in all Bell tests forever more. 1. Dan, I suggest you don't assume what you want to prove. 24. I should have been clearer. I do not seek to prove any aspect of super determination. I believe it to be as extravagant and inaccessible to empirical demonstration as many worlds. All I was trying to say was that super determinism appears to presume that a highly contrived and thus extravagant set of initial conditions existed at the onset of the big bang. But congratulations on an absorbing discussion of a very interesting topic. 1. The universe can into existence as a unitary entity. As the universe expanded and eventually factionalized, its various substructures remained entangled as constrained by its unitary origin. The initial condition of the universe is global entanglement and as such Super determination must be a fallout of primordial global entanglement. 2. Dan, That you "believe" a theory "appears to presume" is not an argument. 3. Axil, With superdeterminism there is no entanglement (at least the version that mostly seems is being discussed here). It's being relied on to explain Bell Inequality violations locally. 4. Just a generic set of initial states for the early universe would do. If you take the physical state in which you do an experiment, then it's obviously not true that you could have made another decision w.r.t. to the experimental set-up while leaving everything else unchanged. Such a different future state in which the experiment is performed would under inverse time evolution not evolve back to an acceptable early universe state. The physical states we can find ourselves in are, as far as inverse time evolution is concerned, specially prepared states whose entropies will have to decrease as we go back in time. But because of time reversal invariance, any generic state will always show an increase in entropy whether or not we consider forward or backward time evolution. 25. Cited above (by Avshalom Elitzur). Very interesting article. Thanks. "The Weak Reality That Makes Quantum Phenomena More Natural: Novel Insights and Experiments" Yakir Aharonov, Eliahu Cohen, Mordecai Waegell, and Avshalom C. Elitzur November 7, 2018 26. Hi Sabine, I am not sure where Cramer stands in those pictures. Please could you tell me? 1. I don't know. I haven't been able to make sense of his papers. 27. If this is still about physics and not personal religion, there's no experimental basis for either super-determinism, or "classical determinism" for that matter. Instead, determinism was always a baseless extension in Newtonian mechanics and easily disproved by chaos experiments. If there was any lingering doubt, Quantum Mechanics should have killed determinism for good, but, zombies never die, hence the super-determinism hypothesis. Now, how about some experimental evidence? 1. What do you mean by "no experimental basis"? If no one is doing an experiment, of course there isn't an experimental basis. You cannot test superdeterminism by Bell-type tests, as I have tried to get people to understand for a decade. Yet the only thing they do are Bell-type tests. 2. Chaos theory is classical determinism. It is called deterministic chaos, where the path of a particle is completely determined, but due to exponential separation in phase space one is not able to perfectly track the future of the particle. 3. Chaos doesn't disprove determinism; it can be the result of an entirely deterministic evolution. Chaos addresses immeasurability. We have difficulty predicting the weather for precisely this reason, and the "butterfly effect" illustrates the issue: We cannot measure the atmosphere to the precision necessary to pick up every tiny event that may matter. In fact it was discovered because of a deterministic computer program: When the experimenter restarted an experiment using stored numbers for an initial condition, the results came out completely differently; because he had not stored enough precision. He rounded them off to six digits or whatever, and rounding off to one part in a million was enough to change the outcome drastically, due to exponential feedback effects in the calculations. That doesn't make the equations or evolution non-deterministic, it just means it is impossible to measure the starting conditions with enough precision to actually predict the evolution very far into the future, before feedback amplification of errors will overwhelm the predictions. I believe there are even theoretical physics reasons for why this is true, without having to invoke any quantum randomness at all. There's just no way to take measurements on every cubic centimeter of air on the earth, and its physically impossible to predict the future events (and butterfly wing flaps) that will also have influence. All of it might be deterministic, but the only way WE can model it is statistically. And I'll disagree with Dr. Hossenfelder on one point, namely that "shut up and compute" is not dead; often that is the best we humans can do, in a practical sense. Speculating about "why" is well and good, it might lead to testable consequences; but if theories are devised to produce exactly the same consequences with zero differences, that is a waste of time and brains; it is inventing an alternative religion. IMO The valid scientific response to "why" is "I don't know why, not yet." And if we already have a predictive model that always works -- we might better serve ourselves and our species working on something else more pressing. 28. Sabine, do you have any specific suggestions for how to build a superdeterminism test? I think this is an intriguing idea. One reason why I love the way John Bell's mind worked was that he pushed he pilot model (which I should note I do not accept) to the extreme, and found that it produced clear, specific models that led him to his famous inequality. Bell felt he might never have landed upon his inequality if he had instead relied on the non-intuitive, mooshy-smooshy (technical term) fuzzy-think of Copenhagen. Thus similarly, I am intrigued by your assertion that pushing superdeterminism to the limit may also lead to specificity in how to test for it. (Also, my apologies in advance if you have already done so and I simply missed it.) 1. So you discard Bohmian Mechanics? What are youre objections? 29. Is there a way to distinguish whether quantum mechanics is truly stochastic, or merely chaotic (in the mathematical chaos theory sense that the outcome is so sensitive to initial conditions that it is impossible to predict outcomes more than statistically because we can't measure the initial conditions with sufficient sensitivity)? 1. Can you help me understand that comment, please? I mean Schroedinger's equation yields a wave function - nothing chaotic or stochastic there. However when the wave function is used to determine (say) the position of an electron, using the Born rule, the result is a probability density. How would that be reformulated in terms of chaos? 30. Sabine, Bell's inequality has been tested to a level of precision that most people in the field are accepting the results. To my knowledge this has been done with entangled particles, are you aware of anyone trying to do an inverted experiment using maximally decohered particles (Quantum discord?). Is this such an obvious move that all the Bell experimenters are already doing it as a control group? If Suskin is right and ER == EPR then can we use that knowledge to build thrusters that defy conservation of momentum locally (transmit momentum to the global inertial frame a-la Mach effect?) Just an engineer looking for better engines. 31. Is superdeterminism compatible with the assumption of measurement settings independence? 32. I think that the implications of this (possibly the most overlooked in 20th C QM IMO) paper "Logic, States, and Quantum Probabilities" by Rachel Wallace Garden exposes a flawed assumption in the argument that violation of Bell inequalities necessitates some form of non-locality. A key point being that the outcomes of quantum interactions do not produce the same kind of information as a classical measurement, and this has clear-cut mathematical implications as to the applicability of the Bell inequality to quantum interactions. Rachel points out that the outcome of a quantum measurement on Hilbert space (such as a polarization test) is a denial (e.g. "Q: What is the weather? A: It is not snowing". In effect, all you can say about a photon emerging from channel A of a polarizing beamsplitter is that it was not exactly linearly aligned with B - all other initial linear and circular states are possible. As Rachel shows, the proof of the Bell inequality depends on outcome A implyng not B. That is, it relies on an assumption that there is a direct equivalence between the information one gets from a classical measurement of a property (e.g. orientation of a spinning object) and the interaction of a photon with a polarization analyzer. It is clear, while Bell inequalities do show that a measurement that determines a physical orientation in classical system is subject to a limit, nevertheless one cannot reasonably conclude that quantum systems are somehow spookily non-local when the experiments produce a different kind of information (a denial) to the classical determination. As Rachel's mathematical analysis shows, when a measurement produces a denial, then violation of the Bell inequality is an expected outcome. International Journal of Theoretical Physics, Vol. 35, No. 5, 1996 33. What are your thoughts on validation of different models using experiments in fluid dynamics? 34. A layman like me sees it this way: The detectors are observers, and there is reality. This reality is in turn an interpretation or description of the actuality. In quantum mechanics the detectors are observing a reality that came about by the subject of the experiment i.e., the experiment itself is a program, an interpretation, a description of say actuality because given the conditions of the double slit experiment the results are invariable the same up until this day. It is the output of this program that the detectors observe, and they in turn interpret this output or reality according to the their settings--like mind set. This interpretation is the influence of the detectors on the output which we see as a collapse of a number of possibilities to a single possibility i.e., the absence of interference. The detector is itself a program, a hardware program in that. We can go on wrapping in this way, interpretation after interpretation, until we arrive at something well defined and deterministic. That is, it is movement from infinite possibilities or indeterminism tending towards a finite possibility or determinism or clear definition with each interpretation along the way. And all along the way there is an increasing degree of complexity and with complexity come form and definition. Lets say there is the uninterpreted primordial to begin with. Then there are programs which interpret. Among the programs there is a common factor: the least interpreted primordial lets say light or photons. Starting with the least interpreted primordial if we unravel or reverse engineer the program, we get to the program. Once we have got the program then cant we predict the output, which is the interpretation, the description or "the observed"? 35. If the big bang produced a singular primordial seed, then the number of ways that the subsequent universe evolved from that seed can only be partitioned in a finite number of ways. The universe must then possess a finite dimensional state space. There is a finite number of ways that the universe can be partitioned. This constraint imposed on the universe by this finite dimensional state space mean that all these various states can be knowable. This ability to know all the states that the universe can assume is where Superdeterminism is derived from. There was a period at the very earliest stages of the universe’s differentiation; it was possible to know the states of all its component parts. At that time the universe was in a state of Superdeterminism. This attribute of the universe is immutable and therefore cannot change with the passage of time. It follows that the state of Superdeterminism must still be in place today. 36. It is true that determinism kills free will. But as David Hume pointed out, the inverse case is also relevant: free will requires some kind of determinism to work. You need a causal connection between decisión and action, ideally we expect the same decision to cause the same action...Not an easy problem indeed. 37. The easily overlooked energy cost of ab initio superdeterminism: Set up your pool table with a set of end goals. Set your launcher angles for each ball, eg to 5 digits of precision, and see how many generations of bounce you can control. Let's naively say it's roughly linear, 5 generations of bounce control for 5 digits of launch precision. Thus controlling a million generations of bounce requires (naively) a million digits of launch precision, _for each particle_. The actual ratio will be more complex, but inevitably will have this same more-means-more relationship. What is easily overlooked is that since information is a form of entropy, it always has a mass-energy cost. Efficient coding can make that cost almost vanishingly small for classical systems such as computers, but for subatomic particles with quantum-small state spaces, figuring out even how to represent such a very large launch precision number per particle becomes deeply problematic, particularly in terms of the mass-energy cost of the launch number in comparison to the total mass-energy of the particle itself. This in a nutshell is the hidden energy cost of any conceivable form of ab initio superdeterminism: The energy cost of a launch with sufficient precision to predetermine the entire future history of the universe quickly trends towards infinity. In another variant, this same argument is why I don't believe in points of any kind, except as succinct summaries of the limit behavior of certain classes of functions. 38. @Sabine Perhaps one reason why people do not like superdeterminism (SD) is that, as to giving us a comprehensible picture of reality, it scores even worse than standard QM. It amounts to saying that the strange correlations we observe (violations of Bell's inequalities) are due to some mysterious past correlations that arose somewhere at the beginning of the universe. By some magical tour de force, such correlations turn out to be exactly those predicted by QM. Except that, without the prior development of QM, we would be totally incapable of making any predictions based on SD alone. Yes, SD is a logical possibility, but so far-fetched and so little fruitful that it mainly reveals how contemptors of Copenhagen are running out of arguments. 1. opamanfred, Yes, that's right, that is the reason they don't like it. As I have explained, however, that reason is just wrong. Look how you pull up words like "mysterious" and "magical" without justifying them. You are assuming what you want to argue, namely that there is no simple way to encode those correlations. This is obviously wrong, of course, because you can encode them on a future boundary condition in a simple way, that simple way being what's the qm outcome. Of course you may say now, all right, then you get back QM, but what's the point of doing that? And indeed, there is no point in doing that - if that's the only thing you do. But of course what you want is to make more predictions than what QM allows you to. 2. @sabine Of course, you can. But you must admit it is a strange way to proceed: You observe the present, and then carefully tailor the past so that the present is a consequence of that past. If that's not fine tuning... Besides, I see another problem. QM clearly shows a lot of regularity in Nature. How would you accommodate such a regularity using past correlations, if not by even more fine tuning? Like, imagine a roulette that yields with clockwise regularity red-black-red-black-r-b-r-b-r-b-r-b and so on. Would you be comfortable with ascribing that to some past correlations? 3. opamanfred, (a) If you think something is fine-tuned, please quantify it. (b) I already said that superdeterminism trivially reproduces QM, so you get the same regularities for the same reason. Sounds like straightforward deduction to me; if Sherlock finds a dead man skewered by an antique harpoon, he deduces a past that could lead to the present, e.g. somebody likely harpooned him. There's no fine-tuning, if anything is "fine-tuned" it is the present highly unusual event of a man being harpooned on Ironmonger Lane. 5. (a) Perhaps I misunderstand SD. But how do you fix the past correlations in such a way that QM is correctly reproduced now? I had the impression this is done completely ad hoc. Slightly different correlations would yield predictions vastly different from QM, hence "fine tuning". (b) For the same reason?? In one case the regularities are the result of a consistent theory (QM) that is rather economical in its number of basic postulates. In the other, you choose an incredibly complex initial condition for the sole purpose of reproducing the results of the aforementioned theory. 6. opamanfred "Slightly different correlations would yield predictions vastly different from QM, hence "fine tuning"." You cannot make such a statement without having a model that has a state space and a dynamical law. It is an entirely baseless criticism. If you do not understand why I say that (and evidently you do not), then please sit down and try to quantify what is fine-tuned and by how much. 7. Sabine, “... superdeterminism trivially reproduces QM, so you get the same regularities for the same reason.” In QM the reason for the non-local correlation of measurement results of EPR pairs is their (e.g. singlet) state. Is this also the reason in SD? So far, I thought the reason for the correlation in SD is sacrificing statistical independence ... I am confused ... Maybe the prediction might be the same (or more), but the explanation or reason is a completely different one. (Remark: I find how TSVF trades non-locality for retrocausality quite interesting. And since the unitary evolution is deterministic, retrocausality works. But as you said this does not solve the measurement problem.) Maybe I am misunderstanding something, but naively it seems easy to come up with a model that does this. Let's suppose Alice and Bob prepare N spin-1/2 particles in a singlet state. They each have a measuring device that can measure the projection of the spin of a particle along the axis of the device. For simplicity, the relative alignments are fixed to be either perfectly anti-aligned, or else misaligned by a small angle theta. Quantum mechanics predicts the expected cross correlation of the spins will depend on theta like -1+theta^2/2. So for large enough N, by measuring the cross correlation for a few different values of theta, Alice and Bob can measure the expected quantum mechanical correlation function. A super-determinist could also explain the correlation function by imposing a future boundary condition on each particle. Given the mis-alignment of the detectors which the pair will experience -- the relative alignment of the detectors is assumed known since we are imposing this condition on the particles in the future after the measurement is made -- we pick the final spin values from a distribution which reproduces the quantum mechanical behavior. I will also assume a simple evolution rule, that the final spin we impose as a final boundary condition does not change when we evolve backwards in time -- for example, imagine a electron moving through a region with no electric or magnetic fields. Here's why I think a super-determinist would have a fine-tuning problem. From the super-determinist's point of view, there are 2N future boundary conditions to choose (2N numbers in the range [-1,1]) -- the value of the spin projected on the measuring device for each particle in each of the N entangled pairs. The number of possible future boundary conditions is exponential in N. Given the statistical error associated with N measurements, there will be an exponential in sqrt(N) possible configurations which will be consistent with the quantum mechanical correlation function. The super-determinist knows nothing about quantum mechanics (otherwise what is the point), so I will assume it's natural to use a distribution that does not privilege any particular choice of initial conditions, like a uniform distribution from -1 to 1. In that case, it's exponentially unlikely (in sqrt(N)) that the drawn distribution of spin components will be consistent with the quantum mechanical prediction. If the point is that the super-determinist should choose future boundary conditions according to a distribution which is highly peaked around values that will reproduce the quantum mechanical correlations, I agree this is mathematically self consistent, but I don't see what is gained over using ordinary quantum mechanics. It seems that one has just moved the quantum mechanical correlations, into a very special distribution that (a) is naturally expressed as a future boundary condition (which raises some philosophical issues about causality, but I'm willing to put those aside) and (b) one has to use a probability distribution which (as far as I can tell) can't be derived from any principles except by reverse engineering the quantum mechanical result. Maybe I am just missing the point... 9. Reimond, Statistical dependence is not the reason for the correlation in the measurements; statistical dependence is just a property of the theory that prevents you from using Bell's theorem to falsify it. Just where the correlations come from depends on the model. Personally I think the best way is to stick with quantum mechanics to the extent possible and just keep the same state space. So the correlations come from where they always come from. I don't think that's what 't Hooft and Palmer are doing though. 10. Andrew, Thanks for making an effort. Superdeterminism reproduces QM and is therefore equally predictive for the case you mention. It is is equally fine-tuned or not fine-tuned for the probabilistic predictions as is quantum mechanics. The probabilistic distribution of outcomes in QM, however, are not the point of looking at superdeterministic models. You want to make more predictions than that. Whether those are fine-tuned or not (and hence have explanatory power or not) depends on the model, ie on the required initial state, state space, and dynamical law. 39. ... ... let the paper speech by itself. 40. Sabine Hossenfelder (9:05 AM, July 30, 2019) wrote: This a very interesting comment that I think is right—that retrocausality would require a modified [current quantum formulation]* to make sense. Perhaps there will be a future post on this? * Schroedinger equation, or path integral 1. Philip, What I meant to say, but didn't formulate it very well, is that to solve the measurement problem you need a non-linear time evolution. Just saying "retrocausal" isn't sufficient. Yes... will probably have a post on this in the future, but it will take more time for me to sort out my thoughts on this. (As you can tell, it took me a long time to write about superdeterminism for the same reason. It's not all that easy to make sense of it.) 2. I have never thought this time reversal idea was worth much. QM is time reversal invariant, so as I see things it will not make any difference if you have time reversed actions. The idea of nonlinear time is interesting, and this is one reason of the quantum interpretations I have a certain interest in the Montevideo interpretation. This is similar to Roger Penrose's idea that gravitation reduces waves. I have though potentially the GRW might fit into this. Quantum fluctuations of the metric which result in a g_{tt} fluctuation and nonlinear time evolution might induce the spontaneous collapse. If one is worried about conservation of quantum information, the resulting change in the metric is a very weak gravitational wave with BMS memory. 3. Lawrence Crowell: I read this summary of Montevideo interpretation ( That seems like a promising route. But what I got was that the nonlinear evolution of (env + detector + system) exponentially diminishes all but one eigenstate, to the point that it is impossible to measure anything else but that one state. So 'collapse' isn't an instantaneous event but more of reaching an asymptotic state, so it would require a detector larger than the universe to distinguish any but the dominant eigenstate. It also makes the "detection" entirely environmental, no observer required; which fits with my notion that a brain and consciousness are nothing special, just a working matter machine and part of the environment, so 'observation' is just one type of environmentally caused wavefunction collapse. I like it! 4. The Montevideo Interpretation proposes limitations on measurement due to gravitation. Roger Penrose has proposed that metric fluctuations induce wave function collapse. Gambini and Pullen are thinking in part with a quantum clock. It has been a long time since I read their paper, but their proposal is that gravitation and quantum gravitation put limits on what can be observed, which in a standard decoherence setting is a wave function reduction. The Bondi metric, written cryptically as ds^2 = (1 - 2m/r)du^2 - dudr - γ_{zz'}dzdz' + r^2 C_{zz}dz^2 + CC + D_zC_{zz} + CC gives a Schwarzschild metric term, plus the metric of a 2-sphere, plus Weyl curvature and finally boundary terms on Weyl curvature. We might then think of concentrating on the first term. Think of the mass as a fluctuation, so we have m → m + δm, where we are thinking of a measurement apparatus as having some quantum fluctuation in the mass of one of its particles or molecules. With 2m = r this is equivalent to a metric fluctuation, and we have some variance δL/L for L the scale of the system. For the system scale ~ 1meter and δL = ℓ_p = √(Għ/c^3) = 1.6×10^{-35}m δL/L ≈ p or the probability for this fluctuation and p ≈ 10^{-35}. Yet this is for one particle. If we had 10^{12} moles then we might expect some where a molecule has a mass fluctuation approaching the Planck scale. If so we might then expect there to be this sort of metric shift with these Weyl curvatures. Does this make sense? It is difficult to say, for clearly no lab instrument is on the order of billions of tons. On the other hand the laboratory is made of states that are entangled with the building, which are entangled with the city that exists in, which is entangled with the Earth and so forth. Also there is no reason for these fluctuations to be up to the Planck scale. The Weyl curvatures would then correspond to very weak gravitational waves produced. They can be very IR and still carry qubits of information. If we do not take these into account the wave function would indeed appear to collapse, and since these gravitational waves are so weak and escape to I^+ there is no reasonable prospect for a recurrence. In this way Penrose's R-process appears FAPP fundamental. This is unless an experiment is done to carefully amplify the metric superposition, such as what Sabine refers to with quantization of large systems that might exhibit metric superpositions. We have the standard idea of the Planck constant intertwining a spread in momentum with a spread in position ħ ≤ ΔxΔp, and the same for time and energy. Gravitation though intertwines radial position directly with mass r = 2GM/c^2, and it is not hard to see with the Gambini-Pullen idea of motion we can do the same with r = r0 + pt/√(p^2 + m^2) that we can include time. The variation in time, such as in their equation 2 due to a clock uncertainty spread can just as well be due to the role of metric fluctuations with a mass. 41. Sabine: I know what you mean because you have reiterated sufficiently often in previous posts :) You are technically correct, but your position precludes any meaningful way to do science. You can "explain" everything by positing initial conditions. Fine. But it leads nowhere. Come on, don't be disingenuous. Sherlock's deduction would be entirely reasonable of course. Much less so if Sherlock had maintained: "We have found this dead mean skewered by an antique harpoon because of some correlations that arose 13.8 billion years ago at the birth of the universe". Sherlock was notorious for substance abuse, but not to that point... 1. Opamanfred, You are missing the point. (a) You always need to pose initial conditions to make any prediction. There is nothing new about this here. Initial conditions are always part of the game. You do not, of course, explain everything with them. You explain something with them if they, together with the dynamical law, allow you to describe observations (simpler than just collecting data). Same thing for superdeterminism as for any other theory. (b) Again, the point of superdeterminism is not to reproduce quantum mechanics. The point is to make more predictions beyond that. What do you mean by "it leads nowhere". If I have a theory that tells you what the outcome of a measurement is - and I mean the actual outcome, not its probability - how does this "lead nowhere"? 42. I think the discussion above overlooks the possibility that hidden variables based on IMPERFECT FORECASTS by nature are behind "quantum" phenomena. I provide convincing arguments for this hypothesis in two papers. I copy the titles and abstracts below. You can search for them online. Both papers cite experimental publications. The second paper departs from an experimental observation made by Adenier and Khrennikov, and also makes note of an experiment performed on an IBM quantum computer which I believe provides further evidence pointing towards forecasts. Title: A Prediction Loophole in Bell's Theorem Abstract: We consider the Bell's Theorem setup of Gill et. al. (2002). We present a "proof of concept" that if the source emitting the particles can predict the settings of the detectors with sufficiently large probability, then there is a scenario consistent with local realism that violates the Bell inequality for the setup. Title: Illusory Signaling Under Local Realism with Forecasts Abstract: G. Adenier and A.Y. Khrennikov (2016) show that a recent ``loophole free'' CHSH Bell experiment violates no-signaling equalities, contrary to the expected impossibility of signaling in that experiment. We show that a local realism setup, in which nature sets hidden variables based on forecasts, and which can violate a Bell Inequality, can also give the illusion of signaling where there is none. This suggests that the violation of the CHSH Bell inequality, and the puzzling no-signaling violation in the CHSH Bell experiment may be explained by hidden variables based on forecasts as well. 43. "How much detail you need to know about the initial state to make predictions depends on your model." Better, on your particular experiment. If the knob on the left device is controlled by starlight from the left side, and that of the right side controlled by the right side, you need quite a lot of the universe to conspire. No, this decision is quite trivial. If there is superdeterminism, no statistical experiment is able to falsify anything, thus, one has to give up statistical experiments as useless. No. The original state of the device, say, a0, may be as dependent from the measured state as you like. If you use the polarization of incoming starlight s together with your free will decision f simply as additional independent inputs, (both evenly distributed between 0 and 360 degrees), so that the resulting angle is simply a = a0 + s + f, then the statistical independence of f or s is sufficient to lead also to the statistical independence of a. That a0 is correlated does not disturb this at all. So, at least one source of independent random numbers would be sufficient - all you have to do is Bell's experiment with devices controlled (just influenced in a sufficiently heavy way is sufficient) by these independent random numbers. Thus, the whole world has to conspire completely. Non-Einstein-local. A return to the Lorentz ether, that's all. As it is, quantum theory is indeed nonlocal, but it could be easily approximated by a local (even if not by an Einstein-local) theory. Nothing more seriously non-local than the situation with Newtonian gravity. The configuration space trajectory q(t) is, like in Newtonian theory, continuous in all the realist interpretations of QT, all the real change is a local, continuous one. Only how it is changed possibly depends on the configuration far away too. So, there is nothing worse with the non-locality than in NT. What's the problem with such a harmless revival of a problem scientists have lived with already in classical physics is beyond me. 1. Ilja, Which is why it's hard to see evidence for it. You better think carefully about what experiment to make. Patently wrong. You cannot even make such a statement without having an actual model. "superdeterminism" is not a model. It's a principle. (And, as such, unfalsifiable, just like determinism.) 2. Ilja, This is completely false. The only claim a superdeterministic theory has to make is that a particular type of electromagnetic phenomena (emission of an entangled particle pair and the later behavior of those particles in a magnetic field, such as the one in a Stern-Gerlach device) are not independent. There are plenty of examples of physical systems that are not independent, an obvious one being the motion of any of two massive objects due to gravity. Such objects will orbit around their common center of mass, regardless of how far they are. Some other examples are stars in a galaxy, electrons in an atom, synchronized clocks, etc.). Superdeterminism maintains that the particles we call "entangled" are examples of such systems. The only difference is that we can directly intervene and "mess-up" with the correlations due to gravity, we can de-synchronize clocks but we cannot control the behaviour of quantum particles because we ourselves are built out of them. We are the result of their behavior so whatever we do is just a manifestation of how they "normally" behave. So, superdeterminism brings nothing new to physics, it is in complete agreement with all accepted physical principles, including the statistical ones. A slightly different way to put this is to observe that a superdeterministic interpretation of QM will have the same predictions as QM. So, as long as you think that statistical experiments are not useless if QM is correct, those experiments will also not be useless if a superdeterministic interpretation of QM is correct. On the other hand if you do believe that statistical experiments are useless if QM is true they will also be useless if a non-local interpretation (such as de-Broglie-Bohm theory) of QM is true. As you can see, the most reasonable decision between non-locality and superdeterminism is superdeterminism because this option does not conflict with any currently accepted physical principle and it does not require us to go back to the time of Newton. 44. The quantum world is fully deterministic based on vacuum quantum Jumps: God plays dice indeed with the universe ! However living creatures seem to be able to influence initiatives suggested by RP I to Veto actions by RP II at all levels of consciousness according to Benjamin Libet. ( Trevena and Miller) 45. Sabine, Do you know sbout the Calogero conjecture ( and do you think it is related to SD? 1. Opamanfred, Thanks for bringing this to my attention. No, I haven't heard of it. I suppose you could try to understand the "background field" as a type of hidden variable, in which case that would presumably be superdeterministic. Though, if the only thing you know about it is that it's "stochastic", it seems to me that this would effectively give you a collapse model rather than a superdeterministic one. 46. Sabine, "It’s not like superdeterminism somehow prevents an experimentalist from turning a knob." why not? How can an experimentalist possibly turn the knob other then how she/he has been determined to turn it? If hidden variables exist their evolution would have to be highly chaotic. What's the likelihood that any predictions could be made, even if we could somehow find those variables and ascertain their (approximate) initial values? 1. i aM wh, She cannot, of course, turn the knob other than what she's been determined to do, but that's the case in any deterministic theory and has nothing to do with superdeterminism in particular. That's a most excellent question. You want to make an experiment in a range where you have a reasonable chance to see additional deterministic behavior. This basically means you have to freeze in the additional variables as good as possible. The experiments that are currently being done just don't probe this situation, hence the only thing they'll do is confirm QM. (I am not sure about chaotic. May be or may not be. You clearly need some attractor dynamics, but I don't see why this necessarily needs to be a chaotic one.) 2. Is this here (chapter 4) still the actual experiment you want to be performed? 3. Reimond, No, it occurred to me since writing the paper that I have made it too complicated. You don't need to repeat the measurement on the same state, you only need identically prepared states. Other than that, yes, that's what you need. A small, cold, system where you look for time-correlations. 4. Sabine, “...don't need to repeat the measurement on the same state...” This means you remove the mirrors? But then there is no correlation time left, only the probability that e.g. a photon passes the two polarizers. SD, if true, would then give a tiny deviation from what QM tells us. Is it like this? 5. Reimond, You can take any experiment that allows you to measure single particle coherence. Think, eg, double slit. The challenge is that you need to resolve individual particles and you need to make the measurements in rapid succession, in a system that has as few degrees of freedom as possible. (The example in the paper is not a good one for another reason, which is that you need at least 3 pointer states. That's simply because a detector whose states don't change can't detect anything.) 47. Why not just accept Lagrangian mechanics, successfully used from QFT to GR, which is not only deterministic (Euler-Lagrange), but additionally time/CPT-symmetric, well seen in equivalent action optimization formulation. In contrast, "local realism" is for time-asymmetric "evolving 3D" natural intuition. Replacing it with time-symmetric spacetime "4D local realism" like in GR: where particle is its trajectory, considering ensemble of such objects, Feynman path ensemble is equivalent with QM. Considering statistical physics of such basic objects: Boltzmann distribution among paths in euclidean QM or simpler MERW ( ) we can see where Born rules come from: in rho ~ phi^2 one psi comes from past ensemble (propagator from -infinity), second psi from future ensemble (propagator from +infinity), analogously to Two State Vector Formalism of QM. Hence we directly get Born rules from time-symmetry, they allow not to satisfy inequalities derived in standard probabilistics: without this square. I have example construction of violation of Bell-like inequality for MERW (uniform path ensemble) on page 9 of 48. Pure determinism doesn’t work as an explanatory system because it doesn’t have the basis for explaining how new (algorithmic) relationships could come into existence in living things. Equations can’t morph into algorithms, except in the minds of those people who are religious believers in miraculous “emergence”. 1. @Lorraine Ford, these days when I encounter the word "emergence" in an article, I lose interest. Because I know the hand waving has begun. 2. jim_h: "emergence" is not hand waving. Cooper pairs and the low-energy effective field theory of superconductivity are emergent. Classical chaotic behavior (KAM) is emergent from quantum mechanics, and easily demonstrated by computation. 3. "Equations can’t morph into algorithms" Perhaps then equations (from a Platonic realm of Forms) should be left behind by physicists, and replaced by algorithms. 4. dtvmcdonald: "emergence" is a vapid truism. All phenomenological entities are emergent. 49. Well if the sacred cow random is sacrificed and reconize that nature does not play fair and is not playing with a double sided coin but a double head coin where the ohsevation point is always unpredictible and dump the cosmos as a machine and a more organic perspective yeah it works. It's consistent with weather, forest fires, earthquakes etc. In regards Free will we have an infinite choice of wrongs and one right. That's neither deterministic nor freewill. That's also inspite of ourselves When we get it right. 50. I believe that there may be a way to “circumvent the initial conditions of the universe” quandary but more to the point the more complicated case dealing with the “initial conditions that affect the process that the observer wants to examine” can be dealt with. In a takeoff of the classic Einstein though experiment the rider/observer can observe what is happening on the relativistic train that he is traveling on. The observer is synchronized in terms of initial conditions that apply to both he and what he wants to observe. He is also completely synchronized in space-time with the condition that he wants to observe. Now if the observer becomes entangled with the condition that he wants to observe, he now becomes totally a part of that condition since he has achieved complete simpatico with it. In an illustrative example of how this could be done, the person who wants to observe what a photon does when it passes through a double slit, both the double slit mechanism, the photon production mechanism, and the sensors sampling the environment in and around each slit would need to be entangled as a system. This entanglement would also include the data store that is setup to record the data. After the data has been recorded, the observer decoherers the entanglement condition of the system. After this step, the observer can examine the data store and is free from quantum mechanical asynchrony to finalize his observations and make sense of it. 51. Even if one could construct a superdeterministic quantum theory in Minkowski space, what would be the point? We do not know what the geometry of the spacetime is that we live in and if it is deterministic in the first place, and I don't think that we will ever know. 52. Reading about the umpteen "interpretations" of QM is interesting, e.g. Review of stochastic mechanics Edward Nelson Department of Mathematics, Princeton University Abstract. "Stochastic mechanics is an interpretation of nonrelativistic quantum mechanics in which the trajectories of the configuration, described as a Markov stochastic process, are regarded as physically real. ..." but all together it reminds be of the witches' song in Macbeth: Double, double toil and trouble; Fire burn and caldron bubble. Fillet of a fenny snake, In the caldron boil and bake; Eye of newt and toe of frog, Wool of bat and tongue of dog, 53. If one accepts the block-universe picture (BU) entailed by relativity theory (and one should on consistency grounds) then Bell's theorem becomes ill formulated. Ensembles in the BU are those of 4D `spacetime structures', not of initial conditions. QM then becomes just a natural statistical description of a classical BU, with wave functions encoding various ensembles; no measurement problem either. I have recently published a paper on the topic which made little to no impact. The reason for this is (I think, and would be happy to be corrected) that physicists are jailed in IVP reasoning (Initial Value Problem). They say: we accept the BU, but ours is a very specific BU; complete knowledge of data (hidden variables included) on any given space-like surface, uniquely determines the rest of the BU (a ridiculously degenerated BU). In this case, QM being a statistical description of the BU, though still consistent, becomes truly strange. IVP reasoning started with Newton's attempt to express local momentum conservation in precise mathematical terms. However, there are other ways to do so without IVP. In fact, moving to classical electrodynamics of charged particles, one apparently MUST abandon IVP to avoid the self-force problem, and similarly with GR. Without IVP, QM becomes just as `intuitive' as Relativity. Moreover, QM in this case is just the tip of the iceberg, as certain macroscopic systems as well should exhibit `spooky correlations' (with the relevant statistical theory being very different from QM if at all expressible in precise mathematical terms). arXiv:1804.00509v1 [quant-ph] 54. Sabine, I've heard an argument that if superdeterminism is true quantum computers won't work the way we expect them to work. I guess the idea is that if many calculations are done, correlations would show up in places we wouldn't expect to find them. Is there anything to that? Would it be evidence in favor of SD if quantum computers don't turn out to be as "random" as we expect them to be? 1. i aM wh, That's a curious statement. I haven't heard this before. Do you have a reference? By my own estimates, quantum computing in safely in the range where you expect quantum mechanics to work as usual. Superdeterminism doesn't make a difference for that. 2. 't Hooft made that suggestion here: see page 12 and 13. I'm not sure if 't Hooft has changed his opinion on this matter. 55. Dear Sabine, I have a high opinion of your book, and consequently have a low opinion of myself, since I don't understand superdeterminism at all. I hope you or some other kind person can give me a small start here. I am not a physicist but do have a background of research in mathematical probability. I skimmed through the two reference articles you cited but didn't see the kind of concrete elementary example I needed. Here is the sort of situation I am thinking about. A particle called Z decays randomly into two X-particles with opposite spins. Occasionally a robot called Robot A makes a spin measurement for one X-particle , after its program selects a direction of spin mesaurement using a giant super-duper random number generator. Perhaps Robot A chooses one of a million containers each containing a million fair coins, or takes some other pseudo-random steps, and finally tosses a coin to select one of two directions. On these occasions Robot B does the same for the other X-particle, also using a super-super random number generator to find its direction of measurement. A clever physicist assumes that three things are statistically independent: namely the result of the complex random-number generator for A, the result of complex random-number generator for B, and the decay of particle Z. With some other assumptions, the physicist cleverly shows that the results should satisfy a certain inequality, and finds that Nature does not agree. If I understand the idea of superdeterminism, somewhere back at the beginning of time a relationship was established that links two giant super-duper random number generators together with particle Z in a special way, resulting in a violation of the inequality, while preserving the all the statistical independence that we constantly observe in so many places, and at so many scales. Is this the idea? Probably I have missed something. Of course there is no free will in what I described. I will be very happy to understand a different approach to this! 1. jbaxter, Several misunderstandings here. First, the phrase "a relationship was established" makes it sound as if there was agency behind it. The fact is that such a relationship, if it exists at one time, has always existed at any time, and will continue to exist at any time, in any deterministic theory. This has absolutely nothing to do with superdeterminism. All this talk about the initial conditions of the universe is therefore a red herring. No one in practice, of course, ever writes down the initial conditions of the universe to make a prediction for a lab experiment. Second, regarding statistical independence. What we do observe is that statistical independence works well to explain certain classical observations. One could argue that violations of Bell's inequality in fact demonstrate that it is not fulfilled in general. This is a really important point. You are taking empirical knowledge from one type of situation and apply it to another situation. This isn't justified. Am I correct to understand this as saying that every two points in space-time are "correlated"? IIRC, Susskind's EP = EPR combined with the holographic universe (the bulk can be described by a function over the surface) implies that there should be a lot of entanglement between any two points in the universe. Such entanglements would mean that there is, indeed, no statistical independence between points in space (-time). 3. Thanks Sabine for taking the time to respond. Somewhat related to your second point, I would say that no one really understands how the usual independence which we observe, ultimately arises. We do understand how independence _spreads_ though, e.g. small shaking in my arm muscles leads to a head rather than a tail when I try to toss a coin. All this for our classical lives, of course. The infectious nature of noise is part of the reason that avoiding statistical independence is a tough problem, I suppose. Obviously it would be great to get progress in this direction. Classical mechanics is as "emergent" from quantum mechanics as Cooper pairs and a low energy limit field theory of superconductivity are emergent from quantum mechanics. And thus chaos, in the KAM sense, is emergent as a low energy / long distance scale phenomenon from QM. This chaos can be fed back into QM through, e.g. deciding whether to have your polarizer at 0 or 45 or 90 degrees from Brownian motion. 57. Superdeterminism can only be an explanation for entanglement if the universe was created by a mind who followed a goal. And that is religion in my understanding. 1. antooneo, Any model with an evolution law requires an initial condition. That's the same for superdeterminism as for any theory that we have. By your logic all theories in current science are therefore also "religion". 2. Sabine, what do you mean by "REQUIRES an initial condition"? In a Lagrangian formulation, for example, initial and final conditions are required, and only on position variables (this, BTW, is the case for retro-causality arguments). The universe, according to Relativity (block-universe) couldn't have been created/initiated at some privileged space-like three-surface, and then allowed to evolve deterministically. This is the IVP jail I was talking about. 3. Sabine, think about Lagrangian mechanics - it has 3 mathematically equivalent formulations: Euler-Lagrange equation to evolve forward in time, or backward just switching sign of time, or action optimizing between two moments of time. Theories we use are fundamentally time/CPT-symmetric, we should be extremely careful if enforcing some time asymmetry in our interpretation - what leads to paradoxes like Bell violaition. Spacetime view (block universe) as in general relativity is a safe way, repairing such paradoxes. 4. Sabine: Any model has initial conditions. Particularly of course a superdeterministic system. But that is not sufficient to have the results which are supposed to be explained by it, e.g. the entanglement of particles in experiments where also the setup has to be chosen by experimenters influenced in the necessary way. So, even if Superdeterminism is true, the initial conditions have to be set with this very special goal in mind. The assumption of such a mind is religion; what else? 5. antooneo, That is correct, the initial conditions are not sufficient. You also need a state space and a dynamical law. Your statement that "the initial conditions have to be set with this very special goal in mind" is pure conjecture. Please try to quantify how special they are and you should understand what I mean: You cannot make any such statement without actually writing down a model 6. antooneo, Sabine, If you want the reason for initial conditions being prepared for future measurnments, instead of enforcing time-asymmetric way of thinking to time/CPT symmetric theories, try mathematically equivalent time-symmetric formulations, e.g. action optimizing for Lagrangian mechanics: history of the Universe as the result of Big Bang in the past and e.g. Big Crunch in the future, or using path-ensembles: like Feynman's equivalent with QM - leading to Born rule from time symmetry ( one amplitude from the past (path enesemble/propagator), second from the future. 58. A schema (e.g. one consisting of nothing but a set of deterministic equations and associated numbers), that doesn’t have the potential to describe the logical information behind the evolution of life, is not good enough these days. The presence of algorithmic/ logical information in the world needs to be acknowledged, but you can’t get algorithmic/ logical information from equations and their associated numbers. 59. When we say that there is a Renaissance going on in QM, do we mean the community actually going to take a serious look at new ideas? Is there any experimental evidence that points to one new flavor over another one? The last time we revisited the subject a few years ago I remember hearing that very few professionals were willing to consider alternatives. I know there is a fledgling industry of quantum computing and communications, but getting industry to pursue and finance research into new flavors of QM could be harder than getting academia to do it, even if one flavor does offer tantalizing features. 60. This comment has been removed by the author. 61. Do you have any reference on how "Pilot wave theories can result in deviations from quantum mechanics"? Moreover, what's your opinion on a stochastic interpretation of quantum mechanics? 62. Taking up Lorraine Ford's thread, is life deterministic? It is deterministic as long as the pattern continues, the moment you break out of the pattern as a matter of insight that insight may be inventive and therefore we come upon something totally new. Evolution talks about the animal kingdom and the plant kingdom let us say the evolution of the animate world. Then I think there is also the evolution of the inanimate world. Let us start with a big bang and arrive at iron, thereafter, matter evolved into gold and so on up to uranium. Now the radioactive elements are unstable because they are recent species, they have not stabilized or rather fully fallen into a pattern. What I am trying to say is life is only a branch of inanimate evolution i.e., the evolution of the universe branched of into inanimate evolution and animate evolution. And animate evolution is not deterministic it was a quirk. 63. @Sabine There is a weird version of Creationism that appears to be worryingly similar to SD. To put it shortly, some creationists accept that all the evidence from the archeological record is real and that the conclusion that life forms have evolved is a logical deduction. However, these creationists believe that all this evidence was put there on purpose by God to make us believe that the world is several billion years old, while in fact it was created only a few thousand years ago. It's just that God put evry stone and every bone in the right place to make us jump to the Darwinian conclusion. So I wonder. How's that different from SD when you replace God by Big Bang, and Darwinian evolution with QM? Boshaft ist der Herrgott, after all... 1. Opamanfred, I explaiend this very point in an earlier blogpost. The question is whether your model (initial condition, state space, dynamical law) provides a simplification over just collecting data. If it does, it's a perfectly fine theory. Superdeterminism, of course, does that by construction because it reproduces quantum mechanics. The question is whether it provides any further simplification. To find out whether that's the case you need to write down a model first. 64. Think, just think please... is evolution possible without memory? Is life possible without memory? Certainly not. Is memory possible without "recording"? Certainly not. Could life on earth have been possible if "recording" was not already going on in the inanimate world, If "memory" was not there in the inanimate world? Taking this questioning further, I suspect that "recording" takes place even at the quantum level, and "memory" is part of the quantum world. And it seems that this capacity to "record" and hold in "memory" gradually led to the appearance of gold and life on earth. Comment moderation on this blog is turned on.
f0740af201f51052	It is often said that interacting electron systems (say on a lattice, like the Hubbard model for example) are difficult to solve because of the exponential size of the Hilbert space. However, I am not sure I find this argument very compelling. Non-interacting systems live in the same Hilbert space and yet are easily solved. There are also cases of interacting models that can be solved and it is not by brute force, I'm thinking about the Bethe ansatz for example. I can also quote this interesting paper : Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space: As an application, we showed that the set of quantum states that can be reached from a product state with a polynomial-time evolution of an arbitrary time-dependent quantum Hamiltonian is an exponentially small fraction of the Hilbert space. This means that the vast majority of quantum states in a many-body system are unphysical, as they cannot be reached in any reasonable time. As a con- sequence, all physical states live on a tiny submanifold, and that manifold can easily be parametrized by all poly-sized quantum circuits. So, are these models difficult to solve because no one has found a suitable ansatz yet, or is there a fundamental reason to give up the hope of finding such an ansatz? Or at least a more convincing argument than the size of the Hilbert space? Related : Proving that the electronic Schrödinger equation has no closed analytic solutions for >1 electron The fact that weakly-interacting systems are easy to solve is a red herring. Those systems do indeed live in the same Hilbert space, but they can be confined to an excellent approximation to a very 'thin', and very well-characterized, slice of that space, i.e. to the non-interacting ground state in direct sum with only a few variations of single-particle excitations, without involving the full brunt of the exponential sea of entangled states. Your second argument, on the other hand, is much more compelling, and it cannot be easily set aside. However, it is only an argument on the lowest possible bound on the computational complexity of algorithms to solve for the eigenstates of local hamiltonians ─ if we are clever enough to find those algorithms. As an example, the computational difficulty of preparing entangled states is often used as a suggested justification for why Matrix Product States, and similar representations, have a good shot at being able to capture most of the fraction of Hilbert space that the system actually occupies while using only polynomial resources. However, just because there are potentially polynomial-complexity algorithms somewhere out there in algorithm-space that can effectively capture the nontrivial submanifold of the exponential Hilbert space that's occupied by the ground states of local hamiltonians ─ that doesn't make the problem easy to solve, because you need to put in the human intellectual and conceptual work to understand how to get those algorithms and what makes them tick. And, unlike the weakly-interacting case, you can't just make a blind guess at the subspace of interest and trundle off in that direction hoping that brute force will solve everything, because that brute-force approach would require you to play on a sandbox that was exponentially larger than the slice of interest, and that exponential size completely trumps your pretended brute force. • $\begingroup$ I would just add that "polynomial-time" is not the same as actually tractable in practice. I think people generally believe that tensor network algorithms should allow you to solve the quantum many-body problem in polynomial time (at least for suitably "nice" Hamiltonians), the problem is that the polynomial exponent in the known algorithms is just too large to be that useful. $\endgroup$ – Dominic Else Apr 29 '18 at 20:50 If every electron lives in its own Hilbert space $\mathcal{H}_e$ of size $D$, and we have $N$ electrons, then the total many-body system lives indeed in an exponentially bigger space $\mathcal{H}_s=\mathcal{H}_e^{\otimes N}$ of size $D^N$, which becomes rapidly intractible. The surprising thing is not that interacting systems are hard to solve, but rather that non-interacting systems are easy to solve. For non-interacting systems, the problem factorizes entirely, all copies of $\mathcal{H}_e$ are truly independent and can be solved separately (or in fact, one of them is enough if they are identical), so that the exponential complexity of $\mathcal{H}_s$ is not important. If there is weak interaction, a factorization can be made aproximately (Hartree, Gutzwiller, Mean-field ansatzes). A good example is the Gross-pitaevskii equation which assumes a problem factorized for individual particles. To get improvement, one should add Bogoliubov fluctuations. More generally, the systematic way to do this is a cumulant expansion of correlations (see also: BBGKY hierarchy ) that must be truncated. For a non-interacting system, there are no correlations. Your Answer
b62e044aa9156297	It is well known that a quantum computer is reversible. This means that it is possible to derive an input quantum state $\|\psi_0\rangle$ from an output $\|\psi_1\rangle$ of an algorithm described by a unitary matrix $U$ simply by applying transpose conjugate to $U$, i.e. \begin{equation} \|\psi_0\rangle = U^\dagger\|\psi_1\rangle \end{equation} In article Arrow of Time and its Reversal on IBM Quantum Computer an algorithm for a time reversal and going back to an input data $\|\psi_0\rangle$ is proposed. Steps of the algorithm are following: 1. Apply a forward time unitary evolution $U_\mathrm{nbit}\|\psi_0\rangle = \|\psi_1\rangle$ 2. Apply an operator $U_\psi$ to change $\|\psi_1\rangle$ to $\|\psi_1^\rangle$, where the new state $\|\psi_1^\rangle$ is complex conjugate to $\|\psi_1\rangle$ 3. Apply an operator $U_R$ to get "time-reversed" state $\|R\psi_1\rangle$ 4. Finally, apply again $U_\mathrm{nbit}$ to obtain the input state $\|\psi_0\rangle$ According to the paper, the algorithm described above simulates reversal of the time arrow. Or in other words, it simulates a random quantum fluctuation causing a time reversal. Clearly, when the algorithm is run on a quantum computer, it returns back to initial state but without application of an inverse to each algorithm step. The algorithm simply goes forward. My questions are these: 1. Why it is not possible to say that an application of $U^\dagger$ on output of algorithm $U$ is reversal of time arrow in general case? 2. It is true that above described algorithm returns a quantum computer to an initial state but it seems that the algorithm simply goes forward. So where I can see the a reversal of time arrow? 3. The authors of the articles have found out that when a number of qubit involved in the time reversal algorithm is increasing, the effect of time reversal diminishes: • How is it possible to reverse time for few qubits and concurently to preserve flowing of time in forward direction for another qubits? • Does this mean that time flows differently for different qubits? • When do the qubits return to commnon time frame to be possible to use them in another calculation? • $\begingroup$ The arrow of time is a statistical concept (a law of large numbers really). It's not a meaningful notion at the level of qubits. See Emilio Pisanty's answer here. 2 and 3 are non-questions in the sense that they're based on faulty premises and misunderstandings of the notion of the time arrow as a thermodynamic concept. $\endgroup$ – Sanchayan Dutta Jan 4 at 7:46 Of course if we have unitary evolution $$\|\psi_1\rangle = U\|\psi_0\rangle$$ then $$\|\psi_0\rangle = U^\dagger\|\psi_1\rangle$$ I did not read the paper, but evidently the authors do something different, based on the following: the Schrödinger equation $$i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi$$ changes its form if we substitute $t\rightarrow -t$ to complex conjugate: $$-i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi$$ and its solution is also complex conjugate. So time reversal is antiunitary operator $U_RK$ where $U_R$ is unitary and $K$ is complex conjugation. \|improve this answer\|\|\|\|\| • $\begingroup$ Thanks. However, could you (or anybody else) please answer my other questions (no. 2 and 3)? $\endgroup$ – Martin Vesely Dec 27 '19 at 20:50 Your Answer
c96b2267c2136afa	lördag 21 januari 2017 The Origin of Fake Physics Peter Woit on gives on Not Even Wrong a list of fake physics most of which can be traced back to the fake physics character of Schrödinger's linear multi-dimensional equation, as exposed in recent posts. Woit's list of fake physics thus includes different fantasies of multiversa all originating from the multi-dimensional form of Schrödinger's equation giving each electron its own separate 3d space/universe to dwell in. But the linear multi-d Schrödinger equation is a postulate of modern physics picked from out of the blue as a ready-made and as such like a religious dogma beyond human understanding and rationality. Why modern physics has been driven into such an unscientific approach remains to be understood and exposed, and discussed... The standard view is presented by David Gross as follows: • Quantum mechanics emerged in 1900, when Planck first quantized the energy of radiating oscillators. • Quantum mechanics is the most successful of all the frameworks that we have discovered to describe physical reality. It works, it makes sense, and it is hard to modify. • Quantum mechanics does make sense, although the transition, a hundred years ago, from classical to quantum reality was not easy. • The freedom one has to choose among different, incompatible, frameworks does not influence reality—one gets the same answers for the same questions, no matter which framework one uses. • That is why one can simply “shut up and calculate.” Most of us do that most of te time. • By now...we have a completely coherent and consistent formulation of quantum mechanics that corresponds to what we actually do in predicting and describing experiments and observations in the real world. • For most of us there are no problems. • Nonetheless, there are dissenting views. So, the message is that quantum mechanics works if you simply shut up and calculate and don't ask if it makes sense, as physicists are being taught to do, but here are dissenting views... Note that the standard idea ventilated by Gross is that quantum mechanics somehow emerged from Planck's desperate trick of "quantisation" of blackbody radiation 1900 when taking on the mission of explaining the physics of radiation while avoiding the "ultra-violet catastrophe" believed to torpedo classical wave mechanics. Planck never believed that his trick had a physical meaning and in fact the trick is not needed because an explanation can be given within classical wave mechanics in the form of computational blackbody radiation with the ultraviolet catastrophe not showing up. This is what Anthony Leggett, Nobel Laureate and speaker at the 90 Years of Quantum Mechanics Conference, Jan 23-26, 2017, says (in 1987): • If one wishes to provoke a group of normally phlegmatic physicists into a state of high animation—indeed, in some cases strong emotion—there are few tactics better guaranteed to succeed than to introduce into the conversation the topic of the foundations of quantum mechanics, and more specifically the quantum measurement problem. • I do not myself feel that any of the so-called solutions of the quantum measurement paradox currently on offer is in any way satisfactory. • I am personally convinced that the problem of making a consistent and philosophically acceptable 'join' between the quantum formalism which has been so spectacularly successful at the atomic and subatomic level and the 'realistic' classical concepts we employ in everyday life can have no solution within our current conceptual framework; • We are still, after three hundred years, only at the beginning of a long journey along a path whose twists and turns promise to reveal vistas which at present are beyond our wildest imagination. • Personally, I see this as not a pessimistic, but a highly optimistic, conclusion. In intellectual endeavour, if nowhere else, it is surely better to travel hopefully than to arrive, and I would like to think that the generation of students now embarking on a career in physics, and their children and their children's children, will grapple with questions at least as intriguing and fundamental as those which fascinate us today—questions which, in all probability, their twentieth-century predecessors did not even have the language to pose. The need of a revision, now 30 years later, of the very foundations of quantum mechanics is even more clear, 90 years after conception. The starting point must be the wave mechanics of Schrödinger without particles, probabilities, multiversa, measurement paradox, particle-wave duality, complementarity and quantum jumps with atom microscopics described by the same continuum mathematics as the macroscopic world. PS Is quantum computing fake physics or possible physics? Nobody knows since no quantum computer has yet been constructed. But the hype/hope is inflated: perhaps by the end of the year... Inga kommentarer: Skicka en kommentar
c0ddd514e4d3a6f6	Viewpoint: Dissipative Stopwatches • Christine Muschik, Institute of Photonic Sciences, Av. Carl Friedrich Gauss, 3, 08860 Castelldefels, Barcelona, Spain Physics 6, 29 Contact with the environment usually destroys the delicate operations of a quantum computer, but engineered dissipative processes may allow for a more robust preparation and processing of quantum states. APS/C. Muschik Figure 1: (Top) In conventional time evolution of quantum systems, different input states \|ψin lead to different output states \|ψout if a unitary operation U is applied. (Bottom) Quantum state preparation by reservoir engineering takes a different approach. If the interaction of the system with a bath is engineered such that \|ψfin is the unique steady state of the dissipative process, then the system is driven into this state irrespective of the initial state initiating the evolution paths ρ1, ρ2, etc.(Top) In conventional time evolution of quantum systems, different input states \|ψin lead to different output states \|ψout if a unitary operation U is applied. (Bottom) Quantum state preparation by reservoir engineering takes a different approach. ... Show more When riffle-shuffling a deck of 52 cards, seven shuffles are necessary to arrive at a distribution of playing cards that is, to a large degree, independent of the initial ordering. The fact that initial correlations survive if the deck is only shuffled a few times and disappear suddenly after seven shuffles is well known by magicians, who use this phenomenon in card tricks to amaze their audience. In a paper in Physical Review Letters [1], Michael Kastoryano of the Free University of Berlin, Germany, and colleagues show how this effect can be leveraged for quantum information processing. Quantum information science uses phenomena such as superpositions and entanglement to devise quantum devices capable of performing tasks that cannot be achieved classically. These applications are typically based on unitary dynamics (that is, time evolutions that are governed by the Schrödinger equation). One big practical problem hindering the operation of such devices in the quantum regime is dissipation caused by the interaction of the system with its environment. In the last several years, a new approach to quantum information processing has led to a rethinking of the traditional concepts that rely on unitary dynamics alone and avoid dissipation unconditionally: instead, these new protocols harness dissipative processes for quantum information science. Actively using dissipation in a controlled way opens up interesting new possibilities and has important advantages: dissipative protocols are robust and, as explained below, allow one to prepare a desired quantum state, irrespective of the initial state of the system. However, the underlying processes are intrinsically probabilistic and time independent. In general, it is therefore not clear how to incorporate them in the existing framework of unitary quantum information processing. One route around this difficulty is to embrace the probabilistic and time-independent nature of these processes and use specifically designed dissipative architectures (see, for example, Ref. [2]), but so far there are very few, and these schemes are conceptually very different from unitary protocols. Protocols based on unitary dynamics typically require precise timing and operations, which are conditioned on previous ones. The work by Kastoryano et al. shows how dissipative processes can be timed and used in a conventional way without losing the specific advantages of dissipative schemes [1]. The unitary time evolution (Fig. 1) of a pure quantum state \|ψ under a Hamiltonian Ĥ is governed by the Schrödinger equation iħ\|Ψ·(t)=Ĥ\|Ψ(t). Accordingly, a unitary time evolution U(t)=e-iĤtħ transforms a pure state \|Ψin always into another pure state \|Ψout=U(t)\|Ψin. Consider, for example, a system with spin states \| and \|. The Hamiltonian Ĥ=ħκ(\|ge\|+\|ge\|) causes the spin to flip. If acting for a time t=π/2κ, it transforms \| into \| and \| into \|. Figure 2: Using the protocols developed by Kastryano et al. [1], dissipative quantum information processing can be performed by engaging Markov processes in a precisely time-ordered fashion. Imagine a relay race in which one runner hands a baton to a second runner. In a similar way, quantum states evolving according to Liouville operators L1 and L2 can trigger each other. In this example runner L1 starts at t0 and ends at t1, where runner L2 takes over until t2 (shown as the finish line, but this process could be repeated over more sequences). Normally, dissipative evolutions cannot be timed in that way, but the protocols developed by Kastoryano et al. allow one to perform dissipative operations sequentially at specific points in time during well-defined time windows.Using the protocols developed by Kastryano et al. [1], dissipative quantum information processing can be performed by engaging Markov processes in a precisely time-ordered fashion. Imagine a relay race in which one runner hands a baton to a second ru... Show more Dissipative processes, in contrast, can turn a pure state into a mixed one that is described by a density matrix ρ, representing a statistical mixture. A dissipative time evolution is governed by a master equation (i.e., an equation of motion for the reduced density operator of a subsystem that interacts with an environment; the dynamics of the system is obtained by tracing out the degrees of freedom of the environment). Here we consider Markov processes (that is, “memoryless” ones) that are described by a time-independent Liouvillian master equation ρ·=L(ρ) with L(ρ)=Γ(2âρâ+-â+âρ-ρâ+â), with jump operator â and rate Γ. It is instructive to consider, for example, the jump operator â=\|\|. The corresponding master equation causes a system in state \| to relax to \| at a rate Γ. A dissipative process of this type can be understood as applying the jump operator â with certain probability ρ, which is determined by Γ. If we prepare our system in state \| and wait a short while, we don’t know whether there has already been a quantum jump \|\|⟩ or not, hence the resulting quantum state is a mixed one ρmixed=p\|\|+(p-1)\|\|. (The steady state can be still a pure one, \| in this example.) Because of this intrinsically probabilistic feature, dissipative processes are difficult to time. Kastoryano and colleagues develop new tools that allow one to use dissipative processes in such a way that the desired transitions occur at very well-defined points in time (Fig. 2). The key to dissipative protocols is to tailor the interaction between the system and a bath such that a specific desired jump operator â is realized. This jump operator is chosen such that the target state ρfin is the unique steady state of the dissipative evolution L(ρfin)=0. For dissipative quantum computing [3], the result of the calculation is encoded in ρfin, and if the goal is quantum state engineering, ρfin can be, for example, an entangled [4] or topological state [5]. This state is reached regardless of the initial state of the system. Should the system be disturbed, the dissipative dynamics will bring it back to the steady state ρfin. This is impossible for unitary dynamics. In the unitary case, imperfect initialization inevitably leads to deviations from the desired final state. Therefore “reliable state preparation,” the second of the five criteria that DiVincenzo established for a scalable quantum computer [6], was for a long time considered a fundamental requirement for quantum information processing. Kastoryano et al. extend the toolbox of dissipative quantum information processing by introducing devices for timing this type of process exactly. More specifically, they introduce schemes for (i) preparing a quantum state during a specified time window and (ii) for triggering dissipative operations at specific points in time. The authors use these tools for demonstrating a dissipative version of a one-way quantum computation scheme [7]. The central ingredient that is used here is a very interesting mechanism called the “cutoff phenomenon.” Stochastic processes that have this cutoff quality exhibit a sharp transition in convergence to stationarity. They do not converge smoothly to the stationary distribution during a certain period (if the initial state is far away from the stationary state) but instead converge abruptly (exponentially fast in the system size) at a specific point in time. This behavior was first recognized in classical systems [8]. An intriguing classical example is the shuffling of playing cards outlined above. This type of mechanism has now been investigated in the quantum setting. In earlier work [9] by some of the authors of the new paper, the cutoff phenomenon is introduced for quantum Markov processes using the notions of quantum information theory. This provides a quantitative tool to study the convergence of quantum dissipative processes. In Ref. [1], Kastoryano et al. employ this quantum-version of the cutoff phenomenon to start and end dissipative processes at specific points in time, thus allowing for the integration in the framework of regular quantum information architectures. In future, these kinds of tools might also become important for new dissipative quantum error correcting schemes and may point towards new insights regarding passive error protection. Quantum reservoir engineering is a young and rapidly growing area of research. Several protocols have been developed including dissipative schemes for quantum computing [3], quantum state engineering [10], quantum repeaters [2], error correction [11], and quantum memories [12]. Dissipative schemes for quantum simulation [13] and entanglement generation [4] have already been experimentally realized. Actively using dissipative processes is a conceptually interesting direction with important practical advantages. The results presented in Ref. [1] add new tools for exploiting and engineering dissipative processes and provide the linking element for incorporating dissipative methods into the regular framework of quantum information processing. 1. M. J. Kastoryano, M. M. Wolf, and J. Eisert, “Precisely Timing Dissipative Quantum Information Processing,” Phys. Rev. Lett. 110, 110501 (2013) 2. K. G. H. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement Distillation by Dissipation and Continuous Quantum Repeaters,” Phys. Rev. Lett. 107, 120502 (2011) 3. F. Vertraete, M. Wolf, and J. I. Cirac, “Quantum Computation and Quantum-State Engineering Driven by Dissipation,” Nature Phys. 5, 633 (2009) 4. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. Ignacio Cirac, and E. S. Polzik, “Entanglement Generated by Dissipation and Steady State Entanglement of Two Macroscopic Objects,” Phys. Rev. Lett. 107, 080503 (2011) 5. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, “Topology by Dissipation in Atomic Quantum Wires,” Nature Phys. 7, 971 (2011) 6. D. P. DiVincenzo, “The Physical Implementation of Quantum Computation,” Fortschr. Phys. 48, 771 (2000) 7. R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer,” Phys. Rev. Lett. 86, 5188 (2003) 8. P. Diaconis, “The Cutoff Phenomenon in Finite Markov Chains,” Proc. Natl. Acad. Sci. U.S.A. 93, 1659 (1996) 9. M. J. Kastoryano, D. Reeb, and M. M. Wolf, “A Cutoff Phenomenon for Quantum Markov Chains,” J. Phys. A 45, 075308 (2012) 10. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum States and Phases in Driven Open Quantum Systems with Cold Atoms,” Nature Phys. 4, 878 (2008) 11. J. Kerckhoff, H. I. Nurdin, D. S. Pavlichin, and H. Mabuchi, “Designing Quantum Memories with Embedded Control: Photonic Circuits for Autonomous Quantum Error Correction,” Phys. Rev. Lett. 105, 040502 (2010); J. P. Paz and W. H. Zurek, “Continuous Error Correction,” Proc. R. Soc. London A 454, 355 (1998) 12. F. Pastawski, L. Clemente, and J. Ignacio Cirac, “Quantum Memories Based on Engineered Dissipation,” Phys. Rev. A 83, 012304 (2011) 13. J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An Open-System Quantum Simulator with Trapped Ions,” Nature 470, 486 (2011) About the Author Image of Christine Muschik Subject Areas Quantum Information Related Articles Focus: Intercontinental, Quantum-Encrypted Messaging and Video Quantum Information Focus: Intercontinental, Quantum-Encrypted Messaging and Video Synopsis: Quantum Circulators Simplified Quantum Information Synopsis: Quantum Circulators Simplified Synopsis: A Lens to Focus Spins Quantum Information Synopsis: A Lens to Focus Spins More Articles
c1d53c9b22257f87	Category Archives: Mathematics All you wanted to know about Hybrid Orbitals… … but were afraid to ask 2017-08-09 20.29.45 Taken from “Quantum Mechanics in Chemistry” by Jack Simmons ζ1a12s + b12px + c12py + d12pz ζ2a22s + b22px + c22py + d22pz ζ3a32s + b32px + c32py + d32pz ζ4a42s + b42px + c42py + d42pz ζ= 1/√4(2s) + d12pz 1/4 + d12 = 1; d1 = √3/2 Therefore the first hybrid orbital looks like: ζ1\|ζ2〉 = δ1,2 = 0 1/4 + d2√3/2 = 0 d2 = –1/2√3 again we make use of the normalization condition: Finally, our second hybrid orbital takes the following form: ψ2s = (1/4π)½R(r) ψ2px = (3/4π)½sinθcosφR(r) ψ2pz = (3/4π)½cosθR(r) sinθ/cosθ = tanθ = -√8 θ = -70.53°, No, seriously, why can’t orbitals be observed? Dealing with Spin Contamination Atoms in Molecules (QTAIM) – Flash lesson Figure 1 Figure 1 Figure 2 Figure 2 Figure 3 Figure 3 Figure 4 Figure 4 Article in ‘Ciencia y Desarrollo’ (Science and Development) Chemistry without flasks? What is theoretical chemistry? What is it good for? How do you calculate a molecule? The Chuck Norris of chemistry • “Chuck Norris doesn’t eat honey, he chews bees” • “Chuck Norris counted to infinity; twice!” • “Parallel lines meet where Gauss tells them to”. • Roald Hoffmann drinks AlLiH4 aqueous cocktails • Roald Hoffmann can stabilize a tertiary carbanion and a primary carbocation • Roald Hoffmann always gets a 100% yield • Le’Chatellier’s principle first asks for Hoffmann’s permission • Roald Hoffmann once broke the Periodic Table with a roundhouse kick • Roald Hoffmann’s blood is a stronger acid than SbF5 It’s that time of the year again… The Nobel Prizes Teaching QSAR and QSPR at UAEMex Teaching has never been my cup of tea. Karl Friederich Gauss said “Good students do not need a teacher and bad students, well, why do they want one?” I once read this quote somewhere, and although I don’t know if he actually said it or not, there is some truth to it. It is known that Gauss didn’t like teaching, still spent most of his life doing it. Anyway, teaching is important and it has to be done! Therefore as part of my duties as researcher at CCIQS I will have to teach a class at the Faculty of Chemistry of the Mexico State’s Autonomous University (UAEMex). Obviously they want you to teach a class on a subject you are an expert on; I could teach organic chemistry for sure, despite the fact that I haven’t touched a flask in years. My colleague, Dr. Fernando Cortés-Guzmán and I seem to be two of the very few theoretical chemists around so it is up to us to teach all classes within the range of theoretical chemistry, computational chemistry and their applications. This year someone, I still need to find out who, came up with the idea that an interesting application would be QSAR which of course is a very relevant model for drug discovery. Thus, starting today, I will be the first teacher of this subject at UAEMex’s Chemistry Faculty. Although to be quite frank, I think I would have felt better teaching calculus or differential equations, since those already have a syllabus. On the other hand, those subject wouldn’t get me in touch with students in their final years who are the ones to be attracted as potential students for my incipient research group. It has been interesting so far, building the syllabus from scratch; finding all the topics that are worth covering in a semester as well as a proper way to illustrate and teach them. It will be a work in progress all the time and I intend to expand it somehow beyond the classroom; my first thought was to record all the lessons for a podcast. I’m still not sure how to include this blog into the equation or if I should open a new one for the class but I guess I’ll figure it out along the way. I’m not an expert on QSAR or QSPR but I know a good deal about it, mostly because of Dr. Dragos Horvath whom I met in Romania years ago. Perhaps I could persuade him of leaving Strasbourg for a couple of weeks and giving a few lectures. Wish me luck, or maybe I should say: “wish my students luck”! Basis sets In this new post I will address some issues regarding the correct use of the terminology used about basis sets in ab initio calculations. One of the keys to achieve good results in ab initio calculations is to wisely select a basis set; however this requires some previous insight about the specific model to use, the system (molecule/properties) to be calculated and the computational resources at hand. Most of the basis sets available today remain in our codes due to historical reasons more than to their real practical use. We know the Schrödinger equation is not analytically solvable for an interestingly big enough molecule, so the Hartree-Fock (HF) approach approximates its solution in terms of MOs but these MOs have to be constructed of smaller functions, ideally AOs but even these are constructed as linear combinations of simpler, linearly independent, mutually orthogonal functions which we call Basis Sets. For true beginners: Imagine the 3D vector space as you know it. The position vector corresponding to any point in this space can be deconstructed in three different vectors: R =ax+by+cz In this case x, y and z would be our basis vectors which comply with the following rules: A) They are linearly independent; none of them can be expressed in terms of the others. B) They are orthogonal; their pairwise scalar product is zero. C) Their pairwise vector product yields the remaining one with its sign defined by the range three tensor epsilon. In a vector space with more than three dimensions we can always find a basis which has the same properties described above with which we are able to uniquely define any other vector belonging to this hypothetical space. In the case of Quantum Mechanics we are dealing with function spaces (since our entity of interest is the Wavefunction of a quantum system) instead of vector ones, so what we look for are basis functions that allow us to generate any other function belonging to this space. This is one of those examples that survive for historical reasons. Its value relies on the fact that is a good first start to obtain the properties of small systems. EXPAND minimal basis: This term refers to the fact of using a single STO for each occupied atomic orbital present in the molecule. double zeta basis: Here each STO is replaced by two STO’s wich differ in their zeta values. This improves the description of each orbital at some computational cost. A single STO is used to describe core orbitals (a minimal core basis set) while two or more are used to describe the valence orbitals. A plane wave is a wave of constant frequency whose wavefronts are described as infinite parallel planes. When dealing with -tranlational- symmetrical systems (such as crystals) the total wavefunction can be deconstructed as a combination of plane waves. This kind of basis set is suitable for Periodic Boundary Conditions (PBC) computations if a suitable code is available for it, since plane wave solutions converge slowly. Softwares such as CRYSTAL make use of plane wave solution to find the electronic properties of crystaline solids. As usual I hope this post is of help. Please rate or comment on this post just to know we are working on the right path! Wheel? I think knot! Once again an awful title. This post follows my previous one on graphs and chemistry, and it addresses an old idea which I have shared in the past with many patient people willing to listen to my ramblings. It is a common conception/place to state that the wheel was the invention that made mankind spring from its more hominid ancestors into the incipient species that would eventually become homo sapiens; that it was the wheel, like no other prehistoric invention or discovery, what made mankind to rise from its primitive stage. I’ve always believed that even if the wheel was fundamental in the development of mankind, man first had to build tools to make wheels out of something; otherwise they would have been just a good theoretical conception. But even despite the fact that building tools was in itself a pretty damn good start, I strongly believe that mankind’s first groundbreaking invention were knots. For even a wheel was a bit useless until it was tied to something. From my perspective, the invention of the wheel was an event bound to happen since there are many round shaped things in nature: from the sun and the moon to some fruits and our own eyes. Achieving the mental maturity of taking a string (or a resembling equivalent of those days) and tie it, whether around itself or to something, was, in my opinion, the moment in which the opposable thumbs of mankind realized they could transform it’s surroundings. Furthermore, at that stage the mental maturity achieved made it possible for man to remember how to do it over again in a consistent way. The book ‘2001 – a space odissey’ by A. C. Clarke, describes this process in the first chapter when a group of hominids bumps into the famous monolith. Their leader (i think his name was moonlight), under the spell of this strangely straight and flat thing takes two pieces of grass and ties them together without knowing or understanding what he is doing. I was pleased to read that I was not alone in that thought. The concept of a knot keeps on amazing me given their variety and the different purposes they serve according to their properties. These were known to ancient sailors who have elevated the task of knot-making to a practical art form. The mathematical background behind them has served to lay one of today’s most fundamental (and controversial) theories about the composition of matter: string theory. Next time when you make the knot of your necktie think about this tedious, obnoxious little habit was based on something groundbreaking that truly makes us stand out from the rest of the species in the animal kingdom. %d bloggers like this:
be7bc6dbc7de5303	Document Type Publication Date The Schrödinger theory for a system of electrons in the presence of both a static and time-dependent electromagnetic field is generalized so as to exhibit the intrinsic self-consistent nature of the corresponding Schrödinger equations. This is accomplished by proving that the Hamiltonian in the stationary-state and time-dependent cases {\hat{H}; \hat{H}(t)} are exactly known functionals of the corresponding wave functions {\Psi; \Psi(t)}, i.e. \hat{H} = \hat{H}[\Psi] and \hat{H}(t) = \hat{H}[\Psi(t)]. Thus, the Schrödinger equations may be written as \hat{H}[\Psi]\Psi = E[\Psi]\Psi and \hat{H}[\Psi(t)]\Psi(t) = i\partial\Psi(t)/\partial t. As a consequence the eiegenfunctions and energy eigenvalues {\Psi; E} of the stationary-state equation, and the wave function \Psi(t) of the temporal equation, can be determined self-consistently. The proofs are based on the 'Quantal Newtonian' first and second laws which are the equations of motion for the individual electron amongst the sea of electrons in the external fields. The generalization of the Schrödinger equation in this manner leads to additional new physics. The traditional description of the Schrödinger theory of electrons with the Hamiltonians {\hat{H}; \hat{H}(t)} known constitutes a special case. This is the peer-reviewed version of the following article: V. Sahni. J. Comput. Chem. 2017, DOI: 10.1002/jcc.24888, which has been published in final form at http://dx.doi.org/10.1002/jcc.24888. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Available for download on Friday, March 01, 2019
8d40d618f69b519e	Dismiss Notice Join Physics Forums Today! I Why psi squared instead of psi? 1. Sep 3, 2016 #1 Why should we square the abs value of the wave function to get probabilities? Why don't just forget the imaginary part, and (in order to get positive values) square the real part of the function? Last edited by a moderator: Sep 4, 2016 2. jcsd 3. Sep 3, 2016 #2 If you take the Schrödinger equation as given, you can derive that the absolute square of the wave function is conserved but not the square of the real or imaginary part. (You can move around the real and imaginary parts by multiplying the wave function by a global phase ##e^{i \phi}## without changing anything physical.) Last edited by a moderator: Sep 4, 2016 4. Sep 4, 2016 #3 User Avatar Science Advisor 2016 Award The possibility to interpret ##\|\psi(t,\vec{x})\|^2## as a probability distribution hinges on the fact that $$\\|\psi \\|^2=\int_{\mathrm{R}^3} \|\psi(t,\vec{x})\|^2=\text{const}$$ due to the Schrödinger equation, $$\mathrm{i} \partial_t \psi(t,\vec{x})=-\frac{\Delta}{2m} \psi(t,\vec{x}) + V(\vec{x}) \psi(t,\vec{x}).$$ Of course, also ##\|\psi\|^2 \geq 0## is also important for having a probability distribution. 5. Sep 4, 2016 #4 6. Sep 4, 2016 #5 Have something to add? Draft saved Draft deleted Similar Discussions: Why psi squared instead of psi? 1. Why psi is complex! (Replies: 8) 2. Why psi^2? (Replies: 17)
daad554cf154f439	tisdag 8 oktober 2013 From Impossible Fiction to Possible Reality in Quantum Mechanics The wave function $\Psi$ as a solution to Schrödinger's equation supposedly describing the quantum mechanics of an atom with $N$ electrons, depends on N three-dimensional spatial variables $x_1,..., x_N$ (and time), altogether $3N$ spatial dimensions, with $\vert \Psi (x_1,..., x_N)\vert^2$ interpreted as the probability of the configuration with electron $j$ located at position $x_j$. The wave function $\Psi (x_1,...,x_N)$ is thus supposed to carry information about all possible electron configurations and as such contains an overwhelming amount of information, which however is not really accessible because of an overwhelming computational cost already for small N with difficulties starting already for $N = 2$. To handle this difficulty drastic reductions in complexity are being made by seeking approximate solutions as wave functions with drastically restricted spatial variation based on heuristics. There are claims that this is feasible using structural properties of the wave function (but full scale computations seem to be missing). An alternative approach would be to seek $N$ wave functions $\Psi_1,..., \Psi_N$, depending on a common three-dimensional space coordinate $x$, with $\Psi_j(x)$ carrying information about the presence of particle $j$, as a form of Smoothed Particle Mechanics (SPM) as a variant of classical particle mechanics. The corresponding Schrödinger equation consists of a coupled system of one-electron equations in the spirit of Hartree, and is readily computable even for N large. The solution $\Psi_1(x),..., \Psi_N(x)$, of the system would give precise information about one possible electron configuration. If this is a representative configuration, this may be all one would like to know. As an example, SPM for the Helium atom with two atoms appears to give a configuration with two half-spherical electron lobes in opposition, as a representative configuration with other equally possible configurations obtained by rotation, as suggested in a previous post and in the sequence Quantum Contradictions. Instead of seeking information about all possible configurations by solving the 3N dimensional many-electron Schrödinger equation, which is impossible, it may be more productive to seek information about one possible and representative configuration by solving a system of 3 dimensional one-electron wave functions, which is possible. Inga kommentarer: Skicka en kommentar
a9fb24bd19d167ad	Thursday, May 31, 2012 Psychic Contributions to Physics Extrasensory Perception of Subatomic Particles (PDF), (HTML) Wednesday, May 30, 2012 Skeptiko Interview with Dr. Melvin Morse In the skeptiko podcast interview with Dr. Melvin Morse, Morse tells of the case of a child drowning victim who had been underwater for 17 minutes and after being rescued had no heartbeat for an additional 45 minutes. Dr. Morse never saw any sign of life in the patient. At the time he thought she had died. It was only long afterward that he found out she survived. Yet during that time, the patient experienced floating out of her body and remembered being intubated, hearing a phone conversation the doctor had, and hearing the nurses talking about a cat that died. When she regained consciousness she asked the nurses where her friends from heaven were. She also remembered that heaven was "fun". The full interview is here. So by chance or coincidence or fate or whatever, I happened to be in Pocatello, Idaho and there was a child there who had drowned in a community swimming pool. She was documented to be under water for at least 17 minutes. It just so happened that a pediatrician was in the locker room at the same community swimming pool and he attempted to revive her on the spot. His intervention probably saved her life but again, he documented that she had no spontaneous heartbeat for I would say at least 45 minutes, until she arrived at the emergency room. Then our team got there. She was really dead. All this debate over how close do these patients come to death, etc., you know, Alex, I had the privilege of resuscitating my own patients and she was, for all intents and purposes, dead. In fact, I had told her parents that. I said that it was time for them to say goodbye to her. This was a very deeply religious Mormon family. They actually did. They crowded around the bedside and held hands and prayed for her and such as that. She was then transported to Salt Lake City. She lived. She not only lived but three days later she made a full recovery. Alex Tsakiris: And what did she tell you… Dr. Melvin Morse: Her first words, the first words she said when she came out of her coma, she turned to the nurse down at Primary Children’s in Salt Lake City. She says, “Where are my friends?” And then they’d say, “What do you mean, where are your friends?” She’d say, “Yeah, all the people that I met in Heaven. Where are they?” [Laughs] The innocence of a child. So I saw her in follow-up, another one of these odd twists of fate. I happened to be in addition doing my residency and just happened to be working in the same community clinic in that area. My jaw just dropped to the floor when she and her mother walked in. I was like, “What?” I had not even heard that she had lived. I had assumed that she had died. She looked at me and she said to her mother, “There’s the man that put a tube down my nose.” [Laughs] Alex Tsakiris: What are you thinking at that point when she says that? Dr. Melvin Morse: You know, it’s one of those things—I laughed. I sort of giggled the way a teenager would giggle about sex. It was just embarrassing. I didn’t know what to think. Certainly, I’d trained at Johns Hopkins. I thought when you died you died. I said, “What do you mean, you saw me put a tube in your nose?” She said, “Oh, yeah. I saw you take me into another room that looked like a doughnut.” She said things like, “You called someone on the phone and you asked, ‘What am I supposed to do next?’” She described the nurses talking about a cat who had died. One of the nurses had a cat that had died and it was just an incidental conversation. She said she was floating out of her body during this entire time. I just sort of laughed. And then she taps me on the wrist. You’ve got to hear this, Alex. After I laughed she taps me on the wrist and she says, “You’ll see, Dr. Morse. Heaven is fun.” [Laughs] I was completely blown away by the entire experience. I immediately determined that I would figure out what was going on here. This was in complete defiance of everything I had been taught in terms of medicine. Tuesday, May 29, 2012 Meditation Music Here is some music that may help with the meditation I discussed yesterday: • Put a Little Love in Your Heart Think of your fellow man lend him a helping hand put a little love in your heart. • Get Together C'mon people now, Smile on your brother Ev'rybody get together Try and love one another right now • Shower the People Shower the people you love with love Show them the way that you feel If you click on the links above you will go to another site which has the lyrics of the song. If you search the internet, you may be able to find mp3 downloads or youtube videos of this music. Monday, May 28, 2012 How to Tap into Universal Love Update: I have added this post to my web site and made a few updates to it. Please refer to Tapping into Universal Love on the meditation page on my web site for the most recent version of this information. God is love. People who experience being in the presence of God during near death experiences describe having an overwhelming feeling of being loved. God is omnipresent. You can tap into this source of universal love without having a near death experience. To do it you use your spiritual capabilities - the capabilities that all spirits have and that as an incarnated spirit you have access to even though you are incarnated. Spirits interact with their world through their mind. They think of a place they want to go to and they start moving there. They are telepathic, they think of someone and their thoughts go off to that person. Spirits use their mind the way an incarnated person uses tools. Spirits create by using their mind. We use the same word "create" to describe how people use their imagination because it is the same thing. To create a tap into universal love, use your imagination. Imagine a light beam of love coming down to you from above. Hold your hands in front of you with your palms facing upward to receive it. Relax any tension or tightness you may feel in your chest, open your heart, and let the love flow out into the world. Try this meditation: Step 1: Imagine a light beam of love coming down upon you from above. Hold your hands in front of you with your palms facing upward to receive it. Say to yourself, "Love is all around why don't you take it?" (If you know the tune, you may sing it to yourself). Step 2: Relax any tension or tightness you may feel in your chest, open your heart, and imagine love emanating from your heart and flowing out into the world, or to a situation you don't like (to desensitize yourself to the situation), or to someone who might be a problem for you (to develop forgiveness and tolerance). Say to yourself, "Love is all around why don't you make it?" (If you know the tune, you may sing it to yourself). Repeat these two steps for the duration of the meditation session. If you feel like smiling while you do this, go ahead and smile. It is probably an indication that you are doing it right. Also, sometimes smiling a little bit can help you reenter this state. I recommend some music to play during this meditation in this post. Friday, May 25, 2012 The Experience of Oneness I made a minor change to my web page on Varieties of Mystical Experiences in the section Kensho and Kundalini. I added a link to a web page by Christine Farrenkopf discussing scientific research on the changes in brain activity that occur when meditators experience a sense of oneness sometimes referred to as a nondual state. (UPDATE: I changed the link on my web site to go to this post.) From Farrenkopf's web report: The "peak" of meditation is clearly a subjective state, with each individual attaining it in different manners and having different time requirements. However, the sensation and meaning behind this moment is consistent among all who reach it. At the peak, the subjects indicate that they lose their sense of individual existence and feel inextricably bound with the universe. "There [are] no discrete objects or beings, no sense of space or the passage of time, no line between the self and the rest of the universe" (Newberg 119). The subjects then meditated. When they reached the peak, they pulled on a string attached at one end to their finger and at the other to Dr. Newberg.2 This was the cue for Newberg to inject the radioactive tracer into the IV connected to the subject. Because the tracer almost instantly "locks" onto parts of the brain to indicate their activity levels, the SPECT gives a picture of the brain essentially at that peak moment (Newberg 3). The results revealed a marked decrease in the activity of the posterior, superior parietal lobe and a marked increase in the activity of the prefrontal cortex, predominantly on the right side of the brain (Newberg 6). Such changes in activity levels demonstrated that something was going on in the brain in terms of spiritual experience. The next step was to look at what these particular parts of the brain do. Studies of damage suffered to a region of the brain have enabled us to draw conclusions about its role by observing loss of function. It has been concluded that the posterior, superior parietal lobe is involved in both the creation of a three-dimensional sense of self and an individual's ability to navigate through physical space (Journal 216). The region of the lobe in the left hemisphere of the brain allows for a person to conceive of the physical boundaries of his body (Newberg 28). It responds to proprioceptive stimuli, most importantly the movement of limbs. The region of the lobe in the right hemisphere creates the perception of the matrix through which we move. From a subjective point of view, when in the nondual state, it seems like the self disappears and the experiencer becomes "one with everything". From an objective point of view, research on meditators shows that the experience is due to a decrease in activity of the posterior, superior parietal lobe in the brain. These results are consistent with other research which shows that region of the brain is responsible for the sense of self. At first glance, this may seem like a materialist explanation for the experience of oneness, but it is consistent with hypothesis that consciousness is non physical and the brain acts as a filter of consciousness. It indicates that the sense of self is not an objective fact. The sense of self is a subjective opinion, an illusion, produced by the brain. It is also interesting that people who have near death experiences also report a sense of oneness which suggests the experience of oneness is a real experience of our true nature when we are not constrained by the physical brain. Whatever the explanation, the experience of oneness does show that the sense of self and separateness we consider to be our normal reality is merely a subjective opinion. Thursday, May 24, 2012 Mario Beauregard on Near Death Experiences Mario Beauregard's recent article on Near Death Experiences, Near death, explained discusses some cases of NDEs and discusses why skeptical explanations for the phenomena are wrong. He concludes NDEs are strong evidence for the afterlife. His response to criticisms of the article are provided in a second article: Near-death, revisited Corroborated veridical NDE perceptions during cardiac arrest (and several other phenomena discussed in “Brain Wars”) strongly suggest that so-called “scientific materialism” is not only limited, but wrong. In line with this, nearly a century ago, quantum mechanics (QM) dematerialized the classical universe by showing that it is not made of minuscule billiard balls, as drawings of atoms and molecules would lead us to believe. In other words, QM acknowledges that the physical world cannot be fully understood without making reference to mind and consciousness, that is, the physical world is no longer viewed as the primary or sole component of reality (this was well explained by Wolfgang Pauli, one of the founders of QM...) Wednesday, May 23, 2012 Dean Radin: Consciousness and the double-slit interference pattern: Six experiments In the recent past, I have posted about the ability of consciousness to influence the physical world at the level of quantum mechanics and in the brain. This shows that consciousness is not an epiphenomenon, nor is it an illusion, and that conscisouness cannot be produced by matter or physical processes. Many of the founders of quantum mechanics such as Erwin Schrödinger, and Max Planck believed this to be true. Now Dean Radin has published a research paper describing more evidence of this. His latest experiments show how meditators concentrating on an experimental apparatus can cause a photon to change from a wave to a particle. He used a double slit apparatus and found that when meditators concentrated on the apparatus, the interference pattern caused by light waves decreased and the pattern more closely resembled that which would be caused by particles rather than waves. Consciousness and the double-slit interference pattern: Six experiments Dean Radin Leena Michel, Karla Galdamez, Paul Wendland, Robert Rickenbach, and Arnaud Delorme, PHYSICS ESSAYS 25, 2 (2012) © 2012 Physics Essays Publication. [DOI: 10.4006/0836-1398-25.2.157] Tuesday, May 22, 2012 Randi's Unwinnable Prize: The Million Dollar Challenge Last month I wrote a couple of posts in a discussion forum explaining why the Amazing Randi's million dollar challenge has not been won. This is important to understand because pseudoskeptics often claim that because no one has won the prize, all claims of paranormal powers must be false. However, the truth is that that a number of paranormal phenomena have been shown to be genuine and the million dollar prize has not been won because it is simply not a good way to demonstrate paranormal powers. The million dollar challenge requires that the applicant has to beat 1 million to one odds. Setting such a high barrier for success makes sense if you are risking a one million dollar prize. However, 1 million to one odds are much higher than the scientific standard of proof so this challenge is not necessarily the best or fairest way to determine if paranormal abilities are genuine. Designing a test that is fair to both the applicant and the challenger requires sophisticated knowledge of statistics (How many trials would be needed for a psychic to have a 95% confidence level that they could beat million to one odds if their accuracy was 75%?) Most psychics don't have the understanding of statistics necessary to look out for their own interests and therefore most applicants will not be able to demand a protocol that gives them a fair chance of winning. This is the most likely reason no one has won the prize. Furthermore, most applicants who know about Randi or understand the details of the challenge would be reluctant to spend the time, effort and expense of applying because they would not trust it to be a fair test or have confidence that they would be judged fairly or rewarded fairly if they succeeded. • Randi supposedly has said, "I always have an out. (Fate, October 1981)", and "I am a charlatan, a liar, a thief and a fake altogether." (This is reported to have been said on PM Magazine, on July 1st, 1982.) Applicants for the prize have legitimate reasons not to trust Randi. An interview in Will Stoor's book The Heretics quotes Randi in making several deceptive statements. • The prize is in bonds but Randi won't say when the bonds mature or who issued the bonds so no one knows what the prize really is. Why won't he say what the prize really is? Applicants are legitimately afraid the prize is some sort of worthless trick. • The applicant has to pay for their own travel expenses involved in attempting the prize. Why would they do that when they have good reasons not to trust Randi and they don't know what the prize really is? • Randi has a history of making mean spirited statements. He has been forced to retract statements in the past. However, applicants have to sign an agreement not to sue Randi even if he makes makes misleading, defamatory, slanderous, or libelous statements about the psychic. • The applicants for Randi's prize have to prove themselves to a very high statistical standard far beyond the level that is generally considered proof in science experiments. An experiment could be designed to satisfy this standard with fewer than ten trials. However, a psychic, depending on their rate of accuracy, might need hundreds of trials to have a fair chance of obtaining such an unlikely result1. Most psychics won't realize this because they don't have the necessary expertise in science or statistics and this may be the primary reason no applicant has ever won the prize. One scientist who did apply for the prize never heard back from Randi. Why would anyone be willing to spend their time and money to try to win the challenge when they don't trust Randi, or believe that the challenge is fair, or that the prize is real? The challenge is not really serious. Most applicants who understand the details would be reluctant to spend the time, effort and expense of applying because they would not trust it to be a fair test or have confidence that they would be judged fairly or rewarded fairly if they succeeded. The most likely reason no one has won is because most applicants do not have the expertise in statistics needed to demand a protocol that will give them a fair chance of winning. The prize is a publicity stunt designed to give materialist pseudoskeptics a one liner: "Why has no one won the prize?!?!" The correct answer is: ... Because it is not a good way to measure paranormal powers and anyone who understands the situation would have very good reasons not to apply. It is sadly ironic that so many of Randi's followers, who pride themselves on their critical thinking skills, are fooled into thinking this prize is a legitimate test of paranormal phenomena. There are many independent forms of empirical evidence for ESP and the afterlife. The entire movement of pseudoskeptics is based on misdirection. Randi's followers believe they are helping to protect people from fraud, but in fact they themselves are victims of many deceptions perpertrated by the leaders of the pseudoskeptic movement. I discuss this in greater detail on my web page on Skeptical Misdirection. (1) In an experiment to measure psychic ability, there are three numbers that need to be considered: • The first number represents the confidence that the outcome is not due to chance. The million dollar challenge requires the psychic perform at a level that would occur by chance only once in a million times. • The second number is the rate of accuracy of the psychic's abilities. For example, a psychic might have an accuracy of 75% in some task where the probability of being correct by chance is only 50%. • The third number is the number of trials which are needed to give the psychic a high level of confidence that they would win the prize given their rate of accuracy. In order to achieve the required confidence that the psychic's performance is not due to chance, the challenge could require two tests of ten or fewer trials. However, the psychic might not be able to pass such a test if they are not 100% accurate. But, if the psychic is given a sufficient number of trials, they may demonstrate a success rate that, while not 100% accurate, still cannot be explained by chance at the level of confidence demanded by the challenge. In order for the psychic to have a 95% confidence level that they could beat million to one odds if their accuracy was 75%, they might need over 100 trials. Most psychics are not well enough versed in statistics to know how to measure their rate of accuracy or how to calculate the number of trials they need to have a good chance of winning the prize. Skeptical Organisations and Magazines A Guide to the Skeptics at Contenders have to pay for their own travelling expenses if they want to go to Randi to be tested: Rule 6: "All expenses such as transportation, accommodation and/or other costs incurred by the applicant/claimant in pursuing the reward, are the sole responsibility of the applicant/claimant." Also, applicants waive their legal rights: Rule 7: "When entering into this challenge, the applicant surrenders any and all rights to legal action against Mr. Randi, against any person peripherally involved and against the James Randi Educational Foundation, as far as this may be done by established statutes. This applies to injury, accident, or any other damage of a physical or emotional nature and/or financial, or professional loss, or damage of any kind." Applicants also give Randi complete control over publicity Flim-Flam Flummery: A Skeptical Look at James Randi at Recently I picked up Flim-Flam again. Having changed my mind about many things over the past twenty years, I responded to it much differently this time. I was particularly struck by the book's hectoring, sarcastic tone. Randi pictures psychic researchers as medieval fools clad in "caps and bells" and likens the delivery of an announcement at a parapsychology conference to the birth of "Rosemary's Baby." After debunking all manner of alleged frauds, he opens the book's epilogue with the words, "The tumbrels now stand empty but ready for another trip to the square" – a reference to the French Revolution, in which carts ("tumbrels") of victims were driven daily to the guillotine. Randi evidently pictures himself as the executioner who lowers the blade. In passing, two points might be made about this metaphor: the French Revolution was a product of "scientific rationalism" run amok ... and most of its victims were innocent. The Psychic Challenge by Montague Keen Source: hosted at Now for the more serious bit: first, the $1million prize. Loyd Auerbach, a leading USA psychologist and President of the Psychic Entertainers Association (some 80% of the members of his Psychic Entertainers' Association believe in the paranormal, according to Dr. Adrian Parker, who was on the programme, but given no opportunity to reveal this) exposed some of the deficiencies in this challenge in an article in Fate magazine. Under Article 3, the applicant allows all his test data to be used by the Foundation in any way Mr. Randi may choose. That means that Mr. Randi can pick and chose the data at will and decide what to do with it and what verdict to pronounce on it. Under Article 7, the applicant surrenders all rights to legal action against the Foundation, or Mr. Randi, no matter what emotional, professional or financial injury he may consider he has sustained. Thus even if Mr. Randi comes to a conclusion different from that reached by his judges and publicly denounces the test, the applicant would have no redress. The Foundation and Mr. Randi own all the data. Mr. Randi can claim that the judges were fooled. The implicit accusation of fraud would leave the challenger devoid of remedy. These rules, be it noted, are in stark contrast to Mr. Randi's frequent public assertions that he wanted demonstrable proof of psychic powers. First, his rules are confined to a single, live applicant. No matter how potent the published evidence, how incontestable the facts or rigorous the precautions against fraud, the number, qualifications or expertise of the witnesses and investigators, the duration, thoroughness and frequency of their tests or (where statistical evaluation is possible) the astronomical odds against a chance explanation: all must be ignored. Mr. Randi thrusts every case into the bin labelled 'anecdotal' (which means not written down), and thereby believes he may safely avoid any invitation to account for them. Likewise, the production of a spanner bent by a force considerably in excess of the capacity of the strongest man, created at the request and in the presence of a group of mechanics gathered round a racing car at a pit stop by Mr. Randi's long-time enemy, Uri Geller, would run foul of the small print, which requires a certificate of a successful preliminary demonstration before troubling Mr. Randi himself. A pity, because scientists at Imperial College have tested the spanner, which its current possessor, the researcher and author Guy Lyon Playfair, not unnaturally regards as a permanent paranormal object, and there is a standing challenge to skeptics to explain its appearance. Randi's dishonest claims about dogs by Rupert Sheldrake - from hosted at Beware Pseudo-Skepticism - The Randi Challenge at The next logical step is to find out what the bonds are really worth. To do that, I e-mailed Randi at the address he provided on his website. I politely pointed out where it said the prize was in bonds in the Challenge rules, and then I asked what corporations issued the bonds, what the interest rates were, and when the maturity dates are. These are the main factors at determining if the bonds are worthless or not. Randi replied with, "Apply, or go away." I explained to him that I wanted clarification on what he was offering. That this had nothing to do with my claim, but they were questions aimed at getting more information about the Challenge. Randi replied with, "Immediately convertible into money. That's all I'm going to get involved in. Apply, or disappear." Obviously that doesn't answer my question at all. Immediately convertible into how much money? Convertible through who? The Myth of the Million Dollar Challenge at While the JREF says that "all tests are designed with the participation and approval of the applicant", this does not mean that the tests are fair scientific tests. The JREF need to protect a very large amount of money from possible "long-range shots", and as such they ask for extremely significant results before paying out - much higher than are generally accepted in scientific research (and if you don’t agree to terms, your application is rejected). As a consequence, you might well say "no wonder no serious researcher has applied for the Challenge." Interestingly, this is not the case. Dr Dick Bierman, who has a PhD in physics, informed me that he did in fact approach James Randi about the Million Dollar Challenge in late 1998 Monday, May 21, 2012 Consciousness is not an Illusion or an Epiphenomenon I have updated my web page on Skeptical Fallacies to include a section Consciousness is not an Illusion or an Epiphenomenon Skeptics will sometimes say that consciousness is an illusion or that consciousness is an epiphenomenon of the brain. An epiphenomenon is caused by some phenomenon but cannot affect the phenomenon that causes it. Saying that consciousness is an illusion or an epiphenomenon does not really explain consciousness. See the section Consciousness cannot be Explained as an Emergent Property of the Brain for an explanation of why giving a scientific name to a phenomenon is not the same as explaining it. The Wikipedia article on Epiphenomenon says, The Wikipedia article of Ephphenomenalism says: If consciousness cannot affect the brain, then consciousness may be an illusion. However, there is significant evidence that consciousness can affect the brain. One form of evidence that consciousness can influence the brain comes from the placebo effect. In certain situations, if a patient is given an inactive substance but is told that he is being given a drug, the patient will experience the effects that the drug is said to cause. One example of this occurs when a patient is given a sugar pill but told it is a pain killer. In this situation, patients report that pain is reduced and in fact studies have indicated that this effect is caused by the production of naturally occurring opioids in the brain. The Wikipedia article on the Placebo Effect says, What is significant about the placebo effect is that it requires the patient to believe they are being given a drug. With a real drug like a pain killer, the patient will experience the effects even if they don't know they are being treated with it. However, for the placebo effect to occur, the patient must be conscious of the fact that they are being treated. This shows that conscious awareness of a medical treatment can cause the brain to produce opioids. It shows that consciousness can affect the brain. Another form of evidence that consciousness can affect the brain comes from the phenomenon of self-directed neuroplasticity. Neuroplasticity refers to the ability of neurons in the brain to change their organization or grow. This can occur when someone learns a skill or recovers from an injury. Self-directed neuroplasticity occurs when neurons in the brain change their organization or grow in response to self observation of mental states. One situation where self-directed neuroplasticity occurs is meditation. During meditation, a person will observe, (ie. be conscious of) their inner state: their mental activity and the sensations in their body. This conscious attention has been found to cause changes in the brain. The article Self-Directed Neuroplasticity: A 21st-Century View of Meditation by Rick Hanson, PhD discusses this: One of the enduring changes in the brain of those who routinely meditate is that the brain becomes thicker. In other words, those who routinely meditate build synapses, synaptic networks, and layers of capillaries (the tiny blood vessels that bring metabolic supplies such as glucose or oxygen to busy regions), which an MRI shows is measurably thicker in two major regions of the brain. One is in the pre-frontal cortex, located right behind the forehead. It’s involved in the executive control of attention – of deliberately paying attention to something. This change makes sense because that’s what you're doing when you meditate or engage in a contemplative activity. The second brain area that gets bigger is a very important part called the insula. The insula tracks both the interior state of the body and the feelings of other people, which is fundamental to empathy. So, people who routinely tune into their own bodies – through some kind of mindfulness practice – make their insula thicker, which helps them become more self-aware and empathic. This is a good illustration of neuroplasticity, which is the idea that as the mind changes, the brain changes, or as Canadian psychologist Donald Hebb put it, neurons that fire together wire together. The article Mind does really matter: evidence from neuroimaging studies of emotional self-regulation, psychotherapy, and placebo effect. (Beauregard M. Prog Neurobiol. 2007 Mar;81(4):218-36. Epub 2007 Feb 9) says, The scientific evidence from the placebo effect and from self-directed neuroplasticity shows that consciousness cannot be an illusion or an epiphenomenon produced by the brain because consciousness can affect the brain. Friday, May 18, 2012 Consciousness Cannot be Explained as an Emergent Property of the Brain I have updated my web page on Skeptical Fallicies to include a section explaining why Consciousness Cannot be Explained as an Emergent Property of the Brain Some skeptics, when asked to explain how consciousness is produced by the brain will say it is an emergent property. They may say the complexity of the brain somehow causes consciousness to emerge. This is not an actual explanation, it is just a scientific sounding way to say that they cannot explain it. It creates the impression of an explanation without offering any actual explanation. An emergent property is a property that is not necessarily caused by the individual parts of a system but emerges when they are arranged in a certain fashion. For example, a wheel rolls. This is not necessarily a property of matter. Matter might be formed into a solid cube which does not roll. But when matter is arranged in a wheel, it will roll. However, merely stating something is an emergent property is not an explanation. Saying consciousness is an emergent property of the brain does not explain consciousness. When you examine a wheel you can understand why it will roll. The laws of physics explain how the ability to roll is caused by a particular arrangement of matter. When you examine a brain you cannot tell how it produces the subjective experiences of consciousness. Physics cannot explain how the subjective experiences of consciousness, what it is like to feel happy, or what it is like to see blue, or what it is like to feel pain, will arise from a particular arrangements of neurons in the brain. When skeptics say consciousness is an emergent property of the brain, that is not an explanation of consciousness. It is a rhetorical trick used because they cannot explain how consciousness is produced by the brain. They are only applying a scientific sounding name to fool people, including themselves, into thinking it is an explanation. Thursday, May 17, 2012 Another Cause of Skepticism Wednesday, May 16, 2012 Old Nordic Mediums I've previously blogged about photographs of ectoplasm produced by the physical medium Jack Webber here and here. Here are some interesting Photographs from physical seances demonstrating levitation. An English translation is here. More photos by the same photographer, Sven Türck, are here. The seances were conducted by Nordic Mediums. Tuesday, May 15, 2012 Why Darwinism is False Last week I posted about why belief in the afterlife is not compatible with the belief that natural selection is solely responsible for the diversity of life (Darwinism). Here is an article that discusses some of the other flaws Darwinism. It is a criticism of the book Why Evolution is True by Jerry Coyne. Why Darwinism Is False by Jonathan Wells Darwin called The Origin of Species “one long argument” for his theory, but Jerry Coyne has given us one long bluff. Why Evolution Is True tries to defend Darwinian evolution by rearranging the fossil record; by misrepresenting the development of vertebrate embryos; by ignoring evidence for the functionality of allegedly vestigial organs and non-coding DNA, then propping up Darwinism with theological arguments about “bad design;” by attributing some biogeographical patterns to convergence due to the supposedly “well-known” processes of natural selection and speciation; and then exaggerating the evidence for selection and speciation to make it seem as though they could accomplish what Darwinism requires of them. Faced with such evidence, any other scientific theory would probably have been abandoned long ago. Judged by the normal criteria of empirical science, Darwinism is false. Its persists in spite of the evidence, and the eagerness of Darwin and his followers to defend it with theological arguments about creation and design suggests that its persistence has nothing to do with science at all.50 The article gives detailed explanations in support of these statements. I don't believe in Creationism. I don't believe Intelligent Design should be taught in schools. However, I do believe there are serious flaws in Darwinism and those should be taught in schools. I also believe scientists should be free to look for empirical and theoretical evidence of Intelligent Design without being persecuted or ostracized. A major source of problems with Darwinism comes from the fact that it is based on belief in methodological naturalism, the philosophy that only natural phenomena should be studied by science. This is a wrong view. Science should be about uncovering the truth and should not have any built in philosophical bias. However, many mainstream scientists, because of their bias towards methodological naturalism, can't objectively assess the evidence for and against Darwinism. Every bit of evidence is interpreted to be consistent with Darwinism in exactly the same Creationists interpret every bit of evidence to agree with the Bible. This is one reason Darwinists are so hostile to their critics. Their philosophical beliefs are threatened by criticism of Darwinism. This philosophical bias in favor of methodological naturalism is similar in the effects of reductionism on scientific thinking. It so strongly influences the thinking of scientists that they cannot conceive of possibilities outside of their preconceived ideas. This cripples their ability to understand consciousness and psychic phenomena. Monday, May 14, 2012 I have updated my web site to include a section on philosophical arguments that the mind is not produced by the brain. There are very good philosophical reasons to believe the mind is not produced by the brain and therefore the mind is non-physical. Peter Williams discusses several reasons for this in his article: Why Naturalists Should Mind about Physicalism, and Vice Versa, (Quodlibet Journal: Volume 4 Number 2-3, Summer 2002.) Williams explains that if the mind and the brain were the same, then all the properties of the mind would be properties of the brain. He then demonstrates that the mind cannot be identical to the brain by giving several examples of properties of the mind that are not properties of the brain. Gary R. Habermas and J.P.Moreland argue against physicalism from the ‘qualia’ of imagined sensory images. Qualia is the subjective feel or texture of conscious experience: "Picture a pink elephant in your mind. Now close your eyes and look at the image. In your mind, you will see a pink property. . . There will be no pink elephant outside you, but there will be a pink image of one in your mind. However, there will be no pink entity in your brain; no neurophysiologist could open your brain and see a pink entity while you are having the sense image. The sensory event has a property – pink – that no brain event has. Therefore, they cannot be identical." [19] To put this another way, the subjective feel of mental experiences such as the feeling of pain, the hearing of sound or the taste of chocolate seems very different from anything that is purely physical: "If the world were only made of matter, these subjective aspects of consciousness would not exist. But they do exist! So there must be more to the world than matter." [20] Williams gives several more examples. These include intentionality, the ability to reason, free will, and moral responsibility. See the linked article for an explanation of why these phenomena demonstrate that the mind cannot be made of matter. Williams concludes: At the very least, the mind has several immaterial properties ... It follows that no merely physical explanation of the mind is possible. More information, including links to further reading, can be found on my web site. Friday, May 11, 2012 Retrocausality demonstrated with quantum entanglement. The article is here. Thursday, May 10, 2012 I believe in ESP because I have seen psychic miracles day after day in our government-sponsored investigations. It is clear to me, without any doubt, that many people can learn to look into the distance and into the future with great accuracy and reliability. This is what I call unobstructed awareness or remote viewing (RV). To varying degrees, we all have this spacious ability. There are presently four classes of published and carefully examined ESP experiments that are independently significant, with a probability of chance occurrence of less than one time in a million... The full article is here. Wednesday, May 9, 2012 Karl "Falsifiability" Popper believed the soul was nonmaterial. Many skeptics say theories that contradict materialism are unscientific because those theories are not falsifiable. For a theory to be scientific, it must be testable. For a theory to be testable, it must be falsifiable: there must be a situation, where if the theory is wrong, you can demonstrate it is wrong. For example, you can test the theory of gravity by measuring how objects fall. If they don't accelerate the way the theory of gravity predicts they should, then the theory is wrong, it is falsified. If objects do fall the way the theory predicts, then the theory is right. Skeptics often say belief in spirits or psi is unscientific because any unexplained phenomena can be said to be caused by a spirit or by psi and there is no way to disprove it. What may be surprising to many skeptics is that Karl Popper, who first proposed that falsifibility is necessary for a theory to be scientific, did not believe in materialism. He believed in dualism which holds that the mind is nonmaterial. The wikipedia article on Karl Popper explains falsifiability: The wikipedia article on Philosophy of Mind describes Popper as a defender of interactionist dualism espoused by Descartes. The wikipedia article on Rene Descartes explains that dualism as espoused by Descartes holds that the soul is nonmaterial and does not follow the laws of nature. Is belief in psi or spirits unscientific? It depends. It depends on what those beliefs are theorized as an explanation of. For example, if you theorize that spirits are an explanation of mediumship, that can be tested. If a medium is communicating with a spirit, the medium should be able to obtain information about the spirit that medium could not otherwise know. If the medium could not obtain any information about the spirit such as their appearance, their personality traits, the things they did in life etc., then the theory that the medium is communicating with a spirit would not pass the test. There is, in fact, a lot of evidence that mediums do communicate with spirits. The medium Mrs Piper passed many such tests. Tuesday, May 8, 2012 Consciousness is not Produced by the Brain I have updated the description of the filter model of the brain on my web site: There is no doubt that the brain and the conscious mind interact. Brain damage can cause loss of some functions of consciousness. Amnesia after a head injury or poor memory due to aging are two examples. Neurological activity can be measured and shown to be associated with mental activity. Nerve impulses from sensory organs result in brain activity, and the conscious mind has awareness of the sensations perceived. When the mind generates the impulse to move, nerve impulses are carried from the brain to the muscles to cause movement. Consciousness is affected by brain activity and it is able to influence brain activity. However this is only a correlation, it is not proof that neurological activity causes consciousness. The correlation between consciousness and brain activity should also exist if the brain is an interface between a nonphysical mind and the physical body. One way to think of this is that the brain is like a filter of consciousness. This is called the filter model of the brain. In the filter model, consciousness is a nonphysical phenomena and the brain filters consciousness while we are incarnated in our physical bodies. The brain could filter some aspects of consciousness the way a colored glass can filter out some wavelengths of light. What passes through the brain filter is a restricted set of conscious faculties that we have while in the physical body. The filter model is superior to the hypothesis that the brain produces consciousness because the filter model explains more evidence. You can damage a filter in two ways. You can clog it or you can punch a hole in it. When brain damage causes loss of function like amnesia, that is like a clog in the filter. When brain injury results in increased function, that is like a hole punched in the filter. An example of increased function is when people have increased psychic abilities after a brain injury. In the filter model one of the functions of the brain is to restrict consciousness. In that case, if you release the conscious mind from the brain as happens during a near death experience you should have expanded, unfiltered, consciousness. This is exactly what happens during a near death experience. People who have NDE's are able to perceive more than they do when in the body. They report seeing in 360 degrees and seeing colors that they do not see when in the body. Blind people report seeing during NDE's. Some near death experiencers report being able to communicate telepathically with other beings. Some report understanding that time is just an illusion or that they seem to have access to all the knowledge in the universe. More at my web site. Monday, May 7, 2012 No, it isn't. There are three reasons. Friday, May 4, 2012 Evidence for the Afterlife from Quantum Mechanics I added a section about the evidence for the afterlife that comes from quantum mechanics to my web site: When physicists study matter at the atomic level, they find that the properties of matter are not determined until a conscious being percieves that matter. This demonstrates that physical matter depends on consciousness for its existence and therefore consciousness cannot arise from matter. The brain, which is composed of matter, cannot produce consciousness so consciousness must have an existence independent from matter. While this interpretation of quantum mechanics in not universally held by all physicists, some of the original founders of quantum mechanics, including Nobel Prize winners in physics such as Max Planck and Erwin Schrödinger, believed this. More here. Thursday, May 3, 2012 Medium Jack Webber In a previous post, Ectoplasm and Materialization, I linked to photographs of Jack Webber producing ectoplasm during a seance. Since that time I had a chance, in the comment section of Michael Prescott's blog, to ask Zerdini his opinion of their authenticity. His reply was very informative ... The response:: Yes they were: Leon Isaacs, who took the photographs at Webber’s circles, used two cameras placed at different angles…shots using this two-camera technique showed the disposition of trumpets and other objects, establishing that they were not held aloft by any material agency. Isaac’s pictures were taken by flashlight, the source of the light being screened by an infrared filter which suppressed practically all visible light rays and only permitted infrared emanations to pass. In effect there was a brief glow at the instant of exposure which had no harmful effects on the medium. Many of the photos of Jack Webber were taken by a 'Daily Mirror' photographer. Harry Edwards can be seen in some of the photographs as one of the sitters. THE following report occupied the best part of the two centre pages of the Daily Mirror on February 28th, 1939. "Cassandra" is the pen-name of a gentleman on the staff of the Daily Mirror who writes a daily pertinent review on matters in general. He is well known for his cryptic and biting sarcasm, and has, on numbers of occasions, given full vent to his opposition to spiritualism. The séance in question was held in North London at a place to which the medium had never been before, and the people present were complete strangers. Mr. Leon Isaacs had been asked to take infra-red photographs. The problem arose as to the means of transporting the equipment, and since "Cassandra" had a car, he was asked to help this way. Thus the only reason why "Cassandra" was present was because he possessed a car. The article was illustrated by a photograph (Plate No. 20), with the following description beneath it "The medium in a trance, lashed to the chair, while a table leaves the ground and books fly through the air ... a photograph taken during the séance attended by 'Cassandra.' The heading was "Cassandra got a surprise at Séance," and his report, in his caustic manner, reads as follows: "I claim I can bring as much scepticism to bear on spiritualism as any newspaper writer living, and that's a powerful load of scepticism these days. I haven't got an open mind on the subject--I'm a violent, prejudiced unbeliever with a limitless ability to leer at the unknown. At least, I was till last Saturday. And then I got a swift, sharp, ugly jolt that shook most of my pet sneers right out of their sockets. "Picture to yourself a small room in a typical suburban house. In one corner a radio-gramophone. In the centre a ring of chairs. At the far end an armchair." "About a dozen people filed in and sat in the circle. I hope they won't mind my saying it, but they struck me as a credulous collection that would have brought tears of joy to a sharepusher's eyes." "Almost everyone a genuine customer for a lovely phony gold brick." "They sat down and the medium, a young Welsh ex-miner, was then roped to the arm-chair. The photographer and I stood outside the circle. The lights went out and we sailed rapidly into the unknown." "The medium gurgled like water running out of a bath, and we opened up with a strangled prayer." "The circle of believers answered with 'All Hail the Power of Jesu's Name,' and I was told that we were 'on the brink.' I thought we were in Cockfosters, Herts, but I soon began to doubt it when trumpets sprayed with luminous paint shot round the room like fishes in a tank. They hovered like pike in a stream, and then swam slowly about. "The medium snored and struggled for breath." "Hymns, Trumpets" "Somebody put a record on, and we were soon bellowing 'Daisy, Daisy, give me your answer, do.' The trumpets beat time and hurled themselves against the ceiling." "A bell rang." "There was considerable excited laughter, and in a slight hysteria we sang 'There is a green hill far away,' followed by the profane, secular virility of 'John Brown's body.' "A tambourine with 'God is Love' written on it became highly unreasonable, and flew up noisily round our heads. "The rough stertorous breathing of the medium continued, and a faint tapping sound heralded a voice speaking from one of the trumpets that was well adrift from its moorings. A faint, childish voice said in a voice of deep melancholy that it was 'Very, very happy.' More voices spoke." "Water was splashed about (there was none in the room when we started) and books took off from their shelves." "Table moved." "The medium remained lashed to his chair." "A clockwork train ran across the floor." "Suddenly a heavy table slowly left the ground. The man who was sitting next to it said calmly 'The table's gone !' The photographer released his flash-you see the result on the right." "At no time did the medium move from his chair. I swear it." "The table landed with a thump in the middle of the circle. A book that was on it remained in position." "I'll pledge my word that not a soul in the room touched it. It was so heavy that it needed quite a husky fellow to lift it. I felt the weight of it afterward." "What price cynicism ? What price heresy?" "Don't ask me what it all means, but you can't tell me now that these strange and rather terrifying things don't happen." "I was there. I saw them. I went to scoff." "But the laugh is sliding slowly round to the other side of my face." (Signed) 'CASSANDRA.' And this: THE séance reported took place on May 24th, 1939, and occupied two pages of the Sunday Pictorial dated May 28th, 1939. Mr. Gray prefaced his report with an affidavit as follows: "I, BERNARD GRAY, of 27, Barn Rise, Wembley Park, in the County of Middlesex, journalist, make Oath and say as follows - "1. That my description of the incidents enumerated in the Article written by me hereunto annexed and marked 'B.G.' to appear in the issue of the Sunday Pictorial of the Twenty-eighth day of May One thousand nine hundred and thirty-nine under the heading of' I Swear I Saw This Happen' is true. "2. I further make Oath and say that the incidents so described in such Article did occur in my presence." This oath was sworn before a solicitor yesterday. I bound him to his chair, hand and foot, with knots and double knots which a sailor once taught me. Just to make sure he couldn't wriggle out and back without my knowing it, I tied lengths of household cotton from the ropes to the chair legs. And I sewed up the front of his jacket with stout thread. So began my second investigation into the mysteries of Spiritualism. The man I had trussed up was Jack Webber, formerly a Welsh miner. He's now a medium - a man for whom such remarkable claims are made that I selected him for my first test. Through him, I was told, are performed some of the most astonishing miracles of spirit power, physical demonstrations intended to prove the reality of life after death. And in this, my second adventure into Spiritualism during my association with the Sunday Pictorial, I want physical phenomena. Startling deeds, not words, as proof. Not testimonies of people claiming to be healed, not messages from the dead. Just material facts which a materially minded man like me can grasp. I want final and complete conviction. That is more important to me than Hitler, the Axis, or even the threat of war. And that is why I have asked the Editor to allow me-for a while-to leave politics, and go in search of Truth. So we sat, fourteen of us, a cheerful, talkative group of very ordinary people, in a plainly furnished room at Balham, London. There was a Metropolitan policeman. A consulting engineer. A waiter. A postman. A foreman plumber. Several women of various ages. And next to me, between the medium and me, Mr. Harry Edwards, leader of the Balham Psychic Society, by trade a printer. We all held hands loosely, Mr. Webber settled himself back as comfortably as my knots would allow, and out went the light, leaving only a red bulb gleaming dully through the darkness from the middle of the room. Things began to happen immediately. They went on happening with remarkable rapidity, with startling variety, for ninety minutes. But I do not want to recount them in order. For I want to describe first two astonishing happenings which make the rest seem small in contrast. Happenings which I, personally, can only compare with the miracles of the New Testament. I am sitting, remember, only one removed from the medium. An hour of the séance has gone by. The early tenseness, the trace of excitement, which perhaps affected me at the start has disappeared. I am my normal, cool, and vigilant self-alert for any sign of deception, accustomed to the eerie glimmer of light we get from the red bulb near the ceiling. In the corner, so near I can touch him, the medium is breathing heavily, gulping occasionally, moaning uneasily at times, like a man with a nightmare. Suddenly, he gurgles alarmingly, as if making some still greater effort. Before me rises a kind of tablet, rather like a slate, and from the upper surface it sheds a luminous white light. I watch it intently, not in the least perturbed. I saw it in its normal state before the séance started. An ordinary piece of four-ply wood, about a foot long and nine inches wide. Now it hovers in front of the medium's face, its soft radiance lighting his features so clearly I can see the closed eyes and the twitching lips. It moves gently down to his hands and I see quite clearly that the arms are still bound to the chair. The glowing tablet has moved over to me. It hangs motionless so close to my face I feel that if I breathe hard I shall blow it away. "Watch!" says Mr. Edwards, giving my hand a squeeze. Then above the tablet I begin to see something white emerging from the darkness. Almost invisible at first, it grows stronger every moment, like a motor car headlamp advancing through fog ; until I can clearly see it as a diaphanous ellipse, standing on its end, as it were, on the tablet. "Ectoplasm," says Mr. Edwards. "Watch closely in the centre of it !" No need to tell me. My eyes are glued on it, though, I want to emphasize, I'm still cool and unemotional. Now, framed in this luminous halo, I can perceive dimly what appear to be features. They are becoming clearer, easier to trace. There's the nose, and - yes the mouth. The eyes, and, my God ! The eyelids are moving. The tablet moves still closer. The eyes, soft and natural, are looking directly into mine. I jerk myself back to a detached, inquisitive state of mind, examine the thing in front of me closely and searchingly. It's not like the pictures of spirit faces many of us have seen in Spiritualist papers. It's not white and unearthly, like the frame in which it is set. RATHER IS IT A HUMAN FACE - BUT SOFTER, FINER, AND SOMEHOW DIFFERENT. I can trace the cheek-bones fading back from the eyes. The lips, they are quite clear. The chin, rounded and delicate, is silhouetted against the lower rim of the halo. I recognize it suddenly as the face of a very old lady. Just like a lovely miniature - for it is much smaller, now I come to think, than the face of any human adult. "Try and speak to us," says Mr. Edwards, encouragingly. I am watching the lips. They part a little, move with an effort. There's a whisper. What is she saying ? Who is she speaking to ? Yes - I've got it. "Who's she speaking to ?" I ask, without taking my eyes off the face for a second. "You," replies Edwards."Speak to her!" "Who are you ?" I ask, gently. "I am--," she answers, and whispers a name I shall not repeat - it is personal. "I cannot stay," she goes on. "I just want you all to see me. God bless you, my boy ..." The tablet and its burden move away. I can see it floating around our circle. Other sitters are exclaiming that they can see it, quite plainly, that it's wonderful. The tablet returns to me. The features in the miniature are fading, like outlines yielding to the dusk of a summer evening. Now the halo is going too. Only the tablet is left. Its gleam disappears with the suddenness of a light being extinguished. The tablet falls with a clatter at my feet. "Lights on," says a voice instantly. There's the click of a switch. In less than five seconds the whole room is bathed in electric light. Everybody is in his or her place, holding hands. The medium is bound just the same in his chair unconscious in his trance. The deep voice which comes from the medium's corner - they call it the voice of Black Cloud, Webber's Indian spirit "guide" says: "I want the gentleman sitting next to Mr. Edwards to hold the medium's right hand. I want the lady on the left of the medium to hold his left hand." Edwards guides my hand over his knees to the hand of the medium. I feel my fingers seized in a powerful grasp. The pressure tightens till it hurts. I set my teeth and wait. The medium is moaning like a man in pain. I can feel a soft fabric rubbing against my wrist. "Can you feel his coat ?" asks the deep voice in the corner. "I can feel some kind of material on my wrist," I answer, readily. "I am dematerializing his coat and taking it off." Now the coat is rubbing the other side of my wrist. Something drops to the floor with a light, rustling impact. Simultaneously, it seems, somebody presses the switch. The medium is in his shirt-sleeves. He is no longer wearing his coat. Round his arms, over his shirt now, are the ropes, still fastened by my patent knots. The thin strands of cotton from the ropes to the chair are unbroken. On the floor, the medium's jacket. Not a stitch holding the edges together broken. My twisted thread round the button just as I had left it. "That is merely intended to prove to you that the spirit world exists and has power to dematerialize," says the deep voice in the corner, when the lights are off again. "Later I hope to replace the medium's coat." Half an hour later the lady on the other side and I are asked to hold Webber's hands a second time. Again the grip is firm enough to be painful. A rustling. Cloth rubbing against my wrist again. Yes, and now the other side. Lights. Webber is wearing his coat once more. Over and round each arm, the bonds. The cotton intact. The thread just as before. BUT THE BONDS AND THE COTTON ARE OVER THE COAT. "My hand was gripped by his all the time," says the girl across from me, rubbing her fingers. "And I felt the coat go through my wrist. Didn't you ?" Well, those two happenings, or miracles - call them what you like, take a bit of explaining away. There were other things too. Heaps of them. "I can feel a hand on my head," said Mr. Edwards, casually, just as if it were quite a natural thing for a hand to emerge from nowhere. "I can feel something on my head," I said a moment later, and gripped Edwards's hand more tightly to make sure it hadn't been raised. Something was pulling my hair pretty hard. I realized then with a sense of shock that the "something" was definitely fingers, yet rather different from human fingers. They felt sharper, more like claws, seemed almost metallic at the tips. My neighbour chuckled. "I know what they're doing," he said, highly amused. The fingers pulled me firmly by the hair in Edwards's direction, till my head was touching his. My hair was pulled and twisted about for fully a minute. "We're being tied together," said my neighbour, laughing. "Can't you feel your hair being twisted with mine ?" We were tied together, too ! We couldn't separate, and the séance was held up for a moment or two while the lights were put on so that we could be unraveled. "A mischievous trick," said everybody else, laughing at our plight. Mischievous, all right. Inexplicable, too. I'll swear nobody moved before, during, or after the knots were tied in our hair. Frequently throughout the proceedings the luminous trumpets were shooting about the room three at a time, with the speed and accuracy of swallows in flight. "I should like to be absolutely sure nobody is holding them," I said boldly, though I myself considered it impossible. One of the trumpets shot straight at my head with the speed of an express train, pulled up sharp just as it touched my temple, and I cringed expecting a knock-out blow. That tin cone proceeded to run itself on my face and round my head, pressing first the broad end, then the narrow end, against my lips to prove it had no earthly connection at any point on its surface. A bell which I'd seen on a table in a corner rose into the air and rang a rhythmic accompaniment to our singing. A pair of clappers, similar to those used by a dance band drummer, floated about clacking merrily in time with the music. In a powerful bass voice, which has been recorded on gramophone discs, "Reuben" led some of the singing. Toys in the room, illuminated by a strange incandescent glow, leapt from the table and sailed about near the ceiling. A boy, I was told, plays with the toys - a boy who died some years ago. As something moved off the table and began to dart about the room, Mr. Edwards explained that it was a doll. Whatever it was, it settled on my knee, and frolicked up and down my leg. I could feel it as well as see it glowing, like an outsize glowworm. It came to rest finally on my knee. And when the lights came on, I found that it was indeed a toy elephant, such as any child would use in play. You see, therefore, it wasn't a gloomy gathering by any means. The strange pranks with the toys - a clockwork engine wound itself up and ran itself down near the ceiling - distinctly enlivened the proceedings. All these little things, however, paled into insignificance beside a remarkable demonstration of furniture removing by unseen hands. I saw it in passage, because it was outlined against the red light. And of course there were spirit messages for some of the sitters. I do not want to write about them. In this series of articles I am concerned more with incidents. Well, that is my testimony. I cannot explain anything I saw. And although many of my friends will think I've gone crazy - I say again : I SAW IT HAPPEN. Wednesday, May 2, 2012 David Bohm one of the top theoretical quantum physicists believed in parapsychology. I have updated the Eminent Researchers page on my web site to include David Bohm. David Bohm was one of the best quantum physicists of all time and one of the most significant theoretical physicists of the 20th century (wikipedia). David Joseph Bohm FRS[1] (20 December 1917 – 27 October 1992) was an American-born British quantum physicist who contributed to theoretical physics, philosophy of mind, neuropsychology. David Bohm is widely considered to be one of the most significant theoretical physicists of the 20th century.[2] In the article, David Bohm and Jiddo Krishnamurti which appeared in the Skeptical Inquirer, July 2000, Martin Garner wrote that Bohm was favorably impressed with parapsychology including Rupert Sheldrake's morphogenetic fields. Bohm took Uri Geller's psychic phenomena seriously and carried with him a key bent by Geller. Bohm believed in panpsychicsm, in one interview he said, "Even the electron is informed with a certain level of mind," Tuesday, May 1, 2012 Erwin Schrödinger (Nobel Prize in Physics) believed consciousness was not prodced by the brain and could not be explained in physical terms. I have updated the Eminent Researchers page on my web site to include Erwin Schrödinger. Erwin Schrödinger received the Nobel Prize in Physics in 1933. He believed consciousness was not produced by the brain and could not be explained in physical terms. Erwin Rudolf Josef Alexander Schrödinger ( 12 August 1887 – 4 January 1961) was an Austrian born physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933. In 1935 he proposed the Schrödinger's cat thought experiment.[2]ödinger Schrödinger wrote: Other quotes by Schrödinger: The observing mind is not a physical system, it cannot interact with any physical system. And it might be better to reserve the term "subject" for the observing mind. ... For the subject, if anything, is the thing that senses and thinks. Sensations and thoughts do not belong to the "world of energy."
df571b7ddd9ce7a9	Tuesday, November 28, 2006 Chemistry (from Greek χημεία khemeia[1] meaning "alchemy") is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms, such as molecules, crystals, and metals. Chemistry deals with the composition and statistical properties of such structures, as well as their transformations and interactions to become materials encountered in everyday life. Chemistry also deals with understanding the properties and interactions of individual atoms with the purpose of applying that knowledge at the macroscopic level. According to modern chemistry, the physical properties of materials are generally determined by their structure at the atomic scale, which is itself defined by interatomic forces. Chemistry is often called the "central science" because it connects other sciences, such as physics, material science, nanotechnology, biology, pharmacy, medicine, bioinformatics, and geology.[2] These connections are formed through various sub-disciplines that utilize concepts from multiple scientific disciplines. For example, physical chemistry involves applying the principles of physics to materials at the atomic and molecular level. Chemistry pertains to the interactions of matter. These interactions may be between two material substances or between matter and energy, especially in conjunction with the First Law of Thermodynamics. Traditional chemistry involves interactions between substances in chemical reactions, where one or more substances become one or more other substances. Sometimes these reactions are driven by energetic (enthalpic) considerations, such as when two highly energetic substances such as elemental hydrogen and oxygen react to form the less energetic substance water. Chemical reactions may be facilitated by a catalyst, which is generally another chemical substance present within the reaction media but unconsumed (such as sulfuric acid catalyzing the electrolysis of water) or a non-material phenomenon (such as electromagnetic radiation in photochemical reactions). Traditional chemistry also deals with the analysis of chemicals both in and apart from a reaction, as in spectroscopy. All ordinary matter consists of atoms or the subatomic components that make up atoms; protons, electrons and neutrons. Atoms may be combined to produce more complex forms of matter such as ions, molecules or crystals. The structure of the world we commonly experience and the properties of the matter we commonly interact with are determined by properties of chemical substances and their interactions. Steel is harder than iron because its atoms are bound together in a more rigid crystalline lattice. Wood burns or undergoes rapid oxidation because it can react spontaneously with oxygen in a chemical reaction above a certain temperature. Substances tend to be classified in terms of their energy or phase as well as their chemical compositions. The three phases of matter at low energy are Solid, Liquid and Gas. Solids have fixed structures at room temperature which can resist gravity and other weak forces attempting to rearrange them, due to their tight bonds. Liquids have limited bonds, with no structure and flow with gravity. Gases have no bonds and act as free particles. Another way to view the three phases is by volume and shape: roughly speaking, solids have fixed volume and shape, liquids have fixed volume but no fixed shape, and gases have neither fixed volume nor fixed shape. Water (H2O) is a liquid at room temperature because its molecules are bound by intermolecular forces called Hydrogen bonds. Hydrogen sulfide (H2S) on the other hand is a gas at room temperature and standard pressure, as its molecules are bound by weaker dipole-dipole interactions. The hydrogen bonds in water have enough energy to keep the water molecules from separating from each other but not from sliding around, making it a liquid at temperatures between 0 °C and 100 °C at sea level. Lowering the temperature or energy further, allows for a tighter organization to form, creating a solid, and releasing energy. Increasing the energy (see heat of fusion) will melt the ice although the temperature will not change until all the ice is melted. Increasing the temperature of the water will eventually cause boiling (see heat of vaporization) when there is enough energy to overcome the polar attractions between individual water molecules (100 °C at 1 atmosphere of pressure), allowing the H2O molecules to disperse enough to be a gas. Note that in each case there is energy required to overcome the intermolecular attractions and thus allow the molecules to move away from each other. Scientists who study chemistry are known as chemists. Most chemists specialize in one or more sub-disciplines. The chemistry taught at the high school or early college level is often called "general chemistry" and is intended to be an introduction to a wide variety of fundamental concepts and to give the student the tools to continue on to more advanced subjects. Many concepts presented at this level are often incomplete and technically inaccurate, yet they are of extraordinary utility. Chemists regularly use these simple, elegant tools and explanations in their work because they have been proven to accurately model a very wide array of chemical reactivity, are generally sufficient, and more precise solutions may be prohibitively difficult to obtain. History of chemistry The roots of chemistry can be traced to the phenomenon of burning. Fire was a mystical force that transformed one substance into another and thus was of primary interest to mankind. It was fire that led to the discovery of iron and glass. After gold was discovered and became a precious metal, many people were interested to find a method that could convert other substances into gold. This led to the protoscience called Alchemy. Alchemy was practiced by many cultures throughout history and often contained a mixture of philosophy, mysticism, and protoscience (see Alchemy). Alchemists discovered many chemical processes that led to the development of modern chemistry. As history progressed the more notable alchemists (esp. Geber and Paracelsus) evolved alchemy away from philosophy and mysticism and developed more systematic and scientific approaches. The first alchemist considered to apply the scientific method to alchemy and to distinguish chemistry from alchemy was Robert Boyle (1627–1691); however, chemistry as we know it today was invented by Antoine Lavoisier with his law of Conservation of mass in 1783. The discoveries of the chemical elements has a long history culminating in the creation of the periodic table of the chemical elements by Dmitri Mendeleyev. The Nobel Prize in Chemistry created in 1901 gives an excellent overview of chemical discovery in the past 100 years. In the early part of the 20th century the subatomic nature of atoms were revealed and the science of quantum mechanics began to explain the physical nature of the chemical bond. By the mid 20th century chemistry had developed to the point of being able to understand and predict aspects of biology spawning the field of biochemistry. • Physical chemistry is the study of the physical and fundamental basis of chemical systems and processes. In particular, the energetics and dynamics of such systems and processes are of interest to physical chemists. Important areas of study include chemical thermodynamics, chemical kinetics, electrochemistry, statistical mechanics, and spectroscopy. Physical chemistry has large overlap with molecular physics. Physical chemistry involves the use of calculus in deriving equations. It is usually associated with quantum chemistry and theoretical chemistry. • Theoretical chemistry is the study of chemistry via fundamental theoretical reasoning (usually within mathematics or physics). In particular the application of quantum mechanics to chemistry is called quantum chemistry. Since the end of the Second World War, the development of computers has allowed a systematic development of computational chemistry, which is the art of developing and applying computer programs for solving chemical problems. Theoretical chemistry has large overlap with (theoretical and experimental) condensed matter physics and molecular physics. Essentially from reductionism theoretical chemistry is just physics, just like fundamental biology is just chemistry and physics. Other fields include Astrochemistry, Atmospheric chemistry, Chemical Engineering, Chemo-informatics, Electrochemistry, Environmental chemistry, Flow chemistry, Geochemistry, Green chemistry, History of chemistry, Materials science, Medicinal chemistry, Molecular Biology, Molecular genetics, Nanotechnology, Organometallic chemistry, Petrochemistry, Pharmacology, Photochemistry, Phytochemistry, Polymer chemistry, Solid-state chemistry, Sonochemistry, Supramolecular chemistry, Surface chemistry, and Thermochemistry. Fundamental concepts The most convenient presentation of the chemical elements is in the periodic table of the chemical elements, which groups elements by atomic number. Due to its ingenious arrangement, groups, or columns, and periods, or rows, of elements in the table either share several chemical properties, or follow a certain trend in characteristics such as atomic radius, electronegativity, electron affinity, and etc. Lists of the elements by name, by symbol, and by atomic number are also available. In addition, several isotopes of an element may exist. An ion is a charged species, or an atom or a molecule that has lost or gained one or more electrons. Positively charged cations (e.g. sodium cation Na+) and negatively charged anions (e.g. chloride Cl−) can form neutral salts (e.g. sodium chloride NaCl). Examples of polyatomic ions that do not split up during acid-base reactions are hydroxide (OH−) and phosphate (PO43−). Chemical bond States of matter Chemical reactions A Chemical reaction is a process that results in the interconversion of chemical substances. Such reactions can result in molecules attaching to each other to form larger molecules, molecules breaking apart to form two or more smaller molecules, or rearrangement of atoms within or across molecules. Chemical reactions usually involve the making or breaking of chemical bonds. For example, substances that react with oxygen to produce other substances are said to undergo oxidation; similarly a group of substances called acids or alkalis can react with one another to neutralize each other's effect, a phenomenon known as neutralization. Substances can also be dissociated or synthesized from other substances by various different chemical processes. Quantum chemistry Solutions of the Schrödinger equation for the hydrogen atom gives the form of the wave function for atomic orbitals, and the relative energy of say the 1s,2s,2p and 3s orbitals. The orbital approximation can be used to understand the other atoms e.g. helium, lithium and carbon. Chemical Laws Interpersonal chemistry In the fields of sociology, behavioral psychology, and evolutionary psychology, with specific reference to intimate relationships or romantic relationships, interpersonal chemistry is a reaction between two people or the spontaneous reaction of two people to each other, especially a mutual sense of attraction or understanding.[4] In a colloquial sense, it is often intuited that people can have either good chemistry or bad chemistry together. Other related terms are team chemistry, a phrase often used in sports, and business chemistry, as between two companies.[5] Recent developments in neurochemistry have begun to shed light on the nature of the "chemistry of love", in terms of measurable changes neurotransmitters such as oxytocin, serotonin, and dopamine. Saturday, November 25, 2006 INTRODUCTION: The fungal functional type is comprised of sessile heterotrophs with cell walls. Rather than ingesting food as animals do, fungal organisms absorb food across the cell wall. The assemblage of organisms termed fungi are classified into two general categories. First, are the true fungi (Kingdom Fungi) which evolved from motile, aquatic protozoa that are also ancestors to the animal kingdom. True fungi first evolved as the chytrids (Phylum Chytridiomycota) who produce an enlarged globular cell from which numerous filaments grow into the food source. Chytrids produce motile spores and gametes, and the vegetative cells are ceonocytic (many nuclei float around in one big cell). Chytrids gave rise to the Zygomycetes (Phylum Zygomycota), which produce no motile cells but form ceonocytic hyphae. From the Zygomycetes, the advanced fungi arose and in time formed the Phylum Dikaryomycota. Organisms in the Dikaryomycota produce hyphae comprised of individual nucleated cells separated by walls (septate hyphae) with each cell having two haploid nuclei in what is called an N + N configuration. The two main groups of dikaryotic fungi are the ascomycetes (sac-forming fungi) and the basidiomycetes (club-forming fungi). The majority of fungi affecting humans are ascomycetes and basidiomycetes. The second category of fungal organisms is the pseudofungi, made up of various unrelated protista groups. The pseudofungi were formerly classified into the catch-all kingdom Protista but have recently been reclassified into more-specific kingdoms that reflect genetic relationships. Important pseudofungi are the Oomycetes (egg fungi and water molds), and the slime molds. Oomycetes are closely related to the stramenopilous algae - the brown algae, golden-brown algae and diatoms of the Kingdom Stramenopila. The close relationship between the oomycetes and the brown algae is evident in that both have cellulose walls, and they share the same type of flagella. Oomycetes descend from algae that lost their chloroplasts, and hence have adopted a heterotrophic life form. Oomycetes also produce filamentous hyphae to better absorb nutrients from the food source. Because they have no relationship with the chytrids, it is clear that the oomycetes and chytridiomycetes independently evolved the mycelial life form. Slime molds evolved from various ancient protozoa and have little genetic affinity to other fungi or algae groups. They are animal-like in that they ingest food early in the life-cycle, but are fungal-like in that they produce walled sporangia and spores. In today’s lab, you will have a chance to examine the diversity of both false and true fungi. The lab will begin with the false fungi and then follow an evolutionary sequence through the true fungi. Many specimens illustrate important reproductive phases in the life-cycles of these organisms. You should examine these carefully because reproductive features are important to understanding and distinguishing the various groups of fungi. Although not required, we urge you to draw and label the specimens you examine. Our experience has been that drawings with appropriate labels are the best way to learn the features emphasized in lab and lecture. Drawings are also an excellent study tool to refresh your memory just prior to the exam A) Slime molds (Kingdoms Myxomycota and Dictyosteliomycota): Slime molds are largely saprophytic and are typically found on decaying wood in moist forests. During the vegetative phase of the life cycle (see figure 16-6 on page 353 of your text), they begin life as independent amoebae, ingesting microscopic bits of organic debris. The free-living amoebae eventually swarm together to produce a multicellular blob called a plasmodium. After a while, the plasmodium forms cellulose walls around the nuclei and produces sporangia (or fructifications). The sporangia release large numbers of air-borne spores which germinate in the presence of water to form free-living amoebae, thus completing the life cycle. Slime molds defy simple categorization. They are animal-like in that they ingest food during the amoeboid- and plasmodial-phase. They are plant-like in their formation of cell walls and sporangia during reproduction. Because of the cell-walls formed during the reproductive phase they are considered fungal in nature. There are two major types of slime molds: A) the Myxomycota are the acellular, or true plasmodial slime modes: The plasmodium stage of this group is made up of large blobs of ceonocytic protoplasm with many nuclei inside. B) the Dictyosteliomycota are the cellular clime molds, where the plasmodium is made of individual cells separated by membranes (but not cell walls). Observations on Display: Fruiting bodies of slime molds are readily found in Ontario forests in the autumn. Some will be on display for you to examine, along with a diagram of the life cycle. B) Oomycetes: the water molds, or egg fungi (Kingdom Stramenopila) Oomycetes are characterized by the formation of large egg-bearing cells on the tips of specialized hyphae termed oogonia (see figure 17-4 on page 374 of your text). Large, non-motile eggs form inside the oogonia and are fertilized by male-like hyphae termed antheridia. The antheridia grow into the egg and deposit the male gametes, which then fuse with the egg to form a zygote, termed the oospore. The oospore undergoes mitosis and forms a sporangia. The spores that are produced disperse to infect leaves, seedlings, fish and dead organisms. Oomycetes are important saprophytes in aquatic habitats. In terrestrial habitats, they are generally parasitic. Important diseases caused by water molds are downy mildews (Peronospora), potato late blight (Phytophthora infestans), and damping off disease (Pythium spp.). We will examine three species, Achlya, a saprophytic water mold; Albugo candida, the white rust of mustard plants and Phytophthora infestans. Examine the following cultures with the dissecting microscope: 1. Achlya whole mounts: Achlya is a water mold that grows on organic debris in lakes and rivers. It forms floating mycelia mats arising from the food material, and in these mats sexual reproduction occurs. On your bench are prepared slides for you to examine with the light microscope. Focus on the blue-green material in the center of the slide. This is a hyphal mat with oogonia. Observe the large conspicuous oogonia within the mycelium. Inside the oogonia you can see zygotes (oospores), that will later divide to form sporangia. Examine the oogonia closely to see if additional hyphae are attached to it. These would be antheridia, which present the sperm nucleus to the egg cells within the oogonia. Note the properties of the vegetative hyphae. Do you see cross walls, or are the cells continous within a filament? 2. Phytophthora infestans: This organism causes potato late blight, one of the worst crop diseases in the history of humanity. In the 1840’s, Phytophthora infestans was introduced to Europe from Peru. It rapidly spread across the continent, destroying much of the potato crop. In Ireland, peasants were particularly dependent on the potato for survival, and the crop losses in 1845-1848 killed millions of Irish and forced the migration of millions more to North America. In continental Europe, the loss of the potato crop led to widespread economic failure and social revolt. Many of the radical worker movements that would later influence world history, such as Marxism-Leninism, arose in the wake of the food crisis caused by the Phytophthora outbreak. Examine the Phytophthora slide prepared fresh from a culture stained with cotton blue. Identify the round oogonia in the slide and examine the hyphae for cross walls between the cells. Are there any oospores within the Oogonia? Can you see any antheridial hyphae attached to the oogonia? If you do, show other students in your group. 3. Albugo candida (White Rust of Mustards): Albugo infects plants of the mustard family, forming white pustules on the leaves. These pustules are comprised of many asexual conidiospores bursting through the plants epidermis. Inside the plant, fungal hyphae form oogonia and antheridia, which will mate and form oospores. The oospores develop sporangia which disperse genetically-distinct spores. The two prepared slides show the extent of the infection by the parasitic white rust fungus. Examine the prepared cross section showing Albugo conidiaspores bursting through the epidermis of a mustard fruit. Note the strings of conidiospores forming under the epidermis. The bulge formed by the mass of conidia produce the rust pustule. Upon rupturing, the conidiospores are released on the wind to start a series of new infections. Examine the prepared slide of the Albugo sexual organs (oogonia and antheridia). The oogonia are evident as dense, red-staining circles among the cells of the leaf tissue. Notes and Drawings: We have assembled a range of specimens from the Kingdom Fungi, and they are arranged for you to examine beginning with the most primitive (the Chytrids) and then progressing through the Zygomycetes to the Ascomycetes and Basidiomycetes. Chytrids are mainly aquatic or parasitic. They are important decomposers of pollen, dead insects and seeds that fall into ponds and rivers. Others are parasites of algae, higher fungi, mosquitoes, rotifers, and water molds. The general body plan is to form a large, central ceonocytic globule that either directly invades the body of the host, or produces diminutive hyphae termed rhizomycelium, which invade the surface cells of the host. The rhizomycelium grows into the food source and absorbs nutrients across the chitenous cell wall. Chytrids are difficult to maintain in culture and have to be baited from natural sources. We have purchased slides showing a common parasitic chytrid, Synchytrium, invading a plant host, and have prepared a fresh culture of chytrids on snake skin. Chytrids are important decomposers of animal epithelial tissue in aquatic habitats. To capture them, we have placed strips of snake skin in pond water. Chytrid spores swim to the snake skin, and then germinate on the skin, forming a simple globular body with rhizomycelium. Synchytrium: Examine the prepared slide of Synchytrium that has infected either leaves or potato tubers. Note the simple globular body (the sorus) of the chytrid embedded in the tissue of the plant. Some of the globules may have matured into sporangia, and you may see spores inside. Chytrids on snakeskin: In the dissecting scope, you can see the rhizomycelia extending from the snake skin, along with many protozoa. Note the pinhead-like cells within the mycelia. These are either the central globules from which multiple filaments of rhizomycelium arise, or they are sporangia. In the light microscope, examine the globular cells and the hyphae growing away from them. The globular cell and rhizomycelium form the basic body plan of the saprophytic chytrids. You will also see many tiny creatures swimming about the mycelium. Some of these may be zoospores released from sporangia. Examine the mycelium for any evidence of sporangia. Sporangia will be apparent as they will have only one hyphae attached to them. The Zygomycetes are important saprophytes, including species that are major decomposers of dung and food. Members of this group have a zygotic lifecycle (see figure 15-11 on page 316 of your text). The gametes are non-motile, and are born on the tips of specialized, fertile hyphae termed gametangia. The gametangia contain many haploid nuclei. In zygomycetes, the sexual act consists of two fertile hyphae growing towards each other. As they approach each other the ends of the hyphae form gametangia. The gametangia come in contact, after which the end walls disintegrate, releasing the haploid gamete nuclei into one common space. Pairs of haploid nuclei then fuse, creating many diploid nuclei. The nucleate cell formed from the residuals of the two gametangia is termed a zygosporangium. Zygosporangia typically develop a thick wall that protects the diploid nuclei from harsh conditions, forming a many nucleate (ceonocytic) resting cell. Zygosporangia germinate when the diploid nuclei undergo meiosis to produce many new haploid nuclei. The haploid nuclei are walled off into distinct spores, which are released from a dispersal sporangium that grows out of the zygosporangium. The most commonly encountered zygomycetes are the bread molds, which are important saprophytes that grow on carbohydrate-rich foods, including bread. The mycelium formed on the surface of the bread is a cottony mass that is initially white but soon darkens as the mycelium forms asexual sporanagia. Large numbers of mitospores are released, allowing the fungus to quickly spread. Examine the following, first under the dissecting scope and then with the light microscope: 1. Zygorrhynchus moelleri: This mold growing on an agar plate shows the major stages of a typical Zygomycete life cycle. First examine the culture under the dissecting microscope, and then take a very small scraping of the agar with a dissecting needle or knife. Place this on a microscope slide, stain with the cotton blue stain on your bench, and examine at medium power with both phase contrast and normal visible light. With the dissecting scope, examine the cottony matrix of the mycelium, and the sporangia rising above it, forming dark spheres on elongated stalks. These are mostly mitosporangia used in asexual reproduction. You may be able to see zygosporangia mixed in amongst the mycelium. They will be dark, barrel-shaped granules on the surface of the agar. With the light microscope, observe the slide of agar, note the clear tubular nature of the hyphae and the absence of cross walls. Next observe any mitosporangia you may have opened while pressing down the cover slip. Finally, observe the zygosporangia. These are dark, barrel-shaped structures with rough walls. Note the hyphae attached to the zygosprangia. These are the stalks of the gametangia, and are termed suspensors. 2. Rhizopus stolonifera: This is the common bread-mold, a regular feature of most pantries. We have provided you with a Petri plate of Rhizopus to examine. Note the following with the dissecting scope under high power: Examine the mycelium and sporangia. You may be able to see elongated, horizontal hyphae connecting the sporangial stalks. Rhizopus spreads by these elongated hyphae, termed stolons (after the strawberry runners of the same name). Where stolons settle onto a food source, they produce anchoring hyphae that penetrate the food. Sporangia form above this contact point. This habit allows for rapid spread of Rhizopus over a loaf of bread. Typically, Zygomycetes reproduce asexually by mitospores when conditions are good, allowing for rapid spread over a new food source. They switch to sexual reproduction when the food is exhausted and conditions deteriorate. 3. Ungulate dung: We may display some moose or cow feces which may show a range of dung-zygomycetes, possibly including the hat-throwing fungus Pilobus. If anything of interest appears to be present, examine it with the dissecting scope and note the nature of the sporangia. The dikaryomycota were formerly classified as the phyla Ascomycota and Basiodiomycota (for example, see chapter 15 in your text), but recent advances in the systematic understanding have led to the merging of these two groups in a single phylum of higher fungi, the Dikaryomycota, with the ascomycetes and basiodiomycetes being separated into subphyla termed Ascomycotina and Basiodiomycotina. We will focus on these two groups. The common feature of these groups is the formation of dikaryotic hyphae. The dikaryon arises when the protoplast of haploid hyphae fuse (they undergo plasmogamy). The nuclei do not initially fuse, and the resulting mycelium is made up of cells that are dikaryotic, or in the N + N state. Fusion of the two nuclei occurs in the fruiting body of the fungus, forming a diploid cell that immediately undergoes meiosis and mitosis to produce four to eight spores. The spores are released from sporangia formed in the fruiting body. In the ascomycetes, eight spores are released from sacs termed asci (singular is ascus, from the Latin word for sac). In the basiodiomycetes, the spores are formed on the end of a club-like sporangia termed a basidium (from the Latin word for club). The fruiting bodies of each fungus are termed an ascocarp (ascoma in your text), and the basidiocarp (basidioma in your text). The mushroom cap is a basidiocarp. We have a number of specimens for you to examine today from each subphylum. 1. Subphyllum Ascomycotina a. Unicellular forms: the yeasts: These are single-celled fungi that typically live within the food medium. Most are saprophytic, although some can become parasitic. The yeast fungus Candida albicans is an important pathogen in humans, forming diaper-rash, vaginal and urethral tract infections, and the potentially deadly sexually-transmitted disease candidiasis. The common yeast Saccharomyces cerviseae is the yeast of baking, brewing and enology (wine making). This yeast is preferred in fermentations as it rapidly grows, produces pleasant as opposed to noxious or toxic waste-products, and is tolerant of high (>10%) concentrations of ethanol. Saccharomyces cereviseae: The common brewers yeast is growing on agar. Take a very small portion off the culture and smear into a drop of water on a microscope slide. Add a cover slip, and examine at low and then high power with the compound scope. Look for budding cells amongst the large numbers of indistinct single yeast cells. These are apparent from the blob-like cellular extensions, termed buds that arise from mature yeast cells. Rather than simply dividing in two as most algae and plant cells do, yeasts divide by extruding protoplasm into a bud. This extrusion is then encapsulated in a wall and split off to form a new independent cell. Occasionally, you may see some yeast forming asci: Yeasts live in both a haploid and diploid state. When conditions are harsh, two diploid yeast nuclei merge to form a zygote, which then undergoes meiosis to produce a four celled sac, the ascus. The ascus splits open to release the four cells, which then bud to start a new population of yeast cells. You may be able to see four-celled asci floating among the many cells in your slide. If so, show your classmates. b. Filamentous Ascomycetes: Multicellular ascomycetes produce hyphae and mycelium, and form ascocarps. Three types of ascocarps are produced by these fungi, cleistothecia (enclosed spheres), perithecia (vase-like) and apothecia (cup shaped). You should examine examples of each. b.1 Cleistothecial species: i. Powdery mildew (Uncinula spp.): Powdery mildews are common pathogenic fungi that infect leaves, forming a powdery mycelium on the surface. Powdery mildews reproduce asexually by forming chains of spores (conidiospores) on special hyphae termed conidophores. During sexual reproduction, they form a simple enclosed ascocarp, the cleistothecia. Cleistothecia are completely enclosed, with no opening for the developing spores to escape. When mature, the ascocarp wall ruptures, allowing enclosed asci with their ascospores to spill out and disperse. Often, cleistothecia have barbs and hooks, which can help disperse the entire ascocarp by clinging onto the fur of passing animals. Examine the following: Dissecting scope: Scan across the leaf infected with Uncinula to note the powdery mycelium, with chains of conidia rising above it. Periodically, you will see a pepper grain-like object with multiple elongated hooks attached. This is the cleistothecia. Light microscope: Scrape some cleistothecia onto a microscope slide and cover gently with a cover slip. Examine at low power. Next press of the cover slip to rupture the ascocarp and release the spores inside. ii. Powdery mildew on leaves: We also have specimens of unknown powdery mildews collected on leaves from around Toronto. Examine these under the dissecting scope for cleistothecia and conidia. b.2 Perithecial species: The perithecium is a vase-shaped ascocarp with a narrow, open neck. Inside are multiple asci with spores. When mature, the asci protrude from the neck of the perithecia and forcibly eject the spores into the air. Sordaria is a dung saprophyte that is closely related to Neurospora, the fungus that has become one of the leading model organisms in genetic research. Dissecting scope: Sordaria is growing on agar plates, and the perithecia can be seen as dark pepper-like grains mixed in a mass of conidia-forming hyphae. Examine the perithecia closely and note the pear-like shape of the ascocarp. Are any asci protruding from the perithecia? Light microscope: Scoop some perithecia onto a slide, cover and examine at low-to medium power. Gently push on the cover slip to squash open the perithecia. Note any football-shaped spores and asci that emerge. b.3. Apothecial species: The apothecium is an ascocarp where the asci are directly exposed to the air in a cup, dome or invaginated surface. The fungi are commonly called the saucer, or cup fungi, and they include many beautiful, brightly colored forest species. The delectable morel is an apothecial ascocarp. To demonstrate the apothecium structure and form, we have a set of prepared slides and live specimens from a number of species. i. Bispora centrina (Yellow-fairy cups – live specimens): These are wood decomposing fungi that form small, brightly yellow cup shaped apothecium. Examine the stick with the fairly cups closely. You may take the stick to you bench to examine with a dissecting scope. The asci are formed on the inner surface of the cup. Return the stick to the display area when finished as we have only a few specimens. ii. Peziza (prepared slides): Peziza is a cup fungus that grows on wood and is similar in form to Bispora. A life cycle of Peziza is shown in Fig. 15-14 on page 318 of your textbook. Examine the prepared cross sections of the Peziza ascocarp under medium power with your light microscope. You will note the sac-like structure of the ascus with 8 haploid spores inside. These arose from meiosis and one subsequent round of mitosis. Note the zone of fertile tissue where the asci form. Below this are fattened vegetative cells that form the support structure of the ascocarp. b.4 Ascomycetes of special note i. Claviceps purpurea (Ergot of Rye) We have display specimens of rye shoots infected with Ergot, caused by the perithecia-forming ascomycete Claviceps purpurea. Claviveps is an example of an endo-parasite, a fungus that grows within the stem and leaves of grasses. The fungus retards growth, but does not kill the host plant. In many instances, toxins produced by the fungus deter herbivory, and so the grass host can actually show superior performance relative to a non-infected plant that is eaten. In the case of ergot, the toxin produced is lysergic acid amide, from which the hallucinogenic drug lysergic acid diethyamide (LSD) was derived. Examine the infected rye and note the grain heads with enlarged, dark-colored protrusions extending out from the grass stalk. These are sclerotia, which occur where the fungus has completely infected a developing grain and replaced the grain with a tight mat of interwoven mycelium. As the growing season ends, the sclerotia fall to the ground and overwinter. In the spring, they produce perithecia and in turn, large numbers of spores that infect the new rye crop. Sclerotia break free and mix with the rye grain at harvest. People eating rye contaminated with ergot sclerotia experience severe poisoning, called ergotism. Symptoms include wild hallucinations coupled with extreme burning sensations in the extremities. Constriction of minor veins is common, leading to limbs dying and falling off. The pain is severe, and a typical victim would scream in agony while madly hallucinating. Before modern science explained the cause, people in the past would interpret the symptoms as an attack of demons, and in regions affected by ergot outbreaks, the citizens often turned to extreme religious practices to exorcise the devil. Throughout history, witch hunts, new religious movements, and mass hysteria have been attributed to ergot outbreaks. Today, ergot poisoning is rare, and rye grain is routinely screened to filter out the larger sclerotia. Sclerotia are now intentionally grown as a source of drugs to control internal bleeding, migraine headaches, and to alter mental states in psychiatric patients. ii. Peach Brown Rot (Monilinia fructicola): Many ascomycetes are severe pathogens of fruit crops. One of the worst is Peach Brown rot, which stunts trees and destroys mature peaches, apricots, cherries and related fruit. Infected trees form cankers on the twigs and leaves. Conidia erupting from the cankers are dispersed to infect other trees by asexual means. Fruits are infected as they near maturity. After infection, lesions develop on the fruit and it prematurely rots, falls to the ground and dries to a mummified carcass. Peach mummies are completely infected with the mycelium and in this form, the fungus will overwinter. In the spring, the fungus in the mummy form apothecia, from which spores will be released in huge numbers to infect new trees. Examine the Monilinia cultures on agar with the dissecting microscope and note the lemon-shaped conidiospores arising from the mycelium. You can examine these more closely by taking a small piece of agar and preparing it on a slide for examination with the light microscope. iii. Penecillium: Many ascomycetes are saprophytes that infect food and building materials in the home. Some are also sources of important drugs, while other produce powerful carcinogens. Penicillium is one of the most common molds in the household pantry, where it infects bread, fruits and milk products. Penicllium species are also important in making strong-flavored cheeses such rouquefort, gorgonzola, chamenbert, brie and Danish Blue. The blue-green color is actually the reproductive conidia of the Penicillium mold. Pennicilium is also the source of penicillin, the antibiotic that prevents wall synthesis in gram negative bacteria. Examine the culture with the dissecting scope and note the green-blue broom-like conidiospore masses rising above the mycelium. These masses give Penicillium molds their characteristic color. Take a sample and prepare a microscope slide of it. Examine the conidiophores with conidia under the compound microscope. Note the broom-like structure of the spore-bearing mass. We also have some blue-cheese on display. Examine the Pennicilium colony through the dissecting scope and try to identify the sporangia. iv. Aspergillus: Aspergillus species are common black-colored molds in the household environment. They are frequently found on bread, drywall, and grains. Many species produce aflotoxins, which are powerful carcinogens of the liver found in stored grains, peanuts and cereals, including corn flakes. It is unwise to eat foods contaminated with wild Aspergillus species as they likely contain aflotoxins. (For example, never eat wild peanuts, or musty old grain). Beneficial Aspergillus spp. are used to produce soy sauce, miso (fermented soy paste), and to ferment rice in an early step in sake production. Examine with a dissecting scope the culture on the agar plate and note the fan-shaped mass of conidia arising above the mycelium. Next, examine a piece of the mycelium to see the bulbous conidiophore. The dark masses of conidia give this fungus its particular color and shape. Take a small chunk of infected agar and prepare a slide of the sporangia for the light microscope. Examine the swollen top of the condiophores and the attached fan-shaped array of conidia. 2. The subphylum Basidiomycotina The most familiar fungi are the basidiomycetes. The fruiting bodies of the basidiomycetes (the basidiocarp) are the recognizable features of species of mushrooms, toadstools, coral fungi, shelf fungi and tooth fungi. In each, the main body lives underground or in wood as a dispersed mycelium. Although all basidiomycetes reproduce by forming spores on club-shaped basidia, there are actually two main groups: the homobasidiomycetes and the heterbasidiomycetes. The homobasidiomycetes produce one type of spore, the basidiospore. The heterobasidiomycetes produce two types of spores during the sexual life cycle. We will focus on the homobasidiomycete life cycle as exemplified by the common food mushroom, Agaricus campestris. Heterobasidiomycetes are the pathogenic rusts and smuts. a. Basidiomycete yeast (Rhodotorula ruba): Some basidiomycetes also have evolved the unicellular life form. A common basidiomycete yeast is the red yeast, Rhodotorula ruba, a contaminant of bathroom curtains, tile and grout. The pink scum in filthy bathtubs and showers is caused by Rhodotorula. (You may remember the battle between the Cat-in-the-Hat and pink bathroom scum). Examine the red yeast culture on display. If time permits, you may prepare a microscope slide of the cells from the agar culture. Examine them for budding and basidia, which are distinguished by an elongated shape and horn-like points on one end of the cell. b. Heterobasidiomycetes: The heterobasidiomycetes include the rust diseases of grasses, and smut diseases of maize. Other members of this group are wood decomposers such as the jelly fungi. We may have a jelly fungus in the wild mushroom display. Examine the specimens of grasses infected with wheat rust (Puccinia graminis). Note the rust-colored pustules forming on the blades of the grass. These are where asexual spores are formed to allow for continued infection of healthy plants during the summer. Black pustules appear in the late-summer. These are where teliospores are formed. Teliospores are overwintering spores that form basidiospores in the spring. If available, examine any corn smut (Ustilago maydis) that may be on display. Corn smuts attack developing corn kernels and produce large, grey-colored smutballs that are filled with dark spores. Immature smutballs are served as a delicacy in Latin American cuisines. Rusts and smuts are virulent parasites of grain crops, with the potential to wipe out the production of an entire region in any given year. The primary means of preventing infestation is to breed crop cultivars that are resistant to rust infections. The rusts eventually evolve new ways to infect the cultivar, so government agencies are continuously breeding resistance into varieties to stay ahead of the rust capacity to re-evolve virulence. Should breeding efforts fall behind (for example, via cost-cutting measures by governments and agribusiness), major rust outbreaks could result, ruining grain crops and causing food prices to sky-rocket. c. Homobasidiomycetes: the Mushrooms We have fresh specimens of the common store-bought mushroom for examination, along with cultures of the inky cap mushroom, and a range of wild mushrooms from southern Ontario. To aid in examining fine detail, we have prepared slides showing cross sections of mushroom caps for you to examine under the light microscope. A detailed diagram of the life-cycle of the mushroom is presented in Figure 15-19 of your textbook (page 321). c.1. The common food mushroom Agricus bisporus: Examine a) the mycelium of the spawn blocks on display, b) the young button mushrooms and c) mature-spore producing mushrooms from the collection of fresh mushrooms provided. i. Mycelial stage: sample mycelia of the mushroom spawn that is available. Stain with cotton blue and view with the light microscope under both normal light and phase contrast. Find some isolated hyphae and examine this under high power. Note the septate nature of the hyphae. This is one of the diagnostic features of the basidiomycetes. Each cell contains two haploid nuclei in the N + N configuration. A key feature of Basidiomycetes is the presence of clamp connections, which form after cell division in order to keep the N + N dikaryotic configuration intact (see figure 15-21 in your text). Clamp connections may be visible along the end walls of the hyphal cells, forming bulges or loops around the septate wall. ii. The mushroom button stage: The basidiocarp forms from tightly woven mycelia. Initially, the basidiocarp form a button, or egg stage. Cut open a button and examine a) the immature stalk (or stipe), b) the young, white to pink gills, and c) the developing cap which extends down over the stipe. With a razor blade, cut a thin slice of a gill, and look at the slice with the light microscope. Stain the gill with Melzer’s blue. You may be able to see developing basidia with miniature spores. iii. The mature mushroom: Note the features of the basidiocarp structure. The main parts are a) a well developed stalk, termed the stipe. The cap, termed the pileus, and the gills, where the spore bearing tissues occur. The gills have turn chocolate brown as millions of spores mature. Cut a thin slide of the mature gill and examine for spores and horned basidia. Stain with the Melzers stain placed on your bench. You should be able to isolate one or two good basidia from the mass of tissue. iv. Prepared slide of the mushroom cap: Examine under the light microscope the cross sections prepared of an Agaricus pileaus. The cross sections of gills clearly demonstrate the club shaped basidia arising from the zone of fertile tissue (the hymenium). Examine the basidia under high power and note how the spores are attached to the horn-like basidial tips (the sterigmata). The spores appear as party balloons taped to a club. v. Spore prints: As the spores mature, they are released and drift into the air below the cap. If the cap is place over paper, the spores cannot disperse and settle onto the paper to form a print of the mushroom gills. Spore prints have been prepared for you to examine. Spore prints are often used to identify particular mushrooms, and they are an easy way to verify the color of the spores. c.2. The Inky-cap mushroom, Coprinus cinereus. We have displays of the Inky-cap mushroom for you to examine. The cultures show how the mushroom arises from the mycelium. Prepared slides are also available if you wish to examine the gill structure and the basidia of Corpinus. The cap self-digests upon maturity forming a purple-ink colored mass of goo. c.3. The oyster mushroom Pleurotus ostreatus: Oyster mushrooms are delightful edible mushrooms that grow on logs and decaying stumps. They produce white spores on short gills, and exhibit a large, fan-shaped pileus with a short stipe. They are now commonly cultivated and are available in many supermarkets. Examine the oyster mushrooms for morphological structure, then thin slice a gill and examine for the basidia and white spores under the microscope. c.4. The chanterelle (Chanterellus cineareus): Chanterelles are a wonderful delicacy that is prized for its gentle, buttery flavor. Unlike the gilled mushrooms, chanterelles form their basidia on gently folded tissue underneath a wavy pileus. Assuming we have these available, take a small piece of the pileaus of a chanterelle and examine for basidia and spores. What color are the spores? c.6. The wild mushroom display: We have collected a variety of wild mushrooms for you to examine for variation in form. Examine each carefully, paying attention to the pileus, the stipe (if present) and where the spore bearing tissues are located. In many of the samples, gills are not present. Instead, the spores are produced on elongated tubes that form pores on the underside of the pileus, on teeth-like protrusions that hang below the pileus, on coral-like prongs, or on rumpled folds of tissue that resemble elbow skin. These traits distinguish the major families of mushroom-forming species. You may take some of the specimens and examine them under the dissecting scope in order to better see the pores, teeth or prongs of the spore-bearing tissue. Lichens are structures formed by close symbiotic relationships between an algae and a fungus. Both green and blue-green algae can serve as the algal symbiont, while the fungus is typically an ascomycete. Because the sexual stage of the lichen that is visible is that of the fungal partner (the mycobiont), the lichen is typically named after the fungus. In the symbiosis, the algae provide carbohydrates from photosynthesis while the fungus shelters the algae and gathers water and nutrients. Lichens can completely desiccate with no harm to the organisms inside. Upon wetting they rapidly rehydrate and resume activity. This ability allows them to live in extremely harsh surfaces, such as the branches and trunks of trees, the sides of rocks, and bare ground in deserts. In the boreal zone, lichens are important ground covers on bare soil, fallen branches and the surface of rocks. They also are common as epiphytes on trees. In general, they grow extremely slow, reflecting the harsh conditions of the habitats they live in. Lichens come in three general categories based on morphology. Examine the specimens displayed in the lab room. A. Foliose lichens: Lichens that exhibit a leafy shape are termed the foliose lichens. These are common in wetter habitats, for example, forest interiors in eastern Canada. Foliose lichens are important epiphytes growing on branches of standing trees. B. Fructicose lichens: These lichens are shrubby in appearance, with many narrow, highly branched stem-like structures. Fructicose lichens are common in the boreal forest, forming dense ground covers in spruce forests. When dry, they are extremely flammable and can be used as a fire starter. They also help wildfires spread and thus contribute to some of the severe forest fires that occur every summer in Canada. Fructicose lichens are common on the sides of trees, and often hang from the bra nches in dense growths termed Witch’s hair, or Old Man’s Beard. C. Crustose lichens: Crustose lichens occur in the most extreme terrestrial environments where life is possible. They grown on the sides of rocks, buildings and on bare soil, and are common in arid and polar deserts, including the dry valleys of Antarctica. Crustose lichens form brilliant yellow, orange, red and yellow-green colors on the rock, and are some of the most beautiful features in what are otherwise barren landscapes. Study Guide: You should be familiar with the major categories of fungi and the names of the common species of yeast, store mushrooms, and major disease organisms displayed in the lab. You need to know the terms presented in bold font, and should recognize an organism well enough to classify it to phylum or where relevant, to subphylum. Know and understand the distinguishing characteristics of the major phyla presented in lab, as well as that of the ascomycetes and basidiomycetes. Significance of bryophytes • Mosses (Bryophyta) • Liverworts (Marchantiophyta) • Hornworts (Anthocerotophyta) Scientific Name, Common Name Plantae, Land Plants Embryophytes, Green plants The Bryophyta or mosses, unlike the liverworts, are present in most terrestrial habitats (even deserts) and may sometimes be the dominant plant life. As with the liverworts the plant that we commonly see is the gametophyte. It shows the beginnings of differentiation of stem and leaves - but no roots. Mosses may have rhizoids and these may be multicellular but they do little more than hold the plant down. The stem shows some internal differentiation into hydroids and leptoids which are like xylem and phloem of higher plants but very simply organized with no connection to leaves or branching stems. The leaves are mostly one cell thick; sometimes the midrib is several cells thick but this does not contain conducting tissue so it is not equivalent to the vein of a leaf. Male and female gametophytes look identical except when they produce reproductive structures. The male plant produces clusters of antheridia which contain thousands of ciliate sperm. The female produces archegonia, each containing a single egg. Fertilization is dependent on water - sperm are splashed or swim to the archegonia. The zygote grows into the diploid sporophyte which remains attached to the female gametophyte It is a leafless stem with a seta or foot at one end, drawing nutrients from the gametophyte. At the other end is a capsule in which meiosis occurs to form spores. The archegonium grows around the developing sporophyte for a while but becomes separated from the gametophyte and is carried up to form a cap or calyptra over the sporangium. Curiously, the sporangia of some mosses have stomata much like those on the leaves of vascular plants. Immature moss capsules with calyptra The calyptra is lost when the sporangium is mature as is the operculum or lid on the end of the capsule. Underneath the operculum there are often peristome teeth which open under dry conditions and control spore release A spore germinates to produce a filamentous protonema which sooner or later produces buds that grow into new gametophytes. Ecology of mosses Mosses require abundant water for growth and reproduction. They can tolerate dry spells by drying out or,in the case of mosses like Sphagnum , by holding huge amounts of water in dead cells in the leaves. They look pretty lowly and insignificant, but have become dominant in particular habitats and Sphagnum itself is said to occupy 1% of the earth's surface (half the area of the USA). Because of its ability to soak up blood and its relative freedom from bacterial contamination Sphagnum was used in dressings. The moss itself is used in some horticultural media and it is an important source of peat. Polytrichum commune one of the larger mosses with mature sporophytes If you have tried to grow a lawn in a shady location you have probably been troubled by mosses as weeds. Like many lower organisms they are very sensitive to copper salts and can be controlled in this way. On the other hand mosses are green and better adapted to shade than most grasses, so maybe we should accept them in this situation. Natural Perspective The Plant Kingdom : Mosses and Allies Mosses and their allies are small green plants that are simlutaneously overlooked and deeply appreciated by the typical nature lover. On the one hand, very few people pay attention to individual moss plants and species. On the other hand, it is the mosses that imbues our forests with that wonderful lush "Rainforest" quality which soothes the soul and softens the contours of the earth. These wonderfully soft carpets of green are, in fact, Nature's second line of attack in its war against rocks. After lichens have created a foothold in rocks the mosses move in, ultimately becoming a layer of topsoil for higher plants to take root. The mosses also hold loose dirt in place, thus preventing landslides. Ecologically and structurally, mosses are closer to lichens than they are to other members of the plant kingdom. Both mosses and lichens depend upon external moisture to transport nutrients. Because of this they prefer damp places and have evolved special methods of dealing with long dry periods. Higher plants, on the other hand, have specialized organs for transporting fluid, allowing them to adapt to a wider variety of habitats. Bryophytes used to be classified as three classes of a single phylum, Bryophyta . Modern texts, however, now assign each class to its own phylum: Mosses ( Bryophyta ), Liverworts ( Hepatophyta ), and Hornworts ( Anthoceraphyta ). This reflects the current taxonomic wisdom that the Liverworts and Hornworts are more primitive and only distantly related to Mosses and other plants. Mosses (Phylum: Bryophyta ) All plants reproduce through alternating generations. Nowhere is this more apparent than in the mosses. The first generation, the gametophyte , forms the green leafy structure we ordinarily associate with moss. It produces a sperm and an egg (the gametes) which unite, when conditions are right, to grow into the next generation: the sporophyte or spore-bearing structure. The moss sporophyte is typically a capsule growing on the end of a stalk called the seta . The sporophyte contains no clorophyl of its own: it grows parasitically on its gametophyte mother. As the sporophyte dries out, the capsule release spores which will grow into a new generation of gametophytes, if they germinate. Mosses, the most common, diverse and advanced brypophytes, are categorized into three classes: Peat Mosses ( Sphagnopsida ), Granite Mosses ( Andreaopsida ), and "True" Mosses ( Bryopsida or Musci ) . Shown: Class: Bryopsida ; Order: Hypnales ; Family: Brachythecia ; Homolathecium nutalli (probably) Leafy Liverworts (Phylum: Hepatophyta , Class: Jungermanniidae ) While people typically know what a moss is, few have even heard of liverworts and hornworts. These primitive plants function much like mosses and grow in the same places, often intertwined with each other. The liverworts take on one of two general forms, comprising the two classes of liverworts: Jungermanniidea are leafy, like moss; Marchantiopsida are leaf-like ( thalloid ) similar to foliose lichens . The leafy liverworts look very much like mosses and, in fact, are difficult to tell apart when only gametophytes are present. The "leaves," however, are simpler than moss and dont have a midrib ( costa ). The stalk of the sporophyte is translucent to white; its capsule is typically black and egg-shaped. When it matures, the capsule splits open into four equal quarters, releasing the spores to the air. The liverwort sporophyte shrivels up and disappears shortly after releasing its spores. Because of this one hardly ever sees liverwort sporophytes out of season. Moss sporophtyes, on the other hand, may persist much longer. Shown: Class: Jungermanniidea ; Order: Jungermanniales ; Family: Scapaniaceae ; Scapania spp. (probably) Leaf-like Liverworts (Phylum: Hepatophyta ; Class: Marchantiopsida ) The leaf-like ( thalloid ) liverworts are, on the whole, more substantial and easier to find than their leafy counterparts. The gametophyte is flat, green and more-or-less strap-shaped. The body may, however, branch out several times to round out the form. When the gametophyte has become fertilized and is ready to produce its sporophyte generation it may grow a tall green umbrella-shaped structure called the carpocephalum . The sporophyte grows on the underside of this structure, often completely hidden from view. During the dry season, leaf-like liverworts may shrivel up and completely disappear from view until the rains arrive again. Thalloid liverworts are much easier to identify than their leafy counterparts due to the wider variety of gametophyte shapes. Shown: Class: Marchnatiopsida ; Order: Marchantiales ; Family: Aytoniaceae ; Asterella californica Hornworts (Phylum: Anthoceraphyta ) Hornworts are very similar to liverworts but differ in the shape of the sporophyte generation. Instead of generating spores in a capsule atop a stalk, the hornwort generates spores inside a green horn-like stalk. When the spores mature the stalk splits, releasing the spores. Under the microscope, hornwort cells look quite distinct as well: they have a single, large chloroplast in each cell. Other plants typically have many small chloroplasts per cell. This structure imparts a particular quality of color and translucency to the body ( thallus ) of the plant. Hornworts are all grouped into a single class, Anthocerotae , containing a single order, Anthocerotales . Shown: Class: Anthocerotae ; Order: Anthocerotales ; Family: Anthocertaceae ; Phaeoceros spp. Suggestions for the Use of Keys 1. Select appropriate keys for the materials to be identified. The keys may be in a flora, manual, guide' handbook, monograph, or revision (see Chapter 30). If the locality of an unknown plant is known, select a flora, guide, or manual treating the plants of that geographic area (see Guides to Floras in Chapter 30). If the family or genus is recognized, one may choose to use a monograph or revision. If locality is unknown. select a general work. If materials to be identified were cultivated, select one of the manuals treating such plants since most floras do not include cultivated plants unless naturalized. 2. Read the introductory comments on format details, abbreviations, etc. .before using the key. 3. Read both leads of a couplet before making a choice. Even though the first lead may seem to describe the unknown material, the second lead may be even more appropriate. 4. Use a glossary to check the meaning of terms you do not understand. 5. Measure several similar structures when measurements are used in the key, e.g. measure several leaves not a single leaf. Do not base your decisions on a single observation It is often desirable to examine several specimens. 6. Try both choices when dichotomies are not clear or when information is insufficient, and make a decision as to which of the two answers best fits the descriptions. 7. Verify your results by reading a description, comparing the specimen with an illustration or an authentically named herbarium specimen. Suggestions for Construction of Keys 1. Identify all groups to be included in a key. 2. Prepare a description of each taxon (see Chapter 24 for details for description and descriptive format). 3. Select "key characters" with contrasting character states. Use macroscopic, morphological characters and constant character states when possible. Avoid characteristics that can only be seen in the field or on specially prepared specimens, i.e., use those characteristics that are generally available to the user. 4. Prepare a Comparison Chart (see Figure 25-3). 5. Construct strictly dichotomous keys. 6. Use parallel construction and comparative terminology in each lead of a couple. 7. Use at least two characters per lead when possible. 8. Follow key format (indented or bracketed see Figures 25-1 and 25-2). 9. Start both leads of a couple with the same word if at all possible and successive leads with different words. 10. Mention the name of the plant part before descriptive phrases, e.g., leaves or flowers blue not blue flowers, leaves alternate not alternate leaves. 11. Place those groups with numerous variable character states in a key several times when necessary. 12. Construct separate keys for dioecious plants, for flowering or fruiting materials and for vegetative materials when pertinent. • Shrub or woody vine. o Woody vine; petals 7 or more 3. Decumaria o Shrub; petals 4 or 5.  Leaves alternate or on short spur branches.  Leaves pinnately veined; ovary superior; fruit a capsule 1. Itea  Leaves palmately veined; ovary inferior; fruit a berry 2. Ribes  Leaves opposite.  Petals usually 4;-stamens 20-40; fruit longitudinally dehiscent, not ribbed; 4. Philadelphus  Petals usually 5; stamens 8-10; fruit poricidally dehiscent, 10- to 15-ribbed 5. Hydrangea • Herbs. o Staminodia present; petals more than 10 mm long 6. Parnassia o Staminodia absent; petals less than 10 mm long.  Leaves ternately decompound 7. Astilbe  Leaves simple.  Flowers solitary in leaf axils, or in short, leafy cimes.  Sepals 4; carpels 2 8. Chrysosplenium  Sepals 5; carpels 3 9. Lepuropetalon  Flowers in racemes or panicles.  Petals pinnatifid or fringed; stem leaves opposite 10. Mitella  Petals not pinnatifid or fringed; stem leaves alternate or absent.  Ovary 1-celled.  Inflorescence paniculate; stamens 5 11. Heuchera  Inflorescence racemose; stamens 10 12. Tiarella  Ovary 2-celled.  Stamens 5; leaves palmately lobed 13. Boykinia  Stamens 10; leaves not palmately lobed 14. Saxifraga • 1. Shrub or woody vine 2. • 1. Herbs 6. o 2. Woody vine; petals 7 or more Decumaria. o 2. Shrub; petals 4 or 5 3. • 3. Leaves alternate or on short spur branches 4. • 3. Leaves opposite 5. o 4. Leaves pinnately veined; ovary superior; fruit a capsule Itea. o 4. Leaves palmately veined; ovary inferior; fruit a berry Ribes. • 5. Petals usually 4; stamens 20-40; fruit longitudinally dehiscent, not ribbed Philadelphus • 5. Petals usually 5; stamens 8-10; fruit poricidally dehiscent, 10-15 ribbed Hydrangea. o 6. Staminodia present; petals more than 10 mm long Parnassia. o 6. Staminodia absent; petals less than 10 mm long 7. • 7. Leaves ternately decompound Astilbe. • 7. Leaves simple 8. o 8. Flowers solitary in leaf axils, or in short, leafy cymes 9. o 8. Flowers in racemes or panicles 10. • 9. Sepals 4; carpels 2 Chrysosplenium. • 9. Sepals 5; carpels 3 Lepuropetalon. o 10. Petals pinnatifid or fringed; stem leaves opposite Mitella. o 10. Petals not pinnatifid or fringed; stem leaves alternate or absent 11. • 11. Ovary 1-celled 12. • 11. Ovary 2-celled 13. o 12. Inflorescence paniculate; stamens 5 Heuchera. o 12. Inflorescence racemose; stamens 10 Tiarella. • 13. Stamens 5; leaves palmately lobed Boykinia. • 13. Stamens 10; leaves not palmately lobed Saxifraga. Figure 25-2. Example of a bracketed key. (Modified from Radford, A. E., 11. E. Ahles, and C. R. Bell. 1968. Manual of the Vascular Flora of the Carolinas. University of North Carolina Press. Chapel Hill, North Carolina. Used with permission.) 1. Identification of an unknown. Select an unknown specimen and identify it by keying in an appropriate manual, flora, or monograph. Verify your results by reading a description, by comparing with an illustration or by checking with your instructor. 2. Preparation of a comparison chart. Select 5 or more specimens from the group provided by your instructor. Identify each by keying. Verify your results. Prepare a description of each similar to those in a flora or manual. Be sure characters and character states are in the same order. Select contrasting character states and prepare a comparison chart (see Figure 25-3). 3. Construction of keys. Construct a dichotomous key to these specimens using the information in the comparison chart. Decumaria Itea Ribes Parnassia Heuchera Saxifraga Habit Woody vine Shrub Shrub Herb Herb Herb Leaf arrangement Opposite Alternate Alternate or on spur roots Basal (Rosulate) Basal (Rosulate) Basal Petal Number 7-10 5 5 5 5 5 Locule Number 7-10 2 1 1 1 2 Stamen Number 7+ 5 5 5 (stamonodia 5) 5 10 Fruit Type Capsule Capsule Berry Capsule Capsule Capsule Figure 25-3. A comparison chart used in the construction of keys (for six of the genera in Figures 25-1 and 25-2). In almost every ditch in Holland with reasonably clean water we will in summer find slimy masses of filamentous algae, floating as scum on the surface. It looks rather distasteful, but a ditch like that is not polluted, only eutrophic (rich in nutrients). In spring these filamentous algae grow under water but when there is enough sunlight and the temperatures are not too low, they produce a lot of oxygen, sticking in little bubbles between the tangles of the algae. These come to the surface and become visible as slimy green masses. In these tangles we will find mainly three types of filamentous algae, Spirogyra, Mougeotia and Zygnema. In this article we will mainly write about Spirogyra. From a distance these slimy tangles look perhaps a bit dirty, but under the microscope the filaments are very beautiful and moreover, they have a spectacular way of reproducing. Spirogyra owes its name to a chloroplast (the green part of the cell) that is wound into a spiral, a unique property of this genus which makes it easily to recognise. In The Netherlands up till now there are found more than 60 species of Spirogyra , in the whole world more than 400. For the determination of a species it is necessary to look for reproducing specimens with spores. But a precise determination is not necessary for learning a lot of interesting facts from Spirogyra. It is easy to see that there are many species ; in a clean, eutrophic ditch with hard water in Holland we will find easily 20 different species. If we look at a filament of Spirogyra with the microscope, the first thing that attracts attention is the chloroplast, a narrow, banded spiral with serrated edges. The small round bodies in the chloroplast are pyrenoids, centres for the production of starch. In the middle of the cell we see the transparent nucleus, with fine strands linking it to the peripheral protoplasm. The filaments contain cells of different sizes and it is easy to find a new cell, just formed after a division. The really interesting part comes as Spirogyra reproduces sexually. When two filaments are close together, the process starts. Cell outgrowths form connections between the filaments and a sort of ladder is formed. The contents of the cells in one filament will go through the connection tubes to the cells in the other filament. A zygospore is formed with a thick cell wall, round or oval and with a brownish colour. This conjugation process takes place especially between half May and half June. The spores are liberated, sink to the bottom and germinate in the next spring to form a new filament. It is very worthwhile to look in a sample of algae for the different stages of this conjugation process. It is always a nice surprise to find the conjugating filaments. Spirogyra can also exhibit, apart from the ladder like conjugation, another form of conjugation. Two neighbouring cells in the same filament can connect via a tube. There are several other genera of related filamentous algae; Zygnema and Mougeotia, with respectively star like and plate like chloroplasts. These genera live in general in more acid, soft fresh water. The conjugation figures look different from those in Spirogyra, for instance X-like. Dune pools are a rich biotope for Spirogyra. In ditches the amount of species declines when the water becomes very eutrophic. Other filamentous algae then replace it, like Cladophora, Vaucheria and Enteromorpha. In the end we only will find duck weed. Then a ditch does not receive light, with disastrous consequences for the growths of plants and the production of oxygen. The Filamentous Algae. This gallery includes only the filamentous green algae. The group is a heterogeneous one in which the members, although superficially similar, show a wide diversity in their life cycle and modes of reproduction. Spirogyra, Oedogonium and Cladophora are amongst the varieties most frequently encountered. All blue-green algae are now classified amongst the Bacteria, and will be found in the Cyanobacteria gallery. Spirogyra is a filamentous green alga which is common in freshwater habitats. It has the appearance of very fine bright dark-green filaments moving gently with the currents in the water, and is slimy to the touch when attempts are made to collect it. The slime serves to deter creatures which otherwise attatch themselves to underwater plants, so Spirogyra under the microscope is usually spotless. A field of Spirogyra filaments. Their appearance is not quite typical in that the nuclei are unusually prominent, and the characteristic spiral chloroplasts are so fine and tightly wound that close examination is needed to confirm the identification. In any case the possession of spiral chloroplasts is sufficient to positively identify Spirogyra to genus. Darkfield, x120. The central portion of a cell of Spirogyra showing the nucleus and giving an insight into the way the spiral chloroplast contacts with the wall of the cell. The filament in the background provides another view. Brightfield. x1000. Central portion of a Spirogyra cell showing nucleus and chloroplasts. Brightfield, x1000. This filament of Spirogyra is about to break into two filaments. The wall of each cell (centre of picture) has developed an inward indentation at the junction between the cells. Increase in pressure in each cell will cause the indentation to pop out, forcing separation of the filaments, and leaving them with highly convex ends. Brightfield. x1000. Two filaments of Spirogyra, the lower one clearly showing the nucleus. This picture also gives a good insight into the way the chloroplasts line the wall of the cell. Brightfield. x1000. Conjugation in Spirogyra. In common with other members of its phylum (Gamophyta) Spirogyra lacks a motile variant at all stages of its life history; ie, no motile gametes (ova or sperm), no zoospores etc. Sexual reproduction is by a process called conjugation -- another of the famously remarkable sights available to the microscopist. Although it is not possible to distinguish them visually, certain filaments in a loose parallel bundle of Spirogyra assume the female, and others the male, role in the process which follows. The cells of adjacent filaments develop bumps which grow towards one another and eventually fuse to form a continuous tube between the cells. Meanwhile the contents of each cell have detatched themselves from their respective cell walls and have formed a round ball. Over a relatively short space of time (minutes), the green spheres from the male filament squeeze their way down the connecting tubes to fuse with a similarly contracted female cell in the other filament. The result of this sexual union is the formation of a zygospore with a tough resistant outer covering within the chambers of the female filament. After a dormant period, these zygotes undergo meiosis and germinate, resulting in new filaments of Spirogyra. Once seen never forgotten. The central pair of cells are joined by a conjugation tube which has yet to fuse into form a continuous passage. The cell contents are at a similarly early stage of detatching themselves from the cell wall to form a ball. By contrast, the two cells to the right contain newly formed zygospores as a result of consummated conjugation. Male and female cells now occupy the same space, and are pictured before fusion to form a zygospore has taken place. The filament designated female is the one in which the zygospores have formed. Two mature zygospores of Spirogyra from another part of the specimen which provided the above pictures. In this form, Spirogyra can survive winter or other adverse conditions and germinate in the spring to form new filaments. The hardened outer spore wall can be seen reflecting the light from the darkfield condenser. Darkfield. x400. A zygospore of Spirogyra against a background of decaying plant remains and other algal forms. Darkfield, x400. It's hard to say what is happening here. It looks like the stage in conjugation of Spirogyra in which the contraction of the cell contents to a ball is not quite complete, and the spiral nature of the chloroplast is still discernable. Darkfield, x1000. Cladophora and Microspora. Darkfield, x300. Darkfield, x400. Darkfield, x600. Pteridium aquilinum Bracken Fern Bracken Fern Photo © by Earl J.S. Rook Flora, fauna, earth, and sky... The natural history of the northwoods Name: • Pteridium, from the Greek pteris (pteris), "fern" • aquilinum, from the Latin, "eagle like" Taxonomy: • Kingdom Plantae, the Plants o Division Polypodiophyta, the True Ferns  Class Filicopsida  Order Polypodiales  Family Dennstaedtiaceae  Genus Pteridium • Taxonomic Serial Number: 17224 • Considered a single, worldwide species, although some disagree Description: • A large, deciduous, rhizomatous fern Identification: • Distinguished from other large North Country ferns by the large three part leaf atop a tall stalk. • Field Marks o broad triangular leaf held almost parallel to the ground o smooth, grooved, rigid stalk about as long as the leaf o narrowed tip to leaflets Distribution: • Global; throughout the world with the exception of hot and cold deserts • Light, windborne spores allow colonization of newly vacant areas. Associates: • Shrubs: Bunchberry (Cornus canadensis), Twinflower (Linnaea borealis) • Herbs: Wild Sarsaparilla (Aralia nudicaulis), Large Leaf Aster (Aster macrophyllus), Blue Bead Lily (Clintonia borealis), Gold Thread (Coptis trifolia), Bedstraws (Galium ssp.), Oak Fern (Gymnocarpium dryopteris), Canada Mayflower (Maianthemum canadense), Bishop's Cap (Mitella nuda), One Flowered Pyrola (Moneses uniflora), One Sided Pyrola (Pyrola secunda), Rose Twisted Stalk (Streptopus rosea), Starflower (Trientalis borealis), Kidney Leaf Violet (Viola renifolia), Violets (Viola spp.) • Mammals: Palatability is usually nil to poor History: • Considered so valuable during the Middle Ages it was used to pay rents. • Also used as a green mulch and compost • Facts and Folklore: Reproduction: • Reproduces by spores and vegetatively by rhizomes • Shaded plants produce fewerspores than plants in full sun Propagation: • Division most successful method Cultivation: • Hardy to USDA Zone 3 (average minimum annual temperature -40ºF) • Characteristically found on soils with medium to very rich nutrients. Population Genetics and Evolution In 1908, G.H.Hardy and W. Weinberg independently suggested a scheme whereby evolution could be viewed as changes in frequency of alleles in a population of organisms. In this scheme, if A and a are alleles for a particular gene locus and each diploid individual has two such loci, then p can be designated as the frequency of the A allele and q as the frequency of the a allele. For example, in a population of 100 individuals ( each with two loci ) in which 40% of the alleles are A, p would be 0.40. The rest of the alleles would be ( 60%) would be a and q would be equal to 0.60. p + q = 1 These are referred to as allele frequencies. The frequency of the possible diploid combinations of these alleles ( AA, Aa, aa ) is expressed as p2 +2pq +q2 = 1.0. Hardy and Weinberg also argued that if 5 conditions are met, the population's alleles and genotype frequencies will remain constant from generation to generation. These conditions are as follows: • The breeding population is large. ( Reduces the problem of genetic drift.) • Mating is random. ( Individual show no preference for a particular mating type.) • There is no mutation of the alleles. • No differential migration occurs. ( No immigration or emigration.) • There is no selection. ( All genotypes have an equal chance of surviving and reproducing.) The Hardy-Weinberg equation describes an existing situation. Of what value is such a rule? It provides a yardstick by which changes in allelic frequencies can be measured. If a population's allelic frequencies change, it is undergoing evolution. Estimating Allele Frequencies for a Specific Trait within a Sample Population: Using the class as a sample population, the allele frequency of a gene controlling the ability to taste the chemical PTC (phenylthiocarbamide) could be estimated. A bitter taste reaction is evidence of the presence of a dominant allele in either a homozygous (AA) or heterozygous (Aa) condition. The inability to taste the PTC is dependent on the presence of the two recessive alleles (aa). Instead of using the PTC paper the trait for tongue rolling may be substituted. To estimate the frequency of the PTC -tasting allele in the population, one must find p. To find p, one must first determine q ( the frequency of the non tasting allele). 1. Using the PTC taste test paper, tear off a short strip and press it to your tongue tip. PTC tasters will sense a bitter taste. 2. A decimal number representing the frequency of tasters (p2+2pq) should be calculated by dividing the number of tasters in the class by the total number of students in the class. A decimal number representing the frequency of the non tasters (q2) can be obtained by dividing the number of non tasters by the total number of students. You should then record these numbers in Table 8.1. 3. Use the Hardy-Weinberg equation to determine the frequencies (p and q ) of the two alleles. The frequency q can be calculated by taking the square root of q2. Once q has been determined, p can be determined because 1-q=p. Record these values in Table 8.1 for the class and also calculate and record values of p and q for the North American population. Table 8.1 Phenotypic Proportions of Tasters and Nontasters and Frequencies of the Determining Alleles Phenotypes Allele Frequency Based on the H-W Equation Tasters (p2+2pq) Non Tastes(q2) p q Class Population #= %= #= %= North American Population 0.55 0.45 Topics for Discussion: 1. What is the percentage of heterozygous tasters (2pq) in your class? ______________________. 2. What percentage of the North American population is heterozygous for the taster allele? _____________ Case Studies: Case 1 ( Test of an Ideal Hardy-Weinberg Community) The entire class will represent a breeding population, so find a large open space for its simulation. In order to ensure random mating, choose another student at random. In this simulation, we will assume that gender and genotype are irrelevant to mate selection. The class will simulate a population of randomly mating heterozygous individuals with an initial gene frequency of 0.5 for the dominant allele A and the recessive allele a and genotype frequencies of 0.25AA, 0.50Aa, and 0.25aa. Record this on the Data page at the end of the lab. Each member of the class will receive four cards. Two cards will have A and two cards will have a. The four cars represent the products of meiosis. Each "parent" will contribute a haploid set of chromosomes to the next generation. 1. Turn the four cards over so the letters are not showing, shuffle them, and take the card on top to contribute to the production of the first offspring. Your partner should do the same. Put the cards together. The two cards represent the alleles of the first offspring. One of you should record the genotype of this offspring in the Case 1 section at the end of the lab. Each student pair must produce two offspring, so all four cards must be reshuffled and the process repeated to produce a second offspring. 2. The other partner should then record the genotype of the second offspring in the Case 1 section at the end of the lab. Using the genotypes produced from the matings, you and your partner will mate again using the genotypes of the two offspring. That is , student 1 assumes the genotype of the first offspring, and student 2 assumes the genotype of the second offspring. 3. Each student should obtain, if necessary, new cards representing their alleles in his or her respective gametes after the process of meiosis. For example, student 1 becomes the genotype Aa and obtains cards A,A,a,a; student 2 becomes aa and obtains cards,a,a,a,a. Each participant should randomly seek out another person with whom to mate in order to produce offspring of the next generation. You should follow the same mating procedure as for the first generation, being sure you record your new genotype after each generation in the Case 1 section. Class data should be collected after each generation for five generations. At the end of each generation, remember to record the genotype that you have assumed. Your teacher will collect class data after each generation by asking you to raise your hand to report your genotype. Allele frequency: The allele frequencies, p and q, should be calculated for the population after five generations of simulated random mating. Number of A alleles present at the fifth generation Number of offspring with genotype AA _____________ X 2= _______________ A alleles Number of offspring with genotype Aa _____________ X 1= ________________A alleles Total = ____________ A alleles p = Total number of A alleles = Total number of alleles in the population In this case, the total number of alleles in the population is equal to the number of students in the class X 2. Number of a alleles present at the fifth generation Number of offspring with genotype aa _____________ X 2= _______________ a alleles Number of offspring with genotype Aa _____________ X 1= ________________A alleles Total = ____________ a alleles q = Total number of a alleles = Total number of alleles in the population 1. What does the Hardy-Weinberg equation predict for the new p and q?. 2. Do the results you obtained in this simulation agree? __________ If not, why not? 3. What major assumption(s) were not strictly followed in this simulation? Case 2 ( Selection ) In this case you will modify the simulation to make it more realistic. in the natural environment , not all genotypes have the same rate of survival; that is, the environment might favor some genotypes while selecting against others. An example is the human condition sickle-celled anemia. It is a condition caused by a mutation on one allele, in which a homozygous recessive does not survive to reproduce. For this simulation you will assume that the homozygous recessive individuals never survive. Heterozygous and homozygous dominant individuals always survive. The procedure is similar to that for Case 1. Start again with your initial genotype, and produce your "offspring" as in Case 1. This time, However, there is one important difference. Every time your offspring is aa it does not reproduce. Since we want to maintain a constant population size, the same two parents must try again until they produce two surviving offspring. You may need to get new allele cards from the pool. Proceed through five generations, selecting against the homozygous offspring 100% of the time. Then add up the genotype frequencies that exist in the population and calculate the new p and q frequencies in the same way as it was done in Case 1. Number of A alleles present at the fifth generation Number of offspring with genotype Aa _____________ X 1= ________________A alleles Total = ____________ A alleles p = Total number of A alleles = Total number of alleles in the population Number of a alleles present at the fifth generation Number of offspring with genotype Aa _____________ X 1= ________________A alleles Total = ____________ a alleles q = Total number of a alleles = Total number of alleles in the population 1. How do the new frequencies of p and q compare to the initial frequencies in Case 1? 2. How has the allelic frequency of the population changed? 3. Predict what would happen to the frequencies of p and q if you simulated another 5 generations. 4. In a large population, would it be possible to completely eliminate a deleterious recessive allele? Explain. Hardy-Weinberg Problems 1. In Drosophila, the allele for normal length wings is dominant over the allele for vestigial wings. In a population of 1,000 individuals, 360 show the recessive phenotype. How many individuals would you expect to be homozygous dominant and heterozygous for this trait? 2. The allele for the ability to roll one's tongue is dominant over the allele for the lack of this ability. In a population of 500 individuals, 25% show the recessive phenotype. How many individuals would you expect to be homozygous dominant and heterozygous for this trait? 3. The allele for the hair pattern called "widow's peak" is dominant over the allele for no "widow's peak." In a population of 1,000 individuals, 510 show the dominant phenotype. How many individuals would you expect of each of the possible three genotypes for this trait? 4. In a certain population, the dominant phenotype of a certain trait occurs 91 % of the time. What is the frequency of the dominant allele? Data Page: Case 1 ( Hardy-Weinberg Equilibrium ) Initial Class Frequencies: AA ________ Aa________ aa_________ My initial genotype :_______________ F1 Genotype ______ F2 Genotype ______ F3 Genotype ______ F4 Genotype ______ F5 Genotype ______ Final Class Frequencies: AA ________ Aa________ aa_________ p _________ q __________ Case 2 ( Selection ) Initial Class Frequencies: AA ________ Aa________ aa_________ My initial genotype :_______________ F1 Genotype ______ F2 Genotype ______ F3 Genotype ______ F4 Genotype ______ F5 Genotype ______ Final Class Frequencies: AA ________ Aa________ aa_________ p _________ q __________ Biology 198 Hardy-Weinberg practice questions The Hardy-Weinberg formulas allow scientists to determine whether evolution has occurred. Any changes in the gene frequencies in the population over time can be detected. The law essentially states that if no evolution is occurring, then an equilibrium of allele frequencies will remain in effect in each succeeding generation of sexually reproducing individuals. In order for equilibrium to remain in effect (i.e. that no evolution is occurring) then the following five conditions must be met: 1. No mutations must occur so that new alleles do not enter the population. Obviously, the Hardy-Weinberg equilibrium cannot exist in real life. Some or all of these types of forces all act on living populations at various times and evolution at some level occurs in all living organisms. The Hardy-Weinberg formulas allow us to detect some allele frequencies that change from generation to generation, thus allowing a simplified method of determining that evolution is occurring. There are two formulas that must be memorized: p = frequency of the dominant allele in the population q = frequency of the recessive allele in the population p2 = percentage of homozygous dominant individuals q2 = percentage of homozygous recessive individuals 2pq = percentage of heterozygous individuals Individuals that have aptitude for math find that working with the above formulas is ridiculously easy. However, for individuals who are unfamiliar with algebra, it takes some practice working problems before you get the hang of it. Below I have provided a series of practice problems that you may wish to try out. Note that I have rounded off some of the numbers in some problems to the second decimal place: Remember the basic formulas: p = frequency of the dominant allele in the population q = frequency of the recessive allele in the population p2 = percentage of homozygous dominant individuals q2 = percentage of homozygous recessive individuals 2pq = percentage of heterozygous individuals 1. PROBLEM #1. A. The frequency of the "aa" genotype. Answer: 36%, as given in the problem itself. B. The frequency of the "a" allele. Answer: The frequency of aa is 36%, which means that q2 = 0.36, by definition. If q2 = 0.36, then q = 0.6, again by definition. Since q equals the frequency of the a allele, then the frequency is 60%. C. The frequency of the "A" allele. Answer: Since q = 0.6, and p + q = 1, then p = 0.4; the frequency of A is by definition equal to p, so the answer is 40%. D. The frequencies of the genotypes "AA" and "Aa." Answer: The frequency of AA is equal to p2, and the frequency of Aa is equal to 2pq. So, using the information above, the frequency of AA is 16% (i.e. p2 is 0.4 x 0.4 = 0.16) and Aa is 48% (2pq = 2 x 0.4 x 0.6 = 0.48). E. The frequencies of the two possible phenotypes if "A" is completely dominant over "a." Answers: Because "A" is totally dominate over "a", the dominant phenotype will show if either the homozygous "AA" or heterozygous "Aa" genotypes occur. The recessive phenotype is controlled by the homozygous aa genotype. Therefore, the frequency of the dominant phenotype equals the sum of the frequencies of AA and Aa, and the recessive phenotype is simply the frequency of aa. Therefore, the dominant frequency is 64% and, in the first part of this question above, you have already shown that the recessive frequency is 36%. 2. PROBLEM #2. Sickle-cell anemia is an interesting genetic disease. Normal homozygous individials (SS) have normal blood cells that are easily infected with the malarial parasite. Thus, many of these individuals become very ill from the parasite and many die. Individuals homozygous for the sickle-cell trait (ss) have red blood cells that readily collapse when deoxygenated. Although malaria cannot grow in these red blood cells, individuals often die because of the genetic defect. However, individuals with the heterozygous condition (Ss) have some sickling of red blood cells, but generally not enough to cause mortality. In addition, malaria cannot survive well within these "partially defective" red blood cells. Thus, heterozygotes tend to survive better than either of the homozygous conditions. If 9% of an African population is born with a severe form of sickle-cell anemia (ss), what percentage of the population will be more resistant to malaria because they are heterozygous (Ss) for the sickle-cell gene? Answer: 9% =.09 = ss = q2. To find q, simply take the square root of 0.09 to get 0.3. Since p = 1 - 0.3, then p must equal 0.7. 2pq = 2 (0.7 x 0.3) = 0.42 = 42% of the population are heterozygotes (carriers). 3. PROBLEM #3. There are 100 students in a class. Ninety-six did well in the course whereas four blew it totally and received a grade of F. Sorry. In the highly unlikely event that these traits are genetic rather than environmental, if these traits involve dominant and recessive alleles, and if the four (4%) represent the frequency of the homozygous recessive condition, please calculate the following: A. The frequency of the recessive allele. Answer: Since we believe that the homozygous recessive for this gene (q2) represents 4% (i.e. = 0.04), the square root (q) is 0.2 (20%). B. The frequency of the dominant allele. Answer: Since q = 0.2, and p + q = 1, then p = 0.8 (80%). 4. PROBLEM #4. Within a population of butterflies, the color brown (B) is dominant over the color white (b). And, 40% of all butterflies are white. Given this simple information, which is something that is very likely to be on an exam, calculate the following: A. The percentage of butterflies in the population that are heterozygous. B. The frequency of homozygous dominant individuals. Answers: The first thing you'll need to do is obtain p and q. So, since white is recessive (i.e. bb), and 40% of the butterflies are white, then bb = q2 = 0.4. To determine q, which is the frequency of the recessive allele in the population, simply take the square root of q2 which works out to be 0.632 (i.e. 0.632 x 0.632 = 0.4). So, q = 0.63. Since p + q = 1, then p must be 1 - 0.63 = 0.37. Now then, to answer our questions. First, what is the percentage of butterflies in the population that are heterozygous? Well, that would be 2pq so the answer is 2 (0.37) (0.63) = 0.47. Second, what is the frequency of homozygous dominant individuals? That would be p2 or (0.37)2 = 0.14. 5. PROBLEM #5. A rather large population of Biology instructors have 396 red-sided individuals and 557 tan-sided individuals. Assume that red is totally recessive. Please calculate the following: A. The allele frequencies of each allele. Answer: Well, before you start, note that the allelic frequencies are p and q, and be sure to note that we don't have nice round numbers and the total number of individuals counted is 396 + 557 = 953. So, the recessive individuals are all red (q2) and 396/953 = 0.416. Therefore, q (the square root of q2) is 0.645. Since p + q = 1, then p must equal 1 - 0.645 = 0.355. B. The expected genotype frequencies. Answer: Well, AA = p2 = (0.355)2 = 0.126; Aa = 2(p)(q) = 2(0.355)(0.645) = 0.458; and finally aa = q2 = (0.645)2 = 0.416 (you already knew this from part A above). C. The number of heterozygous individuals that you would predict to be in this population. Answer: That would be 0.458 x 953 = about 436. D. The expected phenotype frequencies. Answer: Well, the "A" phenotype = 0.126 + 0.458 = 0.584 and the "a" phenotype = 0.416 (you already knew this from part A above). E. Conditions happen to be really good this year for breeding and next year there are 1,245 young "potential" Biology instructors. Assuming that all of the Hardy-Weinberg conditions are met, how many of these would you expect to be red-sided and how many tan-sided? Answer: Simply put, The "A" phenotype = 0.584 x 1,245 = 727 tan-sided and the "a" phenotype = 0.416 x 1,245 = 518 red-sided ( or 1,245 - 727 = 518). 6. PROBLEM #6. A very large population of randomly-mating laboratory mice contains 35% white mice. White coloring is caused by the double recessive genotype, "aa". Calculate allelic and genotypic frequencies for this population. Answer: 35% are white mice, which = 0.35 and represents the frequency of the aa genotype (or q2). The square root of 0.35 is 0.59, which equals q. Since p = 1 - q then 1 - 0.59 = 0.41. Now that we know the frequency of each allele, we can calculate the frequency of the remaining genotypes in the population (AA and Aa individuals). AA = p2 = 0.41 x 0.41 = 0.17; Aa = 2pq = 2 (0.59) (0.41) = 0.48; and as before aa = q2 = 0.59 x 0.59 = 0.35. If you add up all these genotype frequencies, they should equal 1. 7. PROBLEM #7. After graduation, you and 19 of your closest friends (lets say 10 males and 10 females) charter a plane to go on a round-the-world tour. Unfortunately, you all crash land (safely) on a deserted island. No one finds you and you start a new population totally isolated from the rest of the world. Two of your friends carry (i.e. are heterozygous for) the recessive cystic fibrosis allele (c). Assuming that the frequency of this allele does not change as the population grows, what will be the incidence of cystic fibrosis on your island? Answer: There are 40 total alleles in the 20 people of which 2 alleles are for cystic fibrous. So, 2/40 = .05 (5%) of the alleles are for cystic fibrosis. That represents p. Thus, cc or p2 = (.05)2 = 0.0025 or 0.25% of the F1 population will be born with cystic fibrosis. 8. PROBLEM #8. You sample 1,000 individuals from a large population for the MN blood group, which can easily be measured since co-dominance is involved (i.e., you can detect the heterozygotes). They are typed accordingly: M MM 490 0.49 MN MN 420 0.42 N NN 90 0.09 Using the data provide above, calculate the following: A. The frequency of each allele in the population. Answer: Since MM = p2, MN = 2pq, and NN = q2, then p (the frequency of the M allele) must be the square root of 0.49, which is 0.7. Since q = 1 - p, then q must equal 0.3. B. Supposing the matings are random, the frequencies of the matings. Answer: This is a little harder to figure out. Try setting up a "Punnett square" type arrangement using the 3 genotypes and multiplying the numbers in a manner something like this: MM (0.49) MN (0.42) NN (0.09) MM (0.49) 0.2401* 0.2058 0.0441 MN (0.42) 0.2058 0.1764* 0.0378 NN (0.09) 0.0441 0.0378 0.0081* C. Note that three of the six possible crosses are unique (*), but that the other three occur twice (i.e. the probabilities of matings occurring between these genotypes is TWICE that of the other three "unique" combinations. Thus, three of the possibilities must be doubled. D. MM x MM = 0.49 x 0.49 = 0.2401 MM x MN = 0.49 x 0.42 = 0.2058 x 2 = 0.4116 MM x NN = 0.49 x 0.09 = 0.0441 x 2 = 0.0882 MN x MN = 0.42 x 0.42 = 0.1764 MN x NN = 0.42 x 0.09 = 0.0378 x 2 = 0.0756 NN x NN = 0.09 x 0.09 = 0.0081 E. The probability of each genotype resulting from each potential cross. Answer: You may wish to do a simple Punnett's square monohybrid cross and, if you do, you'll come out with the following result: MM x MM = 1.0 MM MM x MN = 0.5 MM 0.5 MN MM x NN = 1.0 MN MN x MN = 0.25 MM 0.5 MN 0.25 NN MN x NN = 0.5 MN 0.5 NN NN x NN = 1.0 NN 9. PROBLEM #9. Cystic fibrosis is a recessive condition that affects about 1 in 2,500 babies in the Caucasian population of the United States. Please calculate the following. A. The frequency of the recessive allele in the population. Answer: We know from the above that q2 is 1/2,500 or 0.0004. Therefore, q is the square root, or 0.02. That is the answer to our first question: the frequency of the cystic fibrosis (recessive) allele in the population is 0.02 (or 2%). B. The frequency of the dominant allele in the population. Answer: The frequency of the dominant (normal) allele in the population (p) is simply 1 - 0.02 = 0.98 (or 98%). C. The percentage of heterozygous individuals (carriers) in the population. Answer: Since 2pq equals the frequency of heterozygotes or carriers, then the equation will be as follows: 2pq = (2)(.98)(.02) = 0.04 or 1 in 25 are carriers. 10. PROBLEM #10. In a given population, only the "A" and "B" alleles are present in the ABO system; there are no individuals with type "O" blood or with O alleles in this particular population. If 200 people have type A blood, 75 have type AB blood, and 25 have type B blood, what are the alleleic frequencies of this population (i.e., what are p and q)? Answer: To calculate the allele frequencies for A and B, we need to remember that the individuals with type A blood are homozygous AA, individuals with type AB blood are heterozygous AB, and individuals with type B blood are homozygous BB. The frequency of A equals the following: 2 x (number of AA) + (number of AB) divided by 2 x (total number of individuals). Thus 2 x (200) + (75) divided by 2 (200 + 75 + 25). This is 475/600 = 0.792 = p. Since q is simply 1 - p, then q = 1 - 0.792 or 0.208. 11. PROBLEM #11. The ability to taste PTC is due to a single dominate allele "T". You sampled 215 individuals in biology, and determined that 150 could detect the bitter taste of PTC and 65 could not. Calculate all of the potential frequencies. Answer: First, lets go after the recessives (tt) or q2. That is easy since q2 = 65/215 = 0.302. Taking the square root of q2, you get 0.55, which is q. To get p, simple subtract q from 1 so that 1 - 0.55 = 0.45 = p. Now then, you want to find out what TT, Tt, and tt represent. You already know that q2 = 0.302, which is tt. TT = p2 = 0.45 x 0.45 = 0.2025. Tt is 2pq = 2 x 0.45 x 0.55 = 0.495. To check your own work, add 0.302, 0.2025, and 0.495 and these should equal 1.0 or very close to it. This type of problem may be on the exam. 12. PROBLEM #12. (You will not have this type of problem on the exam) What allelic frequency will generate twice as many recessive homozygotes as heterozygotes? Answer: We need to solve for the following equation: q2 (aa) = 2 x the frequency of Aa. Thus, q2 (aa) = 2(2pq). Or another way of writing it is q2 = 4 x p x q. We only want q, so lets trash p. Since p = 1 - q, we can substitute 1 - q for p and, thus, q2 = 4 (1 - q) q. Then, if we multiply everything on the right by that lone q, we get q2 = 4q - 4q2. We then divide both sides through by q and get q = 4 - 4q. Subtracting 4 from both sides, and then q (i.e. -4q minus q = -5q) also from both sides, we get -4 = -5q. We then divide through by -5 to get -4/-5 = q, or anotherwards the answer which is 0.8 =q. I cannot imagine you getting this type of problem in this general biology course although if you take algebra good luck XICE MUSIC ON YOU TUBE ...
d776637a0d84fbfd	Saturday, November 29, 2014 Letter to Hom on Christian Salvation Alright, now we do a bit of theology! Aren't you excited? From math to quantum to cosmology to the heavenly Father of Jesus! Quite a healthy wide range of things for the spirit to grow, wouldn't you say? ;-) This specific post/thread is for dear brother Hom on twitter who wrote me to discuss and share my thoughts on the issue of whether a Christian who is saved can lose or keep his/her salvation after numerous sins post their commitment to faith in Christ. In his tweets to me, Hom said: "I was taught that in my early years and I "kinda" maintain that belief. Yet I know NO passages in the Biblical text that make the point clear. Can you help?" The belief that Hom alluded to here is the view I expressed in our twitter discussion that there can be cases of alleged Christians who could 'lose' their salvation, or grace, by the continued sinful lives that they can lead subsequently -- even as replete that life may be with sins of egregious proportions (such as committing genocide, indulging in the major sins stated in the Bible, etc, without naming them all). Some believers are of the view that once you have given your life to Christ as your Savior, you are also saved of your subsequent sins too, no matter how big or small they may be after your pronounced belief. Consequently, even if you live a life of crime, rape, murder, stealing, lying, and any sin you can imagine (so much for the Ten Commandments!), you are still saved because you cannot lose it. So consequently, Hitler, being a Christian would still be saved by grace even after all the inhumanity and genocide that he has caused to tens of millions of human beings. So, you are free to do as you please, sin to whatever degree that you wish (to whatever extreme), and you are assured that you are saved and will go to heaven. Of course, I cannot accept such a view. (And I never have.) That is not at all how I read the Bible, especially the teachings of the New Testament as to how Christians should conduct their lives. (The Hebrew Scriptures already prescribe divine punishment to various of these sins even for alleged believers of the Mosaic community!) Indeed, St Paul has dedicated a fair amount of his time, travels, and writings to some churches in Asia Minor where he heard that numerous egregious sins continue to be committed by (alleged) Christian members who believed that they had been saved so that now they can do whatever they wanted. That is why St Paul explicitly emphasized: The Letter of James also teaches that a Christian is responsible to showing how his/her life conduct is to be exemplified through their behavior, actions, or works -- backed, of course, through their faith in Christ. The Lord Jesus taught us to judge a tree by the fruit that it bears - very eloquently, simply, and without any complicated theology. When the tree produces bad fruit, what is to be done unto it? The Master said "Every tree that does not bear good fruit is cut down and thrown into the fire. 20 Thus, by their fruit you will recognize them." (Matthew 7:19-20, NIV.) Clearly, such a tree cannot then have been saved or ascribed salvation to begin with, even if that tree lead others to believe that it was a good tree. Therefore, in extreme cases like Hitler, or anyone allegedly claiming to be a Christian, such continued bearing of bad fruit would at the very least cast serious doubt on their claims to being believers. Would we believe someone who claims allegiance to the US Constitution only to see that individual violate its articles and laws time and again (even in extreme ways)? I would certainly at least question them. So we're not saying that they had grace and then lost it, but that maybe they didn't have grace in the first place (as we assumed by taking their claim at face value). There are many examples like the above in the Bible that tell me that the view I expressed is far more reasonable (or at least less problematic) than the view that the Christian community could be allowed to harbor such horrific individuals who do such harm to the faith. If Christians are serious about Jesus' teaching, they are responsible to act it out in their hearts and minds as well as with their fellow man, their neighbor. I hope that I have shared my thoughts with you, Hom, in a gentle spirit, even as I am no Bible teacher nor do I have a degree in theology! But I speak as just one believer, sharing my thoughts and experiences. Ultimately, Jesus knows the full precise answers. As St Paul said, we know in part and we prophecy in part, and in another place he says "For now we see through a glass, darkly." Yours in Christ, Saturday, November 8, 2014 A game with $\pi$ Here's an image of something I wrote down, took a photo of, and posted here for you. It's a little game you can play with any irrational number. I took $\pi$ as an example. You just learned about an important math concept/process called continued fraction expansions. With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish. As an example, if you truncate the above last expansion where the 292 appears (so you omit the "1 over 292" part) you get the rational number $\frac{355}{113}$ which approximates $\pi$ to 6 decimal places. (Better than $\frac{22}{7}$.) You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers. Exercise: for the square root of 2, show that the sequence you get is 1, 2, 2, 2, 2, ... (all 2's after the 1). For the square root of 3 the continued fraction sequence is (so it starts with 1 and then the pair "1, 2" repeat periodically forever). Monday, August 18, 2014 3-4-5 complex number has Infinite order This is a good exercise/challenge with complex numbers. Wednesday, July 30, 2014 Multiplying Spaces! Believe it or not, in Math we can not only multiply numbers but we can multiply spaces! We can multiply two spaces to get bigger spaces - usually of bigger dimensions. The 'multiplication' that I'm referring to here is known as Tensor products. The things/objects in these spaces are called tensors. (Tensors are like vectors in a way.) Albert Einstein used tensors in his Special and his General Theory of Relativity (his theory of gravity). Tensors are also used in several branches of Physics, like the theory of elasticity where various stresses and forces act in various ways. And definitely in quantum field theory. It may sound crazy to say you can "multiply spaces," as we would multiply numbers, but it can be done in a precise and logical way. But here I will spare you the technical details and try to manage to show you the idea that makes it possible to do. Q. What do you mean by 'spaces'? I mean a set of things that behave like 'vectors' so that you can add two vectors and get a third vector, and where you can scale a vector by any real number. The latter is called scalar multiplication, so if $v$ is a vector, you can multiply it by $0.23$ or by $-300.87$ etc and get another vector: $0.23v$, $-300.87v$, etc.) The technical name is vector space. A straight line that extends in both directions indefinitely would be a good example (an Euclidean line). Another example is you take the $xy$-plane, 2D-space or simply 2-space, or you can take $xyz$-space, or if you like you can take $xyzt$-spacetime known also as Minkowski space which has 4 dimensions. Q. How do you 'multiply' such spaces? First, the notation. If $U$ and $V$ are spaces, their tensor product space is written as $U \otimes V$. (It's the multiplication symbol with a circle around it.) If this is to be an actual multiplication of spaces there is one natural requirement we would want. That the dimensions of this tensor product space $U \otimes V$ should turn out to be the multiplication of the dimensions of U and of V. So if $U$ has dimension 2 and $V$ has dimension 3, then $U \otimes V$ ought to have dimension $2 \times 3 = 6$. And if $U$ and $V$ are straight lines, so each of dimension 1, then $U \otimes V$ will also be of dimension 1. Q. Hey, wait a second, that doesn't quite answer my question. Are you dodging the issue!? Ha! Yeah, just wanted to see if you're awake! ;-) And you are! Ok, here's the deal without going into too much detail. We pointed out above how you can scale vectors by real numbers. So if you have a vector $v$ from the space $V$ you can scale it by $0.23$ and get the vector $0.23v$. Now just imagine if we can scale the vector $v$ by the vectors in the other space $U$! So if $u$ is a vector from $U$ and $v$ a vector from $V$, then you can scale $v$ by $u$ to get what we call their tensor product which we usually write like $u \otimes v$. So with numbers used to scale vectors, e.g. $0.23v$, we could also write it as $0.23 \otimes v$. But we don't normally write it that way when numbers are involved, only when non-number vectors are. Q. So can you also turn this around and refer to $u \otimes v$ as the vector $u$ scaled by the vector $v$? Absolutely! So we have two approaches to this and you can show (by a proof) that the two approaches are in fact equivalent. In fact, that's what gives rise to a theorem that says Theorem. $U \otimes V$ is isomorphic to $V \otimes U$. (In Math, the word 'isomorphism' gives a precise meaning to what I mean by 'equivalent'.) Anyway, the point has been made to describe multiplying spaces: you take their vectors and you 'scale' those of one space by the vectors of the other space. There's a neat way to actually see and appreciate this if we use matrices as our vectors. (Yes, matrices can be viewed as vectors!) Matrices are called arrays in computer science. One example / experiment should drive the point home: Let's take these two $2 \times 2$ matrices $A$ and $B$: $A = \begin{bmatrix} 2 & 3 \\ -1 & 5 \end{bmatrix}, \ \ \ \ \ \ \ B = \begin{bmatrix} -5 & 4 \\ 6 & 7 \end{bmatrix}$ To calculate their tensor product $A \otimes B$, you can take $B$ and scale it by each of the numbers contained in $A$! Like this: $A\otimes B = \begin{bmatrix} 2B & 3B \\ -1B & 5B \end{bmatrix}$ If you write this out you will get a 4 x 4 matrix when you plug B into it: $A\otimes B = \begin{bmatrix} -10 & 8 & -15 & 12 \\ 12 & 14 & 18 & 21 \\ 5 & -4 & -25 & 20 \\ -6 & -7 & 30 & 35 \end{bmatrix}$ Oh, and 4 times 4 is 16, yes so the matrix $A\otimes B$ does in fact have 16 entries in it! Check! Q. You could also do this the other way, by scaling $A$ using each of the numbers in $B$, right? Right! That would then give $B\otimes A$. When you do this you will get different matrices/arrays but if you look closely you'll see that they have the very same set of numbers except that they're permuted around in a rather simple way. How? Well, if you switch the two inner columns and the two inner rows of $B\otimes A$ you will get exactly $A\otimes B$! Try this experiment with the above $A$ and $B$ examples by working out $B\otimes A$ as we've done. This illustrates what we mean in Math by 'isomorphism': that even though the results may look different, they are actually related to one another in a sort of 'linear' or 'algebraic' fashion. Ok, that's enough. We get the idea. You can multiply spaces by scaling their vectors by each other. Amazing how such an abstract idea turns out to be a powerful tool in understanding the geometry of spaces, in Relativity Theory, and also in quantum mechanics (quantum field theory). Warm Regards, Saturday, July 26, 2014 Bertrand's "postulate" and Legendre's Conjecture Bertrand's "postulate" states that for any positive integer $n > 1$, you can always find a prime number $p$ in the interval $n < p < 2n$. It use to be called "postulate" until it became a theorem when Chebyshev proved it in 1850. (I saw this while browsing thru a group theory book and got interested to read up a little more.) What if instead of looking at $n$ and $2n$ you looked at consecutive squares? So for example you take a positive integer $n$ and you ask whether we can always find at least one prime number between $n^2$ and $(n+1)^2$. Turns out this is a much harder problem and it's still an open question called: Legendre's Conjecture. For each positive integer $n$ there is at least one prime $p$ such that $n^2 < p < (n+1)^2$. People have used programming to check this for large numbers and have always found such primes, but no proof (or counterexample) is known. If you compare Legendre's with Bertrand's you will notice that $(n+1)^2$ is a lot less than $2n^2$. (At least for $n > 2$.) In fact, the asymptotic ratio of the latter divided by the former is 2 (not 1) for large $n$'s. This shows that the range of numbers in the Legendre case is much narrower than in Bertrand's. The late great mathematician Erdos proved similar results by obtaining k primes in certain ranges similar to Bertand's. A deep theorem related to this is the Prime Number Theorem which gives an asymptotic approximation for the number of primes up to $x$. That approximating function is the well-known $x/\ln(x)$. Great sources: [1] Bertrand's "postulate" [2] Legendre's Conjecture (See also wiki's entries under these topics.) Friday, July 25, 2014 Direct sum of finite cyclic groups The purpose of this post is to show how a finite direct sum of finite cyclic groups $\Large \Bbb Z_{m_1} \oplus \Bbb Z_{m_2} \oplus \dots \oplus \Bbb Z_{m_n}$ can be rearranged so that their orders are in increasing divisional form: $m_1\|m_2\|\dots \| m_n$. We use the fact that if $p, q$ are coprime, then $\large \Bbb Z_p \oplus \Bbb Z_q = \Bbb Z_{pq}$. (We'll use equality $=$ for isomorphism $\cong$ of groups.) Let $p_1, p_2, \dots p_k$ be the list of prime numbers in the prime factorizations of all the integers $m_1, \dots, m_n$. Write each $m_j$ in its prime power factorization $\large m_j = p_1^{a_{j1}}p_2^{a_{j2}} \dots p_k^{a_{jk}}$. Therefore $\Large \Bbb Z_{m_j} = \Bbb Z_{p_1^{a_{j1}}} \oplus \Bbb Z_{p_2^{a_{j2}}} \oplus \dots \oplus \Bbb Z_{p_k^{a_{jk}}}$ and so the above direct sum $\large \Bbb Z_{m_1} \oplus \Bbb Z_{m_2} \oplus \dots \oplus \Bbb Z_{m_n}$ can be written out in matrix/row form as the direct sum of the following rows: $\Large\Bbb Z_{p_1^{a_{11}}} \oplus \Bbb Z_{p_2^{a_{12}}} \oplus \dots \oplus \Bbb Z_{p_k^{a_{1k}}}$ $\Large\Bbb Z_{p_1^{a_{21}}} \oplus \Bbb Z_{p_2^{a_{22}}} \oplus \dots \oplus \Bbb Z_{p_k^{a_{2k}}}$ $\Large \vdots$ $\Large\Bbb Z_{p_1^{a_{n1}}} \oplus \Bbb Z_{p_2^{a_{n2}}} \oplus \dots \oplus \Bbb Z_{p_k^{a_{nk}}}$ Here, look at the powers of $p_1$ in the first column. They can be permuted / arranged so that their powers are in increasing order. The same with the powers of $p_2$ and the other $p_j$, arrange their groups so that the powers are increasing order. So we get the above direct sum isomorphic to $\Large\Bbb Z_{p_1^{b_{11}}} \oplus \Bbb Z_{p_2^{b_{12}}} \oplus \dots \oplus \Bbb Z_{p_k^{b_{1k}}}$ $\Large\Bbb Z_{p_1^{b_{21}}} \oplus \Bbb Z_{p_2^{b_{22}}} \oplus \dots \oplus \Bbb Z_{p_k^{b_{2k}}}$ $\Large \vdots$ $\Large\Bbb Z_{p_1^{b_{n1}}} \oplus \Bbb Z_{p_2^{b_{n2}}} \oplus \dots \oplus \Bbb Z_{p_k^{b_{nk}}}$ where, for example, the exponents $b_{11} \le b_{21} \le \dots \le b_{n1}$ are a rearrangement of the numbers $a_{11}, a_{21}, \dots, a_{n1}$ (in the first column) in increasing order. Do the same for the other columns. Now put together each of these rows into cyclic groups by multiplying their orders, thus $\Large\ \ \Bbb Z_{N_1}$ $\Large \oplus \Bbb Z_{N_2}$ $\Large \vdots$ $\Large \oplus \Bbb Z_{N_n}$ $\large N_1 = p_1^{b_{11}} p_2^{b_{12}} \dots p_k^{b_{1k}}$, $\large N_2 = p_1^{b_{21}} p_2^{b_{22}} \dots p_k^{b_{2k}}$, $\large \vdots$ $\large N_n = p_1^{b_{n1}} p_2^{b_{n2}} \dots p_k^{b_{nk}}$. In view of the fact that the $b_{1j} \le b_{2j} \le \dots \le b_{nj}$ is increasing for each $j$, we see that $N_1 \| N_2 \| \dots \| N_n$, as required. $\blacksquare$ Latex on Blogger LaTeX work here if you add the small script below: Short exact sequence: $\large 0 \to H \to G \to G/H \to 0$ and an integral $\large \int \sqrt{x} dx$. Each finite Abelian group is isomorphic to a direct sum of cyclic groups where $m_1\|m_2\|\dots \| m_n$. (One of my favorite results from group theory.) Thanks to a gentle soul's responding at tex.stackexchange: To get LaTeX to work on Blogger, go to Design, then to "edit HTML", then to "edit template". In the HTML file insert the following script right after where it says < head >: extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"], tex2jax: { Tuesday, June 17, 2014 Richard Feynman on Erwin Schrödinger I thought it is interesting to see what the great Nobel Laureate physicist Richard Feynman said about Erwin Schrödinger's attempts to discover the famous Schrödinger equation in quantum mechanics: When Schrödinger first wrote it [his equation] down, He gave a kind of derivation based on some heuristic Arguments and some brilliant intuitive guesses. Some Of the arguments he used were even false, but that does Not matter; the only important thing is that the ultimate Equation gives a correct description of nature. -- Richard P. Feynman (The Feynman Lectures on Physics, Vol. III, Chapter 16, 1965.) It has been my experience in reading physics books that this sort of `heuristic' reasoning is part of doing physics. It is a very creative (sometimes not logical!) art with mathematics in attempting to understand the physical world. Dirac did it too when he obtained his Dirac equation for the electron. Tuesday, June 3, 2014 Entangled and Unentangled States Let's take a simple example of electron spin states. The reason it is 'simple' is that you only have two states to consider: either an electron's spin is 'up' or it is 'down'. We can use the notation ↑ to stand for the state that its spin is 'up' and the down arrow ↓ to indicate that its spin is down. If we write, say, two arrows together like this ↑↓ it means we have two electrons, the first one has spin up and the second one spin down. So, ↓↓ means both particles have spin down. Now one way in which the two particles can be arranged experimentally is in an entangled form. One state that would describe such a situation is a wavefunction state like this: Ψ = ↑↓ - ↓↑. This is a superposition state combining (in some mysterious fashion!) two basic states: the first one ↑↓ describes is a situation where the first particle has spin up and the second spin down, and the second state ↓↑ describes a situation where the first particle has spin down and the second particle has spin up. But when the two particles are in the combined superposition state Ψ (as above), it's in some sort of mix of those two scenarios. Like the case of the cat that is half dead and half alive! :-) Why exactly is this state Ψ 'entangled' -- and what exactly do we mean by that? Well, it means that if you measure the spin of the first electron and you discover that its spin is down ↓, let's say, that picks out the part "↓↑" of the state Ψ! And this means that the second electron must have spin up! They're entangled! They're tied up together so knowing some spin info about one tells you the spin info of the other - instantly! This is so because the system has been set up to be in the state described by Ψ. Now what about an unentangled state? What would that look like for our 2-electron example. Here's one: Φ = ↑↑ + ↓↓ + ↑↓ - ↓↑. Here this state is made up of two electrons that can have both their spins up (namely, ↑↑), both their spins down (↓↓), or they could be in the state Ψ (consisting of the opposite spins). In this wavefunction state Φ (called a "product state" which are generally not entangled), if you measure the spin, say, of the first electron and you find that it is up ↑, then what about the spin of the other one? Well, here you have two possibilities, namely ↑↑ and ↑↓ involved in Φ, which means that the second electron can be either in the up spin or the down spin. No entanglement, no correlation as in the Ψ case above. Knowing the spin state of one particle doesn't tell you what the other one has to be. You can illustrate the same kind of examples with photon polarization, so you can have their polarizations entangeled or unentangled - depending on how the system is set up by us or by nature. Thursday, May 29, 2014 Periodic Table of Finite Simple Groups Chemistry has its well known and fantastic periodic table of elements. In group theory we have an analogous 'periodic table' that describes the classification of the finite simple groups (shown below). (A detailed PDF is available.) It summarizes decades worth of research work by many great mathematicians to determine all the finite simple groups. It is an amazing feat! And a work of great beauty. Groups are used in studies of symmetry - in Math and in the sciences, especially in Physics and Chemistry. A group is basically a set $G$ of objects that can be "combined", so that two objects $x, y$ in $G$ produce a third object $x \ast y$ in G. Loosely, we refer to this combination or operation as a 'multiplication' (or it could be an 'addition'). This operation has to have three basic rules: 1. The operation must associative, i.e. $(x\ast y)\ast z = x\ast(y\ast z)$ for all objects $x, y, z$ in $G$. 2. $G$ contains a special object $e$ such that $e\ast x = x = x\ast e$ for all objects $x$ in $G$. 3. Each object $x$ in $G$ has an associated object $y$ such that $x\ast y = e = y\ast x$. Condition 2 says that the object $e$ has no effect on any other object - it is called the "identity" object. It behaves much like the real number 0 in relation to the addition + operation since $x + 0 = x = 0 + x$ for all real numbers. (Here, in this example, $\ast$ is addition + and e is 0.) As a second example, $e$ could also be the real number 1 if $ast$ stood for multiplication (in which case we take $G$ to be all real numbers except 0). Condition 3 says that each object has an 'inverse' object. Or, that each object could be 'reversed'. It turns out that you can show that the $y$ in condition 3 is unique for each $x$ and is instead denote by $y = x^{-1}$. The commutative property -- namely that $x\ast y = y\ast x$ -- is not assumed, so almost all of the groups in the periodic table do not have this property. (Groups that do have this property are called Abelian or commutative groups.) The Abelian simple groups are the 'cyclic' ones that appear in the right most column of the table. (Notice that their number of objects is a prime number $2, 3, 5, 7, 11, \dots$ etc.) The periodic table lists all of the finite simple groups. So they are groups as we just described. And they are finite in that each group $G$ has finitely many elements. (There are many infinite groups used in physics but these aren't part of the table.) But now what are 'simple' groups? Basically, they are ones that cannot be 'made up' of yet smaller groups or other groups. (More technically, a group $G$ is said to be simple when there isn't a nontrivial normal subgroup $H$ inside of $G$ -- i.e., $H$ is a subset of $G$ and is also a group under the same $\ast$ operation of $G$, and further $xHx^{-1}$ is contained in $H$ for any object $x \in G$.) So a simple group is like a basic object that cannot be "broken down" any further, like an 'atom', or a prime number. One of the deepest results in the theory of groups that helped in this classification is the Feit-Thompson Theorem which says: each group with an odd number of objects is solvable. (The proof was written in the 1960s and is over 200 pages - published in Pacific Journal of Mathematics.) Wednesday, May 28, 2014 St Augustine on the days of Creation In his City of God, St Augustine said in reference to the first three days of creation: "What kind of days these were it is extremely difficult, or perhaps impossible for us to conceive, and how much more to say!" (See Chapter 6.) So a literal 24 hour day seemed not the plain meaning according to St Augustine. He does not venture to speculate but leaves the matter open. That was long long before modern science. Sunday, May 25, 2014 Experiment on Dark Matter yields nothing The most sensitive experiment to date by the Large Underground Xenon (LUX), which was designed to detect dark matter, has not registered any sign of the substance. See Nature article: No sign of dark matter in underground experiment Eugenie Samuel Reich As a result, some scientists are considering other (exotic) possibilities: Dark-matter search considers exotic possibilities, by Clara Moskowitz Saturday, May 17, 2014 This is my first (test) post on this blog. I'm just examining it, maybe to add a few thoughts and ideas now and then. Thank you for reading.
0e8fb1c35744637c	Electronic-Structure Calculation for exciting helium 0. General Preparations Units in exciting: By default, all quantities in the exciting code are given atomic units: Energies in Hartree, lengths in Bohr, etc. In case other units are desirable, they can be converted using templates as a post-processing to exciting's standard output. Very important: Before starting, the following shell variable must be set by the user: EXCITINGROOT = Directory where exciting has been downloaded, e.g.: /home/user/exciting . The setting of this variable can be done in a bash shell by typing (from now on the symbol $ will indicate the shell prompt): $ export EXCITINGROOT=/local_path_to_exciting_root Please note: In the input-files shown on this page, the placeholder EXCITINGROOT always needs to be replaced by the value of the $EXCITINGROOT variable. 1. Electronic structure of silver 1.1 Groundstate calculation The starting point of a groundstate calculation is the crystal structure, only. At the beginning of a groundstate calculation, an initial electron density is generated, which is obtained from a superposition of atomic densities. Thus, this initial electron density lacks electron-electron and electron-ion interactions between atoms and is normally a rather crude approximation of the density. Then, the calculation iteratively goes through the following steps: 1. Determine the potential from the electron density. 2. Solve of the Kohn-Sham (KS) equations to get the eigenfunctions and eigenvalues as well as the total energy. 3. Calculate the electron density from the KS eigenfunctions. 5. Start again with (1). To prepare your calculation, create a new, empty directory named Ag/ somewhere on your filesystem. In this directory, save the following lines as input.xml (do not forget to replace EXCITINGROOT by the value of the $EXCITINGROOT variable!): <?xml-stylesheet href="http://xml.exciting-code.org/inputfileconverter/inputtohtml.xsl" type="text/xsl"?> <input xsi:noNamespaceSchemaLocation="EXCITINGROOT/xml/excitinginput.xsd" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsltpath="EXCITINGROOT/xml/"> <structure speciespath="EXCITINGROOT/species"> <crystal scale="7.7201"> <basevect>0.5 0.5 0.0</basevect> <basevect>0.5 0.0 0.5</basevect> <basevect>0.0 0.5 0.5</basevect> <species speciesfile="Ag.xml"> <atom coord="0.0 0.0 0.0" /> <groundstate ngridk="8 8 8"></groundstate> If the visualization program XCrySDen is set up appropriately (find here how to do this: XcrysdenExcitingSetup) you can visualize the structure in an exciting input.xml executing $ xcrysden --exciting input.xml After this, start the groundstate calculation by executing the following command in the Ag/ directory: $ $EXCITINGROOT/bin/excitingser The calculation should roughly take 1 minute. During the calculation, output files are created, which contain all kind of information on your material system and on the calculation. Some of the output files are already created at the beginning of the calculation and will not be changed anymore during the run. The most important among them are: filename description SYMCRYS.OUT Information on the symmetry operations of the crystal; more symmetry information is found in the files SYMT2.OUT, SYMSITE.OUT, SYMMULT_TABLE.OUT, SYMMULT.OUT, SYMLAT.OUT, SYMINV.OUT, and SYMGENR.OUT. IADIST.OUT Interatomic distances; useful to check the correctness of an input file. EQATOMS.OUT Information on equivalency of atoms due to the crystal symmetry. geometry.OUT.xml Structural information on your system. This will often be identical to the element structure in your input file, but may differ in case you set the groundstate attribute primcell = "true". Other files are updated or extended in each iteration, and contain information about the scf calculation. Here are the most important ones: filename description info.xml "Master" output file in XML format, containing the essential information on the material system, on the parameters of the calculation, on the results (total energy, energy contributions, charge contributions, Fermi energy …) of each iteration, and more … INFO.OUT Extended version of info.xml, displayed as plain text file. TOTENERGY.OUT Total energy (Hartree); each line corresponds to one iteration. EFERMI.OUT Fermi energy (Hartree). FERMIDOS.OUT Density of states at the Fermi level (states/unit cell/Hartree); each line corresponds to one iteration. EVALCORE.OUT Eigenvalues (=energy levels) of the core states. 1. How many iterations did the calculation go through? 3. What was the change in total energy between the • first two iterations? • last two iterations? 4. What is the Fermi energy of the system? 5. How many occupied valence bands are there in the system? 6. How much charge is there inside the muffin-tin sphere, and how much is found in the interstitial? 1.2 Density of states To calculate it, you need to do the following to simple modifications in input.xml: 1. add the attribute do="skip" to the xml-element <groundstate>; 2. add the element <properties> … </properties> after the <groundstate> element; 3. insert the subelement <dos> … </dos> into the element <properties>; 4. add the attribute nsmdos="1" to the element <dos> (see Input Reference for more details). The corresponding part of the input.xml should now look like this: <groundstate ngridk="8 8 8" do="skip"></groundstate> <dos nsmdos="1"> Then execute excitingser again on the command line: $ $EXCITINGROOT/bin/excitingser This time, the program will produce the following files: • TDOS.OUT … total DOS • PDOS_S01_A0001.OUT … partial density of states for each atom, l and m quantum number. • IDOS.OUT … interstitial DOS • dos.xml … all densities of states assembled in XML format. Let's have a look at the total DOS in TDOS.OUT. To visualize it, execute $ xmgrace TDOS.OUT This will open the plotting tool xmgrace and display the total density of states. The units in this plot are • Energy (Hartree) for the x axis • DOS (states/unit cell/Hartree) for the y axis. Inside xmgrace, you can change the plot appearance in any way you want, zoom in to the see details of the DOS, or produce a figure in any format you like (ps, jpg, png, etc. …). See Xmgrace Quickstart for more info. 1. We were using the attributes do="skip" for the element <groundstate> for generating the DOS after the groundstate SCF run. Find out why, by searching for the element groundstate in Input Reference, and then going down to its attribute do. 2. Consider the electronic configuration (e.g., see http://www.webelements.com/silver/atoms.html -> Ground state electron configuration) and the electron binding energies (http://www.webelements.com/silver/orbital_properties.html -> Electron binding energies) of atomic silver: 1. Which states do you expect to form the valence band in the Ag compound? 2. Which states are there with energy greater than 3 Hartree? (The conversion factor between Hartree and eV is found in Input Reference.) 3. What about the silver 4p states? Try to find out where they are located in the Ag compound. To do so, you will need to use the attribute winddos of the element <dos> — check out Input Reference to see why, and how this works. 4. Do the following modifications inside xmgrace (see Xmgrace Quickstart for help …): • change the x-region of the plot to -0.4 — 0.25 • autoscale x axis • add axes labels • add a title • increase the line-width of the DOS curve to 3 • change the colour of the DOS curve to red • add a legend • generate the file Ag_dos.png. 1.3 Band structure To calculate the band structure of silver, insert the subelement <bandstructure> in the element <properties> with the following specifications: <path steps="100"> <point coord="0.75000 0.50000 0.25000" label="W" ></point> <point coord="0.50000 0.50000 0.50000" label="L" ></point> <point coord="0.00000 0.00000 0.00000" label="GAMMA"></point> <point coord="0.50000 0.50000 0.00000" label="X" ></point> <point coord="0.75000 0.37500 0.37500" label="K" ></point> As you may have realized, we have removed the subelement <dos> now. Now execute excitingser again on the command line: $ $EXCITINGROOT/bin/excitingser $ xsltproc $EXCITINGROOT/xml/visualizationtemplates/xmlband2agr.xsl bandstructure.xml $ xmgrace Ag_bandstructure.agr The result should look like this: 1. Use http://www.cryst.ehu.es/ -> Space Groups Retrieval Tools -> KVEC to find out about the location of the special k-points within the Brillouin zone. The spacegroup of Ag is 225, Fm-3m. Select Choose to choose the corresponding spacegroup, and then click [ Brillouin zone ] to see the Brillouin zone with the special k-points. • What trend do you see relating the band width to the energy? 1.4 Convergence issues — crucial parameters in a calculation A groundstate calculation using a DFT code fundamentally depends on 2 parameters: • the mesh of k-points (<groundstate> attribute: ngridk); • the size of the basis set for expanding the wave function (<groundstate> attribute: rgkmax). In order to be able to rely on your calculation, you need to understand what these two parameters mean and make a convergence tests for each new system you are dealing with. Let us first have a closer look at the meaning of these two parameters: The k-point mesh From basic considerations of solid-state theory, it can be shown that the Schrödinger (or, in our case, Kohn-Sham) equation for a periodic system has a bunch of solutions, where each solution is characterized by a vector k in reciprocal space. This k-vector is essentially related to the periodicity of the corresponding solution of the Schrödinger equation: Roughly speaking, the solution will have the same value at coordinates in real space that, along the direction of k, have a distance of 2π/\|k\|. Many properties of a solid, including the total energy, are represented as integrals, performed over all possible k-vectors. Obviously, the direct calculation of these integrals is very demanding (one should calculate the solution of the Schrödinger equation for a huge number of k-points). Therefore, the integrals are approximated by sums on a set of k-points distributed on a finite grid. The spacing of the points on the grid is a measure of the accuracy of the calculation of the integrals which also depends on how fast the integrand (the solution of the Schrödinger equation) varies by changing k. The challenge is to find a good number of k-points: Large enough to capture the physical properties of your system well, but small enough to keep calculations as fast as possible. In fact, the number of k-points in a given direction expresses, over how many unit cells electrons feel each other in this direction. This has two important consequences: 1. The larger the unit cell, the smaller is the required k-point mesh. As a rool of thumb, for a given kind of material, two calculations have the same quality with respect to the k mesh, if the product (no. of atoms)*(no. of k-points) is the same for both of them. For example: If you create a supercell of your starting structure by doubling it in z direction, it is enough to use only half of the k-points along z. 2. Systems with longer-range interactions need larger k-point meshes. In metals you need a large number of k-points, since the conduction electrons are delocalized over the whole system, i.e. the interactions are very long-ranged. Moreover, for a good description of the system it is important to know exactly where the conduction bands cross the Fermi level, which also requires a dense k mesh. In contrast, semiconductors and insulators usually have much more localized electronic states, and a gap between valence and conduction bands. Thus, the number of k-points required for a good calculation is much smaller. While for a metal with 1 atom per unit cell, up to ngridk="20 20 20" or larger may be necessary to get reliable physical properties, for a semiconductor with the same unit-cell size ngridk="4 4 4" may already do a good job. If you want to produce high-quality results for publication, a careful convergence test is indispensable. Please note, that, for metals, this test must go hand in hand with a test of the swidth. Check out the Input Reference for the <groundstate> attributes stype and swidth to find out more about this topic. The basis-set size Most of the DFT codes solve the Kohn-Sham equation in reciprocal space, expanding the wavefunctions in terms of suitable periodic basis functions. In our case, the basis functions are the linearized augmented plane waves (LAPW). The larger the size of the basis set, the more accurate is the calculation. The corresponding «groundstate» attribute in the exciting code is rgkmax (see Input Reference). The default value of rgkmax is 7.0. In many cases (e.g., for a good description of forces or equations of state), larger values (8.0 — 9.5) may be necessary to get reliable results. Therefore, for this parameter it is very important to test how large rgkmax must be in order to get the physical property under investigation with the desired accuracy. Terminology: When speaking about the number of the basis functions, you may hear the expressions "rgkmax", "basis-set size", "energy cut-off", or "plane-wave cut-off" — all of them are a measure (in different units) of the same quantity. Scaling of the calculation time… The calculation time scales • linearly with respect to the number of k-points; • as rgkmax to the power of 9. 1. Convergence of the k-mesh: Repeat the calculation of groundstate and DOS using for ngridk the values "3 3 3" ,"6 6 6", …, "21 21 21" in steps of 3. The best way to do so is to (1) create a subdirectory for each k-mesh (e.g. dir_nkx_3/ for ngridk="3 3 3"), (2) copy the input file there and modify the attribute ngridk appropriately, (3) start the calculation in each subdirectory by executing $EXCITINGROOT/excitingser on the commmand line.(Please note: Since both the groundstate and the DOS should be calculated at once, now, set the attribute do="fromscratch" for the <groundstate> element). • Extract the total energy from the last iteration of each groundstate calculation and plot total energy vs. nkx, where nkx is the number of k-points in x direction (e.g. 3 in the case of ngridk="3 3 3"). Taking the total energy of the calculation with ngridk="21 21 21" as a reference: What k-mesh is needed to get a total energy within 10-3 Hartree from the converged result? • Plot the total DOS (file TDOS.OUT) for all k-meshes (You can do so by invoking xmgrace with all TDOS.OUT files at once, see Xmgrace Quickstart. Sect. 6 ). For which value of ngridk can the DOS be considerd to be converged? 2. Convergence of the basis-set size: Do calculations with a fixed value of ngridk="12 12 12", but varying rgkmax from 5 to 10 in steps of 1. Make the same checks as in exercise about the k-mesh, and try to find out for which values of rgkmax the total energy and DOS, respectively, can be considered as converged.
bce226c2e4216d48	Sunday, November 16, 2014 In Defense of Time There is a very strong ongoing discourse about the nature of time and whole books have been written about the illusion of time, both for time as an illusion [The Illusion of Time, Tolle 2008, The Time Illusion, Wright 2012, A Question of Time: The Ultimate Paradox, Sci.Amer. 2012, The Elegant Universe, Greene 2010] and against time as an illusion[Time Reborn, Smolin 2012]. Although there is an illusion about reality, that illusion is not about time. The illusion that we have about reality is in how we discover continuous space and motion, not in the discrete time delays of objects and action. We first discover space as the lonely dark empty nothing that explains why we no longer see an object that has moved behind another object. We learn about space by about the age of two and then we take space and motion for granted as a basic belief that anchors consciousness. We do not really often dwell on the irony of accepting the nothingness of empty space as a something that makes up most of the universe. We simply realize that objects continue to exist even when we do not sense them and the motion of objects in that nothing of empty space simply hides one object behind other objects. While there are many, many more books and articles written about the illusion of space than the illusion of time, somehow we just don't get it. The vast majority of spatial illusions result from a confusion that we have with the time delays that we sense for objects and their backgrounds, what we call spatial depth dimension in an otherwise two-dimensional image. We know that with each of our two eyes we only perceive a two dimensional reality of object time delays and therefore the third dimension emerges as depth only by perspective. Each of our two eyes sees a slightly different two dimensional space from just the one dimension of time delay between objects. Our brain largely organizes the world with the dual concepts of continuous space and motion and uses space and motion to keep track of where objects are and to help predict where objects are going. We imagine ourselves in a rest frame that does is not moving and that a reality exists for both moving objects outside of our brain and objects at rest with us in space. We seem to have a no trouble imagining space as a lonely empty nothing and it is especially ironic that we infer space from the continuum of sensation of a background of objects and the nothing of space emerges from what we do not see or sense. The object that we imagine as empty space is everywhere the same and in some sense immobile and fixed and it is the lack of sensation of any object that we feel is the lonely empty nothing of space that then defines most of the universe that we imagine. But continuous space and time do not describe all objects in the universe for mainstream science. There are objects called black holes and very small objects at the Planck dimension and neither of these two objects exist in continuous space and time. These objects do exist in the reality of time delay and discrete matter and the Mollweide projection maps the two-dimensional sphere of the sky into the ellipse shown below. Likewise, we can map the two dimensions of time onto a Mollweide ellipse and show how the universe projects back onto itself in time. There are always two different observers for every motion or action; one observer, called the rest frame, is not moving and is usually left behind as a result of motion of a second observer that is moving with some action in the moving frame. In general relativity, GR, each of the rest and moving frames have their own clocks and those different clocks keep different time but still provide a single objective and proper time that completely defines that action. Proper time represents the norm of the displacement of a moving object, i.e. what we really experience, in the four dimensional spacetime of GR. Relativity imagines a proper time that is a continuous spatial dimension and in effect does away with time by making it just a fourth dimension of 4D spacetime. The motion of an object in spacetime is equivalent to time, but then there are motions within that object that must also be equivalent to time, and further motions within those motions as well that also affect time. These recursions of embedded motions and times are an integral part of the recursion of relativity but quantum mechanics handles embedded time somewhat differently than GR. Quantum action always begins with a discrete excitation from a ground state of one matter wave as a source or origin. That source bonds a pair of emanating matter waves of two objects into coherent relative futures. The excitation evolves a ground state into an excited state that is a pair of complementary and coherent matter waves, each with complementary and coherent clocks and directions. In classical ballistics, every action results in a reaction or recoil and a bullet firing results in a recoil of the gun in the opposite direction. Likewise in quantum action, every excitation has two coherent matter wave complements as well. In GR, the rest and moving frame clocks represent a proper time, the time that we experience as the present moment, and the clock amplitudes and phases of the two frames do not affect the path of the object and essentially remain coherent for all time. In quantum action, the rest and moving frame clocks begin together and are coherent for only some characteristic time. As long as two clocks remain coherent, they may interfere with each other and therefore affect each other's path. The quantum rest and moving clocks come into existence after some excitation of one or two sources and quantum clocks merge with discrete jumps or quanta into the same proper time norm that GR reports. We therefore experience the same proper time in both quantum and GR times, and we sense the same matter changes of an object and the same motions emerge along with space as a result of sensation. However, there is an inherent decoherence rate for the two quantum clocks of a rest and moving frame that not only limits what we can know about their paths, that decoherence rate is what determines both gravity and charge forces. What we sense about an object involves exchanges of matter amplitude and phase with the object matter waves and those matter and phase exchanges result in a complex neural packet of aware matter that we call a moment of thought. From all of these complex relations among impulses, the relative simplicity of objects at a particular moment emerges as the three dimensions of space in our brain. What we actually sense about an object is, however, quite a complex set of both coherent and incoherent matter waves that represents a relational reality that comprises both us and the object. While what we imagine about an object represents a very much simpler Cartesian reality of time and objects in a mostly empty space, the relational reality of an object is ever so much more complex. This fundamental dualism is a prominent feature of all models of reality and yet matter time necessarily uses time and matter as primal conjugates and not the space and momentum of mainstream science. Since we do not actually sense the nothing of space and motion, we can deduce and can and do imagine the nothing of space to be just about anything that space needs to be. The fact that there are two distinct clocks for each GR action, a rest clock and a moving clock, is also true for quantum action. However, a quantum action begins in the past with either one or two sources of matter waves that may or may not be coherent. Quantum clocks can be entangled and interfere with each other, which just means that the excited state of a source matter wave pair can remain coherent for a very long time and so can result in correlated and seemingly nonlocal actions. This seeming violation of GR's local causal principle of determinism is simply a characteristic of a quantum time and does not actually violate any quantum causal principle. After all, while quantum clocks show interference effects as long as they are coherent, GR clocks are in a sense always coherent and phase has no meaning. Since the two clocks of GR do not interfere with each other as coherent amplitude and phase, the rest and moving clocks of GR merge smoothly into the proper time norm of experience. There is no role for the phase of coherent clocks in GR and so there are no interference effects in GR either. In contrast to the importance of clock time in GR, time as an independent dimension seems to go away in the four-space of GR since there is no phase and no decoherence rate. There is only one possible future in GR and so GR time has no phase and is simply a dimension of displacement that is equivalent to space. Yet quantum atomic time is not only a progress variable, quantum decoherence time is also an integral part of reality and as a result, there is no quantum time operator like there is a quantum mass operator. Even though time is a prominent feature of general relativity and the mass-energy equivalence principle, E = mc2, quantum's adoption of mass-energy equivalence still means there is no expectation value for quantum time. While momentum and space have long had a warm and cozy quantum relationship as conjugates and are nicely complementary, mass and time are not quantum complements of each other like momentum and space for mainstream science. Since there is not an expectation value for time or duration, this is known as the quantum time problem and this is what leads many to suggest that quantum time is an illusion. These arguments rest on the proposition that there is quantum space and motion as our reality, but not matter and time. Quantum energy simply exists like time as a progress variable and is always a result of motion, not the source of motion. However, matter time plays a role reversal and proffers that instead of space and motion being the reality and time a consequence of motion, quantum matter and time are the reality and space and motion emerge as a mere consequence of the action of matter and time. Space is then just a convenient progress variable and the illusion of space is what allows us to keep track of objects in time. It is matter and time that complement each other and not momentum and space. Key in matter time is the matter-energy equivalence principle (MEE) and Lorentz invariance and the rigor of certain bounding assumptions for matter. For matter and time to complement each other, the universe must be of a finite total mass that is finitely divisible and these two assumptions become the basis for a quantum time operator that complements the quantum matter operator. Time becomes the duration of an action and an integration of changes in an object matter spectrum. Just as action is the integration of an object's changes in matter over time, action is also the integration of an object's changes in the time amplitudes of its matter spectrum. An object's changes in matter over time define the object in the present moment, which is within the time spectrum of the universe. Likewise, an object's changes in time amplitudes as a function of matter define an object's matter spectrum that is embedded within the matter spectrum of the boson universe. Tuesday, September 30, 2014 Pulsar Spin Down Pulsars are the wonderful gifts or time and are the clocks of our galaxy and really of our universe. Many stars just like our sun reach their destinys as rotating pulsars, white dwarfs, and neutron stars. These rotating bodies show us the way of our destiny as well as the way of our past. Pulsar rotation is highly periodic, but because they are every dense, pulsars have very unusual properties as well much like the property of spin of the atomic nucleus. In contrast to the nuclear spin, pulsars radiate energy from their poles and it is the precession of those energy beacons that shines much like the rotating lighthouse of maritime lore. Each time a pulsar pole happens to point to earth, we measure a pulse of that pulsar and these pulses vary from periods of several seconds to several thousandths of a second, milliseconds. The pulsars not only tick at very regular rates, pulsars also decay at very regular rates. Some pulsars actually increase in tick rate, but the vast majority of pulsar rates decay over time. This decay rate conforms to the classical αdot = 0.255 ppb/yr decay of matter time as shown by the red dash line in the figures below. This classical decay is proportional to the ratio of gravity and the square of charge and so is the unification of gravity and charge in matter time. Millisecond pulsars are especially accurate timepieces and their trend in the plots below show an average decay that is very similar to the classical decay, and while this could be just a coincidence, it it perfectly consistent with a shrinking universe. Also maybe coincidentally, the measured earth spin down, earth moon orbit decay, and Milky way/Andromeda separation rates happen to fall on this same line... The larger plot below shows where the hydrogen atom and electron spin frequencies lie on the spin down line...oh, and the earth-moon orbital decay is also well known. Note that orbital decay means frequency decay and that means the orbit increases in distance in order for the period to decrease. The classical electron spin velocity, c/α, defines a period for the electron spin and the matter decay, mdot, defines the slope of the decay line. These are the two axioms that drive all gravity and charge forces. As you can see, both c/α and mdot are simply restatements of constants of science and not new parameters. The only new parameter here is that third axiom, m¥, the mass of the smallest particle, the gaechron. The ratio m¥ / me scales gravity to charge force, but does not show up on this plot since that is the period of the universe matter pulse, 27 Byr, of which we are 3.4 Byr into that matter pulse. However, the Andromeda-Milky Way galaxies separation decays at 0.267 ppb/yr, very close to the universe decay rate. Now added is the Allan deviation noise curves for the 171Yb/87Sr lattice clock ratio. Once the ratio noise is coincident with the universal decay constant, that is to what the clock ratio converges. Quasar Numbers and Luminosities This plot shows 46,000 some odd quasars from the SDSS J dataset in terms of numbers per 250 Myrs as well as luminosity in terms of equivalent sun masses turned into energy. Note that while the quasar number densities peak at 10.25 Byrs, the luminosity keeps going up to one sun mass equivalent energy per year. The time scale assumes a Hubble constant H = 74 km/s/Mpc. And of course, the matter time universe scales differently and below is the matter time equivalent plot. The matter time universe is 3.4 Byrs proper time and quasar luminosity scales much differently in an expanding force and decaying matter universe as opposed to the space and time expansion of the big bang (actually just by1/gamma^2). Thus the luminosity of quasars in the early epoch now is very similar to galaxy luminosity in the current epoch, which is due to starlight and not the SMBH. The H = -288 km/s/Mpc, and of course, the Hubble constant is negative for decay and begins at the edge of the universe shrinking inward, just like one might expect for a gravitational universe. There is also some great work with the number density and luminosities of all galaxies, Nature 469 504–507 (27 January 2011) doi:10.1038/nature09717. Here is a plot of luminosity of all galaxies as well as quasars as a function of Hubble time for the space time expanding universe. and here is the corresponding plot for the matter time collapsing universe. Runiverse = 2401 Mpc, 201 billion galaxies at 3.5 Mpc-3. The luminosity uv is the SDSS uv band while sfr is the star forming rate derived from cited models along with the constant galaxy density of 3.5 Mpc-3 shown below. Since there are 54 galaxies in our local group and diameter of 3.1 Mpc, there is 3.5 galaxies per Mpc3. Here is a plot of the local galaxy number density from PASJ: Publ. Astron. Soc. Japan 55, 757-770, 2003 August 25, There are 500,000 galaxies within z=2 in SDSS-10. Here is the plot that shows it all. The galaxy number density is constant at 3.5 Mpc-3, but in a collapsing universe, the space-time metric evolves and the galaxy number density versus time is more like a quadratic function. It appears that quasar number densities are on the order of 0.47% of galaxy numbers in a collapsing universe. This result is really crazy. What it means is that time lensing of the past affects how we interpret our universe. The idea of a quasar as a composite of a boson star and an eternally collapsing object is very appealing. In this case, the event horizon represents a phase transition between a time-like fermionic matter, i.e. the ordinary matter of our universe, and the boson matter-like time of a boson star. Matter time does seem to provide a coupling between the fermions of a rotating accretion disk and the bosons of a rotating boson star. This entity will accrete fermions into the event horizon, undergo phase transition to bosons and emit the balance of the fermions as light at the jets of the quasar. It is very likely that thermodynamics will provide a useful way to handle this phase transition from two such different states of matter. In fact, there may be something quite similar going on at the centers of large neutron stars. Friday, August 8, 2014 Cosmic Microwave Background as Creation The cosmic microwave background (CMB) is a plasma that appears to be 2.7 K in the present epoch and exists in all directions in the sky. The CMB lies beyond all of the stars and galaxies and the cold, dark hydrogen of the past and the CMB represents the creation of all that exists. The CMB plasma spectrum peaks at 160 GHz and so that means that there is no absolute darkness since the CMB bathes us all in the background of CMB microwaves. The actual CMB temperature is thought to be ~3000 K with a redshift of z = 1089, but that result comes from a specific model that is the CDM (cold dark matter) big bang cosmology. In the big bang, the CMB would be expanding at 99.91%c, but different cosmologies result in a range of predictions and the shrinking universe has a really cold interpretation for the CMB, just 0.64 K at z = 1089. In today's epoch, this temperature would be equivalent to the ionization energy of hydrogen at 13.6 eV, which is 158,000 K...a little bit warmer than the CMB. In addition, very slight CMB temperature difference is called the CMB dipole, points the direction that the earth is moving through the cosmos. This arrow of time shows our path through the cosmos and defines both an origin and a destiny. The CMB arrow shows where we came from, i.e. our origin, as well as where we are going, i.e. our destiny. Hurtling through space at 830,000 mph (371 km/s) means that we quickly leave the space of each moment behind and for each moment of thought, about 0.6 s, we move about 130 miles through the karma of the universe even though we imagine ourselves standing perfectly still. If you ever feel like you are not going anywhere, now you can rest assured that we all are on a grand journey together through the karma of the universe and mom and dad, mother earth and father time, are driving. We note our karmic journey on March 11th and September 10th, the days where the sun and earth are best aligned with the CMB time arrow that points the destiny of the universe. This diagram is called a Mollweide projection of the entire sky that shows how the cold blanket of the finite CMB creation dipole wraps all around us. Up and down are the 180 degrees of up and down from the plane of the galaxy and left and right are the 360 degrees as the Milky Way also wraps all around us. In a shrinking aether universe, interpretations of cosmic objects like the CMB vary h, c, and alpha and a shrinking aether universe is much different from an expanding universe. Aether temperature, Tae = T (1+zae)/(1+zH), is a scaled cosmic temperature from the present epoch. Instead of the CMB rest frame temperature being 3,000 K as 2.7 K * (1 + z= 1090), the aether CMB temperature is just 0.64 K, which is 2.7 K x 260 / 1090 = 0.64 K in the aethertime rest frame for the CMB. The CMB motion is 0.24c towards us, which blue shifts the 0.64 K rest frame CMB to our 2.7 K rest frame all in aethertime. The aethertime creation CMB is very close to the edge of the universe and represents cae = 0.062 c, just 6.2% of c in the current epoch and so force is also just 6.2% of that of the current epoch. Atomic time periods increase as atomic force decreases, and eventually universe time transitions from atomic time to aether decay time. Thus in aethertime, the singularity known as an event horizon simply represents the boundary of the shrinking aethertime universe. This interpretation of an aether event horizon differs sharply from that of spacetime and GR of mainstream science. Correspondingly, the event horizon of a singularity known as a black hole represents a similar transition from atomic time to aether decay time and the boson matter inside of a black hole is simply the same boson matter that makes up all of the universe. The fermion accretion disk of a black hole represents the same kind of boundary for a black hole as the CMB does for the universe, but now shifted from 0.64 K to the temperature 13.6 eV energy of hydrogen today. This energy corresponds to ultraviolet spectra called the Lyman series and Lyman blobs are often seen in the early universe with z > 2. In fact, a very large Lyman alpha blob called Himiko appears in the very early universe at z = 6.6 and is thought to represent a nascent black hole and galaxy. Thursday, July 24, 2014 Gravitational Beamsplitter A beamsplitter is a fundamental tool in optics that prepares a photon of light into two coherent states by dividing a parallel beam of light into two separate beam paths, A and B as in the figure. Usually a beamsplitter is a partially silvered mirror inside of a cube of otherwise transparent material and passes 50% of its light while reflecting the other 50% providing two possible futures for every single matter wave of light. This of course ignores the reflections and losses that occur at the other surfaces. There are two interpretations for the action of a beamsplitter that represent two fundamentally different world views; ballistic and deterministic or relational and probabilistic. In a ballistic and deterministic worldview, the beamsplitter simply diverts each photon of light in the source beam onto either path A or path B. If you observe a photon at A, that means that that photon followed path A and that is why it was not seen at B. In the second relational and probabilistic worldview, light actually follow both paths as matter waves and the beamsplitter introduces a coherence or relation between the two possible futures or states for each of two matter waves within the original source beam. In this relational worldview, light from the source propagates along both coherent paths A and B, but still an observer at A sees only 50% of the light waves as photons and does not see the other 50%, but now it is because of constructive and destructive interferences along both paths. Thus, a relational matter wave propagates from a source, the beamsplitter, along both relational paths A and B as a coherent superposition of paths A and B. In a ballistic and deterministic worldview, seeing a photon at A means that that photon was always on path A and never on path B and that is why that photon as a particle did not appear at B. This is very intuitive and is largely what we experience in our macroscopic ballistic reality. In a relational and probabilistic worldview, though, seeing a photon at A does not mean that the light wave was only on path A. Even though that light wave did not appear as a photon at B, the light wave of that photon propagated on both A and B up until it was observed at A. So it was possible to have seen that photon at B up until actually seeing it at A. In essence, seeing A means not seeing B and the two events are coherent and related to each other instantaneously despite their separation in space and time. This confusing superposition of quantum states A and B means that it is not a lack of knowledge that precludes knowledge of path A or B, it is rather an intrinsic superposition of matter waves that makes the precise path fundamentally unknowable. The result is not dependent on the nature of the observer and the photon can be absorbed by any object with the same result. The existence of unknowable paths is even more confusing to understand given the nature of our intuitive ballistic reality where all objects are located in unique places. It is difficult for us to imagine an object as a matter wave emanating from a source on more than one possible coherent and symmetric path. It is even more difficult for us to imagine any number of objects existing in the same place at one time. We only imagine single objects with single ballistic Cartesian trajectories and those paths are ultimately knowable even if we might not know them at the moment. We find it difficult to imagine an object as a matter wave whose exact path is fundamentally uncertain. The way that light waves interfere with themselves and with each other shows the truth of the relational worldview for light as matter waves. It is therefore true that objects behave as matter waves, including photons of light or people or planets, and an object as a matter wave can exist everywhere in the universe even though we only experience the object in one place. Each matter wave can have coherent relational states superimposed with different phases on other Cartesian worldlines. When a matter wave dephases or decoheres, it behaves like the ballistic and deterministic objects of our intuition. We can cause that dephasing or other objects can dephase a matter wave as well in the single object of our ballistic intuition. We imagine a macroscopic reality that has objects on ballistic trajectories and for this reality, objects as matter waves have long since lost any other possible futures. When two incoherent objects collide, they collide in a ballistic reality. Two coherent objects, though, can pass through each other as matter waves given a gravity or charge mediated matter wave coherence and that coherence confuses our innate ballistic worldview. In principle, a gravity beamsplitter as shown in the figure at right can prepare small objects like atoms or molecules into coherent gravity states. Two massive spheres like the earth and moon are in a binary orbit with each other as in the figure. Two much smaller and identical objects, A and B, are in orbits that intersect at the gravitational Lagrange point between the the earth and moon. For quantum gravity, the identical Lagrange objects A and B can take on a superposition of coherent futures, one orbiting earth and the other in a complementary orbit around the moon and those two orbits interfere with each other. For any macroscopic object emitting and absorbing radiation, there is a fairly short time of dephasing and the object matter waves A and B will quickly dephase into either A or B ballistic orbits and the two incoherent objects may then collide at the orbit crossing. For cold microscopic matter, though, a matter wave can persist for a much longer time as a superposition of two possible futures, orbits around the earth and moon. The result will be that the objects A and B can occupy the same space and will pass through each other and interfere with each other but not collide in the ballistic sense. As a result of this coherence, there is an extra binding energy for A and B due to the quantum exchange force between those paths. Essentially, a coherent matter wave that includes both orbits and appears to act simultaneously across arbitrary separations, either appearing as an object or as a matter wave in orbits around earth and moon. The coherent identical objects {A, B} are part of an oscillating orbital state between earth and moon. If these two objects {A, B} are coherent, they will interfere with each other along a coherent trajectory back onto themselves. The gravity actions of coherent matter waves have many unexpected and nonintuitive effects. For example, there is only a collisional future for {A, B} particles at the Lagrange orbital crossing in general relativity, while for quantum gravity there is also a wavelike exchange future for matter waves of properly phased objects {A, B}. Quantum gravity predicts an additional attractive force beyond just simple gravitational or charge forces for coherent matter as a result of exchange of identical particles. This additional gravity binding force due to quantum exchange and is what science now calls dark matter. Quantum exchange forces are coherent relational actions that appear to act instantaneously over arbitrary separation. As a result, even though the exchange correction for the gravity of a galaxy is actually locally quite small compared to Newtonian gravity, over arbitrary separation, quantum gravity exchange can seem like a large amount of dark matter as an equivalent Newtonian gravity action within a galaxy. A superposition state of hydrogen molecules that begins from a stationary ground state involves excitation with several quanta of infrared photons. Such an excitation has sufficient energy to accelerate a pair of neutral H2 molecules into low earth orbit at 8 km/s as shown. As long as the two H2 molecules remain coherent, they do not collide in the classical sense despite being on the same orbit trajectory. Rather, they form a superposition state of the counterpropagating molecules that in effect interfere with each other. As soon as the hydrogens dephase, they experience a classical collision with any number of possible futures. Note that the inverse process where heat is emitted from two molecules results in bonding states that we call gravity. Instead of bonding by displacement of charge, gravity bonding occurs as a result of displacement of neutral particles and emission of heat as photon pairs as quadrupoles. Typically science interprets the heating of a matter accretion as caused by gravity, but in matter time, it is the emission of heat as photon quadrupoles that causes gravity. There is a very large number of gravity states for a matter accretion and therefore a very large entropy as well. Friday, July 18, 2014 The Pleasure of Discovery We each innately get pleasure in discovering how various parts of the universe work and we then make choices based on those discoveries that are each part of our life’s meaning and purpose. We have innate feelings and emotions and so we get pleasure discovering parts of life’s meaning and purpose that are necessarily beyond conscious rational thought as well. We make choices based on our feelings and emotions and beyond science and rational thought, our purpose in life is a primal belief in discovery and that meaning is a necessary primal belief that we all simply have for the purpose of discovery in each of our lives. We get pleasure discovering how various parts of the universe work, but we cannot test that purpose nor can we further describe that purpose except by the other two primal beliefs; the pleasure of our discovery of origin and destiny. • There are many who get pleasure discovering more than the meaning of life and how the universe works...they get pleasure discovering the meaning of everything. In order to discover the meaning of everything, though, one must first understand their innate anxiety about the dark lonely nothing of empty space. The only way to define a primal axiom like the pleasure of discovery is with the other two complementary primal beliefs in the pleasure of origin and destiny. Even though our innate purpose is the pleasure of discovering how the universe works, if we somehow know where we are going in life, we also get pleasure in finding out how to get from where we are, our origin, to where we are going, our destiny, and we get pleasure discovering that journey. Our pleasure in discovery is an axiom that is in some sense like the axiom of matter; since matter is just the way it is and there is no further definition of matter possible except by the two complementary primal axioms of action and time. Although we can say a matter object is red or is large or shiny, once we reduce an object description to a primal belief like matter itself, matter is just matter, which is an identity or ontology and therefore an axiom. Of course, our life and consciousness are both prerequisites for an awareness of belief or of any other axiom. Just like objects are the accumulation of moments of matter, memory is an accumulation of moments of thought as the brain matter that is a part of consciousness. The matter moments of long-term memory couple with the neural recursion that comprises the moments of thought of a day. The sensation-feeling-action of present thought along with long-term memory and emotion completes the neural recursion that we call consciousness. Our memory is a function of consciousness and memory is a record of the action as moments of thought. The neural recursion of action along with the memory that we have are both objective mechanisms of our mind that together form our consciousness. Our time-like consciousness, though, is a combination of these two objective properties of our brain that result in feeling. Therefore consciousness represents a subjective reality in our mind that complements the objective reality of the world outside of our mind. This dualism of mind and body has a long history in philosophy. The question of our purpose has only one clear answer; our purpose is discovering how the parts of the universe work and so purpose is an identity and a primal belief that is only explicable as the other two complementary primal beliefs of origin and destiny. Ultimately, discovery is all about discovery for its own sake. Once again, we see the replay of the dual representations in the definitions of an axiom. Purpose as discovery is how we get from where we are to our destiny, from yesterday until tomorrow and why we imagine desirable futures and how we make choices in our journey from an origin to a destiny. Therefore, purpose, origin, and destiny as such are always what other qualia (the properties of object, like color, weight, and so on) are like, whereas primal beliefs are not like anything else except combinations of their complementary axioms. If you ask a person in what do they believe, their reply is usually in a religion or in a science or in a metaphysics and we imagine their beliefs are a purpose for their lives. Belief and meaning are essential to every conscious life and we all know how to answer the question of in what do we believe, but we do not often recognize that without belief, we simply cannot be. We have an innate anxiety over the nothing that is the empty void of space and we each must first of all frame our reality to deal with this anxiety over the nothing of empty space and over being alone in that empty space. With our relational reality, instead of a largely empty space with just a few objects of our Cartesian reality, we fill time with the many possibilities of relations with those objects. We frame each of our lives and each of our physical realities with the three primal beliefs of origin, destiny, and purpose for all objects and this trimal is essential for predicting action and indeed trimal beliefs are necessary for survival. We might have a purpose driven by innate anxiety, say about dying or about the empty voids of a lonely life or about what's for lunch, and yet we might not even consciously know why we are are anxious about dying or why we are anxious about being alone or why we are hungry. Such a hard-wired anxiety can drive purpose whereas anxiety is an emotion which we simply have and believe in and yet many want to associate our innate anxiety with a supernatural agent. Innate anxiety has evolved like so many of our other behaviors and is something that we just accept and deal with in a variety of ways. We all are innately anxious about the empty voids of our lives and of the nothing of space and of being alone, but in order to understand anything, we must first believe in the nothing of empty space. Since most of the universe that we imagine is the nothing of empty space, that nothing is the most important part the universe that we imagine and that nothing is the most important thing in our lives as well. But nothing is really not as important as it seems. It is important and often vital to be anxious about nothing since with nothing to eat or drink or without shelter or clothing, we could not survive very long. We look into the nothing of the dark sky at night and wonder about the points of light that we see as part of the universe, but we do not wonder about the empty nothing that separates those distant stars. That dark emptiness is just the same dark empty voids about which we are anxious in our lives. It seems strange to be anxious about nothing since most of our reality is made up of the nothing of space, and yet we are more certain about the infinitely divisible nothing of empty space than we are about the objects that we sense embedded into that space. We sense many objects around us and so we know their directions quite well, but object distances can be very difficult to sense and know without some guide like parallax or a reflected echo or a standard candle or a tape measure. Space then is a very convenient way to keep track of the many objects of our reality and we get many cues about distance from other objects. Religion often claims a special role for a supernatural agent everywhere in all the empty voids, especially for innate beliefs like anxiety, since a supernatural agent is fundamentally a belief in a void as something rather than nothing. Religion provides various supernatural agents that make us anxious about nothing, but there are also supernatural beliefs about nothing in science, albeit a more limited set, called axioms. Where did the big bang come from? What exists inside of a black hole? What is the destiny of the universe? Where do physical laws and their constants come from? The untestable axioms of science provide a very rational framework for prediction of action and a further belief in the science of aethertime allows us to understand our reality. We call the axioms of science natural because even though there is no way to understand why they are the way they are, we can accept science axioms and use them to predict future action by trusting the intercessories of science. Philosophy calls an ontology the axioms that we accept as true while philosophy calls an epistemology the rules that we derive from such axioms or ontology. In a completely analogous way, innate anxiety naturally affects behavior even if we do not necessarily understand the origin of that anxiety. We can call the innate anxiety over nothing natural or we can associate that innate anxiety with a supernatural agent of some kind for the nothing that we firmly believe does exist. Although we see or sense objects in space all around us all of the time, we do not often wonder about the process of sensation, how sensation affects feeling, how feeling results in action, and the recursion of how our action results from sensation then in turn feeds back and affects sensation. While there are clear roles for belief in axioms like matter, time, and action, we also believe firmly in the continuous void of empty space as well. What is space like? It turns out that we can describe objects and predict their futures without knowing the volumes they displace in space, but those volumes, surfaces, lines, and points of space do provide a very convenient frame of reference for action. We further use objects as landmarks in that space to anchor our sense of direction on earth and these landmarks are a prominent feature of conscious thought. If we know the time delay of objects from each other and how aether exchanges relate objects to each other, we can describe an object as full of its possible futures without knowledge of motion or space or volume. Aethertime completely represents objects as superpositions of many possible futures with just matter, time, and action, and the Cartesian locations of objects emerge in our mind from that relational representation. Cartesian space is an innate part of our imagination and is a convenient and also a very useful whiteboard for keeping track object action. Cartesian space is therefore deeply embedded into our consciousness and intuition and is a powerful tool of consciousness, but the limitations of continuous space and time can also blind us to a greater understanding of the universe. We can trust the intercessories of science because science repeatedly tests its propositions against an objective reality, which means that the reality of science is largely of observable Cartesian objects and actions on trajectories in continuous space and time time. Science works best by observing the universe and then making predictions about object actions and then verifying those predictions by careful observations of action. An ongoing discourse of the principles of science provides a means to cull and prune and distill the essence of truth about our material world. In a Cartesian particle-like reality, objects only interact weakly and exist on separate trajectories in continuous space and time. In a complementary relational reality, wave-like objects strongly interact and exist as matter waves that fill time with a large number of possible futures. A relational reality represents an object as a spectrum of matter waves with matter exchanges that relate it to all other objects as matter waves, which is a matter spectrum. For a reality of weakly interacting objects, though, science can measure and project a red object into a single location in Cartesian space. For the reality of strongly interacting objects, though, science cannot always test its predictions. For highly relational objects like people, science is often limited to just observation and even though a person can relate their experiences such as that of seeing a red object, science cannot predict a person’s experience in seeing a red object without knowing everything about that person. An experience of a red color will vary from person to person and each person’s life will relate a different set of experiences of red objects to any new experience of a red object. Science can neither measure nor quantify a person’s relational experience of red even though science does very well predicting and measuring the objective Cartesian experience of a red object. Once science knows enough about a person’s past experiences with red objects, science can then predict fairly well that a person’s experience will be much like those who have similar past experiences with red objects. The qualities of our feelings about objects, though, are what we call subjective or relational experiences of objects and a feeling is not possible to test or falsify except by query and discourse. A red color, for example, is simply one of the many qualia that we use to help identify and classify objects. How we might feel about a red color is our feeling alone although we can use a machine to measure a red color for an object. Our feeling of the same red of an object is simply not possible to measure, although we can relate our experience of red to others. Qualia are the hard wiring of our minds and human qualia are therefore a part of the augmentation of Cartesian space with discrete aether and we can describe our feelings about space to others. We associate certain qualia with objects in the same way that we associate the spoken sounds of language with objects and action in time. For example, the names we give objects and their properties allow us to relate those objects to other objects with similar properties and therefore we can predict action much more precisely and describe our predictions to others as well. As others describe their predictions of action to us, we cooperate and that cooperation provides the basis of civilization by enhancing survival in an objective universe. We project the qualia of space around the objects that we observe. Whether an object is near or far away, whether it is high or low, or in which compass direction it lies, locating objects in space is an absolutely vital means of organizing reality in our mind. In this sense, the infinitely divisible void of empty space that we imagine is just a part of the more general notions of aethertime. In order to imagine objects, we certainly do believe in the null object of empty space and that belief defines how we predict the actions of objects as motion in empty space. But we can also derive any motion in space just from the exchange of an object’s matter in time since all motion is equivalent to a change in inertial mass. The trimal axioms of discrete aether, time delay, and aether exchange are the primal qualia that relate objects to other objects and matter, time, and action are likewise hardwired into our consciousness. Matter-like qualia are red, black, heavy, light, pain, heat, cold, etc., and time-like qualia are fast, slow, quick, hurry, sluggish, etc., while action-like qualia are weak, strong, hard, soft, feeble, mighty, etc. We relate objects with similar qualia to each other to better predict and describe the likely futures of those objects. The meaning, imagination, and feeling of each human life is woven into the fabric of civilization and our meaning becomes a part of the collective beliefs in which many of us share. However, just like there are thousands of languages, there are necessarily thousands of beliefs in the meaning of life as well. Despite the existence of many different languages, Chomsky and Wittgenstein have shown that languages are all rooted in the same machinery of our consciousness and that language machinery is part of what makes us human. In a similar way, the machinery of consciousness provides an innate purpose in finding out how the parts of the world work, but on which part of the world we focus varies just like language varies. Our relational minds have the basic machinery of consciousness that relates objects to other objects as qualia for the purpose of finding out about the world. Just like language is a communication among people about objects, qualia are the relationals of our minds that permit us to find out about the world and describe experience. The machinery of consciousness is present in our minds, but we do need to learn the qualia of consciousness just as we need to learn the words to speak and how to place our feet to walk. Matter, time, and action form a trimal qualia of belief that are hardwired into our consciousness and therefore are a nexus or connection between the objectivity of science and subjectivity of experience. For example, in our subjective experience of religion and philosophy, we often refer to the objectivity of science. But religion and philosophy deal largely with the relational and not the Cartesian world of consciousness. There is irony in that while aethertime reveals our Cartesian reality actually emerges from the actions of objects, we usually presume that our Cartesian reality is the only reality of both our religious and our scientific worlds. There are plenty of indications of the limitations of Cartesian space. The infinite divisibility of a void of space, infinitely dividing nothing at all, has posed a conundrum ever since the philosophy of Aristotle and Zeno. More recently, quantum physics shows a universe that differs from that of Cartesian experience and relativity and evolution likewise show us a universe that is different from ordinary experience. Our Cartesian space, motion, and time are very useful and indeed essential for predictions of action and will always be a very useful and therefore essential parts of consciousness. A Cartesian reality emerges from a multitude of very complex sensations into a few simple imaginings of important objects moving on time trajectories in a void of space. Even though we actually only sense some limited number of an object’s possibilities, with this very limited information, we nevertheless imagine that object and predict its journey through space and in time and sometimes we predict very well. Although we are part of an object’s matter spectrum, we usually presume that the reality that we sense is not affected by our presence. Just as for all life, predictions of action are the basis of our survival as well and our action predictions evolve given the ever more elaborate stories of science. The stories of science have evolved into such complexity that it takes years of advanced study for even scientists to achieve a current understanding of just a tiny slice of the universe. In fact, the enterprise of science divides into pieces that seem more like religions than any other religion has ever been. Today people believe very fervently in scientific concepts that they barely understand and sometimes, they simply do not understand them at all, they just believe them as told by a trusted intercessory. When we do not understand a concept that is nevertheless important to us, we trust an intercessory when they tell us about the actions and objects of that concept. So we now have a new cadre inside of monasteries of science, preaching an everlasting life given the meaning of nothing. Instead of accepting the inexplicable walls of our own universe, some scientists now imagine universes far beyond any testable hypotheses within this universe. Not unlike the mystics of ancient China, India, or Greece, theories of everything abound and propagate and provide a fertile soil for nurturing humanity’s immortal soul. We ask about the meaning of life because it is only with purpose that we discover how the universe works and we imagine desirable futures and choose actions to journey to those futures. The meaning of life really has no unique answer except as a reflection of the purpose of discovery, and without a purpose, we can have no life since there would be no desirable future, i.e., no desire to discover how to survive. We must discover our desire to survive with purpose and meaning and it is a desirable future that we discover as the purpose and meaning further discovery. My dog asks for meaning and purpose from me...every day and many times a day. Of course not in human words, but dogs imagine desirable futures and choose actions to realize those futures just like we do. My dog breathes, drinks, eats, seeks shelter, and constantly searches for scents in the backyard and park, and of course, he revels in the purpose of companionship. My dog loves to walk and smell and leave his scent in the park. He loves to be petted and will sit on any lap for hours and so my dog lives his life imagining and choosing desirable futures and his purpose evolves along with mine. In fact, that is exactly what humans do as well. Purpose is different for different people and purpose evolves with each person over time, but basically is finding out about how the world works. Purpose is embedded within each life and purpose is why we imagine and choose desirable futures, and that purpose is part of what life is and therefore part of life’s meaning as well. The recursion of purpose and meaning with a desirable future is actually deeply embedded into each of our life journeys even though it is not always easy to understand why we are on some of the journeys that we are on. When we ask about purpose, we imagine a desirable future with an answer from someone else that will give us purpose. But purpose comes ultimately from within each of us and it is only when we choose actions for a desirable future that we realize our purpose and meaning by those actions. Our purpose is to understand how various parts of the universe work, imagine the possibilities of desirable futures, select one, and choose actions to journey to that desirable future. Our imagination and our consciousness exist only because of the compassion others and without other's compassion, there is no purpose and no meaning in our own lives either. We see others on very similar journeys in life and we therefore share some purpose and compassion and cooperate with each other on our journeys. Howard Hughes was a very famous recluse who was selfish in his privacy. And yet Hughes was surrounded by a cadre of compassionate caretakers, bodyguards, and servants and so had compassion and selfishness with others. He also watched television and his security monitors relentlessly and sadly through much of his later life. His reclusive nature still gave him a purpose in discovery of the world around him that depended on the compassion of others that were not typical. The Unabomber was a recluse who lived alone in a small cabin in Montana on a very modest fixed income. And yet he built explosive devices and mailed them to unsuspecting strangers as part of a selfish diatribe about the selfishness of technology. He found a very selfish purpose injuring strangers in the world with explosive devices since he had no compassion for people in the world with a more civil discourse. My mother-in-law lived the last several years of her life in the fog of dementia. Unable to completely care for herself, she lived with my wife and me for her last years and we therefore became a part of her purpose. Without us around constantly relating to her, she would get very disoriented and agitated and so we simply could not leave her alone for very long periods of time. She was in some sense alone in her thoughts and increasingly unable to read or watch tv or listen to music. Although she did read and watch tv and listen to music, she could not relate any of those experiences to anyone with whom she spoke. Her conversations became very rote and about things like the weather. At first, she could still talk about things of her past, experiences that she remembered, but like the driftwood of childhood amnesia, even those memories slowly eroded one by one as the dementia took her spirit from her. She would say that lunch tasted good and would enjoy eating lunch, but she could not remember what she had for lunch after lunch was finished. Without the purpose and without belief of conscious desire to understand, she increasingly survived on her primitive desires until there was no sense even in those primitive desires and she passed away in a confusion of primitive meaning and purpose. “I am done,” were her final coherent words and a week later, she passed into the same oblivion into which we all shall pass. We predict futures for isolated Cartesian objects by means of the trimal of matter, time, and action. Although knowing the relational trimal of origin, destiny, and purpose of an object is also helpful, for isolated objects, a Cartesian representation of objects in space is usually sufficient. Our Cartesian reality is a particle-like representation that imagines the isolated behavior of an object interacting with another isolated object. For highly interacting relational objects like people, though, we need to know more about them and their purposes to predict their futures. That is, we can predict an object’s future by knowing its Cartesian properties of matter, time, and action, but to predict a person’s behavior, it is more important to know about that person’s motivation and purpose than about their Cartesian state. In fact our own relational reality is a wave-like representation that is more about the relations of objects with each other than it is about their Cartesian states. We tend to ascribe human-like characteristics like purpose and meaning or compassion and selfishness to other objects as a result of the complexity of object action. This anthropomorphic tendency comes from our relational reality where trimal beliefs of origin, destiny, and purpose interpret the action of a complex system as a purpose. Purpose and meaning are simply a way to bond interacting objects within a complex system, and for people, purpose and meaning take on much more importance compared with other simple objects. The importance of purpose and meaning is with predictions of action and prediction of human behavior also affects human behavior. We teach our children compassion in a complex relational civilization, but we also teach a certain amount of selfishness as well. We then predict their compassion versus selfishness as adults in response to various sensations, feelings, and actions and we are usually pretty good at those predictions. If people are hungry and thirsty, they will selfishly find food and water. If they are cold and wet, they will selfishly find shelter. If they are on a journey, they will selfishly find transportation. If they are sick, they will selfishly find health care. However, we are affected by the actions of others and the nature and purpose of our predictions evolve along with the nature and purpose of other’s predictions. These cooperative relations support a relational humanity that has its own purpose and meaning along with its own origin and destiny. If people are hungry and thirsty, we will compassionately give them food and water. If they are cold and wet, we will compassionately provide them shelter. If they are on a journey, we will compassionately give them transportation. If they are sick, we will compassionately find them health care. As an accretion of matter from a nebula, the sun in one sense is a selfish accident of time, but our sun has a compassionate purpose in warming us on the earth. While the sun as a Cartesian object follows the selfish physical laws of nature, it is the sun’s compassion relations that binds the earth with gravity and warms earth with radiation from its sun. In the sun’s relations with earth, we say there is purpose and meaning since the consequences of the sun’s warmth are the biomes that support life and support us in our purpose. We might also consider the water on earth as an accident of time from the accumulation of comet impacts, but those comets are bound to the sun and earth and planets. Water as ice on a comet is a Cartesian object that follows the selfish physical laws while water as a compassionate relational object determines the destiny of earth’s oceans. The purpose of water is to support life just as our purpose is to support life by finding out about the world. Finally, we might consider ourselves an selfish accident of time and that there is no compassionate reason for our existence, which is part of our innate anxiety about the nothing of space. We surmount that anxiety by finding out how the world works and we do find a purpose and meaning in existence and we do imagine desirable futures and we do act to journey to those futures. Our purpose is an axiom and is how we deal with our anxiety about the nothing of space and how we get from where we are right now to one of the many possible futures that is our destiny. Both the compassion of relations among objects and the selfish loneliness of empty Cartesian space are our destiny. Sunday, June 8, 2014 Milky Way Spirals Correlate with Life's Extinctions and Explosions Since we are inside of our galaxy, it is difficult to imagine how our spiral galaxy, the Milky Way, looks from the outside. People have put together this diagram of what our galaxy would look like from above the plane of its disk. The sun is at the top of a dashed-yellow 225 million year orbit around the galaxy center. The sun's journey through the galaxy spiral features seems to correlate with various explosions and extinctions of life on earth as shown. Although the correlation is not perfect, it is very suggestive that changes in our sun's luminosity occur during spiral transits and those changes result in changes in solar irradiance and therefore in earth's climate. Since spiral wave transits also affects the convective cooling of gravitationally compressed matter, spiral transits also impact earth's magma motion and tectonics and volcanism. There are also more gravitational perturbations within a spiral wave and more young stars with more gamma radiation. Sunday, May 11, 2014 Wonders of Spiral Galaxies The spiral shape of many galaxies is truly one of the more extraordinary mysteries of the universe. Most of the reason for dark matter comes from the constant rotation of all galaxies, however the constant rotation of a galaxy is simply a manifestation of gravitization in quantum gravity. Gravitization is a vector force that couples moving star decays with each other. The product of a start matter decay, kg/s, and star velocity, m/s, is dimensionally a vector force and each star in a galaxy has a mass decay due to its radiation as well as a velocity vector from its motion. So each star has in addition to gravity an extra gravitization force due to its moving matter decay and those force vectors couple with other stars in the galaxy to make galaxy rotation constant. One manifestation of constant galaxy rotation is the spiral density waves that persist as fundamental modes in many galaxies long after the perturbation of colliding galaxies. Gravitization couples angular momentum of inner to outer stars and that coupling explains why galaxies rotate at constant velocity. The persistence of spiral density waves of galaxies due to gravitization means that there is no need for the mystery of dark matter to hold galaxies together. The Whirlpool galaxy (below, M51a) has spiral features with reported pitch angles mA = 16.7°, mB = 15.8°, while the dashed rectangle has the proportions of the golden ratio, 1.62. The golden ratio rectangle is quite well known in the aesthetics of architecture and art and encloses a golden spiral with a pitch of 17.0°. Why average pitch of galaxies is the golden spiral pitch seems more than just a coincidence. The asymmetry of the galaxy central bulge, the arrowed circle, is the dynamo that drives galaxy spiral dynamics due to gravitization. With just Einstein's gravity science needs the mystery of dark matter to keep all galaxies from flying apart. The supermassive black hole at the center is therefore somehow tied to the destiny of both the central bulge and the outer spiral disk in a cosmic ballet of gravitization that science barely understands. The tens of thousands of light years across a galaxy are the time equivalents of tens of thousands of years separating events anywhere within the galaxy. Saturday, May 10, 2014 What Is Time? Defining time seems tricky compared to matter and action, but in many ways, it is overly simplistic views of matter and action that makes time seem rather complex in comparison to matter and action. Time is just a property of a source just like color or size or distance, all of which emerge from matter and action and therefore all sources tell time. While it is clear that clocks tell time, it is perhaps not as clear that all other sources also tell time and time, just like color, is just a property of all sources just like red is a property of an apple. Among the properties of the apple are its color and its ripeness and of course, ripeness tells time for the apple. Just like the amplitude and phase coherence that are the two dimensions of matter, time likewise has two dimensions of amplitude and phase coherence. There is a long history from ancient Greece that defines two different kinds of time; Kronos as a kind of absolute time and Kairos as a kind of relative feeling of time. These two dimensions of time along with the two dimensions of matter provide realities for quantum charge and gravity that relate time and matter with the Planck action constant, h, in units of kg s. An objective atomic Kronos time is an interval dimension while a subjective decoherence Kairos action is the second time dimension. While we think of sources as existing without change in one place in space until something happens, sources are always oscillating out of and back into existence including even the universe itself. Sources therefore change and move and evolve, albeit sometimes very slowly at the limit of 0.26 ppb/yr and time emerges from the action of that change. While we imagine inaction as the complement to action, it is only action that exists as less or more action and so an evolving universe is always matter in action. In a quantum universe, there is never really complete inaction and inaction simply means that a source matches our own action or motion. Inaction means that a source evolves just like time and just like matter and so there is never really inaction, just less or more action. Each of the axioms of matter and action really have the same kinds of trickiness and the definitions of matter and action use words that simply mean the axioms as identities. For example, saying matter is a static dimension then defines matter in terms of time since dynamic is another word for changes in time. That circularity is even more confusing than time was in the first place. Saying action means a sequence of events is likewise circular since sequence is another word that means time. Figure 1 shows Cartesian interval time as a line of either infinitely divisible moments or finite moments running from past to present to future. Time in this sense is just like a Cartesian displacement in space and this Cartesian view of time is part of general relativity where there is a continuous time with a determinate future. Block time is very similar except time is now made up of finite moments or intervals that run like the frames of a movie camera from past to future. The future for block time can be determinate and just waiting for the present to catch up or there can be many futures. Relational source action time is an alternate view that tells time by the way that a source is put together as Fig. 2, which is a more primitive relational observer action time. This fossil view of time means that moments of matter come together with past actions and form a source of the present moment from any number of paths. By sensing the source with a matter spectrometer like our consciousness and knowing about the source’s fossil past, an observer can then tell time with any source. There are then a large number of possible futures associated with that present source, but there is no determinate future since all bonding is subject to quantum uncertainty. Sources with very highly structured and periodic actions tell time as clocks in Fig. 3 as relational source action time. Clocks show very regular action and therefore keep a very precise interval time with moments of matter as long as the moments are very short and periodic. However, source actions are reversible since they are built by quantum bonds and it is therefore necessary to impose an overall decoherence action time in order to point the arrow of interval time. This decoherence or action time can be thermal as in a clock power source running down or a person aging, or indeed the decoherence can be intrinsic and the whole clock shrinks. A universal decoherence points an action time direction as well as a universal quantum force from both gravity and charge. Axioms really defy further definition by any single term and so axioms are self-evident characteristics of the universe. Time emerges from the two axioms of matter and action and the trimal of matter, time, and action closes our universe. Matter is then a naturally more static dimension while action is a naturally more dynamic dimension and time emerges as the differential of action with matter, dS/dm. On the one hand, we think of interval time as a single static dimension of the past, since the past is like the frozen hands of a clock and does not seem to change except in the interval of a present moment. On the other hand, we also think of action time as a dynamic dimension that is all about the present moment, which changes and evolves into any number of possible futures. Just as we watch the second hand intervals of a clock evolve into as seemingly determinate future, we also imagine time in our experience of action that involves many possible futures. Our past is a series of moments or intervals like the frames of a video camera, but the present is a action without a determinate future or fate awaiting us in predetermined future frames. But is a moment of interval time accumulating as a past memory or a moment of action time counting down into a possible future? After all, we know time as both the predictable frames of a DVD movie and the unpredictable moments of a life stage play. A moment of time is like the tick of a clock or the recursive neural cycle of our brain or a heartbeat. Unlike the past events of interval time, a moment of action time is a dimension of the present. Action time allows any number of possible futures and the future is not therefore predetermined. Action time as a dimension is a very intuitive and understandable concept of a slowly changing universe and interval time is likewise rather clear in defining the present moment with the short period of atomic time. While action time is very slow, the intervals of atomic time are very fast and that is a little confusing since we really only seem to know time as a single fast atomic time dimension that is a past experience of action. In other words, time is in some sense two dimensional, but our memory of time is only of a norm or of a proper time. Yet there is both an action time and an interval time for all source change as orthogonal dimensions. Roughly speaking, action time represents the aether decoherence of past, present, and future while interval time represents the dynamic and immediate atomic time for an action in the present. Although we think of matter as largely static, just like time, all matter has both a slowly changing as well as a rapidly changing dimension. Our concept of matter as a single dimension of mass comes from the measurement of the gravity mass for a source, but the mass of a source is also in constant evolution as it exchanges matter with other sources. This second dimension for matter is a more dynamic dimension that is how sources exchange matter with each other. Matter as an axiom, you see, is ultimately defined only by both time and action. Dipole light is an oscillation of charge and light's color and polarization oscillate orthogonal to its propagation direction. Light is therefore a matter wave spectrum that is the dynamic exchange that bonds sources to observers in the universe. When light propagates, there is a complementary quadrupole biphoton exchange that bonds the matter left behind. Propagating light always has an entangled complementary photon and that biphoton quadrupole is in exchange with the boson matter from which space emerges. When an observer absorbs light from a source, that exchange bonds observer to source for some period of time. Pairs of light photons called biphotons also represent a coherent quadrupole of neutral oscillation. The quadrupoles of biphoton light represent the propagation of matter amplitude as neutral gravity force. Each source also exists as a matter wave, both as a propagation of matter amplitude and an oscillation of that amplitude in time orthogonal to its propagation. The oscillations of matter waves from sources are extremely high frequency and therefore do not often impact our prediction of action. We like to think of a source that is not moving as stationary, but even stationary sources are comoving with an inertial frame of reference and undergo constant exchange and action with the boson matter of the universe. Primal axioms or beliefs are a necessary and sufficient basis for closing the laws of the universe and anchoring the spectrometer of consciousness. We each need such primal beliefs to anchor and calibrate the spectrometers of our consciousness. Sometimes people feel as though they have no primal beliefs, but that is simply not true. The matter spectrometer of consciousness measures certain properties of sources, but first of all there must be primal beliefs to anchor the qualia of conscious thought. Qualia are the measured properties of sources, a red color for example, and our memory of sources relates them to other sources according to their common qualia. Consciousness only begins when we calibrate our matter spectrometer with beliefs or axioms, because it is those beliefs that allow us to make sense out of the world. Neural recursion is the basic mechanism of thought, but without a set of primal beliefs, we cannot make sense out of the world. In order for people to engage in a useful discussion about the universe, they must have an understanding and agreement about their primal beliefs. Without some understanding of each other’s primal beliefs along with a common language and how their matter spectrometers are calibrated, people usually end up arguing about their primal beliefs even though the discourse was ostensibly about some other attribute of reality. For example, a discourse about a philosophy of time will not be very useful unless there are complementary and compatible philosophies of the other axioms of matter and action, the other primal beliefs of the universe. Before discussing time, we need some manner of defining the lonely nothing that we call empty space and so there would need to be a philosophy of space as well. Two primal beliefs are the fundamental dimensions or axioms of reality from which a third emerges and it is only possible to define each primal with the other two primals. Since time is primal, time is defined as a combination of the other two primals, matter and action. We often use an action, such as the tick period of a clock, to define interval time but time is also the decoherence of that tick interval over action time and an action of those tick intervals recorded by the hands of a clock. Time as a primal axiom is not really like any single thing and the definition of time is only in terms of other two primal axioms; matter and action. The axiom of time therefore includes both a matter moment such as a tick interval and an action that accumulates those ticks such as on the hands or display of a clock. We can describe the action of a tick interval as a moment of matter that decoheres as an action time, which is the amount of matter that defines the interval of a tick along with a decoherence rate. For an hourglass, a matter moment would quite naturally be the mass of a grain of the sand. For a ticking clock, it would be the matter equivalent energy of the balance wheel resonance of the clock's mechanism. Thus an increment of matter defines a metric for a moment or interval time and it is the integration of those matter moments that becomes action time. The second is our fundamental unit of time and is formally set as 9,192,631,770 or about nine billion cycles of the cesium 133 atom hyperfine resonance. There are then 86,400 seconds in every solar day and each tick of the atomic clock then also represents a very small matter equivalent energy of 1.1e-41 kg as a matter moment. The accumulation of these tiny moments over one year amounts to the action of about three hundred hydrogen atoms. The matter spectrometer that we call consciousness samples reality as matter spectra of single moments that we call the present. We remember matter spectra of present moments that we call the past and use those memories to predict the many possible actions that we call the future. The prediction of many possible futures is a dynamic notion of time called A time while the memory of present moments is a static notion of static or B theory of time. Our notion of static time makes it seem like the future is also also frozen into moments that just wait to be played like a movie already recorded in Kronos time. This is the karma or fate of a determinate universe. Our notion of dynamic time, however, makes it seem like there exist an infinity of infinitely divisible present moments from which emerges an infinity of possible futures. The universe of discrete aether has two dimensions of time that emerge from both memories of past actions along with the emergence from the present moment of a large but finite number of possible futures. A two dimensional A-B time emerges from the discrete action of discrete aether as just a possibility from each moment. An A-B time avoids the knife edge of a present moment that is squeezed by the A time past and future and A-B also avoids the messy infinity of B-time moments. Time is therefore not just a moment of matter, as A time, and time is not just an integration of matter moments, as B time, A-B time is really both matter moments and their action. The matter moment defines an interval and a relationship among the actions that we remember as past experience. This interval might be the discrete ticks of a clock, the discrete sand grains of an hourglass, the discrete pulses of an atomic clock, the passage of discrete days, or the discrete neural recursion of human thought. The discrete memory of action can be in the positions of clock hands, the sand in the hourglass, the count of an atomic clock, the calendar of days, in the memory that we have of events, or in the possible futures that we imagine. But time itself is inextricably both discrete matter moments and the integration of those moments as discrete memories of discrete actions. An hourglass keeps time with the passage of grains of sand as hourglass ticks as well as with an accumulation of those grains as the action of the lower hourglass along with a loss of grains in the upper hourglass. Time is neither action alone nor matter alone, but time has the two dimensions of both action and matter just as an hourglass is the relationship between an amount of sand and the matter of a single grain of sand. Likewise any definition of time necessarily includes the two dimensions of both matter and action. The sand in the hourglass bottom is a memory of the integrated gain or loss, the sand in the top is one of many possible futures, and each grain of sand through the neck defines the matter moment of that clock, its tick. Neural time is how we tell the difference between the sources that we remember as our past and the actions that we imagine as possible futures. Our memories of a past action exist as matter in our minds as does how we imagine the future and the neural packets of consciousness differentiates those memories of action from the actions that we imagine for our possible futures. Just like time, we are conscious of both the matter of our memory and the neural packets of our thought. Once anchored, a time-like consciousness is why we are self-aware and why we believe that we exist with a purpose. The matter of our memory is the action of our past while the action of our neural recursion defines the matter of our neural moment. • Recursion of time: Because we see other people act just like we act, we believe we are conscious, and since we are conscious, we imagine and choose desirable futures by acting just like other people act. The definition of time as a series of moments from past to present to future is quite natural and intuitive. However, similar to the definition of space as a mostly empty void with only the volume of an occasional source, time might then also be mostly a timeless void except for occasional moments. But timeless, arbitrary eternities do not emerge to separate time moments and it is therefore curious that space emerges as an infinitely divisible empty void to separate sources. In contrast to an empty space with occasional sources, moments of time are what connect actions to each other with a common matter moment which gives each moment a composite of past and present as well as possible futures. However defining time as only a series of time moments generates paradoxes and the philosophy of time has a long history of a discourse about exactly what a moment of time means. Is time a forward stream of events with only a present moment, the dynamic A time, or is time a patchwork of separate moments, the static B time? Is there a future action as a mement waiting for us to arrive, the karma or fate of a B time movie, or are there many possible future actions and we choose the futures we like from the moment that we are in, the quantum free will of an A time live play? Although the script of an A time play is determinate, the execution of a live A time play has many possible futures. In aethertime, time is a primal axiom and time is not like any single thing except the other two axioms of matter and action. Time is not just the action of a moment in a live A time play nor is time just a series of frozen moments in a B time movie; rather time is like a series of matter moments within an action. Time is both a moment and an accumulation or loss of those moments and we project moments and remember sources and actions much like the stop action of the freeze frame of a video camera. But unlike a movie, what we play back in our mind is a highly selective and relational memory of an event that also incorporates the fading memory of a lifetime of related experiences into the action of thought. We tie every moment of time to a large number of related memories and possible futures and our experience of time is as much in those related but fading memories and possible futures as it is in the immediate sensations-feeling-action recursion of thought. We playback memories not as a DVD but with a selective focus on making predictions and choosing actions that help us survive and achieve our purpose. The neural recursion of sensation-feeling-action in our minds generates neural packets that become the matter of our memory of an event. Memories of related experiences are an active part of neural recursion and so we relate the immediate neural recursion of the present to many past remembered events and form a relational memory of that new experience. With the power of our mind and memory, we project reality as a series of static moments and interpolate when our sensations cannot resolve an action or there is missing information. Time moments are just a projection of our decohering memory of events and so moments are what we think of as time, but time is more than just the memory of moments and prediction of possible futures. Time is actually both the memory of moments as action and the decoherence of those memories as a matter moment that is the tick of our consciousness clock. The function of consciousness is time-like as is the space around us, but it is quite difficult to think about our homuncular recursion of time. The homunculus is a little person inside of our minds that is looking at what we are looking at and so the homunculus is simply a restatement of the fundamental recursion of consciousness. A homunculus, though, also has a homunculus that is also looking at the same thing, and so on. This neural recursion represents the feedback of our brain and is a basic property of thought. When we look at our own homunculus, though, we engage in a recursion or eternal recursion of sensation-feeling-action, since we look at homunculus, homunculus looks at itself and its homunculus at itself, and so on. If our homuncular recursion does not converge, just like any neural recursion in our brain that does not converge, the homuncular notion of self will make no sense and we simply will not understand and therefore will not learn the homuncular recursion of self as a truth. We will only recognize our self as different from the world if we the homuncular recursion makes sense. We project experience from neural action and memory into a series of moments that we naturally interpret as time, but time is more than just memories. This natural view of time as memories of experience is one where we can overlook the many conundrums and paradoxes of that projection as long as we can adequately predict future action. Prediction of action is, after all, what is really important and a key to our survival and the discovery of the meaning of our lives. Evolution therefore favors any mental devices that permit us to better predict action and therefore imagine the many possible futures. Consciousness correspondingly overlooks a large number of illusions and mistakes in perception of sources as long as consciousness achieves the primary goals of survival and purpose. All sources in the universe are a certain time distance or delay away from their observers and every source relates to the many possible futures of all other sources. Although we remember sources from our past and imagine many of those same sources in our possible futures, we can only ever journey to a future source. Sources in our past are only memories and there is no action that journeys to a memory. We can imagine a Cartesian journey that returns to a source that we visited in the past, but such a return will not be to the same source nor along the same path. A journey to return to a source in our past is a future action with a different path to an evolved and therefore different source. All time paths journey to future sources but there is no time path to a past memory. It is rather the projection of a Cartesian return to a spatial source that misleads us to imagine that we might return to a past time. Although we can imagine that Cartesian space does not evolve and change over time, the reality is that space does continually evolve and change. Any space to which we return at some later time, t, on the surface of our planet is a much different space than when our journey began at time zero. Even if we somehow remained fixed in a place within the cosmic microwave background, our most absolute cosmic reference frame, the very nature of space still evolves because of the universal decoherence of matter. The Cartesian separation is really a time separation and what we imagine as the lonely nothing of empty space is simply a time-like projection of our minds. Similar to the hands and ticks of a clock, a walk through a park is a journey in time that involves exchange of matter and each exchange of a matter particle provides a tick of the integrated action that separates observer from source. Complementary with time, matter and action are what make up the universe and matter and action are what bonds sources together and matter and action are also what we project as the lonely nothing of empty space. Matter and action along with time are the three irreducible properties (or axioms or qualia) of our universe and matter and its action in time are what make up the universe. Time is not like anything and matter is also not like anything and definitions of matter are circular unless they incorporate the other two primal axioms. If we say that matter is a substance, for example, a substance is just another word for matter. Matter is an axiom and therefore only the differential of action and time defines matter. Action completes the trimal of matter, time, and action that together describes our world as a matter pulse in time. An action necessarily involves integration of matter over time and even when we imagine sources as stationary and not moving, those sources still evolve and still change. Just like any source, though, a thought represents a highly relational neural spectrum within the set of 100 billion neurons in our mind and so our thoughts are also sources that are co-moving and evolving through space. Therefore, the trimal of matter, time, and action completely describes the evolution of our reality along with the evolution of our universe. Just as time inexorably advances in matter, matter likewise inexorably decoheres over time and matter’s decoherence complements an increase in the atomic clock tick rate. Although there are many sources that are co-moving with us that we call stationary, no source is ever really static and unchanging. Change and evolution are a part of all existence and part of the nature of the universe. Even though we can imagine an unchanging and static source remaining perfectly still on the surface of earth, that source is nevertheless comoving with the earth’s surface, rotating about earth’s axis, in orbit about the sun and galaxy center, and moving through the universe. And, the source’s matter exchanges and decoheres and that evolution occurs along with the ever-increasing forces that hold sources together. Most words that we use to define time are in fact just synonyms for time and defining a word with a synonym is circular or recursive. For example, among the fourteen definitions of time in Merriam-Webster are these two: b. a non-spatial continuum of events that succeed one another from past through present to future. Although it is often useful to define a word with its synonyms, more typically a definition is a short description or story about what the source is like. However primal axioms are not like anything other than all other primal axioms as a description. The primal definition of time is as the differential of action with matter, matter is as the differential of action with time, and action is the integration or product of matter in time. As with any integration, action necessarily has a constant or offset and that simply means that action can be either bound to a rest frame or free in a moving frame. The Merriam-Webster definitions of time incorporate actions, but only implicitly include matter and a definition of time as an axiom must have both action and matter. The words period or duration or continuum or progression of events or the past through present to future are all pretty much synonyms for time and so definitions with these words are equivalent to defining time as time, which is an identity. The accumulation of those actions as matter is an implicit part of these definitions. When we say that time is a sequence or series of matter actions in space, unlike backing up in space, we cannot back up or go back in time and travel to the past. When we define time as both matter and action, then it is clear that we can only ever choose a future and never a past action. It is typical to describe a word such as time by the sources or ideas that it resembles and time is action divided by matter, which is an integration of matter, action, divided by the tick matter moment. To complete a definition of time as a series of moments, we need the action of consciousness. The neural recursion of sensation-feeling-action is a homuncular recursion that is the action of consciousness. Independent of any mind, time is the action of sources along with the accumulation of those actions as matter, a duration. Every action of time is tied to a large number of related moments just as each tick of a clock is related to the accumulation of those ticks in the action of the hands of the clock or in its display. The natural moment of earth is in the length of the day and the action of a year, properties that are tied to the solar system. The natural tick of matter is the frequency of the atomic clock, some nine billion cycles per second. The natural decoherence of that tick, though, is with classical universal decoherence of 0.26 ppb/yr, and so that means the atomic clock gains about one second every 124 years. The neural recursion of our brain, runs at about the rate of our heartbeat, 1.6 Hz and our lives decohere at about 1.3%/yr for an 80 year lifetime. We naturally project the future and past into opposing Cartesian dimensions and this is where our projection of Cartesian space misleads us about time. Matter time shows that we project a three dimensional Cartesian space from time and not the other way around. We project journeys in opposite directions in each of three Cartesian dimensions as forward and backward, up and down, and left and right. However, our journeys in Cartesian space first of all are actions that involve the exchange of matter over time and the integral of that change is the action that separates us from other sources. So a journey from one source to another is an evolution of our matter spectrum and our relations and interactions with other sources are what separate us from those sources. The empty space that we imagine separates sources is just a projection of time as action divided by matter. Journeys on the surface of the earth have a beginning and duration and when we return to a journey’s starting place on earth’s surface, we naturally imagine that we might go back in time to a past memory of that source and of that place as well. However, when we return to the beginning of a journey, the earth and its surface are actually at very different places about its axis, about the sun, about the galaxy, and through the universe. It is much simpler for us to project that we have returned to the same relative place in a comoving space but that is clearly not really so. The place to which we return is both a different space as well as a future time with evolved sources. In astronomy and cosmology, a light year is the distance that light journeys in a year and is a very common measure of distance in the cosmos. In fact, all distance is equivalent to time because the speed of light does not depend on the relative velocity between two sources. The ticks of an atomic clock are therefore very precise and provide very accurate measures of spatial distance even for quite small distances. In fact, with the much lower velocities of human experience, it is quite common for people to describe distance as the time that a journey takes, like a twenty-minute commute to work or a ten-minute drive to the store. Einstein described time as a fourth spatial dimension in order to explain an odd characteristic of light in space. Einstein showed why the velocity of light for a stationary observer does not depend on the velocity of the source of that light. In fact, Lorentz first derived the equation that showed the contraction of space by time that Einstein used in relativity. It was Michelson and Morley who first measured a constant speed of light that was independent of relative velocity and the Lorentz contraction of space was consistent with this observation. Einstein used Lorentz’s projection and then added time as a fourth dimension to our three dimensional space, thereby deriving a four-dimensional space time that has had many far-reaching consequences. However, to explain the constant velocity of light, we could instead presume that light is in some sense stationary and it is us and our comoving sources that are in motion at the speed of light. In such a reinterpretation of reality, distance and separation would be necessarily time-like and matter exchange would describe all relations among sources. Time would not be just one of four spatial dimensions as it is in GR, time would instead describe all distance and the action of matter would be what we call Cartesian space that would provide Lorentz invariance. The projection of a three-dimensional Cartesian space is a very useful device of our imagination, but space would not therefore be necessary to predict action. We both remember and imagine time by counting and recording actions such as heartbeats or footsteps, which are both about one per second, or we count the ticks of a clock at about two per second. The sand grains from an hourglass fall at several per second and if we count the resonances of an atomic clock, they number about nine billion per second. By counting and remembering past heartbeats we imagine that our future heartbeats will add to a past count, but we know that we decohere at 1.3%/yr on average. Thus a clock as time always includes two fundamental qualia for definition: a matter moment, such as a tick, and the accumulation or loss of those moments as action, such as the hands of a clock face. In a similar manner as a clock, a calendar counts, records, and anticipates the number of days, weeks, months, and years of our lives. Time is a reflection of consciousness and provides order for our past as well as order for the possibilities of our future. We periodically adjust our clocks in order to keep them aligned to the natural cycle of the solar year, but we naturally presume that the tick rate of our atomic clock is otherwise constant. In matter time, our clocks tick 0.26 ppb/yr faster each year. We further interpret the past relics of ancient civilizations and the fossils of past earth as the cycles of eons and epochs of that same constant of atomic time. Our memory or record of past heartbeats and our imagining of the possibility of future heartbeats is what we call time and time is therefore a dimension that we think of in the same way that we think of space. In reality, we should really think of space as a manifestation of time and not the other way around. Time differentiates a memory of a past action, which is simply a fossil record within the matter of our brain, from the imagining of a possible future action, which is an action of neural impulses. Therefore time is the progression or sequence of action in our lives and so just as space is time-like, our consciousness is also time-like. Our memory of the count of moments is how we keep track of time and we project our past and future actions as a calendar of events. In all past action, time was equivalent to a distance between sources as well and when we imagine possible future actions, we also imagine a calendar for future actions in a way very similar to the past. This time order very effectively allows us to remember the past as a progression of events and to imagine and predict future events by projecting that memory of the past. Our science has long known that the speed of light is constant in all frames of reference and this has always been difficult to understand and communicate. If a light source moves, doesn’t the light from that source also then move? In fact, the light from either a traveler or a twin with different relative velocities has the same velocity even though each person’s light will appear as a bluer or redder color depending on whether they are moving closer or apart, respectively. Einstein resolved these various conundrums associated with the speed of light by imagining time as a fourth spatial dimension. According to Einstein, in order for the speed of light to remain constant in all frames of reference, atomic time necessarily varies between two people with different relative velocities. As a result, he also showed that the relative velocity between two people distorts or curves the Cartesian space between them and both of these predictions have been repeatedly verified with observations. But there is another way to resolve the conundrum of constant light speed. In matter time, space essentially shrinks at the speed of light and it is this shrinking of space that determines all force and also makes it appear that the speed of light is constant in a comoving frame of reference. Ironically, the traveler motion decreases the shrinkage of space ahead and increases the shrinkage of space behind the traveler. This means that the traveler cannot detect a change in light’s velocity although the frequency of the light does change. Einstein did not talk about any relation between time and memory and imagination and he also did not discuss what would happen if space was not axiomatic in our reality. What if space where a projection of time and matter and not axiomatic? The past is not only in our biological memory; the past is also in the fossil memories of past action. There is no action that rewinds reality and so we can not go back in time because until the end of the universe, there is no action without a reaction. Action can only create new memories and new fossils for the future. Nevertheless we can imagine actions that rewind time because of our Cartesian projection. Poincaré, in fact, proposed that any system of particles in seemingly chaotic motion will still cycle back to the same initial state with some probability, i.e. all systems show a possible reversal in time. However, Poincare’s proposition assumes that the particles and their space do not evolve or change with time. In matter time, space and matter both evolve in time and that means that Poincaré’s hypothesis is therefore based on different axioms from matter time. Time is just one of the three primal axioms of matter, time, and action and time is simply the quotient of matter by action, the clock count divided by its tick action. Time differentiates memory from imagination and while we remember a past time as a count of actions, we imagine a future time as a distance in space, a collection of matter ticks on a aether clock. Thus, our memory of past time is just the marker of matter while we imagine a future time that has both matter and action. Since we project Cartesian paths with opposing directions as we journey forward and backward, up and down, left and right, it is quite natural to project time with opposing directions as well. We organize time from the present as a past into a future, but we actually project space from time and not the other way around. We project our memory of past sources and actions into a path in space, typically a straight line, and we project a future action into the opposite direction in space from our past. Any future action between sources involves a change in the time distance with action and so there is no sense to a journey to a memory. A journey always involves actions that are positive time distances in a chosen direction and a future action is only one of many possible actions and those possibilities are always in our future and never in our past. Once we experience the single reality of what a source did become, it is that single reality that we remember as a past moment and the many other possible futures for a source simply decoheres away. Time as a dimension is then simply the distance between sources and time is an accumulation of matter, divided by the aether action metric. As our heart beats, the distance between heartbeats defines both a time and what we project as space. Even though we may stand perfectly still on the surface of the earth, the earth rotates about its axis, about the sun, around our galaxy, and within the universe. All action on earth defines time as distance and the loss or gain of matter with action. The events of our past are simply memories or relics or fossils of what did occur even as we imagine the possibilities of what might occur in a possible future action. From the action of sources from time delays, we project a three dimensional universe with sources on continuous time trajectories and we predict action both very precisely and very accurately with continuous space and time. We have an innate notion of a continuous void of empty space and Euclid defined the first geometric axioms some 2,300 years ago in ancient Greece and those same axioms are a fixture of our science and engineering even today. Euclid’s right angle is still the cornerstone of our Cartesian reality even though Cartesian space loses meaning for sources at very small and very large scales. The more primitive dimensions of discrete matter, time delay, and action as matter exchange have meaning for all sources in the universe. The primitive reality of matter time augments our understanding of reality for sources that exist in the realities of frozen space and time. We know about a source in either of two complementary ways. A very common and intuitive understanding of physical reality projects a source on an event path relatively unperturbed by other forces, which is a straight line in Cartesian space or a parabolic trajectory on earth. However, we actually sense or perceive a source by what it might become, i.e., by sensing some of its many possibilities, and not by what that source actually is. Our sensations represent just a very small number of a source’s possible futures and the totality of those possibilities is a complementary representation of that source. Yet even with the very small number of possible futures that we actually sense, we imagine quite a large number of possible futures, even those that do not actually make sense. By seeing, hearing, smelling, tasting, and/or touching, we sense a source as a very large number of possible futures as opposed to what the source actually is. That source might not move or it might be moving, it might change color, or it might even disappear or suddenly change its form. We imagine the reality of a source on the basis of a rather limited number of our sensations of the source’s possibilities, but we relate that source to similar sources from a lifetime of experience with similar sources. Because of our past experience, we do not normally need to sense very many possible futures for a source in order to accurately predict that source’s actual future, but we can be and often are fooled by our sensations. There are in fact many illusions that fool us just as there are also very many unlikely futures for a source that surprise us as well. So our imagining of a source on an event trajectory represents a convenient and succinct way for us to reliably predict that source’s future in our universe. The rigid Cartesian reality that we project in our minds can make it very difficult to understand time since space is a projection of time. When we project a source onto an event trajectory, we also project the context of a Cartesian space as having a forward and a backward and therefore an opposing dimension. Quite naturally we project time backwards as a spatial displacement into the past, but we first projected a Cartesian displacement from time as a useful prediction of action. Action, after all, only ever moves us closer to or further from other sources. We can actually never return to the place where we began a journey because that place no longer exists in the universe. We could imagine getting on a spacecraft and reaching a relative velocity that would maintain a place in a universal or proper space despite the rotation of earth about its axis and about the sun and about the galaxy and through the universe. However, the universe itself is shrinking in size and matter and so the universe would change even if we somehow remained in one place in space. We are so accustomed to return journeys on the surface of earth that we do not realize that every action that we take on earth involves an opposite reaction by the earth. When we jump up from earth, she falls down away from us. When we step in one direction, mother earth backsteps in the opposite direction of our stride. Our forward is her backward and our backward is her forward. The actions of our footsteps and of our heartbeats represent not only the duration or time of a past journey, but actions also represent a Cartesian distance for that past journey. Each footstep is an action for us as an observeron an event trajectory that was a part of a past journey. The memory of footsteps as an accumulation of space allows us to imagine a future journey among a large number possible journeys given a variation of our future footsteps. During a walk or run, we can turn around and change direction or we can speed up or slow down to avoid obstacles, all without any concern for the effect our stride has on earth’s rotation about its axis or earth’s orbit about the sun or earth’s place in the galaxy or indeed earth’s place in the universe. And yet all of our choices during a walk do affect the earth’s rotation as well as earth’s orbit about the sun as well as the sun’s path through the galaxy, not to mention our galaxy’s journey in the universe. Although the impact of our stride on the earth is quite small, we can think of changes in time instead of changes in distance. When we look up in the sky at night, we see only the fossil light of the past. The distance of a source that we see is the time it takes for its light to reach us and so the speed of light as a constant defines all distance as time. Every meter that light travels is about three billionths of a second or three nanoseconds, one nanosecond per foot of light travel. That constant speed of light associates an interval time with a distance and is as if everyone walked with the same speed or had exactly the same heartbeat. A second time dimension, an interval time, represents the perpendicular distance between a source and a reference direction and along with the rotation or phase of the source around that reference direction projects that source into our Cartesian space. Thus, these two dimensions of time and one dimension of phase provide an equivalent representation of our Cartesian space with a time map instead of a Cartesian map. Our earth frame of reference usually provides us with a reference direction along with many other sources as landmarks. Since the perpendicular distance between a source and a reference direction is always positive, it is the second time dimension, interval time, along with the phase or rotation of a source about the reference direction that determines a source’s direction. Cartesian space is, then, just a convenient projection of a two-dimensional time universe with phase. We imagine that there are two opposing directions for each of three Cartesian dimensions when in fact Cartesian space is just a projection of matter, time, and phase. The right angle or 90°of Euclidean geometry is equivalent to the π/2 phase angle between time and matter. The uncertainty principle in quantum mechanics involves a phase relationship between matter and time that is a complex number, -i, which derives from the the same 90° phase angle that is the right angle of Euclidean space. In matter time, Euclidean geometry reduces to a basic action equation of our quantum universe, the Schrödinger equation. A time map of a source involves two time dimensions that we project into a Cartesian plane. There is an interval time as the distance to a source and an interval time as a separation of that source from a reference direction and a phase or rotation of that source about that reference direction. We project a three dimensional Cartesian reality from two dimensions of time, event and interval time distances, and one dimension of phase or angle about a reference direction. Now what exactly does it mean to have two dimensions of time? Very simply put there is an action time and an interval time and action time is the particular time associated with an universe action and interval time is an orthogonal atomic time associated with the co-moving source frame of reference. Matter-time’s reinterpretation of reality with two time dimensions and a phase is very different from Einstein’s approach that begins with Cartesian space and then projects time a fourth spatial dimension. Einstein imagined a four-dimensional reality called space time with only a single time dimension along with our three Cartesian dimensions. The recursion of time in relativity results in a great deal of complex mathematics called tensor algebra. Since Cartesian space is a projection from time, it is distorted by time and adding time as a fourth dimension time mixes back with itself in a recursion that forms the basis of space time. Matter time has instead just three dimensions, two of time and one of phase. Although the dilations of time and therefore of space between sources traveling at different relative velocity are identical between matter time and space time science, the existence of two time dimensions in matter time complements the two dimensions for all matter as well. Given a common phase between time and matter, matter and time exist in a kind of holographic reality that defines our universe as a complex Fourier transform between the universe as a pulse of matter in time and the universe as a spectrum of matter amplitudes. In space time, the speed of light is constant and relative velocity distorts time and space between a traveler in motion with respect to a stationary twin. In matter time, light is in a sense stationary and it is actually sources of matter that move away from a light source at the speed of matter. Sources never move faster than the speed of light because motion in one direction slows the matter collapse along that great circle of the universe. In effect, our co-moving velocity is at the speed of light in all directions and once matter has slowed completely down, it then becomes light. The collapse of matter in time is a constant of matter time called mdot, and determines both gravity and charge force. Along with two other constants and the Schrödinger equation, mdot determines all forces and action for the matter-time universe and amounts to a decoherence of 0.26 ppb/yr or a gain of about one second every 124 years of the 9 billion ticks per second atomic clock. When the accretion of matter is greater than some amount, Einstein’s four-dimensional space collapses into a black hole singularity, which is a very unusual but well accepted characteristic of space time. A black hole represents a singularity of space time where no light can escape, time literally stands still at its surface, and inside of the black hole, the laws of our physical universe no longer apply. It is very clear that astronomers have observed the effects of a number of very large matter accretions that center most galaxies, often termed supermassive black holes. Science has more difficulty observing the much more subtle effects of smaller black holes that should form from the collapse of a class of stars known as supergiants. Moreover, the progression of a collapsed star known as a neutron star into the more massive black holes is very uncertain because of the role of angular momentum. It would appear that heavy, slowly rotating neutron stars might behave like black holes and that lighter, rapidly spinning black holes might behave like neutron stars. Space time physics, then, is still an incomplete story for our universe and we await an improved story that includes the unification of gravity and charge forces. Such a story will likely not only close a chapter in our understanding of time and matter, it will open new disciplines for study. In the prevailing paradigm of space time, space exists as an empty void that separates sources. The past memories and future imaginings that are in our minds represent sources that we separate by space, so consciousness is part of the source that is our mind. Our consciousness would then seem to exist as a source in time and we could then imagine a disembodied timeless mind. Consciousness would then be a convenient projection of space time and the same projection of consciousness would differentiate our memory of the past from our imagination of possible futures. In matter time, the past memories and present thoughts that are in our minds are not just sources of matter, they are time-like. Memories are matter sources of action embedded in our brains, but the neural recursion of sensation-feeling-action is action-like. Our consciousness is therefore not just a matter source or an action, but really consciousness is the time-like differential of action with matter. Just like a two dimensional time, consciousness would then also have two dimensions; event consciousness and action consciousness. Just as we have difficulty defining time and space, for the same reason we also have trouble defining the two dimensions of consciousness. Time and time-like concepts all share the characteristic that they are axiomatic and not really like anything except combinations of other axioms. However, we do not have similar difficulty defining the axioms of matter and action. Matter is the static substance of all sources and so comprises the air, water, stone, soil, and fire of our alchemie. So all sources are like matter, but matter itself is an axiom and is only explicable as the product of action and time. We can easily imagine matter or we can just as easily imagine the empty void of space as not matter or nothing. Action is the evolution of a source over time and so action is a very familiar and intuitive dynamic concept, just like matter is a static concept. We can easily imagine either action or the absence of action as a co-moving source that we think of as immobile or stationary. However, when we imagine time, it is very difficult to imagine a complement to time as timelessness. What is timelessness like? The contrapositives of matter and action are straightforward with the opposite of matter as empty space and the opposite of action as inaction and it is only with these contrapositives that we can define timelessness. Timelessness is then the inaction of empty space, a definition of eternity, and once again, we find empty space linked to the contrapositive of time. Timelessness is the inaction of empty space and represents a kind of eternity where time is the action of empty space. In fact, what we imagine separates sources in time is the aether and action that we project the action of aether as the empty void of space. We do experience timelessness during sleep, for example, or during other unconscious states. There is a rich language associated with timelessness: eternal, immortal, perpetual, everlasting, and so on. In fact, many of our religious traditions are embedded into the semantics of a timeless and perpetual eternity that addresses various transcendental questions. We know that we are conscious because the sources and actions that we remember from our past are different from the sources and actions that we imagine in our future. That is time. The timeless nature of our dreams mixes memories and imaginings and in a final dream, the neural impulses of our conscious mind become progressively slower thereby stretching time out. In effect, the timeless nature of a final dream represents the eternity of a final and fading conscious thought. Our final dream ends in either a point of ecstasy for a life fulfilled or in the circle of despair for a life unfulfilled. All sources that are in the universe are in our possible futures at a certain time distance away from us. Although we remember sources from our past, there is no journey that will take us to those past sources and time only projects sources into our possible futures. One very odd thing about time is embodied in the principle of relativity, which is that atomic clocks tick more slowly as they travel away from or towards a stationary twin clock. If a traveler accelerates to 0.8 c on a journey from a stationary twin, in five years according to the stationary twin’s clock, the traveler will journey four light years away from the twin. However, during that journey, the traveler will only have aged three years and so after the traveler slows down to the twin’s inertial frame, it will seem to the traveler that that journey’s velocity, 1.2 c, was faster than the speed of light. The traveler aged 3 years during a journey of 4 light years and traveled faster than the speed of light in the twin’s frame of reference. A traveler’s distance age can exceed the speed of light relative to the twin’s stationary frame of reference left behind. Once the traveler slows back to the twin’s inertial frame, the traveler has only aged three years while the twin has aged 5 years. Of course it will take 4 years to communicate that information back to the twin and 4 more years for the twin to acknowledge that communication and so the traveler will only know that this has occurred 8 years after reaching the destination. We can and do describe the distance between sources in space by the time it takes light to journey between those sources. Therefore, we are conscious because there is a time distance between all sources for all actions in our universe, including our neural impulses. The sequence of neural impulses in our minds represents a time distance between neurons and therefore time is a part of our consciousness. In space time, a universe of the lonely nothing of empty space is possible even without any matter, but in matter time, there can be no universe without matter. Just as no universe is possible without time, no universe is possible without matter and action, either.
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4903bf5c5c71f400	Quantum Network Theory (Part 1) guest post by Tomi Johnson If you were to randomly click a hyperlink on this web page and keep doing so on each page that followed, where would you end up? As an esteemed user of Azimuth, I’d like to think you browse more intelligently, but the above is the question Google asks when deciding how to rank the world’s web pages. Recently, together with the team (Mauro Faccin, Jacob Biamonte and Piotr Migdał) at the ISI Foundation in Turin, we attended a workshop in which several of the attendees were asking a similar question with a twist. “What if you, the web surfer, behaved quantum mechanically?” Now don’t panic! I have no reason to think you might enter a superposition of locations or tunnel through a wall. This merely forms part of a recent drive towards understanding the role that network science can play in quantum physics. As we’ll find, playing with quantum networks is fun. It could also become a necessity. The size of natural systems in which quantum effects have been identified has grown steadily over the past few years. For example, attention has recently turned to explaining the remarkable efficiency of light-harvesting complexes, comprising tens of molecules and thousands of atoms, using quantum mechanics. If this expansion continues, perhaps quantum physicists will have to embrace the concepts of complex networks. To begin studying quantum complex networks, we found a revealing toy model. Let me tell you about it. Like all good stories, it has a beginning, a middle and an end. In this part, I’ll tell you the beginning and the middle. I’ll introduce the stochastic walk describing the randomly clicking web surfer mentioned above and a corresponding quantum walk. In part 2 the story ends with the bounding of the difference between the two walks in terms of the energy of the walker. But for now I’ll start by introducing you to a graph, this time representing the internet! If this taster gets you interested, there are more details available here: • Mauro Faccin, Tomi Johnson, Jacob Biamonte, Sabre Kais and Piotr Migdał, Degree distribution in quantum walks on complex networks, arXiv:1305.6078 (2013). What does the internet look like from above? As we all know, the idea of the internet is to connect computers to each other. What do these connections look like when abstracted as a network, with each computer a node and each connection an edge? The internet on a local scale, such as in your house or office, might look something like this: Local network with several devices connected to a central hub. Each hub connects to other hubs, and so the internet on a slightly larger scale might look something like this: Regional network What about the full global, not local, structure of the internet? To answer this question, researchers have developed representations of the whole internet, such as this one: Global network While such representations might be awe inspiring, how can we make any sense of them? Or are they merely excellent desktop wallpapers and new-age artworks? In terms of complex network theory, there’s actually a lot that can be said that is not immediately obvious from the above representation. For example, we find something very interesting if we plot the number of web pages with different incoming links (called degree) on a log-log axis. What is found for the African web is the following: Power law degree distribution This shows that very few pages are linked to by a very large number others, while a very large number of pages receive very few links. More precisely, what this shows is a power law distribution, the signature of which is a straight line on a log-log axis. In fact, power law distributions arise in a diverse number of real world networks, human-built networks such as the internet and naturally occurring networks. It is often discussed alongside the concept of the preferential attachment; highly connected nodes seem to accumulate connections more quickly. We all know of a successful blog whose success had led to an increased presence and more success. That’s an example of preferential attachment. It’s clear then that degree is an important concept in network theory, and its distribution across the nodes a useful characteristic of a network. Degree gives one indication of how important a node is in a network. And this is where stochastic walks come in. Google, who are in the business of ranking the importance of nodes (web pages) in a network (the web), use (up to a small modification) the idealized model of a stochastic walker (web surfer) who randomly hops to connected nodes (follows one of the links on a page). This is called the uniform escape model, since the total rate of leaving any node is set to be the same for all nodes. Leaving the walker to wander for a long while, Google then takes the probability of the walker being on a node to rank the importance of that node. In the case that the network is undirected (all links are reciprocated) this long-time probability, and therefore the rank of the node, is proportional to the degree of the node. So node degrees and the uniform escape model play an important role in the fields of complex networks and stochastic walks. But can they tell us anything about the much more poorly understood topics of quantum networks and quantum walks? In fact, yes, and demonstrating that to you is the purpose of this pair of articles. Before we move on to the interesting bit, the math, it’s worth just listing a few properties of quantum walks that make them hard to analyze, and explaining why they are poorly understood. These are the difficulties we will show how to overcome below. No convergence. In a stochastic walk, if you leave the walker to wander for a long time, eventually the probability of finding a walker at a node converges to a constant value. In a quantum walk, this doesn’t happen, so the walk can’t be characterized so easily by its long-time properties. Dependence on initial states. In some stochastic walks the long-time properties of the walk are independent of the initial state. It is possible to characterize the stochastic walk without referring to the initialization of the walker. Such a characterization is not so easy in quantum walks, since their evolution always depends on the initialization of the walker. Is it even possible then to say something useful that applies to all initializations? Stochastic and quantum generators differ. Those of you familiar with the network theory series know that some generators produce both stochastic and quantum walks (see part 16 for more details). However, most stochastic walk generators, including that for the uniform escape model, do not generate quantum walks and vice versa. How do we then compare stochastic and quantum walks when their generators differ? With the task outlined, let’s get started! Graphs and walks In the next couple of sections I’m going to explain the diagram below to you. If you’ve been following the network theory series, in particular part 20, you’ll find parts of it familiar. But as it’s been a while since the last post covering this topic, let’s start with the basics. Diagram outlining the main concepts A simple graph G can be used to define both stochastic and quantum walks. A simple graph is something like this: Illustration of a simple graph where there is at most one edge between any two nodes, there are no edges from a node to itself and all edges are undirected. To avoid complications, let’s stick to simple graphs with a finite number n of nodes. Let’s also assume you can get from every node to every other node via some combination of edges i.e. the graph is connected. In the particular example above the graph represents a network of n = 5 nodes, where nodes 3 and 4 have degree (number of edges) 3, and nodes 1, 2 and 5 have degree 2. Every simple graph defines a matrix A, called the adjacency matrix. For a network with n nodes, this matrix is of size n \times n, and each element A_{i j} is unity if there is an edge between nodes i and j, and zero otherwise (let’s use this basis for the rest of this post). For the graph drawn above the adjacency matrix is \left( \begin{matrix} 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \end{matrix} \right) By construction, every adjacency matrix is symmetric: A =A^T (the T means the transposition of the elements in the node basis) and further, because each A is real, it is self-adjoint: (the \dagger means conjugate transpose). This is nice, since (as seen in parts 16 and 20) a self-adjoint matrix generates a continuous-time quantum walk. To recap from the series, a quantum walk is an evolution arising from a quantum walker moving on a network. A state of a quantum walk is represented by a size n complex column vector \psi. Each element \langle i , \psi \rangle of this vector is the so-called amplitude associated with node i and the probability of the walker being found on that node (if measured) is the modulus of the amplitude squared \|\langle i , \psi \rangle\|^2. Here i is the standard basis vector with a single non-zero ith entry equal to unity, and \langle u , v \rangle = u^\dagger v is the usual inner product. A quantum walk evolves in time according to the Schrödinger equation \displaystyle{ \frac{d}{d t} \psi(t)= - i H \psi(t) } where H is called the Hamiltonian. If the initial state is \psi(0) then the solution is written as \psi(t) = \exp(- i t H) \psi(0) The probabilities \| \langle i , \psi (t) \rangle \|^2 are guaranteed to be correctly normalized when the Hamiltonian H is self-adjoint. There are other matrices that are defined by the graph. Perhaps the most familiar is the Laplacian, which has recently been a topic on this blog (see parts 15, 16 and 20 of the series, and this recent post). The Laplacian L is the n \times n matrix L = D - A where the degree matrix D is an n \times n diagonal matrix with elements given by the degrees \displaystyle{ D_{i i}=\sum_{j} A_{i j} } For the graph drawn above, the degree matrix and Laplacian are: \left( \begin{matrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} \right) \qquad \mathrm{and} \qquad \left( \begin{matrix} 2 & -1 & 0 & -1 & 0 \\ -1 & 2 & -1 & 0 & 0 \\ 0 & -1 & 3 & -1 & -1 \\ -1 & 0 & -1 & 3 & -1 \\ 0 & 0 & -1 & -1 & 2 \end{matrix} \right) The Laplacian is self-adjoint and generates a quantum walk. The Laplacian has another property; it is infinitesimal stochastic. This means that its off diagonal elements are non-positive and its columns sum to zero. This is interesting because an infinitesimal stochastic matrix generates a continuous-time stochastic walk. To recap from the series, a stochastic walk is an evolution arising from a stochastic walker moving on a network. A state of a stochastic walk is represented by a size n non-negative column vector \psi. Each element \langle i , \psi \rangle of this vector is the probability of the walker being found on node i. A stochastic walk evolves in time according to the master equation \displaystyle{ \frac{d}{d t} \psi(t)= - H \psi(t) } where H is called the stochastic Hamiltonian. If the initial state is \psi(0) then the solution is written \psi(t) = \exp(- t H) \psi(0) The probabilities \langle i , \psi (t) \rangle are guaranteed to be non-negative and correctly normalized when the stochastic Hamiltonian H is infinitesimal stochastic. So far, I have just presented what has been covered on Azimuth previously. However, to analyze the important uniform escape model we need to go beyond the class of (Dirichlet) generators that produce both quantum and stochastic walks. Further, we have to somehow find a related quantum walk. We’ll see below that both tasks are achieved by considering the normalized Laplacians: one generating the uniform escape stochastic walk and the other a related quantum walk. Normalized Laplacians The two normalized Laplacians are: • the asymmetric normalized Laplacian S = L D^{-1} (that generates the uniform escape Stochastic walk) and • the symmetric normalized Laplacian Q = D^{-1/2} L D^{-1/2} (that generates a Quantum walk). For the graph drawn above the asymmetric normalized Laplacian S is \left( \begin{matrix} 1 & -1/2 & 0 & -1/3 & 0 \\ -1/2 & 1 & -1/3 & 0 & 0 \\ 0 & -1/2 & 1 & -1/3 & -1/2 \\ -1/2 & 0 & -1/3 & 1 & -1/2 \\ 0 & 0 & -1/3 & -1/3 & 1 \end{matrix} \right) The identical diagonal elements indicates that the total rates of leaving each node are identical, and the equality within each column of the other non-zero elements indicates that the walker is equally likely to hop to any node connected to its current node. This is the uniform escape model! For the same graph the symmetric normalized Laplacian Q is \left( \begin{matrix} 1 & -1/2 & 0 & -1/\sqrt{6} & 0 \\ -1/2 & 1 & -1/\sqrt{6} & 0 & 0 \\ 0 & -1/\sqrt{6} & 1 & -1/3 & -1/\sqrt{6} \\ -1/\sqrt{6} & 0 & -1/3 & 1 & -1/\sqrt{6} \\ 0 & 0 & -1/\sqrt{6} & -1/\sqrt{6} & 1 \end{matrix} \right) That the diagonal elements are identical in the quantum case indicates that all nodes are of equal energy, this is type of quantum walk usually considered. Puzzle 1. Show that in general S is infinitesimal stochastic but not self-adjoint. Puzzle 2. Show that in general Q is self-adjoint but not infinitesimal stochastic. So a graph defines two matrices: one S that generates a stochastic walk, and one Q that generates a quantum walk. The natural question to ask is whether these walks are related. The answer is that they are! Underpinning this relationship is the mathematical property that S and Q are similar. They are related by the following similarity transformation S = D^{1/2} Q D^{-1/2} which means that any eigenvector \phi_k of Q associated to eigenvalue \epsilon_k gives a vector \pi_k \propto D^{1/2} \phi_k that is an eigenvector of S with the same eigenvalue! To show this, insert the identity I = D^{-1/2} D^{1/2} into Q \phi_k = \epsilon_k \phi_k and multiply from the left with D^{1/2} to obtain \begin{aligned} (D^{1/2} Q D^{-1/2} ) (D^{1/2} \phi_k) &= \epsilon_k ( D^{1/2} \phi_k ) \\ S \pi_k &= \epsilon_k \pi_k \end{aligned} The same works in the opposite direction. Any eigenvector \pi_k of S gives an eigenvector \phi_k \propto D^{-1/2} \pi_k of Q with the same eigenvalue \epsilon_k. The mathematics is particularly nice because Q is self-adjoint. A self-adjoint matrix is diagonalizable, and has real eigenvalues and orthogonal eigenvectors. As a result, the symmetric normalized Laplacian can be decomposed as Q = \sum_k \epsilon_k \Phi_k where \epsilon_k is real and \Phi_k are orthogonal projectors. Each \Phi_k acts as the identity only on vectors in the space spanned by \phi_k and as zero on all others, such that \Phi_k \Phi_\ell = \delta_{k \ell} \Phi_k. Multiplying from the left by D^{1/2} and the right by D^{-1/2} results in a similar decomposition for S: S = \sum_k \epsilon_k \Pi_k with orthogonal projectors \Pi_k = D^{1/2} \Phi_k D^{-1/2} I promised above that I would explain the following diagram: Diagram outlining the main concepts (again) Let’s summarize what it represents now: G is a simple graph that specifies A the adjacency matrix (generator of a quantum walk), which subtracted from D the diagonal matrix of the degrees gives L the symmetric Laplacian (generator of stochastic and quantum walks), which when normalized by D returns both S the generator of the uniform escape stochastic walk and Q the quantum walk generator to which it is similar! What next? Sadly, this is where we’ll finish for now. We have all the ingredients necessary to study the walks generated by the normalized Laplacians and exploit the relationship between them. Next time, in part 2, I’ll talk you through the mathematics of the uniform escape stochastic walk S and how it connects to the degrees of the nodes in the long-time limit. Then I’ll show you how this helps us solve aspects of the quantum walk generated by Q. In other news Before I leave you, let me tell you about a workshop the ISI team recently attended (in fact helped organize) at the Institute of Quantum Computing, on the topic of quantum computation and complex networks. Needless to say, there were talks on papers related to quantum mechanics and networks! Some researchers at the workshop gave exciting talks based on numerical examinations of what happens if a quantum walk is used instead of a stochastic walk to rank the nodes of a network: • Giuseppe Davide Paparo and Miguel Angel Martín-Delgado, Google in a quantum network, Sci. Rep. 2 (2012), 444. • Eduardo Sánchez-Burillo, Jordi Duch, Jesús Gómez-Gardenes and David Zueco, Quantum navigation and ranking in complex networks, Sci. Rep. 2 (2012), 605. Others attending the workshop have numerically examined what happens when using quantum computers to represent the stationary state of a stochastic process: • Silvano Garnerone, Paolo Zanardi and Daniel A. Lidar, Adiabatic quantum algorithm for search engine ranking, Phys. Rev. Lett. 108 (2012), 230506. It was a fun workshop and we plan to organize/attend more in the future! 33 Responses to Quantum Network Theory (Part 1) 1. John Baez says: Great post! I especially like how you use quantum versus stochastic walks to organize your treatment of the various Laplacian-like operators associated to a graph! Previously these various operators seemed like a bit of a mess to me. It would be good (and probably not hard) to generalize this whole discussion to weighted simple graphs, i.e., those with a positive number labelling each edge. The idea is that the adjacency matrix of a weighted graph is a matrix A of numbers where A_{ij} is the weight of the edge between i to j, or zero if there’s no edge. Weighted graphs are important because in general, ‘not every link is created equal’. Things in general flow more easily, or more often, through some edges than others! Also, as we let the weight of an edge approach zero, our weighted graph can be seen as approaching a graph where that edge doesn’t exist. So, we get a nice topology on the set of all weighted graphs with a given set of vertices. And if you think about it a while, the resulting space of weighted graphs is just the space of symmetric matrices with nonnegative entries. So the math should get very nice. • Actually, in the paper we work entirely on weighted graphs (which are natural both for stochastic and quantum evolution). However, the tricky thing is about interpretation of the Hamiltonian (i.e. a Hermitian matrix) with real, nonnegative entries. If it were just for real entries, it would be a Hamiltonian with time-reversal symmetry. But do you have any ideas how to interpret the restriction on having only nonnegative entries? • Piotr, can you rephrase the question? Do you mean a ‘physics’ interpretation, or a ‘graph/ranking’ interpretation? For ordinary QM, on typically has H=T+V where T is kinetic part, i.e. laplacian, and V is the potential; some interaction term. In the above post, V=0. For graphs, the entries forming the laplacian must, by definition, sum to zero: the diagonal entries are positive, off-diagonal are negative. So if all matrix entries are positive, then there must be some (strong, non-local) V term that describes some interaction between different nodes. (a local V would be zero off-diagonal) To study V, take your H, subtract whatever you think your Laplacian is, and look what’s left over. It will presumably be recognizable as something: I dunno, maybe exp of adjacency matrix or something … I don’t get your time-reversal comment at all: for time-reversal in QM, you must change the sign of both time and energy (i.e. the energy eigenvalues); so this has little to do with the signs in the matrix. And stochastic systems are essentially never time-reversible (frobenius-perron eigenvalue gives the rate of decay to equilibrium). For stochastic systems, one also has the concept of a “wandering set”; a set of points that wander away from their initial locations, never to return. Here, the analog is, I guess, web pages that have no incoming links (or blocks of web pages that have no incoming links), so that their stochastic probability (page rank) shrinks to zero. If you think in terms of measure, that measure is leaving one location, and it has to go somewhere else: it accumulates somewhere (viz, becomes perfectly uniform on the final equilibrium state.) This doesn’t generalize to quantum systems, instead you get a kind-of poincare recurrance or ringing/beating/interference effect. So: free-associating: I’m guessing that perhaps your all-positive entries are trying to capture some net flow from one part of the graph to another. OK, this last sentence doesn’t actually make sense, I’m thinking aloud. • Piotr knows this, and we’re even working on this topic together, but: Time reversal symmetry should be with respect to a specfic quantity. Assuming no spin, and when considering transfer probabilities, then Hamiltonians with real entries in the site basis generate time-inversion symmetric transition rates. 2. John Baez says: I will be going to China in a few minutes, and will be there until August 20. During this time I may not post to this blog very often, or at all. 3. Almost everything developed here also applies equally well to any homogenous space, right? That is, the vector \psi is replaced by a point p in the homogenous space, and the various matricies above by group elements of whatever group it is that is acting on the homogenous space. This abstraction is very rarely done; I’ve always wondered what I’m missing because of it. I figure there are two reasons for this: 1) the authors are not familiar enough with the general idea of homogenous spaces to be able to make specific claims that they’re confident of (heck, I’m certainly not), and 2) some part of the problem definition fails to generalize. The 1) I can excuse, but 2) leaves me hanging. • Tomi Johnson says: Great comment, interesting point. I think I largely fall into the first category: not knowing much about homogeneous spaces. I agree, there should be no problem mathematically transfering what we’ve done here onto a continuous space. For the quantum case this would just be something like the standard single particle Schroedinger equation in a potential with a modified kinetic energy term to account for the geometry of the network. The classical case would some similar stochastic diffusion equation. I’ll have a think about this! • linasv says: Well, I think I’m saying that both quantum and classical are special cases of a general framework. The classical case uses points that live in a simplex (total probability sums to 1); Markov matrices relate/move the various points. The quantum case uses points that live in CP^n (viz, the n-body wave-function) with U(n) matrices to relate/move the various points. The general case has points on some general manifold, with some matrices to move them around. To keep some amount of symmetry/invariance in the problem, it seems like the manifold should be a homogenous space. The Schroedinger eqn is a thing that lives in the tangent space of the manifold. … Perhaps what I’m saying is that I don’t understand why the Laplacians are what they are. I mean, I understand at the shallow level: the quantum case must be symmetric, to get unitarity; the classical case must decay, to get probabilities that sum to 1. These are given axiomatically: these are the rules of the game. What’s the general case? How is the specific manifold forcing the Laplacian into the specific form its taking? I don’t quite see this ‘big picture’. • Tomi Johnson says: Perhaps the reason that the formalism for unitary quantum and stochastic dynamics above has not been generalized (to our knowledge) is that it is unclear what other physical objects \psi and evolutions d \psi / dt = H \psi that would fall under this generalization. Perhaps one is the generalized quantum dynamics, where a physical object \rho, the density matrix (trace 1, Hermitian, positive), is evolved according to d \rho / dt = L \rho, where L is not a Hamiltonian, but some superoperator that leads to the preserving of the trace, Hermiticity and positivity, e.g. L \rho = [H,\rho], where H is Hermitian. In fact, in this formalism you could simultaneously include both quantum dynamics under the symmetric normalized Laplacian and the stochastic dynamics under the asymmetric normalized Laplacian. When we’re back at ISI after the holidays, I’ll raise your point with the others there, and see what they think. • John Baez says: By the way, I fixed your LaTeX. The correct way to use LaTeX on this blog is described right above the box where you type your comments. You need to include the word ‘latex’ in the manner described. Sorry it took a while to approve your comments—I’ve had intermittent access to the internet. • linasv says: Thanks Tomi; of course, I’m just being lazy and could google up enough to keep me busy to answer my own questions. Here’s maybe another way to ask them: if one studies finite automata, one soon realizes that their state transitions live on a graph, and that the probabalistic finite automata is kind-of like a Markov chain. The other thing one discovers is that a finite automata is just the action of a monoid on a set. If the set is a simplex, and the representation of the monoid element is a Markov matrix, then you’ve got your classic radio signal engineering problem. If the set is CP^n and the representation of monoid elements is U(n) then you’ve got a quantum finite automata. But these are just two special cases; the general case is studied, e.g. google suggests: http://citeseerx.ist.psu.edu/viewdoc/summary?doi= The difference between the finite automata and what you are doing is that in the finite automata, the graph edges are labelled with a symbol (the monoid element), and thus a graph walk corresponds to a sequence of symbols (edges walked). A random walk on a graph induces a measure on the set of strings of symbols (aka ‘the language’). If the random walk is independent of the history of the path, then it is Markovian, and the measure factorizes. (if random walk is not independent of the history, then it must be generated by a push-down automaton (context free language) or even a full Turing machine). If I take a random walk on a graph, and, after coming to a vertex, I assign equal probability to leaving by any edge, then I get your stochastic Laplacian, above. Or, as John Baez suggests, I could twiddle the edge weights, and preferentially leave on some edges. Or I could twiddle the exit probabilities at each vertex (i.e. each edge has two weights not one, depending on whether one is coming or going), and recover a classic Markov chain (minus diagonal) instead of the Laplacian. And here is where I get confused. The word ‘Markov’ in my last paragraph is not really the same as in my first paragraph, and yet its closely related, and so mentally I circle back, and wonder “what other sets can the monoid act on?” … and perhaps I now can answer my question, like so: In measure theory, measures must the real (and positive), and so the measure assigned to a language (the set of all random walks on graph) is real-valued, and thus “stochastic”. Perhaps(?) one can contemplate measures with an additional U(1) in them, and perhaps this is what the quantum walk is providing!? So when I ask “what other sets can the monoid act on, and what would the Laplacian, etc. generalize to in such cases?” then perhaps I am contemplating set-valued measures on languages? Hmmmm. Sorry for the long post. Sometimes, mathematics is like a visit to a candy shop; each treat looks more delicious than the last, and picking out just one to enjoy is just too hard. • linasv says: p.s. I goofed in my last post, I mis-characterized what the definition of the language of an FA is. (It’s the sequence of vertexes, not the sequence of edges; the edge sequence is the coding; the vertexes are the plain-text.) Caveat Emptor. 4. amarashiki says: Fascinating! It seems this is going to be another of your great series, John! I can’t wait to read the next one. BTW, networks are pretty like hypergraphs (I think I told you it before…) and I found very brainy the “map-of-internet”. Off-topic: how did you write the text in boxes! Just curious! I am planning to release my domain soon and this class of tricks could be useful for me LaTeXing article series. Best, JFGH • Tomi Johnson says: Thanks for the comment. John might have changed it a little bit, but my original suggestion was to use the html to create the boxes (I just copied this from what is used on the Azimuth forum). Hope that helps! • John Baez says: Amarashiki wrote: Tomi Johnson wrote this, so he deserves all the credit… even for figuring out how to put text in boxes. It works like this: <div style="background:#fff1f1;border:solid black;border-width:2px 1px;padding:0 1em;margin:0 1em;overflow:auto;"> 5. domenico says: I am thinking, now, to an unification of the two physics description. If the Hamiltonian is a function of the wave functions, and it can be real or complex, then the stocastic and quantum description is the same: it is possible to write the Taylor series with wave function terms and complex number. In other word, the solution can be the same for stocastic, or quantum, evolution. • For example, if a Hamiltonian matrix is symmetric from this property, we can say that all stationary states can be chosen to take only real values. This is a physical (sub) consequence of what Piotr mentioned—sub because it is only a consequence of the fact that the Hamiltonian has real entries. If they’re real non-negative, this additional restriction results in additional mathematical properties. • Tomi Johnson says: I agree that it’s a very nice property that unitary quantum dynamics under Hamiltonian -iH is identical to stochastic dynamics under Hamiltonian H. The methods to solve/simulate one type of dynamics can therefore be transferred to the other. This is something I’ve worked on in the context of (tensor network-based) numerical methods for efficiently near-exactly simulating stochastic dynamics. These types of methods were devised for simulating quantum dynamics, but we applied them to stochastic dynamics in the following paper, • T. H. Johnson, S. R. Clark, and D. Jaksch, Dynamical simulations of classical stochastic systems using matrix product states, Phys. Rev. E 82 (2010), 036702; arXiv:1006.2639. if you’re interested. We plan to publish more work on this very soon. 6. I have found this post to be extremely well written and clearly explained, and overall very easy to follow. Nice work! Looking forward for the rest of the series. 7. Ramsay says: Thanks for the nice post. I am curious as to why the “aysmmetric normalized Laplacian” is defined as it is. Specifically, I am more used to seeing an operator that is the transpose of S, i.e., X = D^{-1}L. Then, at least in the contexts that I have studied, it is natural to introduce the inner product \langle x, y \rangle_{D} = x^{T} D y, and the operator X is self-adjoint with respect to this inner product. The inner product is natural in many contexts, where the elements of D encode a measure of the “importance” or weight associated with each node. • Tomi Johnson says: Thanks for the comment! I think the matter of the transpose is easily resolved. We consider stochastic (or transition) matrices to act to the right on column probability vectors. Others consider them to act to the left on row probability vectors. The difference between the two formalisms is just a transpose. As for the inner product, that is a nice fact, thanks for pointing it out. Do you have a link to any of the contexts in which you came across it? • My thoughts are: i. I think it’s the difference between letting operators act to the left, or in our case, the right, on probability vectors. ii. Even in the wikipedia definition of “random walk normalized Laplacian” it’s defined as mentioned. iii. For the inner product, please provide a link so I can look at how it’s used exactly. iv. We are forced to work in the site basis, of the walker. This is the basis that an operator must be self adjoint with respect to. We can redefine the inner product, to take away the asymmetry, but we could also just multiply by D to accomplish the same goal While it’s true that you define self adjoint with respect to an inner product, it’s not clear how that helps us in any way. In fact, if you try to write the operator with respect to this new inner product, it just removes D as we already mentioned. • Ramsay says: Thanks. I should clarify that “the contexts that I have studied” are quite far removed from the random walker setting being discussed here. I also asked the question naively, without having read your paper. The contexts that I am familiar with are discrete Laplacians modeled on the Laplacian of a Riemannian manifold. Typically the domain would be a geometric simplicial complex, and D^{-1}L would be such That D is a diagonal matrix whose entries are the volumes of closed (Voronoi) cells associated with the vertices: they sum to the volume of the manifold. The same idea is sometimes used with graphs. For example, see the discussion on p. 297 (5th page) of Fujiwara’s paper “Growth and the spectrum of the Laplacian of an infinite graph” There is a lot of development of this idea into a “discrete exterior calculus”, where D can be seen as an instance of a discrete Hodge star operator. For me to really appreciate the point (iv) you made, I would need to study your paper, but I haven’t done that yet. Your point (i) answers my question on why the definition is as it is, (i.e., a convention to use row vectors instead of column vectors), although my first hit at wikipedia contradicts your point (ii): 8. […] Last time I told you how a random walk called the ‘uniform escape walk’ could be used to analyze a network. In particular, Google uses it to rank nodes. For the case of an undirected network, the steady state of this random walk tells us the degrees of the nodes—that is, how many edges come out of each node. […] 9. Arjun Jain says: Nice post. Small typo: phi_k -> phi_k above the diagram in the section on the normalized laplacian. 10. In this blog post I will introduce some basics of quantum mechanics, with the emphasis on why a particle being in a few places at once behaves measurably differently from a particle whose position we just don’t know. It’s a kind of continuation of the “Quantum Network Theory” series by Tomi Johnson about our work in Jake Biamonte’s group at the ISI Foundation in Turin. Leave a Reply to linas vepstas Cancel reply WordPress.com Logo Google photo Twitter picture Facebook photo Connecting to %s
d15903cee336b267	Vol. 63, No. 7 (2014) Damped brachistochrone problem and the relation between constraint and theorem of motion Ding Guang-Tao 2014, 63 (7): 070201. doi: 10.7498/aps.63.070201 Abstract + The damped brachistochrone problem and that with non-zero initial velocity are studied. Based on the discussion of these problems, one may take theorems of motion as constraints for some systems, and whether the constraints are holonomic or nonholonomic is related to the fact that the differential theorems of motion are integrable or non-integrable. Simulation of optimal control of train movement based on car-following model Ye Jing-Jing, Li Ke-Ping, Jin Xin-Min 2014, 63 (7): 070202. doi: 10.7498/aps.63.070202 Abstract + Optimal control of train movement is an important way to reduce transport cost, enhance service level, and realize sustainable development. In this paper, based on traditional optimal velocity car-following model, an improved simulation model is presented, it is used to optimize the velocity control of train movement in urban railway system. The proposed model is established by introducing a new function of objective optimal velocity into the classical optimal velocity model (See Phys. Rev. E 51, 1035, Bando et al, 1995) to realize the optimal control of train movement in complicated conditions. Numerical simulation takes the Beijing City Metro Yi Zhuang line as an example. Here some reality measurement data is used. Results show that the proposed model can well describe the dynamic characteristics of train movement under the complex limited condition. Simulation results are close to reality measurement data. This demonstrates that the proposed model is valid. Further, by analyzing the space-time graph, the change of train velocity and travel time, the evolution characters of train flow under complex conditions are discussed. Symplectic FDTD algorithm for the simulations of double dispersive materials Wang Hui, Huang Zhi-Xiang, Wu Xian-Liang, Ren Xin-Gang, Wu Bo 2014, 63 (7): 070203. doi: 10.7498/aps.63.070203 Abstract + Combined with the Lossy Drude-Lorentz dispersive model, a symplectic finite-difference time-domain (SFDTD) algorithm is proposed to deal with the double dispersive model. Based on matrix splitting, symplectic integrator propagator and the auxiliary differential equation (ADE) technique, with the rigorous and artful formula derivation, the algorithm is constructed, and detailed formulations are provided. Excellent agreement is achieved between the SFDTD-calculated and exact theoretical results when transmittance coefficient in simulation of double dispersive film in one dimension is calculated. As to numerical results for a more realistic structure in three dimensions, the simulation of periodic arrays of silver split-ring resonators using the Drude dispersion model are also included. The transmittance, reflectance, and absorptance of the structure are presented to test the efficiency of the proposed method. Our method can be used as an efficiency simulation tool for checking the experimental data. Research on biomolecule-gate AlGaN/GaN high-electron-mobility transistor biosensors Li Jia-Dong, Cheng Jun-Jie, Miao Bin, Wei Xiao-Wei, Zhang Zhi-Qiang, Li Hai-Wen, Wu Dong-Min 2014, 63 (7): 070204. doi: 10.7498/aps.63.070204 Abstract + In order to enhance the performance of AlGaN/GaN high electron mobility transistor (HEMT) biosensor, millimeter grade AlGaN/GaN HEMT structure have been designed and successfully fabricated. Factors influencing the capability of the AlGaN/GaN HEMT biosensor are analyzed. UV/ozone is used to oxidize GaN surface and then 3-aminopropyl trimethoxysilane (APTES) self-assembled monolayer can be bound to the sensing region. This serves as a binding layer in the attachment of prostate specific antibody (anti-PSA) for prostate specific antigen detection. The millimeter grade biomolecule-gated GaN/AlGaN HEMT sensor shows a quick response when the target prostate specific antigen in a buffer solution is added to the antibody-immobilized sensing area. The detection capability of this biomolecule-gate sensor estimated to be below 0.1 pg/ml level using a 21.5 mm2 sensing area, which is the best result of GaN/AlGaN HEMT biosensor for PSA detection till now. The electrical result of the biomolecule-gated GaN/AlGaN HEMT biosensor suggests that this biosensor might be a useful tool for the prostate cancer screening. Stochastic resonance in an overdamped monostable system with multiplicative and additive α stable noise Jiao Shang-Bin, Ren Chao, Li Peng-Hua, Zhang Qing, Xie Guo 2014, 63 (7): 070501. doi: 10.7498/aps.63.070501 Abstract + In this paper we combine α stable noise with a monostable stochastic resonance (SR) system to investigate the overdamped monostable SR phenomenon with multiplicative and additive α stable noise, and explore the action laws of the stability index α (0 α ≤ 2) and skewness parameter β (-1 ≤ β ≤ 1) of the α stable noise, the monostable system parameter a, and the amplification factor D of the multiplicative α stable noise against the resonance output effect. Results show that for different distributions of α stable noise, the single or multiple low-and high-frequency weak signals detection can be realized by adjusting the parameter a or D within a certain range. For a or D, respectively, there is an optimal value which can make the system produce the best SR effect. Different α or β can regularly change the system resonance output effect. Moreover, when α or β is given different values, the evolution laws in the monostable SR system excited by low-and high-frequency weak signals are the same. The conclusions drawn for the study of single-and multi-frequency monostable SR with α stable noise are also the same. These results will be the foundation for realizing the adaptive parameter adjustment in the monostable SR system with α stable noise. Modeling and simulation analysis of fractional-order Boost converter in pseudo-continuous conduction mode Tan Cheng, Liang Zhi-Shan 2014, 63 (7): 070502. doi: 10.7498/aps.63.070502 Abstract + Based on the fact that the inductor and the capacitor are fractional in nature, the fractional order mathematical model of the Boost converter in pseudo-continuous conduction mode is established by using fractional order calculus theory. According to the state average modeling method, the fractional order state average model of Boost converter in pseudo-continuous conduction mode is built. In view of the mathematical model, the inductor current and the output voltage are analyzed and the transfer functions are derived. Then the differences between the integer order and the fractional order mathematical models are analyzed. On the basis of the improved Oustaloup fractional order calculus for filter approximation algorithm and the model of fractional order inductance and capacitance, the simulation results have been compared between the mathematical model and circuit model with Matlab/Simulink software; the origins of model error are analyzed and the correctness of the modeling in fractional order and the theoretical analysis is verified. Finally, the differences and the relations of Boost converter among the continuous conduction mode, the discontinuous conduction mode, and the pseudo-continuous conduction mode are indicated. Spectrum calculation of chaotic SPWM signals based on double fourier series Liu Yong-Di, Li Hong, Zhang Bo, Zheng Qiong-Lin, You Xiao-Jie 2014, 63 (7): 070503. doi: 10.7498/aps.63.070503 Abstract + Chaotic SPWM control has attracted much interests due to its effectiveness for EMI suppression in power converters. However, most researches focus on the simulation and experiment of power converter under chaotic SPWM control, which is lacking a quantitative method. Based on double Fourier series this paper provides a spectrum calculation method for multi-period SPWM or quasi-random SPWM signals firstly, and the related spectrum calculation and simulation for multi-period SPWM are given to verify the accuracy of the spectrum calculation method; then the calculation method is extended to the spectral analysis of chaotic SPWM signals. To observe the impact on the spectrum of chaotic SPWM signals generated by different mappings and in different variation ranges of carrier period, a spectrum comparison between the Tent and Chebyshev mappings is conducted, in which results indicate that the variation range of the carrier period and the selection of mappings have a great influence on spectrum distribution; in the long term, probability density distribution of chaotic mapping will certainly affect the spectrum, and in the short term the initial value of the mapping will also affect the spread spectrum distribution. In summary, the proposed spectrum calculation method in this paper provides a theoretical foundation for the spread spectrum principle of chaotic SPWM control and for the design reference in practical engineering application. Manipulation of the complete chaos synchronization in dual-channel encryption system based on polarization-division-multiplexing Zhong Dong-Zhou, Deng Tao, Zheng Guo-Liang 2014, 63 (7): 070504. doi: 10.7498/aps.63.070504 Abstract + For the dual-channel encryption system, based on polarization-division-multiplexing, we put forward a new control scheme for complete chaos synchronization by means of linear electro-optic (EO) effect. In the scheme, the chaotic synchronization quality of each linear polarization (LP) mode component varies periodically with the applied electric field. The variation regulation is as follows: Complete chaos synchronization ↔ acute oscillation. With the applied electric field fixed at a certain value, the robustness of the complete chaotic synchronization quality due to the bias current and the feedback strength is improved greatly by EO modulation. Each LP mode can obtain the complete chaos synchronization in a large range of the bias current and the feedback strength. And the encoding message modulated to each LP mode can be almost re-established. A new method of background elimination and baseline correction for the first harmonic Zhang Rui, Zhao Xue-Hong, Hu Ya-Jun, Guo Yuan, Wang Zhe, Zhao Ying, Li Zhi-Xiao, Wang Yan 2014, 63 (7): 070702. doi: 10.7498/aps.63.070702 Abstract + A new method of background elimination and baseline correction is proposed, since there are background signal and larger baseline signal in the first harmonic (1f) of the tunable diode laser absorption spectroscopy (TDLAS). The laser-associated intensity modulation signal, electronic noise, and optical interference fringes of the 1f background are analyzed. Harmonic detection in none absorption spectral region (HDINASR) is used to eliminate the background signal. Then the relationship curve between current and intensity is given in different operating temperatures to design a remaining baseline correction method after eliminating the background. The principle of background signal searching and the LabView software flow chart are also given. The TDLAS experimental system is designed to detect hydrogen fluoride (HF) gas. According to spectral line selection principle, the absorption line -1312.59 nm is selected, whose operating temperature is set at 27.0 ℃ and the background temperature is set at 30.2 ℃. After eliminating the background and correcting the baseline, signal distortion is significantly improved and baseline is corrected. Then it is verified that the method is valid at other operating temperature of the laser (26.7-27.2). And the improvement of HF gas concentration is quantitatively analyzed. It is convenient for the subsequent processing of 1f signal. Thermal-sensitive superconducting coplanar waveguide resonator used for weak light detection Zhou Pin-Jia, Wang Yi-Wen, Wei Lian-Fu 2014, 63 (7): 070701. doi: 10.7498/aps.63.070701 Abstract + Since the last decades, superconducting single-photon technology has been extensively used in the quantum security communication and the linear-optic quantum computing fields. Especially, the device based on the coplanar waveguide resonator has attracted substantial interests due to its evident advantages, including the relatively simple structure, the sufficiently high detection efficiency, and the photon-resolving capability, etc. With the profound investigation in optimizing the depositing methods and the material selections, as well as the the development of the relevant theories, the technology of single photon detection based on the coplanar waveguide resonator has obtained a breakthrough. In this review paper we begin from the basic principle of the coplanar waveguide detector, then interpret the relevant theory and some design details of the devices. Finally, based on some of the recent experimental results measured with the low-temperature devices in our lab, we give a brief perspective on the future development of the superconducting coplanar waveguide single photon detectors. Single zeptosecond pulse generation from muonic atoms under two-color XUV fields Li Zhi-Chao, Cui Sen, He Feng 2014, 63 (7): 073201. doi: 10.7498/aps.63.073201 Abstract + We use the Lewenstein model to study the high harmonic generated for a μp atom exposed to two-color XUV pulses. Calculated results show a super continuum plateau in high harmonic spectrum which is formed when the time delay is 0 and XUV frequencies are 5 and 2.5. By synthesizing the continuous high harmonic spectra, a pulse as short as 130 zeptosecond is obtained. Such a single zeptosecond pulse may work as an ultrafast camera to capture ultrafast processes occurring inside nuclei. Theoretical and experimental study on the multi-color broadband coherent anti-Stokes Raman scattering processes Yin Jun, Yu Feng, Hou Guo-Hui, Liang Run-Fu, Tian Yu-Liang, Lin Zi-Yang, Niu Han-Ben 2014, 63 (7): 073301. doi: 10.7498/aps.63.073301 Abstract + In order to exactly distinguish and quantitatively analyze the different or unknown components in a mixture, the global molecular CARS spectra information should be obtained simultaneously with a broad-band coherent anti-Stokes Raman scattering (CARS) spectroscopy in supercontinuum. In a broad-band CARS spectroscopy, two-and three-color CARS processes are generated due to different functions of effective spectroscopic components in supercontinuum. Firstly, we theoretically analyzed the generation conditions of CARS signals and the relationships between their intensity and power of excitation lights in the two types of CARS process with the broad-band excitation. On this basis, the two types of CARS process are achieved with a home-built broad-band CARS spectroscopic system, respectively. Using the functional fitting analysis of the obtained CARS spectral signals of benzonitrile, the relationships between CARS signals and excitation lights are experimentally verified in two different kinds of CARS process. Further optimizations of broad-band time-resolved CARS spectroscopic and microscopic systems, for simultaneously obtaining the global CARS spectral signals of samples, can be achieved under the guidance of theoretical and experimental results. Ab initio calculation of the potential energy curves and spectroscopic properties of BP molecule Wang Wen-Bao, Yu Kun, Zhang Xiao-Mei, Liu Yu-Fang 2014, 63 (7): 073302. doi: 10.7498/aps.63.073302 Abstract + A high-precision quantum chemistry ab initio multi-reference configuration interaction method with aug-cc-pVQZ basis sets has been used to calculate the four states of BP molecule. The four -S states are X3, 3-, 5 and 5-, which are correlated to the lowest dissociation limit of B(2Pu)+P(4Su). Analysis of the electronic structures of -S states shows that the -S electronic states are essentially multi-configurational. We take the spin-orbit interaction into account for the first time so far as we know, which makes the four -S states split into fifteen states. 30+ state is confirmed to be the ground state. The SOC effect is essential for the BP molecule, which leads to the avoided crossings for 0+ and 1 states from X3 and 3-. Based on the PECs of -S and states, the accurate spectroscopic constants are obtained by solving the radial Schrdinger equation. The spectroscopic results may be conducive to further research on BP molecule in experiment and theory. Optical bottle beam generated by a new type of light emitting diode lens He Xi, Du Tuan-Jie, Wu Feng-Tie 2014, 63 (7): 074201. doi: 10.7498/aps.63.074201 Abstract + A new method for generating a single bottle beam directly by light emitting diode (LED) with a secondary optical lens is proposed for the first time, so far as we know. Firstly, in the aspect of geometrical optics, we analyze the principle of generation of a single bottle beam by the LED spot light with a secondary optical lens. Then, we calculate the expression of the length and the radius of the biggest dark region of the bottle beam. After that, a new type of a secondary optical lens is calculated numerically and simulated by numerical recipes software Matlab, three-dimensional modeling software Solidworks and optical simulation software Tracepro. Meanwhile, the minimum size of the bottle beam and the scattering force for trapping particles are calculated. The result shows that the designed secondary optical lens can produce a single bottle beam, the length and the radius of the biggest dark region of the generated bottle beam are in accordance with the theoretical calculations. This result offers a practical and available method for generating a bottle beam with light emitting diode at a low cost. New transverse Zeeman effect method for mercury detection based on common mercury lamp Li Chuan-Xin, Si Fu-Qi, Zhou Hai-Jin, Liu Wen-Qing, Hu Ren-Zhi, Liu Feng-Lei 2014, 63 (7): 074202. doi: 10.7498/aps.63.074202 Abstract + The accurate background correction can determine the minimum limit of trace mercury measurement in atmosphere by the cold vapor atomic absorption method. This paper studies a new method of mercury detection using the common mercury lamp as sources which correct the background according to the transverse Zeeman effect. The resonance spectral line (253.65 nm) of the meccury lamp generates σ-, σ+, and π linear polarized light in the vertical direction of the magnetic field. This study obtains mercury absorbance of σ-, σ+, and π light in different magnetic field intensity by using ultra-high resolution spectrometer, then gets the minimum field intensity of the method. We discuss the existing possible interference caused by benzene with narrow-band absorption and acetone with broadband absorption under 1.78 T magnetic field intensity. Taking σ- and σ+ as background light, and π as absorption light, we quantify the saturated mercury vapor cell with different lengths. With the accurate background correction, the R value of absorption fitting curve can achieve 0.99. Results indicate that the method can accomplish the job of accurate background correction and can be applied to trace mercury measurement in atmosphere. Theoretical analysis on cavity-enhanced laser cooling of Er3+-doped glasses Jia You-Hua, Gao Yong, Zhong Biao, Yin Jian-Ping 2014, 63 (7): 074203. doi: 10.7498/aps.63.074203 Abstract + In recent years, Er3+ doped CdF2-CdCl2-NaF-BaF2-BaCl2-ZnF2 (CNBZN) glass has become one of the new materials in the field of laser cooling of solids. In this paper, using the theory of laser output and standing wave resonance, intracavity-and extracavity-enhanced laser cooling of Er3+-doped CNBZN glass are theoretically analyzed. Calculated results show that enhancement factor can achieve tens to hundreds of times. Moreover, two schemes are compared with each other, and the results show that for low material absorption, especially when the sample length is less than 0.3 mm, intracavity configuration has the advantage of high pumping power and high absorption. However, for high material absorption, especially when the sample length is longer than 3 mm, the extracavity configuration becomes a more efficient means for laser cooling. Finally, according to the operating wavelength and power requirements of Er3+-doped material, cavity enhancement can be realized experimentally using semiconductor diode laser. Statistical analysis of shot-to-shot variation of laser fluence spatial distribution Han Wei, Zhou Li-Dan, Li Fu-Quan, Wang Fang, Feng Bin, Zheng Kui-Xin, Gong Ma-Li 2014, 63 (7): 074204. doi: 10.7498/aps.63.074204 Abstract + The shot-to-shot variation of laser fluence spatial distribution on a large-aperture high-power laser facility is statistically analyzed. Statistical results show that the maximum fluence spatial distribution to which any location in the optic beam will be exposed after N shots, can be described by Gaussian function, and the average fluence across the beam increases with laser shots while standard deviation is relatively constant, independent of laser shots. This is due to the fact that laser fluence spatial distribution possesses similarity over the whole beam and dissimilarity at local positions for different laser shots. Study on the fabrication of gold electrode by laser assembling Zhang Ran, Lü Chao, Xiao Xin-Ze, Luo Yang, He Yan, Xu Ying 2014, 63 (7): 074205. doi: 10.7498/aps.63.074205 Abstract + We proposed the fabrication of gold micro-electrode and grating electrode through laser assembling of gold nanoparticles and realized the electrical interconnection of the single carbon nanotube and gold nanolines, which can decrease the damage of the functional unit to a great extent. This method can also solve the problem of inadequate mass transport in the fabrication of ions. The microstructure could keep unoxidized in the atomasphere with excellent continuity, integrity, and electrical properties, which made this technique have wide application prospects. Influence of coupling coefficient on sparseness of slope response matrix and iterative matrix Cheng Sheng-Yi, Chen Shan-Qiu, Dong Li-Zhi, Liu Wen-Jin, Wang Shuai, Yang Ping, Ao Ming-Wu, Xu Bing 2014, 63 (7): 074206. doi: 10.7498/aps.63.074206 Abstract + Based on a 529-actuator adaptive optic (AO) system, the sparseness of slope response matrix from deformable mirror to Hartmann wavefront sensor and the sparseness of iterative matrix in wavefront reconstruction are analyzed. The influence of actuator coupling coefficient on the slope response matrix sparseness, the iterative matrix sparseness, and the AO system correction quality are also studied under the condition of constant actuator spacing. Larger coupling coefficient results in a lower sparseness of slope response matrix and an iterative matrix. Too large or too small coupling coefficient will lead to lower stability and correction quality of AO system. Finally, the optimal range of coupling coefficient is provided by the balancing correction quality, sparseness of slope response matrix, and stability. Crystals modulated by two parameters and their applications Li Chang-Sheng 2014, 63 (7): 074207. doi: 10.7498/aps.63.074207 Abstract + In the applications of two external fields, such as stresses and electric fields, the optical modulation properties of some crystals are theoretically analyzed using the method of index ellipsoid. Simple mathematical formulas for the calculations of the field-induced principal refractive indexes of some crystals and corresponding azimuthal angles of their principal axes can be deduced from the equation of index ellipsoid if there exists only one nonzero cross term in the equation, e.g. x1x2. According to these simple formulas, we can find out some crystals exhibiting dual transverse electrooptic effect, e.g. crystals of the 6 symmetry point group. Under two simultaneously applied external stresses, elastooptic birefringence of a crystal is proportional to the difference between the two external stresses, and the orientations of their birefringent axes are unchanged. When a stress and an electric field are simultaneously and perpendicularly applied to some crystals such as cubic crystals of 43m point group, the field-induced birefringence of the crystal is proportional to the weighted geometric mean of the applied stress and electric field, and the orientations of their birefringent axes only depend on the ratio of the applied electric field and stress. The above electrooptic and elastooptic modulation properties are useful to the design of novel optical modulators and sensors. A block-based improved recursive moving-target-indication algorithm Hou Wang, Yu Qi-Feng, Lei Zhi-Hui, Liu Xiao-Chun 2014, 63 (7): 074208. doi: 10.7498/aps.63.074208 Abstract + A new block-based recursive moving-target indication algorithm in velocity domain is proposed to solve the problem which is the rapid detection of dim and small target for infrared search and tracking system. Firstly, the two-dimensional least mean square filter is adopted to filter the infrared image sequence, which extracts small targets and residual errors of image sequence. Then, block-based recursive moving-target indication algorithm is adopted to accumulate small target in image block sequence for the enhancement of small target velocity in velocity domain. Finally, resulting image is obtained by using classical recursive moving-target indication algorithm and target velocity for small target detection. Compared with classical method, the proposed method requires less running time, and can be used to detect dim small target image effectively as demonstrated by several groups of experimental results. Analysis of electron momentum relaxation time in fused silica using a tightly focused femtosecond laser pulse Bian Hua-Dong, Dai Ye, Ye Jun-Yi, Song Juan, Yan Xiao-Na, Ma Guo-Hong 2014, 63 (7): 074209. doi: 10.7498/aps.63.074209 Abstract + The electron momentum relaxation time is studied systematically in order to understand its effect during the excited nonlinear ionization process in fused silica with an irradiation of tightly focused femtosecond laser pulses. According to the analysis of a (3+1)-dimensional extended general nonlinear Schrödinger equation, the electron momentum relaxation time shows a huge effect on peak intensity, free electron density, and fluence distributions in the focal region of the incident pulse, meanwhile a value of 1.27 fs is thought to meet the present experimental result based on the theoretical model. Further research indicates that the change of electron momentum relaxation time can have significant difference among several nonlinear mechanisms, such as the laser-induced avalanche ionization, reverse bremsstrahlung, self-defocusing of plasma, etc. Results show that the electron momentum relaxation time plays an important role in the process of femtosecond laser pulses interaction with materials. Study of 1550 nm low loss single mode all-solid photonic bandgap fibers Cheng Lan, Luo Xing, Wei Hui-Feng, Li Hai-Qing, Peng Jing-Gang, Dai Neng-Li, Li Jin-Yan 2014, 63 (7): 074210. doi: 10.7498/aps.63.074210 Abstract + All-solid photonic bandgap fiber shave attracted great attention of researchers due to their particular band gap and dispersion character as well as the merit of easily splicing the traditional optical fiber. We have fabricated the all-solid photonic bandgap fibers using the plasma chemical vapor deposition (PCVD) and a modified tack and draw technique, and the loss and dispersion characteristics were simulated by the finite-difference frequency-domain (FDFD) method. The fiber obtained by this method has a low-loss region at around 1550 nm and can be used as single-mode; its effective model field area and the dispersion of the fiber at 1550 nm are 191.81 μm2 and 16.418 ps/(km·nm), respectively. Combined with the experimental results, the fiber parameters are further optimized by simulation. Calibration of D-RGB camera networks by skeleton-based viewpoint invariance transformation Han Yun, Chung Sheng-Luen, Yeh Jeng-Sheng, Chen Qi-Jun 2014, 63 (7): 074211. doi: 10.7498/aps.63.074211 Abstract + Combining depth information and color image, D-RGB cameras provide a ready detection of human and associated 3D skeleton joints data, facilitating, if not revolutionizing, conventional image centric researches in, among others, computer vision, surveillance, and human activity analysis. Applicability of a D-RBG camera, however, is restricted by its limited range of frustum of depth in the range of 0.8 to 4 meters. Although a D-RGB camera network, constructed by deployment of several D-RGB cameras at various locations, could extend the range of coverage, it requires precise localization of the camera network: relative location and orientation of neighboring cameras. By introducing a skeleton-based viewpoint invariant transformation (SVIT), which derives the relative location and orientation of a detected humans upper torso to a D-RGB camera, this paper presents a reliable automatic localization technique without the need for additional instrument or human intervention. By respectively applying SVIT to two neighboring D-RGB cameras on a commonly observed skeleton, the respective relative position and orientation of the detected humans skeleton for these two cameras can be obtained before being combined to yield the relative position and orientation of these two cameras, thus solving the localization problem. Experiments have been conducted in which two Kinects are situated with bearing differences of about 45 degrees and 90 degrees; the coverage can be extended by up to 70% with the installment of an additional Kinect. The same localization technique can be applied repeatedly to a larger number of D-RGB cameras, thus extending the applicability of D-RGB cameras to camera networks in making human behavior analysis and context-aware service in a larger surveillance area. Molecular dynamics simulation of the thermal conductivity of silicon functionalized graphene Hui Zhi-Xin, He Peng-Fei, Dai Ying, Wu Ai-Hui 2014, 63 (7): 074401. doi: 10.7498/aps.63.074401 Abstract + Direct non-equilibrium molecular dynamics (NEMD) was used to simulate the thermal conductivities of the monolayer and the bilayer silicon functionalized graphenes along the length direction respectively, with the Tersoff potential and the Lennard-Jones potential, based on the velocity Verlet time stepping algorithm and the Fourier law. Simulation results indicate that the thermal conductivity of the monolayer silicon functionalized graphene decreases rapidly with increasing amount of silicon atoms. This phenomenon could be primarily attributed to the changes of graphene phonon modes, mean free path, and motion speed after silicon atoms are embedded in the graphene layer. Meanwhile, the thermal conductivity of the monolayer graphene is declined in the temperature range from 300 to 1000 K. As for the bilayer silicon functionalized graphene, its thermal conductivity increases as a few silicon atoms are inserted into the layer, but decreases when the number of silicon atoms reaches a certain value. Bifurcation and chaos of some strongly nonlinear relative rotation system with time-varying clearance Liu Bin, Zhao Hong-Xu, Hou Dong-Xiao, Liu Hao-Ran 2014, 63 (7): 074501. doi: 10.7498/aps.63.074501 Abstract + The dynamic equation for the relative rotation nonlinear dynamic system with time-varying clearance is investigated. Firstly, transformation parameter is deduced by using the method of MLP; the bifurcation response equations of 1/2 harmonic resonance then are generated by the method of multiple scales, while singularity analysis is employed to obtain the transition set of steady motion; further more the bifurcation characteristic and the bifurcation of the system under the situation of non-autonomy are analyzed. Finally, numerical simulation exhibits many different motions, such as periodic motion, period-doubling motion, and chaos. It is shown that the change of clearance and damp parameters may influence the motion state of the system. Fractional derivative dynamics of intermittent turbulence Liu Shi-Da, Fu Zun-Tao, Liu Shi-Kuo 2014, 63 (7): 074701. doi: 10.7498/aps.63.074701 Abstract + Intermittent turbulence means that the turbulence eddies do not fill the space completely, so the dimension of an intermittent turbulence takes the values between 2 and 3. Turbulence diffusion is a super-diffusion, and the probability of density function is fat-tailed. In this paper, the viscosity term in the Navier-Stokes equation will be denoted as a fractional derivative of Laplatian operator. Dimensionless analysis shows that the order of the fractional derivative α is closely related to the dimension of intermittent turbulence D. For the homogeneous isotropic Kolmogorov turbulence, the order of the fractional derivatives α=2, i.e. the turbulence can be modeled by the integer order of Navier-Stokes equation. However, the intermittent turbulence must be modeled by the fractional derivative of Navier-Stokes equation. For the Kolmogorov turbulence, diffusion displacement is proportional to t3, i.e. Richardson diffusion, but for the intermittent turbulence, diffusion displacement is stronger than Richardson diffusion. Drag reduction on hydrophobic transverse grooved surface by underwater gas formed naturally Wang Bao, Wang Jia-Dao, Chen Da-Rong 2014, 63 (7): 074702. doi: 10.7498/aps.63.074702 Abstract + Low fluid friction is difficult to obtain on super-hydrophobic surfaces for a large flow velocity, because the entrapped gas within the surface is weakened substantially. Once the gas removed, the friction of the fluid increases markedly due to its own surface roughness. In this study, a hydrophobic transverse microgrooved surface is designed to sustain the air pockets in the valleys for a long time. Direct optical measurements are conducted to observe the entrapped gas when water flows past the surface in the perpendicular direction of grating patterns. More importantly, this hydrophobic transverse micro-grooved surface has been determined to have the capability of automatic formation of gas. Some of the gas is continually carried away from the surface and new gas is continually generated to substitute the lost gas. And the stable slippages at the surface are achieved corresponding to the relative stable gas on this designed surface. A novel lattice Boltzmann method for dealing with arbitrarily complex fluid-solid boundaries Shi Dong-Yan, Wang Zhi-Kai, Zhang A-Man 2014, 63 (7): 074703. doi: 10.7498/aps.63.074703 Abstract + A suitable arbitrarily complex boundary condition treatment using the lattice Boltzmann sheme is developed in the fluid-solid coupling field. The new method is based on a half-way bounce back model. A virtual boundary layer is built with the fluid-solid coupling, and all the properties used on the virtual boundary are inter-/extrapolated from the surrounding nodes combining with the finite difference method. The improved method ensures that the particles bounce the same location as that of the macro-speed sampling point, and considers the offset effect on the accuracy of the calculated results when the actual physical borders and the grid lines do not coincide. And its scope is extended to any static or mobile, straight or curved boundary. The processing power of the method under the classic conditions, such as the Poiseuille flow, the flow around a circular cylinder, the Couette flow, etc. is studied. Results prove that the theoretically calculated values agree well with the experimental data. Compared with the results published in the literature, this method has a greater precision when the actual physical borders and gridlines do not coincide. Investigation of electromagnetic hydrodynamics propulsion and vector control by surfaces based on a rotational navigation body Liu Zong-Kai, Gu Jin-Liang, Zhou Ben-Mou, Ji Yan-Liang, Huang Ya-Dong, Xu Chi 2014, 63 (7): 074704. doi: 10.7498/aps.63.074704 Abstract + Realization of electromagnetic hydrodynamics (MHD) propulsion by surfaces needs an electromagnetic body force generated in a conductive fluid (such as seawater and plasma, etc.) around the navigation body. Furthermore, the reaction force against the electromagnetic body force could be used to propel. Based on the basic control equations of electromagnetic field and fluid mechanics, the vector control effect has been analyzed by virtue of field intensity and force distribution characteristic on the rotational navigation body, under two different force action areas. Results show that the navigation attitude adjustment could be realized by this control method without changing attacks and propulsion directions. An upward force moment could be achieved by the control model A. Accordingly, both of the pitching moment and yaw moment could be changed by the control model B. Thus, as a new way of propulsion, the MHD propulsion by surfaces offers several advantages, such as high speed, high efficiency, easy operation, high payload etc. Additionally, in this paper, the vector propulsion has been proved to be one of the remarkable advantages for MHD propulsion by surface. Study on the gain characteristics of terahertz surface plasma in optically pumped graphene multi-layer structures Liu Ya-Qing, Zhang Yu-Ping, Zhang Hui-Yun, Lü Huan-Huan, Li Tong-Tong, Ren Guang-Jun 2014, 63 (7): 075201. doi: 10.7498/aps.63.075201 Abstract + Based on the developed optically pumped graphene multilayer terahertz surface plasma structures, this paper calculates the real part of propagation index and amplification coefficient in optically pumped graphene multilayer structures, discusses the inluences of momentum relaxation time, temperature, numbers of grapheme layers, and the quasi-Fermi energy in the topmost grapheme layer on the real part of propagation index and amplification coefficient. It is shown that when the real part of dynamic conductivity becomes negative in the terahertz range of frequencies in the optically pumped graphene multilayer structures, the surface plasma of graphene layers can achieve gain. By comparing the peeling-graphene-structure with the graphene structure that has a high conducting bottom graphene layer in optically pumped scheme, it can be said that the surface plasma of the peeling-graphene-structure can get a high efficient amplification. Meanwhile, the structure having properly numbers of graphene layers can get a larger amplification than the simple graphene structure in an optically pumped scheme at low temperatures. Three-dimensional modelling and numerical simulation on segregation during Fe-Pb alloy solidification in a multiphase system Wang Zhe, Wang Fa-Zhan, Wang Xin, He Yin-Hua, Ma Shan, Wu Zhen 2014, 63 (7): 076101. doi: 10.7498/aps.63.076101 Abstract + The three-dimensional mathematical model for a three-phase flow during its horizontai solidification is studied using fluid dynamics method based on Eulerian-Eulerian and volume of fraction methods, in which the mass, momentum, species, and enthalpy conservation equations of the Fe-Pb alloy solidification process are solved simultaneously. Effects of Pb area quadratic gradient (∇ (∇SPb)) and Pb concentration quadratic gradient (∇ (∇CPb)) on the segregation formation are investigated. Results show that the segregation mode is manifested as X-segregates in the upper and V-segregates in the lower part during flow-solidification of liquid phase and gas phase. The X-segregates result from the phase transformation driving force of gas phase and “scattering” is due to the orientation of phase transition. When t >tc the lower ∇ (∇SPb) and ∇ (∇CPb) curves cause a larger yielding rate of Pb with a larger down angle of X-segregates and a smaller up angle of X-segregates and V-segregates. All these are favorable for the formation of a well-dispersed microstructure. In addition, the gas-liquid two-phase flow interaction term has an effect on channel segregation, showing that channels occur only in the region where the flow-phase transition interaction term (ul·∇cl and ug·∇cg) is negative. With a negative flow-phase transition interaction term the increase in flow velocity due to the flow perturbation and flow-phase transition interaction becomes more negative, thus the channel continues to grow and tends to be stable. Calculated results show good agreement with experimental data. Theoretical study on geometry and physical and chemical properties of oligochitosan Li Xin, Zhang Liang, Yang Meng-Shi, Chu Xiu-Xiang, Xu Can, Chen Liang, Wang Yue-Yue 2014, 63 (7): 076102. doi: 10.7498/aps.63.076102 Abstract + By using the density functional theory with B3LYP/6-31G+(d) we compute the optimization, vibration frequencies, electron structures of gg conformation of oligochitosans, and study the average binding energies and the zero-point energy corrections using WB97XD method. We also analyze the thermodynamic properties of oligochitosans. Results show that the hydrogen-bond makes the oligochitosan become spiral; average binding energies tend to decrease and stability tends to improve with the increasing degree of polymerization (DP); the water degradation of oligochitosan is an exothermic reaction, so it is feasible to reduce the temperature to improve the degradation yield in experiment; in addition, the energy gap of oligochitosan quickly converges to 6.99 eV with the increase of DP; furthermore, the value of DP7 oligochitosan is in accordance with the convergence value. The HOMO and LUMO of oligochitosan show that chemical activity is mainly distributed in C2 amino, C6 hydroxyl groups, and both ends of oligochitosan chain. These results have instructive significance on the modeling, and can provide a theoritical basis for degradation process, chemical activity position, and size-dependence in physical chemical properties of oligochitosan. Atomistic simulation study on the local strain fields around an extended edge dislocation in copper Shao Yu-Fei, Yang Xin, Li Jiu-Hui, Zhao Xing 2014, 63 (7): 076103. doi: 10.7498/aps.63.076103 Abstract + The local strain fields around an extended edge dislocation in copper are studied via the quasicontinuum multiscale simulation method combined with the virial strain calculation techniques. Results show that in the regions, tens of nanometers away from the dislocation, atoms are experiencing infinitesimal strain; virial strain calculation results are consistent with the predictions from elastic theory very well. In the regions near the dislocation, the virial strain fields can outline the core areas of Shockley partial dislocations precisely, which are in the shape of ellipse with a longer axis 7b1 and a shorter axis 3b1, where b1 is the length of burgers vector of the partial dislocation. Characterization of thermal conductivity for GNR based on nonequilibrium molecular dynamics simulation combined with quantum correction Zheng Bo-Yu, Dong Hui-Long, Chen Fei-Fan 2014, 63 (7): 076501. doi: 10.7498/aps.63.076501 Abstract + A nonequilibrium molecular dynamics model combined with quantum correction is presented for characterizing the thermal conductivity of graphene nanoribbons (GNR). Temperature effect on graphene nanoribbon thermal conductivity is revealed based on this model. It is shown that different from the decreasing dependence in classical nonequilibrium molecular dynamics simulations, an “anomaly” is revealed at low temperatures using quantum correction. Besides, the conductivity of GNR shows obvious edge and scale effects: The zigzag GNR have higher thermal conductivity than the zigzag GNR. The whole temperature range of thermal conductivity and the slope of thermal conductivity at low temperatures both show an increasing dependence of width. Boltzmann-Peierls phonon transport equation is used to explain the temperature and scale effects at low temperatures, indicating that the model constructed is suitable for a wide temperature range of accurate calculation for thermal conductivity of different chirality and width. Research provides a possible theoretical and computational basis for heat transfer and dissipation applications of GNR. Influences of electrode separation on structural properties of μc-Si1-xGex：H thin films Cao Yu, Zhang Jian-Jun, Yan Gan-Gui, Ni Jian, Li Tian-Wei, Huang Zhen-Hua, Zhao Ying 2014, 63 (7): 076801. doi: 10.7498/aps.63.076801 Abstract + Hydrogenated microcrystalline silicon germanium (μc-Si1-xGex:H) thin films have been prepared by radio frequency plasma-enhanced chemical vapor deposition (RF-PECVD) using a mixture of SiH4 and GeH4 as the reactive gases. Effects of electrode separation on the structural properties of μc-Si1-xGex:H thin films have been investigated. Results show that reduction of the electrode separation can increase the Ge content in the films. Moreover, μc-Si1-xGex:H thin film deposited at a lower electrode separation of 7 mm possesses not only a stronger (220) orientation and a larger grain size, but also a lower microstructural factor. Then, the decomposition characteristics of the reactive gases are analyzed according to the variation of the structural properties of the μc-Si1-xGex:H thin films. It is found that the increase of the Ge content is due to the decrease of the SiH4 decomposition rate in the plasma. While the better film quality obtained at the lower electrode separation is attributed to the enhancement of the diffusibility of the Ge precursors caused by improving the proportion of GeH3 radicals First-principles study on p-type ZnO codoped with F and Na Deng Sheng-Hua, Jiang Zhi-Lin 2014, 63 (7): 077101. doi: 10.7498/aps.63.077101 Abstract + The first-principles calculations based on the density functional theory have been performed to investigate the doping behaviors of Na and F dopants in ZnO. It turns out from the calculated results of the band structure, density of states, and effective masses that in the F mono-doping case, the impurity states are localized and the formation energy is up as high as 4.59 eV. In the Na mono-doping case, the impurity states are delocalized and the formation energy decreases as low as -3.01 eV. One cannot obtain p-type ZnO in both instances On the contrary, in the Na-F codoping case, especially when the ratio of F and Na is 1:2, the Fermi-level shifts to the valence bands, the corresponding effective masses are small (0.7m0) and the formation energy is the lowest (-3.55 eV). These may indicate the formation of p-type ZnO having a good conductivity. Ferromagnetism of Zn0.97Cr0.03O synthesized by PLD Xie Ling-Ling, Chen Shui-Yuan, Liu Feng-Jin, Zhang Jian-Min, Lin Ying-Bin, Huang Zhi-Gao 2014, 63 (7): 077102. doi: 10.7498/aps.63.077102 Abstract + Four Zn0.97Cr0.03O films were deposited on quartz wafers in various oxygen environment (0, 0.05, 0.15 and 0.2 Pa) using pulsed laser deposition (PLD). The films were characterized by XRD, PL, XPS, magnetic and electrical properties. Experimental results indicate that: (1) All the films are well crystallized and display a pure orientation. (2) All the films have ferromagnetism, and the film deposited at 0.15 Pa has the biggest Ms. (3) There exist VZn, Oi, Zni, VZn- and VO defects in the four films above, and the percentage of resonance peak area for VZn to the total area of all defects as a function of oxygen pressure is similar to Ms, which means that the magnetizations of the samples are closely related to Zn vacancy VZn. There is a Cr3+ state in the four films when the content of Cr3+ is the largest at 0.15 Pa. To sum up, the experimental results indicate that the substitutive Cr in the oxidation state of t3 and the neutral Zn vacancy in the Zn0.97Cr0.03O films is the most favorable defect complex to maintain a high stability of ferromagnetic order, which is consistent with the calculated results by the first-principle calculations. Theoretical analysis of carbon nanotube photomixer-generated terahertz power Jia Wan-Li, Zhao Li, Hou Lei, Ji Wei-Li, Shi Wei, Qu Guang-Hui 2014, 63 (7): 077201. doi: 10.7498/aps.63.077201 Abstract + On the basis of mixer circuit model of light, the terahertz power generated by the carbon nanotubes (CNT) photomixer is analyzed. By simulating mixer conductance, impedance of the antenna, and light plus paranoid voltage, it is shown that the improved mixer conductance, antenna impedance and light plus paranoid voltage can improve the output power of terahertz waves. The output power can reach dozens of microwatt level in the small-signal limit. One-dimensional photonic crystal(1D PC)-based back reflectors for amorphous silicon thin film solar cell Chen Pei-Zhuan, Hou Guo-Fu, Suo Song, Ni Jian, Zhang Jian-Jun, Zhang Xiao-Dan, Zhao Ying 2014, 63 (7): 077301. doi: 10.7498/aps.63.077301 Abstract + New-type back reflectors based on one-dimensional photonic crystal (1D PC) for amorphous silicon thin film solar cells have been investigated, designed and fabricated. These 1D PCs consist of alternating amorphous Si (a-Si) and silicon dioxide (SiOx), of which the deposition process is compatible with current silicon thin film solar cells technology. Results indicate that the total reflectance of 1D PCs increases with the increase of period number. An average reflectance over 96% can be achieved in the range from 500 to 750 nm with 4 periods or more. Applying the 4-period 1D PC as back reflector in NIP amorphous silicon thin film solar cell with device-configuration of glass/1D PC/AZO/NIP a-Si:H/ITO, a conversion efficiency of 7.9% can be obtained, which is comparable to the AZO/Ag-based solar cell of 7.7% and is much better than the SS-based solar cell of 6.9% (a relative enhancement of 14.5%). Transition metals encapsulated inside single wall carbon nanotubes：DFT calculations Liu Man, Yan Qiang, Zhou Li-Ping, Han Qin 2014, 63 (7): 077302. doi: 10.7498/aps.63.077302 Abstract + The transport properties of a single wall carbon tube with transition metal atoms embedded in it are studied by using the first principles method based on the density functional theory and the nonequilibrium Green’s function. Different transition metal atoms filled in the carbon tube are investigated, and the respective charge and spin transport properties are studied. The conductance of the nanotube is found to be distinctive for different metal elements encapsulated, and quantized reductions of conductance can be seen by a quantum unit (2e2/h). In particular, nanotubes with two iron atoms encapsulated in display different I-V curves when the spins of the two iron atoms are in parallel and antiparallel states respectively. These results can be explained by spin-dependent scattering and charge transfer. The encapsulation may tailor the doping and add magnetic behavior to the carbon nanotubes, which would provide a new and promising approach to detect nanoscale magnetic activity. Analyses of wavelength dependence of the electro-optic overlap integral factor for LiNbO3 channel waveguides Li Jin-Yang, Lu Dan-Feng, Qi Zhi-Mei 2014, 63 (7): 077801. doi: 10.7498/aps.63.077801 Abstract + Wavelength dependence of the electro-optic overlap integral factor (Γ) for a single-mode LiNbO3 (LN) channel waveguide was analyzed experimentally and theoretically. By measuring the half-wave voltage (Vπ) of the LN waveguide at different wavelengths and then substituting the measured values into a formula that describes the relationship between Vπ and Γ, the quantitative dependence of Γ on wavelength was obtained; and it showed that Γ rapidly decreases with increasing wavelength. On the other hand, numerical simulations of the modulating electric field distribution, the modal field distribution, and Γ at different wavelengths were carried out; the calculated relationship between Γ and wavelength is in good agreement with the measured results. Further simulations indicate that as the wavelength increases, the center of the modal field profile gradually moves toward the weak electric field side from the waveguide surface, thus leading to a smaller Γ at a longer wavelength. Such a relationship between Γ and wavelength is partially responsible for the nonlinear dependence of Vπ on wavelength obtained experimentally. This would be useful for designing and optimization of LN waveguide-based devices. Synthesization and luminescent properties of blue emitting phosphor Ba2Ca(PO4)2：Eu2+ Wang Zhi-Jun, Liu Hai-Yan, Yang Yong, Jiang Hai-Feng, Duan Ping-Guang, Li Pan-Lai, Yang Zhi-Ping, Guo Qing-Lin 2014, 63 (7): 077802. doi: 10.7498/aps.63.077802 Abstract + A blue emitting phosphor Ba2Ca(PO4)2:Eu2+ is synthesized by a high temperature solid state method. Effect of the conditions is inverstigated, such as preparation temperature and time, the ratio of Ba/Ca, and Eu2+ concentration, on the phase and luminescent property. Results show that Ba2Ca(PO4)2 and Ba2Ca(PO4)2:Eu2+ have been achieved by selecting the appropriate conditions, such as the temperature 900/1200 ℃ and the time 4 h. The compound Ba2Ca(PO4)2:Eu2+ produces an asymmetric emission band centered at 454 nm under 343 nm UV excitation. For the 454 nm emission, the excitation spectrum extends from 200 to 450 nm with a peak at 343 nm, and has an obvious excitation band in the range of 350–410 nm. With increasing Eu2+ concentration, there occur the concentration quenching effect and redshift phenomenon. With decreasing ratio of Ba/Ca, there has an obvious enhancement in the green region, and the emission color gradually turns from blue to cyan. It is shown that the Eu2+ ion not only can occupy the Ba2+ site but also the Ca2+ site. Therefore, different luminescence centers of Eu2+ can exist in Ba2Ca(PO4)2, and affect its luminescence. Luminescence property of Ce3+-Tb3+-Sm3+ co-doped borosilicate glass under various ultraviolet excitations Chen Qiao-Qiao, Dai Neng-Li, Liu Zi-Jun, Chu Ying-Bo, Li Jin-Yan, Yang Lü-Yun 2014, 63 (7): 077803. doi: 10.7498/aps.63.077803 Abstract + Ce3+-Tb3+-Sm3+ co-doped white light emitting borosilicate glasses were fabricated by high-temperature melting technique. In this paper, the excitation spectra and the emission spectra of Ce3+, Tb3+ and Sm3+ ions-doped and co-doped samples were measured and the energy transfer mechanism of Ce3+, Tb3+, and Sm3+ were studied by analyzing the fluorescence lifetime of single-doped and co-doped samples. The color coordinate, rendering index, and color temperature of the emission spectra can be adjusted by changing the excitation wavelength of ultraviolet LED. Finally, we have obtained the white light which fits for life, study, and work. Bluish-green high-brightness long persistent luminescence materials Ba4(Si3O8)2：Eu2+Pr3+, and the afterglow mechanism Wang Peng-Jiu, Xu Xu-Hui, Qiu Jian-Bei, Zhou Da-Cheng, Liu Xue-E, Cheng Shuai 2014, 63 (7): 077804. doi: 10.7498/aps.63.077804 Abstract + A bluish-green long persistent luminescence material Ba4(Si3O8)2:Eu2+, Pr3+, was synthesized by traditional solid state method in a reductive atmosphere According to the photoluminescence and afterglow spectra measurement, the emission center is the cation Eu2+ in the photoluminescence and afterglow procedure. The Pr3+ co-doped sample forms new defects which could capture current carriers after excitation. On the basis of thermoluminescence and afterglow decay measurement, the afterglow intensity of Pr3+ co-doped sample sharply enhances as compared with Eu2+ doped one, the reason is that the lower depth traps are generated in the shallow trap areas (T1 region). At the same time, the Pr3+ co-doped sample have longer afterglow decay than that doped with only Eu2+; the reason is that the deep traps concentration decreases in the deep trap areas (T2 region). The afterglow mechanism of Pr3+ co-doped sample have two different excitation paths, path 1: the electron of the host is directly projected to traops at 268 nm excitation; path 2 the electron of the Eu2+ corresponds to the transitions from the ground state to the 5d excited state at 330 nm excitation. Then the different afterglow mechanism of phosphor was produced. Structured analysis of iron-based amorphous alloy coating deposited by AC-HVAF spray Ye Feng-Xia, Chen Yan, Yu Peng, Luo Qiang, Qu Shou-Jiang, Shen Jun 2014, 63 (7): 078101. doi: 10.7498/aps.63.078101 Abstract + The uniform and compact Fe-based amorphous alloy coating was prepared by active combustion high velocity air fuel (AC-HVAF) spray method. By tuning the parameters of AC-HVAF spray process, the influence of the spraying gun length, spraying distance, and powder feed rate on non-crystallization has been studied carefully. Results indicate that spraying gun length is the key factor in forming perfect amorphous coating. Spraying distance and powder feed rate may determine the thickness and formation rate of the coating. The prepared coatings have a tight adhesion with the substrate, low porosity, and good non-crystallization, which would effectively maintain the excellent mechanical properties of the Fe-based amorphous alloy. The coating can provide a good protection for the substrate material. Hu Jian, Qiu Xi-Jun 2014, 63 (7): 078201. doi: 10.7498/aps.63.078201 Abstract + By virtue of a functional scaling, the free energy for cytoskeletal microtubule (MT) solution system in the gravitational field has been proposed theoretically, and on this basis the influence of gravitational field on MT’s self-organization process is studied. A concentration gradient coupled with orientational order characteristic of nematic ordering pattern formation is the new feature emerging in the presence of gravity. Theoretical calculation results show that gravity facilitates the isotropic to nematic phase transition, which is reflected in a significantly broader transition region and the phase coexistence region increases with increasing g or MT concentration. We also discuss the numerical results obtained due to local MT concentration changing with the height of the vessel and some phase transition properties. Double-threshold cooperative spectrum sensing for cognitive radio based on trust Zhang Xue-Jun, Lu You, Tian Feng, Sun Zhi-Xin, Cheng Xie-Feng 2014, 63 (7): 078401. doi: 10.7498/aps.63.078401 Abstract + This paper presents a double-threshold cooperative spectrum sensing algorithm which is based on trust and satisfies both reliability and efficiency. The cognitive nodes that satisfy the request of double-threshold have the priority to participate in cooperative sensing and that satisfy the requirement of trust parameters may participate in cooperative sensing if only the number of the former is smaller than a preset value. The fusion center stores the sensing record of each cognitive node and sets the fusion weights according to the partial detected results. Theoretical analysis and simulation show that the bandwidth required for transmitting the sensing parameters decreases, and the detection performance improves because the unreliable users are reduced. Additionally, the algorithm can be made to adapt to different wireless service by adjusting the parameter nt. A novel frequency selective surface of hybrid-element type with sharply decreased stop-band Wang Yan-Song, Gao Jin-Song, Xu Nian-Xi, Tang Yang, Chen Xin 2014, 63 (7): 078402. doi: 10.7498/aps.63.078402 Abstract + Frequency selective radome is one of the most important applications of frequency selective surface (FSS). In order to obtain better stealth performance, a novel element FSS, based on a regular slot element FSS, is presented in this paper. The novel element consists of a slot element in the center and at least two slot strips placed on the periodic boundary. We call such FSS the “hybrid-element type FSS” because it exhibits characteristics of both slot type and patch type FSS. Simulation and optimization work is carried out by using a period moment method and a discrete particle swarm optimization method based on the application requirements of a missile radome. Simulation results show that the hybrid-element type FSS has much steeper transition section between pass-band and stop-band, and much lower transmittance in stop-band when compared with the corresponding slot type FSS. The new FSS also has much lower insertion loss in pass-band, much thinner thickness, much simple structure and fabrication process when compared with the ordinary two-layer FSS. Equivalent sample plate is fabricated using printed circuit method and tested using the free space method. Good fit between simulation and testing results verify the accuracy and feasibility of this novel FSS design. The hybrid-element type FSS is especially suitable for the stealth radome when woking frequencies of both sides are very close. It provides a simple and feasible approach for developing frequency selective radome. An imaging algorithm for missile-borne SAR with downward movement based on variable decoupling Jiang Huai, Zhao Hui-Chang, Han Min, Zhang Shu-Ning 2014, 63 (7): 078403. doi: 10.7498/aps.63.078403 Abstract + The ordinary SAR imaging algorithms are inapplicable to missile-bo-rne SAR due to the high speed dive and high-squint of missile. Aiming at this problem, this paper firstly sets up the model of echo and analyses of two-dimensional spectrum. By using azimuth nonlinear chirp scaling (NLCS) based on variable decoupling, an imaging algorithm for missile-borne SAR is proposed.It can effectively compensate the scene with longitudinal and transverse Doppler shift and improve the focusing quality while it can also simplify the geometric image correction operation. Simulation results are provided to confirm the effectiveness of the proposed algorithm. Synthesis of nanoparticles in SiO2 by implantation of Cu and Zn ions and their thermal stability in oxygen atmoshphere Xu Rong, Jia Guang-Yi, Liu Chang-Long 2014, 63 (7): 078501. doi: 10.7498/aps.63.078501 Abstract + Cu nanoparticles (NPs) embedded in silica were synthesized by implantation of 45 keV Cu ions at a fluence of 1.01017 cm-2, and then subjected to post irradiation with 50 keV Zn ions at fluences of 0.51017 cm-2 and 1.01017 cm-2, respectively. Zn post ion implantation induced modifications in structures, optical absorption properties of Cu NPs as well as their thermal stability in oxygen ambient have been investigated in detail. Results clearly show that Cu-Zn alloy NPs could be formed in the Cu pre-implanted silica followed by Zn ion irradiation at a fluence of 0.51017 cm-2, which causes an unique surface plasmon resonance (SPR) absorption peak at about 516 nm. Subsequent annealing in oxygen atmosphere results in the decomposition of Cu-Zn alloy NPs, at 450 ℃, and thus, ZnO and Cu NPs appear in the substrate. Further increase of annealing temperature to 550 ℃ could transform all the Zn and Cu into ZnO and CuO. Moreover, results also demonstrate that introduction of Zn into SiO2 substrate could effectively suppress the oxidation of Cu NPs, meanwhile, the existence of Cu could promote thermal diffusion of Zn towards substrate surface, which enhances the oxidation of Zn. The underlying mechanism has been discussed. Response function of angle signal in two-dimensional grating imaging Ju Zai-Qiang, Wang Yan, Bao Yuan, Li Pan-Yun, Zhu Zhong-Zhu, Zhang Kai, Huang Wan-Xia, Yuan Qing-Xi, Zhu Pei-Ping, Wu Zi-Yu 2014, 63 (7): 078701. doi: 10.7498/aps.63.078701 Abstract + In this paper, we derive the response function of angle signal in a two-dimensional X-ray grating interferometry system under the condition of parallel coherent light, and depict the surface of the function with Matlab.Although there are four kinds of commonly used beam splitter gratings and three kinds of analyzer gratings, and there are still different compound modes between them, we may find that the ultimate surface of the response function of angle signal can only be of three kinds: the peak type, the valley type, and the peak-valley symmetry type of shifting surfaces. As there is a numerical complementary ralationship between the peak type and the valley type of shifting surfaces, we can take the two kinds as one; and finally we only need to consider two kinds of shifting surface.This conclusion simplies the common understanding of the two-dimensional X-ray grating interferometry method, and lays the foundation for the research of quantitatively extracting the two-demensional signal in the future. Equivalent source reconstruction in inhomogeneous electromagnetic media Zhao Chen, Jiang Shi-Qin, Shi Ming-Wei, Zhu Jun-Jie 2014, 63 (7): 078702. doi: 10.7498/aps.63.078702 Abstract + In this paper, a method that uses magnetic extreme signals for equivalent source reconstruction is presented. Through simulation of specific current dipoles given as the sources of magnetic field signals, the feasibility of a multi-chamber heart model is investigated and the accuracy analysis of equivalent source reconstruction in inhomogeneous media is conducted. The magnitude of the magnetic extreme signals is indicative of the influence of volume conductor on the cardiac magnetic field is analyzed. The method is compared with other four methods which are the method of magnetic gradient extreme signals, the Nelder-Mead algorithm, the trust region reflective algorithm, and the particle swarm optimization algorithm against the criteria in terms of accuracy of source reconstruction and computation time of the algorithm. Results show that the method is practically useful for solving inverse cardiac magnetic field problems. Mass segmentation in mammogram based on SPCNN and improved vector-CV Han Zhen-Zhong, Chen Hou-Jin, Li Yan-Feng, Li Ju-Peng, Yao Chang, Cheng Lin 2014, 63 (7): 078703. doi: 10.7498/aps.63.078703 Abstract + Mass segmentation plays an important role in computer-aided diagnosis (CAD) system. The segmentation result seriously affects classifying mass as benign and malignant. By combining the simplified pulse coupled neural network (SPCNN) and the improved vector active contour without edge (vector-CV), a novel method of mass segmentation in mammogram is proposed in this paper. First, the parameters and termination conditions of SPCNN are obtained through mathematical analysis and the initial contour is segmented by SPCNN. Then, the vector CV model is accordingly modified to overcome the shortcomings of traditional CV model. Finally, combined with the initial contour, the improved vector-CV is used to segment the mass contour. The experiments implemented on the public digital database for screening mammography (DDSM) and the clinical images which are provided by the Center of Breast Disease of Peking University People’s Hospital indicate that the proposed method is better than the existing methods, especially when dealing with the dense breasts of Oriental female. Multiscale permutation entropy analysis of electroencephalogram Yao Wen-Po, Liu Tie-Bing, Dai Jia-Fei, Wang Jun 2014, 63 (7): 078704. doi: 10.7498/aps.63.078704 Abstract + We carried out a detailed analysis and a comparison between normal and epileptic electroencephalogram (EEG) based on multiscale permutation entropy. The relationship between multiscale permutation entropy values of EEG and age, and the effect of scale factor on multiscale permutation entropy value were also discussed. By analyzing normal and epileptic EEG based on multiscale permutation entropy, we found that, at the same age, multiscale permutation entropy value of the normal group’s EEG is higher than that of the epileptic group by an average of 0.19, about 7.9%. In addition, for people of age 3 to 35, their multiscale permutation entropies are clearly maximum. When scale factor is smaller than 15, the value of their entropy would reduce no matter whether the age increases or decreases. The results indicate that multiscale permutation entropy can distinguish between normal and epileptic EEG and reflect the general process of human brain development. Effects of NPB anode buffer layer on the performances of inverted bulk heterojunction polymer solar cells Gong Wei, Xu Zheng, Zhao Su-Ling, Liu Xiao-Dong, Yang Qian-Qian, Fan Xing 2014, 63 (7): 078801. doi: 10.7498/aps.63.078801 Abstract + Inverted configuration bulk heterojunction polymer solar cells based on ITO/ZnO/P3HT:PCBM/NPB/Ag were fabricated, with the donor material being poly(3-hexylthiophene)(P3HT), and the acceptor material being [6, 6]-phenyl-C60-butyric acid methyl ester(PCBM). N, N’-diphenyl-N, N’-bis(1-naphthyl)-1, 1’-biphenyl-4, 4’-diamine(NPB) thin anode buffer layers with different thicknesses, which were used to improve the performances of the devices; and the effects of NPB anode buffer were investigated. The insertion of 1 nm thick NPB improves charge collection of the device, both of the short circuit current and open circuit voltage are enhanced. When the thickness of NPB reaches 25 nm, the series resistances are significantly increased, leading to reduced device performances. Effects of different thicknesses of NPB on charge injection and collection are investigated by capacitance-voltage measurements. NPB with 1 nm thickness improves charge collection of the device but without improving charge injection, and the charge recombination mechanism is dominant if the NPB layer is too thick. NPB thin layer with appropriate thickness could be used to enhance the performances of bulk heterojunction polymer solar cells. Study on cascading invulnerability of multi-coupling-links coupled networks based on time-delay coupled map lattices model Peng Xing-Zhao, Yao Hong, Du Jun, Ding Chao, Zhang Zhi-Hao 2014, 63 (7): 078901. doi: 10.7498/aps.63.078901 Abstract + The couplings among different networks facilitate their communications, while at the same time they also bring the risk of enhancing the wide spread of cascading failures to the coupled networks. Given that there is usually the time-delay during the spread of failures and more than one coupling link a node might possess, a cascading failure model for scale-free multi-coupling-link coupled networks is built in this paper, based on time-delay coupled map lattices (CML) model, which may be wider representative than previous models. Our research shows that in BA (Barabási-Albert) scale-free coupled networks, there is a threshold hT ≈ 3: when the coupling strength is bellow this threshold, the stronger coupling strength corresponds to a lower invulnerability; and vice versa, the stronger coupling strength would bring a higher invulnerability. In addition, our studies show that the presence of time-delay not only prolongs the failure spreading time during which measures can be taken to suppress cascading failures, but also has a significant influence on the eventual cascading size, for detail, if intra-layer time-delay τ1 and inter-layer time-delay τ2 can have any values, then the multiples of the two numbers will cause larger cascading size. We hope our research can provide a reference for building high-invulnerable coupled networks or the increase of the invulnerability of the coupled networks. Virtual trajectory model for lane changing of a vehicle on curved road with variable curvature Ren Dian-Bo, Zhang Jing-Ming, Wang Cong 2014, 63 (7): 078902. doi: 10.7498/aps.63.078902 Abstract + In this paper, a virtual trajectory planning method for vehicle lane changing in automated highway system is studied, and a trajectory model for lane changing on variable curvature road is established with odd-order polynomial constraints. Assuming that the starting lane and the target lane have the same instantaneous center, the motion for lane changing of vehicle on the curved road can be decomposed into a linear centripetal motion and a circular motion around the instantaneous centre of the curved road. If the centripetal motion displacement and the rotational angular displacement meet the requirement of odd-order polynomial constraints, the boundary condition of the above two kinds of motion may be obtained from the constraints, such as time, location, and desired state of vehicle at the start and end of the lane changing behavior. By applying the boundary conditions, the polynomial coefficient is deduced, and the mathematical model of virtual trajectory for lane changing can be designed based on the polynomial models of centripetal displacement and angular displacement. Compared with the existing trajectory planning method for lane changing on curved road, the curvature change has been taken into consideration, and the trajectory model for lane changing has been generalized. Simulation results verify the feasibility of the trajectory planning method proposed in this paper for lane changing on a curved road with variable curvature. Ensemble variational data assimilation method based on regional successive analysis scheme Wu Zhu-Hui, Han Yue-Qi, Zhong Zhong, Du Hua-Dong, Wang Yun-Feng 2014, 63 (7): 079201. doi: 10.7498/aps.63.079201 Abstract + The ensemble variational data assimilation method may be subject to significant uncertainties due to the size of forecast ensemble. We found that this problem occurs because the analysis increment of this method is expressed as a linear combination of ensemble perturbation vectors or expansion of the orthogonal basis vectors. Though this method avoids introducing adjoint model while calculating the gradient of object function, the number of physical control variables is much larger than the sample size of forecast ensemble, which causes the assimilation results to be sensitive to the number of ensemble members. For this reason, the regional successive analysis scheme of ensemble variational method is proposed. By this scheme, the ratio between the number of physical control variables in analysis region and the sample size is decreased, so that it is expected that the problem can be solved. The results of numerical experiments using shallow water model show that the regional successive analysis scheme can give better assimilation results than traditional method, and the analysis precision is improved appreciably. Relationship between the quasi-linear diffusion coefficients and the key parameters of spatial energetic electrons Zhang Zhen-Xia, Wang Chen-Yu, Li Qiang, Wu Shu-Gui 2014, 63 (7): 079401. doi: 10.7498/aps.63.079401 Abstract + It has been proved that the ground-based electromagnetic wave can transfer into ionosphere and interact with high-energy particles. By changing the pitch angle and momentum, the particles are imposed to enter the bounce loss cone and drift loss cone, then electron precipitation takes place and the particle bursts form. In recent decades, the relationship has been observed among electromagnetic disturbance and particle bursts and seismic activity based on satellite data. Here, by wave-particle cyclotron resonant interaction combined with the observation range of LEO satellite (about 350–1000 km), the evolvement trend of the pitch angle quasi-linear diffusion coefficients induced by field-aligned electromagnetic waves, is studied with the change of VLF electromagnetic wave frequency, band width, energies of electron (0.1–20 MeV) and L shell (L=1.1–3). We also show the relationship between VLF electromagnetic wave frequency and minimum energy of precipitation electron induced by it, under certain pitch angle value. The relationship among these quantities may be used to provide theoretical explanation for satellite observations of energetic particle precipitation examples, to provide guidance for extracting information associated with earthquakes from the detection of high-energy particles on the satellite, and to lay the foundation on the data analysis of China seismo-electromagnetic satellite planned to launch at about the end of 2016.
d3159337ceb80df5	Why does plutonium have more oxidation states than samarium? Electron configuration of Pu: $\ce{[Rn] 5f^6 7s^2}$ Electron configuration of Sm: $\ce{[Xe] 4f^6 6s^2}$ I thought that only the valence electrons affected the oxidation states, so why does plutonium have more oxidation states (6,5,4,3) than samarium (2,3) whilst they both have the same valence electrons (including f-orbitals): ..$\ce{f^6}$..$\ce{s^2}$ ? Elements such as scandium (Sc),Yttrium (Y) and Lanthanum (La) all have the same valence electrons (including d-orbitals): ..$\ce{d^1}$..$\ce{s^2}$ , but these elements have the same oxidation state : 3 and that doesn't change, even when going down that group and looking at actinium. Does this have to do with the ionization energy, or the energy level or ....? This is yet another interesting effect of the anomalous compactness of orbitals in the first appearance of each type of subshell ($1s$, $2p$, $3d$, $4f$, $5g$, etc). The solutions to the Schrödinger equation for electron wavefunctions in hydrogen-like atoms are such that these subshells are composed of orbitals with no radial nodes. This means the electrons in these subshells are closer to the nucleus than you might expect, and therefore they are more strongly bound. The striking chemical similarity of the lanthanides comes from the fact that the $4f$ subshell is both anomalously compact and subjected to a high effective nuclear charge. The electrons end up being held so closely to the nucleus that they effective behave as core electrons, and participate very little in chemical interactions; lanthanides do not form coordination compounds using their $4f$ orbitals, and it is very difficult to go past an oxidation state of +3 in most because few conditions can compensate the large energy required to ionize a fourth electron coupled with the relatively weakly bound compounds they would form. In the actinides, the $5f$ orbitals do not suffer from the same lack of radial node, and therefore are more chemically available. The first half of the actinides show many compounds with high oxidation numbers, and have significant coordination chemistry using $f$ orbitals. Curiously, as you go form curium to berkelium, there is a sudden increase in reluctance to display high oxidation numbers, and the actinides beyond curium tend to behave similarly (though of course exploration of their chemical properties is significantly hindered by their rarity and instability). This is likely because even though the $5f$ orbitals have a radial node, in the second half of the actinides they are subjected to such a high effective nuclear charge that become too strongly bound to display significant participation in chemistry. This same effect explains why the metals in the first row of transition metals do not display as many stable compounds with high oxidation states compared to the second and third rows, as the $3d$ orbitals are much smaller than $4d$ or $5d$ orbitals. Your Answer
d611d1afc0a61c0c	torsdag 31 oktober 2013 What Is Wrong with (IPCC) Climate Models? CO2 alarmism is now falling apart because the global warming predicted by the IPCC ratified climate models show little similarity with observations of no warming the last 17 years. This is commonly viewed as evidence that mathematical modeling of global climate is impossible because weather and climate is "chaotic" and thus unpredictable. It is true that weather is not predictable over more than a week, but climate as averaged weather may well be mathematically predicatble over long time. For example, the simple climate model of no change of global temperature, may well be a good model over centuries, until the next ice age. The IPCC climate models predict global warming (at variance with observations) because they are so constructed, that is to show warming even if there is none, not because it is impossible to construct climate models which fit observations. The defunding of mathematical climate modeling to be expected because of the failure of the IPCC models thus lacks rationale in a world in which prediction is needed and may be possible by clever use of computational mathematics. In any case the question of the role of climate modeling is now on the table. tisdag 29 oktober 2013 NADAs Begravning Firas Stort på KTH KTH inbjuder till firande av NADAs 50-årsdag enligt nedanstående affisch. Vad som inte omnämns i inbjudan är att NADA är nedlagt och inte längre finns med oss: Att det handlar om en begravning av något dött och inte ett födelsdagsfirande av något levande. Att Numerisk Analys (NA) inte längre är en egen institution utan har inlemmats i Matematik och nu går samma öde till mötes som Numerisk Analys vid Chalmers, som inte längre är ett undervisningsämne och därmed förlorat sin uppgift. Så motsvarar då NADA till slut sin betydelse på portugisiska, nämligen "ingenting", "nothing". Att Numerisk Analys har gått i graven beror inte på att ämnet saknar aktualitet, hela vårt IT-samhälle bygger ju på Numerisk Analys i olika former, utan på att NADA stelnat i den form och med det innehåll NADA hade under pionjärtiden för 50 år sedan. Så kan framgång byggd på initiell kompetens och efterfrågan (jfr med Nokia) utvecklas till en tvångströja som kväver nytänkande och utveckling och därmed bäddar för undergång. I inbjudan omskrivs detta som "lite nostalgi" och firandet fokuseras på "musik och sång" och "Demonstration av Historisk Klenod", kanske bysten av Germund Dahlquist, grundaren av NADA (en mycket sympatisk person som både kunde räkna och spela piano): Informationsbehandling 1963 Numerisk Analys - Datalogi Tid: Ti 2013-12-17 kl 13.30 Plats: E1 och Sing-Sing För 50 år sedan installerades på KTH den första professorn i Informationsbehandling, särskilt Numerisk Analys, och institutionen grundades. Tisdag 17 december blir det tillfälle för oss på CSC och NA att fira. Preliminärt program från kl. 13.30 (Obs!) i sal E1: • Presentationer, framtidsorienterade och blandade med lite nostalgi av bland andra Anna-Karin Tornberg, Anders Flodström, Jan Gulliksen, Ingrid Melinder, Bertil Gustafsson & Björn Engquist, Haibo Li & Kia Höök & Ylva Fernaeus, Patrik Fältström & Anders Hillbo, Johan Hoffman, Svante Littmarck, Leo Giertz, Jin Moen & Marie Björkman, Yngve Sundblad & Peter Graham • Jubileumspublikationer • Demonstration av historisk klenod • Avslutande middag med musik och sång c:a kl.18.30 i Sing-Sing Alla på CSC och NA är inbjudna. Övriga deltagare efter särskild inbjudan. till Ann-Britt Isaksson Öhman senast den 29 november. (ange om du är med också på middagen) Lennart Edsberg, Ingrid Melinder, Yngve Sundblad Hans Rosling: Consider the Facts! Help the World! Below are the answers to my questions to Hans Rosling posed in the previous post, delivered by Rosling with the following introductory comment: • Thanks for the questions, it is sad for me to learn that I was so unclear that you missed what I tried to say on so many points, but here are my answers to your not so easy to understand questions. My general message to Hans Rosling is as follows: • You tell us that the living conditions for the poor people in the world are improving and you know that this is directly related to the increasing use of fossil fuels. • You have uncritically accepted the IPCC founding dogma that CO2 emissions from burning of fossil fuels is causing dangerous global warming and thus have to be drastically reduced. This dogma is not fact-based since there is no scientific evidence that this effect is real. • What you should do is to use your own Gapminder principle of fighting devastating ignorance with a fact-based worldview, and you should then start by reading the reports from the Nongovernmental International Panel on Climate Change (NIPCC). • You will then find that there is no scientific fact-based reason to limit CO2 emissions, a message which you will greet with great joy and satisfaction since it allows poor people to improve their living conditions. My Question 1: You say that you are not at all involved in climate science, yet you assure the world that IPCC science is truely, truely good. Is this a fact-based message or does it rather reflect ignorance? Roslings Answer 1: As you rightly write I commented on the summary for policy makers. I was explicit about not being a climate scientist, but my judgement is that this summary for policy maker is as good a summary of research as policy makers, public and researchers in other fields can ever hope to get. My Comment 1: You make a judgement about climate science based on ignorance in climate science admitted by yourself. Doing so you violate your own Gapminder principle to fight devastating ignorance with a fact-based worldview. My Question 2: You claim to know that humans are changing the climate, but not how much, how fast, in which way, or where this or that will happen. What do you mean by this statement? Does it have a content? Is that not an example of the devastating ignorance you are fighting? Rosling's Answer 2: No it is not me claiming, it is IPCC claiming that humans are changing the climate, I accept their consensus on this point, but pointed out that their projections have a very wide range of uncertainty. I think I was very clear about the distinction between accepting the consensus that humans are changing the climate and my observations that the uncertainties in the projections are so wide that they range from almost negligible changes to potentially catastrophic. My Comment 2: You accept the consensus of IPCC which you interpret to range from almost neglible change to catastrophic. According to NIPPC it is negligible. Why not accept NIPCC instead of IPCC? My Question 3: You know that massive use of fossil fuel is required to improve the living conditions for billions of poor people, yet you support the idea that the use of fossil fuels must be reduced, drastically reduced. How are you going to resolve this contradiction? Rosling's Answer 3: No I do not know how to solve this contradiction, but it do exist! My suggestion in the end is that the richest, that have by far the highest emission must lower their emission first before they demand restrictions by those with much lower emission. My Comment 3: You admit to pose a contradiction, which you cannot solve as long as you uncritically accept IPCC. If you critically accept NIPCC after looking at the facts, the contradiction will disappear. Why are you stuck with a contradiction? My Question 4: You compare with smoking and cancer and HIV. What is the connection to climate science? Rosling's Answer 4: The connection is that the a scientific consensus regarding a causal link between smoking and cancer as well as between sexual transmission of HIV and Aids was of greatest importance for human behavior, as well as for health, economic and trade policy. The ways those consensus were reached and communicated were far more haphazardly done then IPCC, but then again I said that IPCC is in a more difficult position as they are making predictions rather than concluding about causal links in the present time. My comment 4: You don't answer my question about the connection between human emissions of CO2 and smoking/cancer and HIV. I take it that you mean that there is no connection. My Question 5: You say we have to get things done. What do we have to do? Rosling's Answer 5: If we should avoid a possible catastrophic climate change we, that is we humans, have to reduce or at least stop to increase the CO2 emission. My comment 5: Again, if you look at the facts, then you will understand that it is better for humanity to use its limited resources in a fact-based meaningful way to improve living conditions for the poor, instead of wasting these resources on meaningless non-fact-based restriction of CO2 emissions. måndag 28 oktober 2013 Hans Rosling compares AGW to Smoking and Cancer Hans Rosling and Bill Gates working together on the Global Poverty Project. The Swedish media mega-star and Gapminder edutainer Hans Rosling releases all of his media authority (and Swedish home-made charm) in his talk 200 Years of Global Change in support of IPCC at the presentation in Stockholm September 27 of the IPCC AR5 Summary for Policymakers: • I am not at all involved in the climate research, heh, just standing at the side, looking at the factors that contribute to the climate change and the impact it will have on humans. • I can assess really as an outsider that this IPCC Summary for Policymakers is as good as science can do. It is a truely, truely, good report! (Applause) • I've been through quite a lot of similar things...smoking and cancer...bottlefeeding of young children...HIV... • I am completely independent...they didn't even pay me to come here...(laugh). • IPCC have spent all their effort up to the press conference just to see that the entire content is as correct, meticulously correct and well weighed as it could be. • When I read through the report as an experienced researcher and scientist, I like very much the way the arguments are weighed, and the different wordings: this we know, this is very sure, this we don't know so well, heh, this is is as good we can... And it is especially difficult this report: • Smoking and cancer was easier because it was a practice here and now and a disease here and now...this (report) is about future and future is is almost impossible to do research about, heh? • I think the work now being done in climate science will have importance and implications for science in many other fields that need to do predictions, how you should do and work. • I am not so much professor, I am mainly edutainer at Gapminder, an independent foundation, that fights devastating ignorance with a fact-based worldview that everyone can understand. • It is very clear that the sea level is rising......the uncertainties are shown so clearly... • As I read it, today the question, does human change climate, yes or no, that's over! That's over, that's set! • The big question now is, how much will it change, how fast will it change, in which way will it change and where will this or that happen? The background was already done by Arrhenius in 1900.... • We must take this IPCC report seriously and get things done... • We think we have done more than we have and we haven't understood how much we have to do, thank you! My questions to Hans Rosling are: 3. You know that massive use of fossil fuel is required to improve the living conditions for billions of poor people, yet you support the idea that the use of fossil fuel must be reduced, drastically reduced. How are you going to resolve this contradiction? 4. You compare with smoking and cancer and HIV. What is the connection to climate science? 5. You say we have to get things done. What is it we have to do? I invite Hans to answer as a comment. Watch the shocking alarming increase of atmospheric CO2 as humanity is lifted out of poverty 1800 - 2013. PS1 Will Rosling answer my questions? I bet 100 Skr that he will not. My bet is based on long experience and many observations... PS2 Rosling shows only one graph to support the CO2 alarmism he is selling , not over global temperature which does not rise anymore, but over global sea level which continues a steady slow rise since long before CO2 emissions started to grow: PS3 In talks during 2012 Rosling supported his alarm message with the following picture of a quickly shrinking arctic ice cap: But in 2013 Rosling keeps silent about the arctic ice cap since it is quickly growing and the reason for alarm is gone. What remains of alarm is a steady very slow rise of the sea level as the world is slowly recovering from the Little Ice Age. It is the alarm which is shrinking to zero, not the arctic ice cap. lördag 26 oktober 2013 Quantum Contradictions 29: Elegance and Enigma The book Elegance and Enigma: The Quantum Interviews gives a shocking account of the present state of quantum mechanics as foundation of modern physics: Caslav Brukner: • Quantum theory makes the most accurate empirical predictions. Yet it lacks simple, comprehensible physical principles from which it could be uniquely derived. Without such principles, we can have no serious understanding of quantum theory and cannot hope to offer an honest answer—one that’s different from a mere “The world just happens to be that way”—to students’ penetrating questions of why there is indeterminism in quantum physics, or of where Schrödinger’s equation comes from. • The standard textbook axioms for the quantum formalism are of a highly abstract nature, involving terms such as “rays in Hilbert space” and “selfadjoint operators.” And a vast majority of alternative approaches that attempt to find a set of physical principles behind quantum theory either fall short of uniquely deriving quantum theory from these principles, or are based on abstract mathematical assumptions that themselves call for a more conclusive physical motivation. Jeffrey Bub: • We don’t really understand the notion of a quantum state, in particular an entangled quantum state, and the peculiar role of measurement in taking the description of events from the quantum level, where you have interference and entanglement, to an effectively classical level where you don’t. Christoffer Fuchs: • John Wheeler would ask, “Why the quantum?” To him, that was the single most pressing question in all of physics. You can guess that with the high regard I have for him, it would be the most pressing question for me as well. And it is. But it’s not a case of hero worship; it’s a case of it just being the right question. The quantum stands up and says, “I am different!” If you really want to get to the depths of physics, then that’s the place to look. GianCarlo Ghirardi: • I believe that the most pressing problems are still those that have been debated for more than eighty years by some of the brightest scientists and deepest thinkers of the past century: Niels Bohr, Werner Heisenberg, John von Neumann, Albert Einstein, Erwin Schrödinger, John Bell. • To characterize these problems in a nutshell, I cannot do better than stressing the totally unsatisfactory conceptual status of our best theory by reporting the famous sentence by Bell: “Nobody knows what quantum mechanics says exactly about any situation, for nobody knows where the boundary really is between wavy quantum systems and the world of particular events.” Daniel Greenberger: • I don’t think the measurement problem will be solvable soon, or possibly ever. Lucien Hardy: • The most well-known problem in quantum foundations is the measurement problem—our basic conception of reality depends on how we resolve this. The measurement problem is tremendously important. • But there is another problem that is even more important—and that may well lead to the solution of the measurement problem. This is to find a theory of quantum gravity. The problem of quantum gravity is easy to state: Find a theory that reduces to quantum theory and to general relativity in appropriate limits. It is not so easy to solve. The two main approaches are string theory and loop quantum gravity. Both are deeply conservative, in the sense that they assume it will be possible to formulate a theory of quantum gravity within the quantum formalism as it stands. I do not believe this is the right approach. Tim Maudlin: • The most pressing problem today is the same as ever it was: to clearly articulate the exact physical content of all proposed “interpretations” of the quantum formalism. This is commonly called the measurement problem, although, as Philip Pearle has rightly noted, it is rather a “reality problem.” • Physics should aspire to tell us what exists (John Bell’s “beables”), and the laws that govern the behavior of what exists. “Observations,” “measurements,” “macroscopic objects,” and “Alice” and “Bob” are all somehow constituted of beables, and the physical characteristics of all things should be determined by that constitution and the fundamental laws. • What are commonly called different “interpretations” of quantum theory are really different theories—or sometimes, no clear theory at all. Accounts that differ in the beables they postulate are different physical theories of the universe, and accounts that are vague or noncommittal about their beables are not precise physical theories at all. Until one understands exactly what is being proposed as the physical structure of the universe, no other foundational problem, however intriguing, can even be raised in a sharp way. David Mermin: • In the words of Chris Fuchs, “quantum states: what the hell are they?” Quantum states are not objective properties of the systems they describe, as mass is an objective property of a stone. Given a single stone, about which you know nothing, you can determine its mass to a high precision. Given a single photon, in a pure polarization state about which you know nothing, you can learn very little about what that polarization was. (I say “was,” and not “is,” because the effort to learn the polarization generally results in a new state, but that is not the point here.) • But I also find it implausible that (pure) quantum states are nothing more than provisional guesses for what is likely to happen when the system is appropriately probed. Surely they are constrained by known features of the past history of the system to which the state has been assigned, though I grant there is room for maneuver in deciding what it means to “know” a “feature.” Lee Smolin: • The only interpretations of quantum mechanics that make sense to me are those that treat quantum mechanics as a theory of the information that observers in one subsystem of the universe can have about another subsystem. This makes it seem likely that quantum mechanics is an approximation of another theory, which might apply to the whole universe and not just to subsystems of it. The most pressing problem is then to discover this deeper theory and level of description. Antony Valentini: • The interpretation of quantum mechanics is a wide open question, so we can’t say in advance what the most pressing problems are...What’s important is that we leave the smoke screen of the Copenhagen interpretation well behind us, and that talented and knowledgeable people think hard about this subject from a realist perspective. David Wallace: • Just how are we to understand the apparently greater efficiency of quantum computers over classical ones? Anton Zeilinger: • We have learned from quantum mechanics that naive realism is not tenable anymore. That is, it is not always possible to assume that the results of observation are always given prior to and independent of observation. To me, the most important question is to find out what exactly the limitations are. • This can only be found out by carefully exploring quantum phenomena in more complex situations than we do today. Summary: Everybody asks fundamental questions, and nobody even hints at any answers. That is the present state of quantum mechanics. PS1 With measurement being replaced by computation in the model (some form of Schrödinger's equation), physics can be studied and inspected without interference from the observer. This changes the game completely by reducing the importance of the unsolvable measurement problem. Quantum Contradictions 28: Schrödinger's Cat Schrödinger presents his Cat Paradox in the article The Present Situation in Quantum Mechanics published in 1935: • One can even set up quite ridiculous cases. Schrödinger questions the central idea of the Copenhagen Interpretation that reality arises from measurement: • The rejection of realism has logical consequences. • In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has. But then what does it mean? • There must still be some criterion as to whether a measurement is true or false, a method is good or bad, accurate, or inaccurate - whether it deserves the name of measurement process at all. • Any old playing around with an indicating instrument in the vicinity of another body, whereby at any old time one then takes a reading, can hardly be called a measurement on this body. Schrödinger concludes with: • The simple procedure provided for this by the non-relativistic theory is perhaps after all only a convenient calculational trick, but one that today, as we have seen, has attained influence of unprecedented scope over our basic attitude toward nature. torsdag 24 oktober 2013 Quantum Contradictions 27: Schrödinger's Nobel Lecture Schrödinger, Dirac and Heisenberg ready for the Nobel Banquet (Bunin left) Erwin Schrödinger expresses in his 1933 Nobel Prize lecture The Fundamental Idea of Wave Mechanics his view that atomistic physics is better described by continuous wave mechanics than by discrete particles and discontinuous jumps. Schrödinger thus attacks the by then dominating Copenhagen Interpretation represented by Dirac and Heisenberg with whom he shared the Nobel Prize. After all, Schrödinger got a part of the Prize and thus could ask for a part of the truth, instead of nothing. Schrödinger makes a parallel with optics: • The new undulatory (wave quantum) mechanics corresponds to the wave theory of light. • It is, however, easy to surmise that the neglected phenomenon may in some circumstances make itself very much felt, will entirely dominate the mechanical process, and will face the old system with insoluble riddles, if the entire mechanical system is comparable in extent with the wavelengths of the "waves of matter" which play the same part in mechanical processes as that played by the light waves in optical processes. In the end Schrödinger pays lip service to some form of complementary wave-particle theory in order in order not to turn the Nobel Banquet into a turmoil of shouting: • Only in extreme cases...we think we can make do with the wave theory alone or with the particle theory alone. What Schrödinger meant was that the wave picture should cover 99% of the scene, and thus have 99% of the Prize. Dirac, Heisenberg and Schrödinger with ladies in Stockholm in 1933. onsdag 23 oktober 2013 Talk at NSCM26: Bluff Body Drag The slides to my talk at NSCM26 at Simula Oslo Oct 23 - 25 to the honor of my friends and collegues Juhani Pitkäranta (65) and Rolf Stenberg (60), is now available as tisdag 22 oktober 2013 Quantum Contradictions 26: There are No Particles! There are only waves and resonances. There are no Quantum Jumps, nor are there Particles! (H D Zeh) Niels Bohr brainwashed a whole generation of physicists into believing that the problem had been solved... (Murray Gell-Mann) Schrödinger objected to the ruling Copenhagen Interpretation of quantum mechanics based on the concept of particle as a discrete pointlike object and quanta as a discrete packet of energy: • Particles are just schaumkommen (appearances). • There have been ingenious constructs of the human mind that gave an exceedingly accurate description of observed facts and have yet lost all interest except to historians. I am thinking of the theory of epicycles. • I confess to the heretical view that their modern counterpart in physical theory are the quantum jumps. • There is a difference between a shaky or out-of-focus photograph (waves and resonances) and a snapshot of clouds and fog banks (particles and quanta). Schrödinger claimed that only a wave picture can make sense on microcopic level and that discrete particles and quanta brings in classical macroscopic concepts into a supposedly revolutionary microscopic quantum mechanics, which is contradictory. Schrödinger saw modern physics being shaped from a lack of historical connectedness with an old primitive macroscopic concept like discrete particle suddenly being transformed into a new microscopic concept, which was so revolutionary that it was (and is) beyond grasp to everyone except a small inner circle pretending to understand (Gongorism): • The disregard for historical connectedness, nay the pride of embarking on new ways of thought, of production and of action, the keen endeavour of shaking off, as it were, the indebtedness to our predecessors, are no doubt a general trend of our time. • In the fine arts we notice strong currents quite obviously informed by this vein; we witness its results in modem painting, sculpture, architecture, music and poetry. • There are many who look upon this as a new buoyant rise, while others regard it as a flaring up that inaugurates decay. It is not here the place to dwell on this question, and my personal views on it might interest nobody. • But I may say that whenever this trend enters science, it ought to be opposed. There obviously is a certain danger of its intruding into science in general, which is not an isolated enterprise of the human spirit, but grows on the same historic soil as the others and participates in the mood of the age. • There is, however, so I believe, no other nearly so blatant example of this happening as the theories of physical science in our time. I believe that we are here facing a development which is the precise counterpart of that in the fine arts alluded to above. • The most appropriate expression to use for it is one borrowed from the history of poetry: Gongorism. It refers to the poetry of the Spaniard Luis de Gongora (1561-1627), very fine poems, by the way, especially the early ones. Yet also his later poems (to which the term more particularly refers) are well sounding and they all make sense. But he uses all his acuity and skill on making it as difficult as possible to the reader to unravel the sense, so that even natives of Castile use extended commentaries to grasp the meaning safely. The modernity of the Copenhagen Interpretation was to combine the old idea of discrete particle with the new idea of statistics borrowed from statistical mechanics. Schrödinger objected (but was silenced): • God knows I am no friend of probability theory, I have hated it from the first moment when our dear friend Max Born gave it birth. For it could be seen how easy and simple it made everything, in principle, everything ironed and the true problems concealed. Everybody must jump on the bandwagon. And actually not a year passed before it became an official credo, and it still is. Bohr, Born and Heisenberg behind the Copenhagen Interpretation are all gone, but Schrödinger is alive and waiting for the right moment to open his mouth again... måndag 21 oktober 2013 Quantum Contradictions 25: Damned Quantum Jumps! Schrödinger 1955: I am moving against the stream. But the tide will change. Schrödinger, the founder of quantum mechanics based on Schrödinger's wave equation, did not like the pointlike "particles" supposed to make "quantum jumps" as postulated in the Copenhagen Interpretation of quantum mechanics formed by of Bohr, Born and Heisenberg: But Bohr won the battle in the 1930s and so modern physics was born from an old primitive concept of particles and quantum jumps, instead Schrödinger's educated sophisticated concept of waves and resonances. The result today is a physics in grave crisis with the Nobel Prize this year to the Higgs particle as the final nail in the coffin of a Standard Model of Particle Physics now abandoned by educated physicists: • The Higgs’s boson provides us with one of the worst cases of unnatural fine-tuning. A surprising discovery of the 20th century was the realization that empty space is far from empty. The vacuum is, in fact, a broiling soup of invisible “virtual” particles, constantly popping in and out of existence. Primitivism may be strong initially but runs out of steam over time. Maybe the time for a change of tide is now approaching...may finally the primitive idea of particle will be replaced by the educated idea of wave.... The book Schrödinger's Philosophy of Quantum Mechanics by Michel Bitbol gives an illuminating account of Schrödinger's continued struggle to maintain rationality in an increasingly weird world of elementary particles emerging from the Copenhagen Interpretation. Read and think! Quantum Contradictions 24: Against Measurement John Bell questions in Against Measurement Bohr's and Born's Copenhagen Interpretation of quantum mechanics (allowing physicists to speak only about what can be be measured and not what "is", and then in statistical terms): • Surely, after 62 years, we should have an exact formulation of some serious part of quantum mechanics? • It would seem that the theory is exclusively concerned about "results of measurement", and has nothing to say about anything else. • What exactly qualifies some physical systems to play the role of 'measurer'? • Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system . . . with a PhD? • If the theory is to apply to anything but highly idealised laboratory operations, are we not obliged to admit that more or less "measurement-like" processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time? This is the same as asking if there is no sound in the desert if there is nobody there to listen? To claim that we are not allowed to speak about (and mathematically model) sound as an air vibration unless there is a physicists ear drum reacting to the vibration, would seem absurd. Bell gives the reason driving modern physics into this form of absurdity: • He tried to think of an electron as represented by a wavepacket - a wavefunction appreciably different from zero only over a small region in space. The extension of that region he thought of as the actual size of the electron - his electron was a bit fuzzy. • At first he thought that small wavepackets, evolving according to the Schrodinger equation, would remain small. But that was wrong. Wavepackets diffuse, and with the passage of time become indefinitely extended, according to the Schrodinger equation. • So Schrödinger's "realistic" interpretation of his wavefunction did not survive. Bell here expresses the idea that we have to give up reality because we are supposed to believe that a pointlike blip on a screen cannot be generated by an electron wave extended in space. We are thus only allowed to speak about (and mathematically model) the blip effect and not the cause of the blip. But an extended sound in the desert can give rise to a pointlike "oh" from the listener, and thus the logic of requiring a pointlike input for pointlike output is missing. Anyway, this missing logic is behind the Copenhagen Interpretation, which Bell cannot embrace: • Probability of what, exactly? Not of the electron being there, but of the electron being found there, if its position is 'measured'. • Why this aversion to 'being' and insistence on 'finding'? Bell tells us: • We do not yet have an exact formulation of some serious part of quantum mechanics. Bell thus gives modern physics a bad grade. Bell supports his insistence to ask for an exact formulation (physically meaningful understandable formulation) with a quotation by Feynman: • We do not know where we are stupid until we stick our necks out. Bell did that but nobody wanted to listen to his questions as if the desert was empty. söndag 20 oktober 2013 Quantum Contradictions 23: The Truth • Quantum theory represents one of the great and most beautiful structures in all of physics. • Nonetheless, despite its uncontrovertible experimental successes, the theory has a very shaky philosophical foundation. • The standard Copenhagen interpretation(whatever that is) requires us to accept so many assumptions that defy common sense that ever since the theory was first developed it has led to enormous debates concerning its interpretation. • Most modern physicists accept it without qualification and, indeed, one can develop a creative intuition for using it. • The fact that many of its founding fathers turned against the standard interpretation, whereas their followers have tended to accept it without second thoughts can only partly be ascribed to the circumstance that anything tends to grow more familiar with repeated use. • Part of the explanation must be related to the fact that those very founders were much moreculturally well rounded than most modern physicists. • They were philosophically trained and philosophically inclined and did not like what they saw. • In spite of their doubts, the subject grew rapidly and it became fashionable to avoid questions concerning the foundations. • This attitude only started to change after Bell’s famous theorem in 1964. He showed that one could pose some of one’s intuitive doubts experimentally. • Since then, a number of alternate interpretations have grown and new experimental tests devised. • Today, we know that the strange predictions of the theory hold up experimentally (even though the foundations remain shaky). • We will never go back to classical physics - we must learn to accept and live with the world as it actually is. • What makes quantum mechanics so much fun is that its results run so counter to one’s classical intuitions, yet they are always predictable, even if unanticipated. • That is why I like to say that quantum mechanics is magic, but it is not black magic. This may well be the truth about quantum mechanics as one of the two pillars of modern physics: • A magic perfect theory counter to classical intuition, which we all have to accept without understanding its foundations and without asking the questions the founding fathers posed without ever giving any answers. • A magic perfect theory which (self-proclaimed) physics experts (like Lubos Motl of the Reference Frame) pretend to understand perfectly well, but refuse to answer any question with the excuse that all questions were answered by the founding fathers. This is the truth also of the other pillar of relativity theory: a magic theory which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by its founding father Albert Einstein (who explained very little), The same phenomenon has come to dominate climate science with a magic counter-intuitive "greenhouse effect" which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by the founding fathers (Tyndall and Arrhenius, who explained very little). The same phenomenon has come to dominate modern fluid mechanics with a magic counter-intuitive boundary layer theory which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by the founding father Ludwig Prandtl (who explained very little). fredag 18 oktober 2013 Quantum Contradictions 22: The Scandal N G van Kampen identifies The Scandal of Quantum Mechanics to be: • The article Would Bohr be born if Bohm were born before Born? by N Nikolić with its catchy title is a reminder of the scandalous fact that eighty years after the development of quantum mechanics the literature is still swamped by voluminous discussions about what is called its “interpretation". Van Kampen finds discussions about the interpretation of quantum mechanics scandalous because • Actually quantum mechanics provides a complete and adequate description of the observed physical phenomena on the atomic scale. What else can one wish? • It is true that the connection with gravity is still a problem, but that is outside this discussion. Van Kampen argues from a false premise or non sequitur logical fallacy (complete and adequate description of observed reality), where the real scandal that quantum mechanics is still not understood eighty years after its development, is covered up by the non-scandal that people are still seeking to understand what is not understood. Van Kampen twists the real effect (the discussion) of the real cause (the scandal that QM is not understood) into the cause of an invented scandal (the scandal of still discussing) based on a false premise. Van Kampen argues as if quantum mechanics is adjusted to observed reality, when the truth is the opposite, that quantum mechanics dictates what observed reality can be. torsdag 17 oktober 2013 Quantum Contradictions 21: The Reference Frame Feynman at the Nobel Banquet 1965: I think I can safely say that nobody understands quantum mechanics...I don't know anything about the Nobel Prize, I don't understand what's it all about or what's worth what. If the people in the Swedish academy decides that X, Y or Z wins the Nobel Prize, then so be it. I won't have anything to do with it. It's a pain in the neck...ha..ha..hah...If someone tells you they understand quantum mechanics then all you’ve learned is that you’ve met a liar. Lubos Motl behind The Reference Frame does not like the questions about the physical meaning (foundation) of quantum mechanics being posed by the Spanish physicist Pablo Echenique-Robba in the article: The title of the article connects to the famous "Shut up and calculate!" outburst by Dirac upon similar questions in the 1930s, which Lubos now follows up with: • Richard Feynman, Niels Bohr, and the authors of almost all textbooks of quantum mechanics are victims of his ad hominem attacks... Lubos wants to silence anyone (including me) who dares to question "the universal postulates of quantum mechanics, its probabilistic, proposition-based character, and its intimate connection with linear algebra" and throws invectives instead of simply using the scientific method he prides himself to follow. But the questions remain and "shut up" is not an answer, in particular not a scientific answer. PS For more on my views on quantum mechanics see Dr Faustus of Modern PhysicsMany-Minds Quantum Mechanics and Quantum Contradictions 1-20. onsdag 16 oktober 2013 The Higgs: Searching for an Elephant by Microscope Modern physics is based on two supposedly incompatible theories for the four forces of physics acting on different scales 1. gravitational force: macroscopic: relativity theory: cosmology 2. electromagnetic, weak and strong forces: microscopic: quantum mechanics: atoms The incompatibility has made a unified theory impossible and has driven modern physics into absurdities such as searching for the origin of macroscopic gravitation on microscopic atomistic scales. The 2013 Nobel Prize for the Higgs particle falls into this tradition: The idea is that mass or matter as the subject of macroscopic gravitation is generated from subatomic interactions through the Higgs particle with the Higgs field as an endless ocean in which the Universe is floating. Since the manifestation of mass is gravitation, it means to search for macroscopics in microscopics, that is searching for an Elephant using a microscope (quantum loop gravity and string theory) . It does not seem to me to be a constructive approach. A different approach is sketched in my pet theory described in Newtonian Gravitation of Matter and Antimatter exploring the possibility that • the basic element of the Universe is a gravitational field $\phi (x,t)$ depending on a space coordinate $x$ and time coordinate $t$ • matter and antimatter of density $\vert\rho (x,t)\vert $ is created by differentiation with respect to $x$ of the field $\phi (x,t)$ through the Laplace operator $\Delta$: $\rho =\Delta\phi$ with matter where $\Delta\phi >0$, antimatter where $\Delta\phi < 0$ and vaccum where $\Delta\phi =0$ • a gravitational force $F$ arises as the space gradient $\nabla$ of the field $\phi$: $F = \nabla\phi$. It is thus the gravitational field $\phi$ which • creates visible matter where $\Delta\phi$ is positive and singular • creates visible antimatter where $\Delta\phi$ is negative and singular • creates dark matter and antimatter where $\Delta\phi$ is smooth • separates matter and antimatter by gravitational repulsion • concentrates matter by gravitational attraction. What about that? Hint: The Hen and the Egg of (Dark) Matter. This is something completely different from the Higgs as an "explanation" of the origin of mass. söndag 13 oktober 2013 Lukewarmers and Cooling The climate debate can be divided as follows depending on estimated global warming upon doubled atmospheric CO2: 1. Warmers (IPCC): 1.5 - 4.5 C. 2. Lukewarmers (Lindzen, Spencer): 0.5 - 1 C. 3. Skeptics: 0 C. The observed "hiatus" of warming (slight cooling since 1998) presents a problem to both Warmers and Lukewarmers, since the absence of an expected steady warming under observed steadily increasing CO2, has to be explained as a cancellation effect from unknown natural variations. The gap between Warmers and Lukewarmers is 0.5 C, which is within measurement error. A Skeptic has an easier job and does not have to explain anything different from expectation to rationalize the observations, and in this sense seems to be better off according to Ockham's razor. lördag 12 oktober 2013 Roy: Does More CO2 Cause Warming, Cooling or Nothing? Roy Spencer confesses his deep inner conviction: • I’m far from a political moderate, but I’ve been tagged as a “lukewarmer” in the climate wars. That’s because, at least from a theoretical perspective, it’s hard (but not impossible) for me to imagine that increasing CO2 won’t cause some level of warming. • Some level of warming can probably be expected, but just how much makes a huge difference. • Lindzen and I and a few other researchers in the field think the IPCC models are simply too sensitive, due to positive feedbacks that are either too strong, or even have the wrong sign. But we still believe more CO2 should cause some level of warming. • If the current lack of warming really is due to a natural cooling influence temporarily canceling out CO2-induced warming, what happens when that cooling influence goes away? We are going to see rather rapid warming return… So Roy believes that more CO2 should cause some warming, and if it doesn't as in the present slightly cooling "hiatus" since 17 years, then it is because it is temporarily cancelled by natural cooling and thus may well return with rapid warming... But Roy also states that he (and Lindzen) believes that the warming effect of CO2 is smaller than that postulated by IPCC (which is already quite small and smaller in AR5 than in AR4), and thus may well be too small to ever be observed. Roy is not so alone to have this idea as he thinks. Roy thus can easily imagine a broad spectrum from warming to nothing, but can hardly imagine slight cooling under increasing CO2 like in the present observed hiatus (which he admits is not impossible). A very moderate standpoint, I would say. I write this as a comment to Roys article since Roy does not want my comments on his blog, but Roy is welcome to comment with hopefully some answer to the title question on this blog. PS Roy sends the warning:: • The Danger of Hanging Your Hat on No Future Warming while with respect to the present hiatus, the more relevant warning would be: • The Danger of Hanging Your Hat on Future Warming. fredag 11 oktober 2013 Abstract Nordic Seminar in Computational Mechanics Oslo Oct 23-25: Breaking the Spell of Prandtl Movie is here The abstract for my upcoming talk at NCSM26 in Oslo Oct 23-25 is now available as The talk presents in particular the first ab initio direct computational simulation of the flow of air around an airplane at large angle of attack and low velocity at landing, in close accordance to measured pressure distributions. This represents a major breakthrough in Computational Fluid Dynamics blocked for hundred years by a spell of Prandtl requiring impossible computational resolution of thin boundary layers demanding trillions of mesh points. We use a slip boundary condition modeling the small skin friction of slightly viscous air flow which does not generate and thus does not require resolution of boundary layers. We obtain pressure distributions using three millions of mesh points matching observations, and conclude that boundary layers have little impact (for slightly viscous flow) and thus do not have to be resolved. The spell of Prandtl is thereby broken. onsdag 9 oktober 2013 Nobel Prize in Chemistry Awarded for Not Solving Schrödinger's Equation Picture from presentation of Nobel Prize in Chemistry 2013: Multiscale Models Quantum mechanics based on Schrödinger's equation as the core of modern physics, is presented as an almost perfect mathematical model of the atoms and molecules making up the world. But there is one hook: Schrödinger's equation can be solved analytically only for the Hydrogen atom with one electron and computationally only for atoms with few electrons; already Helium with two electrons poses difficulties. The reason is that the Schrödinger equation is formulated using 3N spatial dimensions for an atom with N electrons with each electron demanding its own three-dimensional space. If each dimension takes 10 pixels to be resolved (very low accuracy), 10^100 = googol = 1 followed by 100 zeros pixels would be required for the Arsenic atom with 33 electrons, which is deadly poison for any thinkable computer, since the required number of pixels would be much larger than the number of atomic particles in the Universe! Schrödinger's equation thus has to be replaced by an equation which is simpler to solve, and this is what computational chemistry is about. The first Nobel Prize in this field was given in 1998 to Kohn and Pople for so called density functional theory which reduces Schrödinger's equation to computable three spatial dimensions with say 100 pixels in each dimension. The next one was given today to Karplus, Lewitt and Warshel for • the development of multiscale models for complex chemical systems: • Karplus, Levitt and Warshel...managed to make Newton’s classical physics work side-by-side with the fundamentally different quantum physics. These prizes were thus awarded for not solving the Schrödinger equation, while the very formulation of the equation (and the similar Dirac equation) was awarded in 1933. In any case the prize was awarded to Computational Mathematics as an expression of The World as Computation. Slinky as Alternative to Higgs Mechanism of Giving Mass to a Body The popular description of Higgs' mechanism supplying mass to a body awarded the 2013 Noble Prize in Physics, goes as follows: Imagine a celebrity (e.g. Brad Pitt) moving through a crowd of people drawing attention from inescapable interaction with the crowd, which can be imagined to generate some kind of resistance to the motion of the celebrity connecting to the amount of fame, or mass, of the star: A more technical description from Wikipedia goes as follows: The idea is evidently that a body acquires mass from interaction with some form of background field or crowd.Is this credible? Maybe. Maybe not. In any case, here is a reprint of a different mechanism that I reflected on some time ago as a possible resolution of Zeno's paradox of the impossibility of motion with a Slinky as mental image: The basic idea is that the slinky moves so to speak by itself and not by interacting with a background field. The kinetic energy of the slinky = 1/2 x mass x velocity^2 equals the energy invested to compress or extend the slinky before letting it go. Mass can then be defined in terms of velocity and invested energy stored as kinetic energy through the motion. Mass is then something carried by the slinky through motion related to the stored energy, which can be released by letting the slinky run into a wall. Is this credible? Maybe. Maybe not. Will Slinky get a Nobel Prize? Maybe not. The motion of a slinky suggests a resolution of Zeno’s Arrow Paradox as a combination of compression-release and switch of stability, where the the slinky appears as a soliton wave, which itself generates the medium through which it propagates. Zeno of Elea (490-430 BC), member of the pre-Socratic Eliatic School founded by Parmenides, questioned the concept of change and motion in his famous arrow paradoxHow can it be that an arrow is moving, when at each time instant it is still? In Resolution of Zeno’s Paradox of Particle Motion I argued that the paradox still after 2.500 years lacks a convincing resolution, and suggested a resolution based on wave motion. A fundamental question of wave propagation is the nature of the medium through which the wave propagates: Is it material as in the case of sound waves in air, or is it immaterial as in the case of light waves in vacuum? If the flying arrow is a wave, which is the medium through which it propagates? It is not enough to say that it is air, because an arrow can fly also in vacuum. We are led to the following basic question: can a wave itself act as the medium through which it propagates? It turns out that a slinky can serve as an answer! To see this take a look at this movie . We see that the motion of a slinky can be described as follows: • oscillation between two forms of energy: elastic energy and kinetic energy compression stores elastic energy • elastic energy is transformed into kinetic energy when the slinky expands • there is a critical moment with the slinky fully compressed in which the downward forward motion of the top ring is reflected in upward forward (and not upward backward motion which would lead to motion on the spot) • the slinky forms itself the medium through which it as a wave propagates • the slinky acts like a soliton wave. We understand that the slinky offers a model for resolution Zeno’s paradox as a wave which itself generates the medium through which it propagates. What is Mass? You can take this model one step further, and view the work required to compress the slinky from an uncompressed rest state, as an investment into kinetic energy of motion, just as a body can be accellerated from rest by the action of a force and gain kinetic energy. This would mean that the slinky has inertial mass and that it can move with different velocities depending on the amount of work invested in the initial compression. We may compare with the propagation of massless electromagnetic waves with given fixed speed of light. This connects to the question Does the Earth Rotate? suggesting to define mass as inertial mass M in terms of kinetic energy K and velocity V from the formula K = 1/2 x M x V x V. PS1 The difference between Higgs and Slinky is a bit like the difference between environment and genetics for living body, with Higgs only exterior environment and Slinky only interior genetics. PS2 There is a connection to Wittengenstein's ladder which the user successively pulls up behind as the climbing advances. PS3 The Higgs mechanism is described in the above picture to "slow down" the motion of an electron as an effect of some kind of viscosity. This seems strange since electrons are not "slowed down" by the mere fact that they have mass. Acceleration is "slowed down" by mass but not velocity. tisdag 8 oktober 2013 From Impossible Fiction to Possible Reality in Quantum Mechanics The wave function $\Psi$ as a solution to Schrödinger's equation supposedly describing the quantum mechanics of an atom with $N$ electrons, depends on N three-dimensional spatial variables $x_1,..., x_N$ (and time), altogether $3N$ spatial dimensions, with $\vert \Psi (x_1,..., x_N)\vert^2$ interpreted as the probability of the configuration with electron $j$ located at position $x_j$. The wave function $\Psi (x_1,...,x_N)$ is thus supposed to carry information about all possible electron configurations and as such contains an overwhelming amount of information, which however is not really accessible because of an overwhelming computational cost already for small N with difficulties starting already for $N = 2$. To handle this difficulty drastic reductions in complexity are being made by seeking approximate solutions as wave functions with drastically restricted spatial variation based on heuristics. There are claims that this is feasible using structural properties of the wave function (but full scale computations seem to be missing). An alternative approach would be to seek $N$ wave functions $\Psi_1,..., \Psi_N$, depending on a common three-dimensional space coordinate $x$, with $\Psi_j(x)$ carrying information about the presence of particle $j$, as a form of Smoothed Particle Mechanics (SPM) as a variant of classical particle mechanics. The corresponding Schrödinger equation consists of a coupled system of one-electron equations in the spirit of Hartree, and is readily computable even for N large. The solution $\Psi_1(x),..., \Psi_N(x)$, of the system would give precise information about one possible electron configuration. If this is a representative configuration, this may be all one would like to know. As an example, SPM for the Helium atom with two atoms appears to give a configuration with two half-spherical electron lobes in opposition, as a representative configuration with other equally possible configurations obtained by rotation, as suggested in a previous post and in the sequence Quantum Contradictions. Instead of seeking information about all possible configurations by solving the 3N dimensional many-electron Schrödinger equation, which is impossible, it may be more productive to seek information about one possible and representative configuration by solving a system of 3 dimensional one-electron wave functions, which is possible. måndag 7 oktober 2013 Quantum Contradictions 21: Atomic Radii? Quantum Mechanics (QM) based on Schrödinger's equation is supposed to give a very accurate description of the atomic world, yet theoretical predictions of atomic radii presented in the literature based on solving Schrödinger's equation, agree with observations to only about 5%. This is excused by saying that the radius of an atom is not well defined, and so a precise number cannot be given. In any case, QM is postulated to be very precise, and a precise answer will be delivered if only the question is the right one: Asking about the radius of an atom is not a good question and thus will have no good QM answer. In contrast, the spin factor on an electron is claimed to be predicted by Quantum ElectroDynamics (QED) to one part in a trillion, which following Feynman is presented as ultimate evidence of the correctness of QED and thus also of its a little (less precise but still very precise) cousin QM. So we have to live with the fact that asking for the radius of an atom will get no QM answer: The precision is about the precision that can be obtained from the above chart. söndag 6 oktober 2013 Whereof the Royal (Academy) Can(not) Speak The Swedish King Carl XVI Gustaf is, according to the Swedish Constitution from 1975, not allowed to speak out on anything of importance and thus has only a ceremonial role. However, the King being a devoted environmentalist with a sharp brain has found a canal to reach out to the Swedish people through his Royal Swedish Academy of Sciences, in charge of the Noble Prizes (delivered from the hands of the King) with the 2013 prizes to be announced the coming week. In particular, the King made a statement in support of the IPCC Assessment Report 4 from 2007 sending the world a global warming alarm in the form of The Scientific Basis for Climate Change presented under Science in Society on the web site of the Academy. Accordingly the Academy acted as the forum for the presentation to the world of the new IPCC Assessment Report 5 with even more alarm, supported by Climate Change: the state of science and a tribute to the first chairman of IPCC: Climate Change: Bert Bolin and beyond. The King is now working on an update of his statement The Scientific Basis for Climate Change to make it accurately represent the increased alarm (from 90% to 95%) expressed in the new IPCC Assessment Report 5. The Republican Party, which finds no scientific evidence of alarming global warming from human CO2 emissions, claims that the support of IPCC alarm expressed by the King and his Royal Academy violates the Swedish constitution and a constitutional crisis now seems inevitable. In the worst scenario the Royal Academy will have to be shut down along with all Royal Castles, which is alarming (96%) to Swedish interests all over the globe and thus to the world. torsdag 3 oktober 2013 Quantum Mechanics Contradictions 20: Addendum The smoothed particle model of atomic physics to be explored computationally in coming posts leads to the following simple model for the ground state energy of an atom with Z + 2 electrons (at most 10 to start with) with 2 electrons in a first inner shell of radius and Z electrons in a second outer shell of radius R, the same for all Z. We assume the ground state energy is half of the kernel potential energy (in correspondence with the virial theorem) and motivate the constant radius of the outer shell from the idea that the "width" of an electron in the second shell scales like 1/Z and thus Z electrons could fill up a shell of constant radius, in accordance with the observation that atomic radii vary surprisingly little with the number (beyond one or two) of electrons in the outer shell. The energy contribution e_Z from the second shell will then be Z x Z / D with D = 2 x R, which gives the following contributions with R = 1 with exact values in parenthesis: • Lithium: e_1 = - 0.5 ( - 0.2) • Beryllium: e_2 = - 2 ( - 1) • Boron: e_3 = - 4.5 ( - 1.9) • Carbon: e_4 = - 8 (- 5.25) • Nitrogen: e_5 = - 12.5 (- 9.8) • Oxygen: e_6 = - 18 (- 15.91) • Fluorine: e_7 = - 24.5 ( - 24.2) • Neon: e_8 = - 32 (- 34) We see a fairly good agreement with respect to the simplicity of the model (complemented by the energies for the inner shell) , which gives good promise to the coming computational exploration. PS Continuing with a third shell of radius R = 2 and D = 2 x R = 4, we get with e_Z = (Z -10) x (Z - 10) /D: • Sodium: e_11 = - 0.25 (- 0.18) • Magnesium: e_12 = - 1 (- 0.80) • Aluminium: e_13 = - 2.25 (- 1.9) • Silicon: e_14 = - 4 (- 3.65) • Phosphorus: e_15 = - 6.25 (- 6.2) • Sulfur: e_16 = - 9 (- 9.82) • Chlorine: e_17 = - 12.25 (- 14.5) • Argon: e_18 = - 16 (- 20.5) Again a fairly good agreement. Taking into account that the atomic radius decreases with increasing number of electrons within the band, improves the fit between model and observation,
80bd2be2bc3fe2ec	The variational principle in quantum mechanics, lecture 2 1. Lecture 2: The helium atom In this lecture we are going to apply the variational method to a more complicated example, the helium atom. The simplified example we consider only involves two particles but still exhibits many of the complications and subtleties that arise in the more general {N}-body problem. Lecture notes can be found in pdf format here. 1.1. The hamiltonian A helium atom consists of a nucleus of charge {2e} surrounded by two electrons. Let the nucleus lie at the origin of our coordinate system, and let the position vectors of the two electrons be {\mathbf{r}_1} and {\mathbf{r}_2}, respectively. The Hamiltonian of the system thus takes the form \displaystyle H = -\frac{\hbar}{2m}(\nabla_1^2 + \nabla_2^2) - e^2\left(\frac{2}{\|\mathbf{r}_1\|} + \frac{2}{\|\mathbf{r}_2\|} - \frac{1}{\|\mathbf{r}_1-\mathbf{r}_2\|}\right), \ \ \ \ \ (1) where we are using units so that {4\pi \epsilon_0 = 1}. We write {H} as {H = H_0 + H_{\text{int}}}, where \displaystyle H_0 = -\frac{\hbar}{2m}(\nabla_1^2 + \nabla_2^2) - e^2\left(\frac{2}{\|\mathbf{r}_1\|} + \frac{2}{\|\mathbf{r}_2\|}\right) \ \ \ \ \ (2) \displaystyle H_{\text{int}} = -\frac{e^2}{\|\mathbf{r}_1-\mathbf{r}_2\|}. \ \ \ \ \ (3) The only(!) term which causes problems here is the interparticle interaction term {H_{\text{int}}}, which is the same order as the single-particle potentials. So, unfortunately, we cannot neglect it. However, let’s do so nonetheless: our strategy is to simply solve {H_0} first to get an idea of the shape of ground-state wavefunctions. Then, inspired by the form of the solution to {H_0}, we design a trial wavefunction which we subsequently use with the variational method. 1.2. Solving {H_0} Notice that that hamiltonian {H_0} is separable: \displaystyle H_0 = H_1\otimes \mathbb{I}_2 + \mathbb{I}_1\otimes H_2, \ \ \ \ \ (4) \displaystyle H_j = -\frac{\hbar}{2m}\nabla_j^2 - \frac{2e^2}{\|\mathbf{r}_j\|}, \quad j = 1, 2. \ \ \ \ \ (5) This means that the eigenfunctions of {H_0}, in the case where the particles are distinguishable, are products {\psi(\mathbf{r}_1, \mathbf{r}_2) = \phi_k(\mathbf{r}_1)\phi_{l}(\mathbf{r}_2)} of the eigenvectors {\phi_k(\mathbf{r}_1)} and {\phi_l(\mathbf{r}_2)} of {H_1} and {H_2}. We can obtain an eigenfunction {\phi(\mathbf{r}_j)} of {H_j}, {j = 1, 2}, by solving the hydrogenic Schrödinger equation \displaystyle H_{j} \phi(\mathbf{r}_j) = E\phi(\mathbf{r}_j), \ \ \ \ \ (6) which reads \displaystyle \left(-\frac{\hbar}{2m}\nabla_j^2 - \frac{2e^2}{\|\mathbf{r}_j\|}\right)\phi(\mathbf{r}_j) = E\phi(\mathbf{r}_j), \ \ \ \ \ (7) which is nothing other than the hamiltonian for a “hydrogen” atom with nuclear charge {2e}. The lowest-energy eigenstate of {H_j} is given by \displaystyle \phi(\mathbf{r}_j) = \psi_0(\mathbf{r}_j) = \frac{Z^{\frac32}}{\sqrt{\pi}a^{\frac32}}e^{-\frac{Zr_j}{a}}, \ \ \ \ \ (8) where {r_j = \|\mathbf{r}_j\|} and {Z = 2} is the nuclear charge. The proof of this is left as an exercise (find a reference which solves the Schrödinger equation for the hydrogen atom and replace the nuclear charge {Z = 1} with {Z = 2} throughout; spherical coordinates are helpful here). The corresponding ground-state eigenvalue is {E_0 = -Z^2 \times 13.6 \text{eV}}. \subsubsection{A note on particle statistics} Electrons are indistinguishable fermions. This means that their wavefunction must be antisymmetric under exchange of particles, i.e., \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2) = -\Psi(\mathbf{r}_2, \mathbf{r}_1). \ \ \ \ \ (9) The problem is that if we say the wavefunction for two electrons is a product: \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2) ``='' \phi(\mathbf{r}_1)\psi(\mathbf{r}_2) \ \ \ \ \ (10) then we run into problems: this wavefunction isn’t antisymmetric under interchange of the two particles! In order to ensure that the overall wavefunction for the two electrons is antisymmetric we write \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt{2}}\left(\phi(\mathbf{r}_1)\psi(\mathbf{r}_2) - \phi(\mathbf{r}_2)\psi(\mathbf{r}_1)\right). \ \ \ \ \ (11) (This is a Slater determinant.) This expression vanishes when {\phi = \psi}, and is the manifestation of the Pauli exclusion principle. But the trial wavefunction we are going to study has exactly the form of a product of identical wavefunctions, which is incompatible with the particle statistics. In our special case, however, we can largely avoid consideration of the particle statistics by realising that there are two species of electrons, namely, spin-up and spin-down electrons: when magnetic fields and spin-orbit couplings are disregarded we can obtain a perfectly legitimate antisymmetric wavefunction for two — now distinguishable by their spin — electrons in the same wavefunction by writing \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt{2}}\left(\phi_{\downarrow}(\mathbf{r}_1)\phi_{\uparrow}(\mathbf{r}_2) - \phi_{\uparrow}(\mathbf{r}_1)\phi_{\downarrow}(\mathbf{r}_2)\right), \ \ \ \ \ (12) i.e., the antisymmetry is carried by the spin degree of freedom. The system is said to be in a singlet state with total overall spin. But in this case we can do all our calculations, i.e., calculations of energy expectation values, as though {\Psi(\mathbf{r}_1, \mathbf{r}_2)} was a product (exercise!); the spin never features in the subsequent derivations. 1.3. The variational method, take 1: no variation Here we take for our “trial wavefunction” the product of the solutions (8) of the hydrogenic single-particle hydrogenic Schrödinger equations for {H_j}: \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2) = \frac{Z^{3}}{\pi a^{3}}e^{-\frac{Z(r_1+r_2)}{a}}. \ \ \ \ \ (13) Of course, this isn’t a variational wavefunction: there are no variational parameters! The energy expectation of value of {\Psi} is given by \displaystyle \langle H\rangle = E_0 + E_0 + \int d\mathbf{r}_1d\mathbf{r}_2 \|\Psi(\mathbf{r}_1, \mathbf{r}_2)\|^2 \frac{e^2}{\|\mathbf{r}_1-\mathbf{r}_2\|} \ \ \ \ \ (14) \displaystyle = 2\times -54.4\text{eV} + 34\text{eV} = -74.8 \text{eV}. \ \ \ \ \ (15) The observed value is {E_{\text{obs}} = -78.9\text{eV}}, so this isn’t too bad. 1.4. The variational method, take 2: an improved trial wavefunction We content ourselves with the simplest possible application of the variational method where we take the variational wavefunction to have the same form as (13) but we replace the nuclear charge {Z} with {Z_\text{eff} = Z-\sigma}. The physical motivation here is that each electron is partially screened from the nucleus by the presence of the other. Thus our trial wavefunction is \displaystyle \Psi(\mathbf{r}_1, \mathbf{r}_2; \sigma) = \frac{(Z-\sigma)^{3}}{\pi a^{3}}e^{-\frac{(Z-\sigma)(r_1+r_2)}{a}}. \ \ \ \ \ (16) If {\sigma} is left arbitrary then the energy expectation value is \displaystyle E(\sigma) = \langle H \rangle = -\frac{e^2}{a}\left(Z^2 - \frac{5}{8}Z + \frac58 \sigma - \sigma^2\right). \ \ \ \ \ (17) Exercise: work out this result. The minimum of {E(\sigma)} is obtained for {\sigma = 5/16}, which corresponds to a screening effect where one electron reduces the nuclear charge by approximately one-third of an electron charge. The minimum value is given by \displaystyle E(5/16) = -\left(Z - \frac6{16}\right)\frac{e^2}{a}. \ \ \ \ \ (18) Exercise: compare this with the observed value. How much of an improvement over the crude estimate is there? 1.5. Conclusions The agreement between theory and experiment can be improved by using a better trial function than the simple products we employed here. The design of such trial functions has occupied researchers since the early days of quantum mechanics. The helium atom is a testbed problem and offers a basic challenge because the two-electron system is mathematically tractable, although solutions in closed form cannot be found. The test of an improved trial state is measured by how closely the variational minimum lies above the precisely measured energy. 8 Responses to The variational principle in quantum mechanics, lecture 2 1. Oblrmyxb says: I’d like to cancel a cheque underage girls naked 2. Evwlprju says: Could you transfer $1000 from my current account to my deposit account? illegal pedo pics 3. Vgvkfzva says: I was made redundant two months ago Pink Teen Models 4. Owqujinr says: How do you spell that? Lolita Teen Leave a Reply WordPress.com Logo Twitter picture Facebook photo Google+ photo Connecting to %s %d bloggers like this:
4a84b711b3d13e21	Rogue waves \| Day 347 On 5 May of 1916 British explorer Ernest Shackleton was on a small boat in the Southern Ocean. The skies were dark – until he noticed a white line on the horizon. Later in his journal Shackleton wrote: “I called to the other men that the sky was clearing, and then a moment later I realized that what I had seen was not a rift in the clouds but the white crest of an enormous wave. During twenty-six years’ experience of the ocean in all its moods I had not encountered a wave so gigantic. It was a mighty upheaval of the ocean, a thing quite apart from the big white-capped seas that had been our tireless enemies for many days.”1 What Shackleton experienced that day was a ‘rogue wave’. These spontaneous surface waves tend to occur out on the open seas; a rogue wave is always exceptionally large in comparison to the waves happening at the same time. This makes them not just dramatic – the rogue waves are so sudden and overwhelming, they can gravely endanger ships and ocean liners. “We felt our boat lifted and flung forward like a cork in breaking surf. We were in a seething chaos of tortured water; but somehow the boat lived through it, half full of water, sagging to the dead weight and shuddering under the blow,” wrote Shackleton. He got away, but many boating disasters have been attributed to rogue waves. Stealthy as they are, rogue waves don’t appear out of nowhere. A team of physicists at the Australian National University has just come forward with a mathematical theory that helps explain the causation of rogue waves and, in conjunction with real-life data from buoys and other devices, potentially can predict when a rogue wave will hit. According to lead author, Professor Nail Akhmediev, at any one time there are about 10 rogue waves scattered throughout the world’s oceans. The surface of the ocean is a chaotic place, and these waves naturally emerge when the circumstances are just right; the researchers have developed a special solution to the nonlinear Schrödinger equation, a theoretical physics equation which is normally used to describe the propagation of light in nonlinear optical fibers. The new theory could lead to the development of better detection devices. “A device on the mast of a ship analysing the surface of the sea could perhaps give a minute’s warning that a rogue wave is developing,” said Akhmediev in a media statement. Show 1 footnote 1. The Voyage of the James Caird, see here. Leave a Reply Your email is perfectly safe with me.
0ca3963f2f0ebade	söndag 30 augusti 2015 Quantum Information Can Be Lost fredag 28 augusti 2015 Finite Element Quantum Mechanics 4: Spherically Symmetric Model I have tested the new atomic model described in a previous post in setting of spherical symmetry with electrons filling a sequence of non-overlapping spherical shells around a kernel. The electrons in each shell are homogenized to spherical symmetry which reduces the model to a 1d free boundary problem with the free boundary represented by the inter-shell spherical surfaces adjusted so that the combined wave function is continuous along with derivates across the boundary. The repulsion energy is computed so as to take into account that electrons are not subject to self-repulsion, by a corresponding reduction of the repulsion within a shell. The remarkable feature of this atomic model, in the form of a 1d free boundary problem with continuity as free boundary condition and readily computable on a lap-top, is that computed ground state energies show to be surprisingly accurate (within 1%) for all atoms including ions (I have so far tested up to atomic number 54 and am now testing excited states). Recall that the wave function $\psi (x,t)$ solving the free boundary problem, has the form • $\psi (x,t) =\psi_1(x,t)+\psi_2(x,t)+...+\psi_S(x,t)$ (1) with $(x,t)$ a common space-time coordinate, where $S$ is the number of shells and $\psi_j(x,t)$ with support in shell $j$ is the wave function for the homogenized wave function for the electrons in shell $j$ with $\int\vert\psi_j(x,t)\vert^2\, dx$ equal to the number of electrons in shell $j$. Note that the free boundary condition expresses continuity of charge distribution across inter-shell boundaries, which appears natural. Note that the model can be used in time dependent form and then allows direct computation of vibrational frequencies, which is what can be observed. Altogether, the model in spherical symmetric form indicates that the model captures essential features of the dynamics of an atom, and thus can useful in particular for studies of atoms subject to exterior forcing. I have also tested the model without spherical homogenisation for atoms with up to 10 electrons, with similar results. In this case the the free boundary separates diffferent electrons (and not just shells of electrons) with again continuous charge distribution across the corresponding free boundary. In this model electronic wave functions share a common space variable and have disjoint supports and can be given a classical direct physical interpretation as charge distribution. There is no need of any Pauli exclusion principle: Electrons simply occupy different regions of space and do not overlap, just as in a classical multi-species continuum model. This is to be compared with standard quantum mechanics based on multidimensional wave functions $\psi (x_1,x_2,...,x_N,t)$ typically appearing as linear combinations of products of electronic wave functions • $\psi (x_1,x_2,...,x_N,t)=\psi_1(x_1,t)\times \psi_2(x_2,t)....\times\psi_N(x_N,t)$ (2) for an atom with $N$ electrons, each electronic wave function $\psi_j(x_j,t)$ being globally defined with its own independent space coordinate $x_j$. Such multidimensional wave functions can only be given statistical interpretation, which lacks direct physical meaning. In addition, Pauli's exclusion principle must be invoked and it should be remembered that Pauli himself did not like his principle since it was introduced ad hoc without any physical motivation, to save quantum mechanics from collapse from the very start... More precisely, while (1) is perfectly reasonable from a classical continuum physics point of view, and as such is computable and useful, linear combination of (2) represent a monstrosity which is both uncomputable and unphysical and thus dangerous, but nevertheless is supposed to represent the greatest achievement of human intellect all times in the form of the so called modern physics of quantum mechanics. How long will it take for reason and rationality to return to physics after the dark age of modern physics initiated in 1900 when Planck's "in a moment of despair" resorted to an ad hoc hypothesis of a smallest quantum of energy in order to avoid the "ultra-violet catastrophe" of radiation viewed to be impossible to avoid in classical continuum physics. But with physics as finite precision computation, which I am exploring, there is no catastrophe of any sort and Planck's sacrifice of rationality serves no purpose. PS Here are the details of the spherical symmetric model starting from the following new formulation of a Schrödinger equation for an atom with $N$ electrons organised in spherical symmetric form into $S$ shells: Find a wave function as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x\in R^3$ and time coordinate $t$ with non-overlapping spatial supports $\Omega_1(t)$,...,$\Omega_N(t)$, filling 3d space, satisfying • $i\dot\psi (x,t) + H\psi (x,t) = 0$ for all $(x,t)$, (1) where the (normalised) Hamiltonian $H$ is given by where $V_k(x)$ is the potential corresponding to electron $k$ defined by • $V_k(x)=\int\frac{\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dy$, for $x\in R^3$, and the wave functions are normalised to correspond to unit charge of each electron: • $\int_{\Omega_j}\vert\psi_j(x,t)\vert^2 =1$ for all $t$ for $j=1,..,N$. Assume the electrons fill a sequence of shells $S_k$ for $k=1,...,S$ centered at the atom kernel with $N_k$ electrons on shell $S_k$ and • $\int_{S_k}\vert\psi (x,t)\vert^2 =N_k$ for all $t$ for $k=1,..,S$, • $\sum_k^S N_k = N$. The total wave function $\psi (x,t)$ is thus assumed to be continuously differentiable and the electronic potential of the Hamiltonian acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\vert\psi_k\vert^2$ for $k\neq j$, with total electronic repulsion energy • $\sum_{k\neq j}\int\frac{\vert\psi_k(x,t)\vert^2\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dxdy=\sum_{k\neq j}V_k(x)\vert\psi_k(x)\vert^2\, dx$. Assume now that the electronic repulsion energy is approximately determined by homogenising the $N_k$ electronic wave function $\psi_j$ in each shell $S_k$ into a spherically symmetric "electron cloud" $\Psi_k(x)$ with corresponding potential $W_k(y)$ given by • $W_k(y)=\int_{\vert x\vert <\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert y\vert}\, dx+\int_{\vert x\vert >\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert x\vert}\, dx$, and $R_k(x)=\frac{N_k-1}{N_k}$ for $x\in S_k$ is a reduction factor reflecting non self-repulsion of each electron (and $R_k=1$ else): Of the $N_k$ electrons in shell $S_k$, thus only $N_k-1$ electrons contribute to the value of potential in shell $S_k$ from the electrons in shell $S_k$. We here use the fact that the potential $W(x)$ of a uniform charge distribution on a spherical surface $\{y:\vert y\vert =r\}$ of radius $r$ of total charge $Q$, is equal to $Q/\vert x\vert$ for $\vert x\vert >r$ and $Q/r$ for $\vert x\vert <r$. Our model then has spherical symmetry and is a 1d free boundary problem in the radius $r=\vert x\vert$ with the free boundary represented by the radii of the shells and the corresponding Hamiltonian is defined by the electronic potentials computed by spherical homogenisation in each shell. The free boundary is determined so that the combined wave function $\psi (x,t)$ is continuously differentiable across the free boundary. torsdag 27 augusti 2015 Finite Element Quantum Mechanics 3: Explaining the Periodicity of the Periodic Table According to Eric Scerri, the periodic table is not well explained by quantum mechanics, contrary to common text book propaganda, not even the most basic aspect of the periodic table, namely its periodicity: • Pauli’s explanation for the closing of electron shells is rightly regarded as the high point in the old quantum theory. Many chemistry textbooks take Pauli’s introduction of the fourth quantum number, later associated with spin angular momentum, as the foundation of the modern periodic table. Combining this two-valued quantum number with the ear- lier three quantum numbers and the numerical relationships between them allow one to infer that successive electron shells should contain 2, 8, 18, or $2n^2$ electrons in general, where n denotes the shell number. • This explanation may rightly be regarded as being deductive in the sense that it flows directly from the old quantum theory’s view of quantum numbers, Pauli’s additional postulate of a fourth quantum number, and the fact that no two electrons may share the same four quan- tum numbers (Pauli’s exclusion principle). • However, Pauli’s Nobel Prize-winning work did not provide a solution to the question which I shall call the “closing of the periods”—that is why the periods end, in the sense of achieving a full-shell configuration, at atomic numbers 2, 10, 18, 36, 54, and so forth. This is a separate question from the closing of the shells. For example, if the shells were to fill sequentially, Pauli’s scheme would predict that the second period should end with element number 28 or nickel, which of course it does not. Now, this feature is important in chemical education since it implies that quantum mechanics can- not strictly predict where chemical properties recur in the periodic table. It would seem that quantum mechanics does not fully explain the single most important aspect of the periodic table as far as general chemistry is concerned. • The discrepancy between the two sequences of numbers representing the closing of shells and the closing of periods occurs, as is well known, due to the fact that the shells are not sequentially filled. Instead, the sequence of filling fol- lows the so-called Madelung rule, whereby the lowest sum of the first two quantum numbers, n + l, is preferentially oc- cupied. As the eminent quantum chemist Löwdin (among others) has pointed out, this filling order has never been derived from quantum mechanics. On the other hand, in the new approach to atomic physics I am exploring, the periodicity directly connects to a basic partitioning or packing problem, namely how to subdivide the surface of a sphere in equal parts, which gives the sequence $2n^2$ by dividing first into two half spheres and then subdividing each half spherical surface in $n\times n$ pieces, in a way similar to dividing a square surface into $n\times n$ square pieces. With increasing shell radius an increasing number of electrons, occupying a certain surface area (scaling with the inverse of the kernel charge), can be contained in a shell. In this setting a "full shell" can contain 2, 8, 18, 32,.., electrons, and the observed periodicity 2, 8, 8, 18, 18, 32, 32, with each period ended by a noble gas with atomic numbers 2 (He), 10 (Neon), 18 (Argon), 36 (Krypton), 54 (Xenon), 86 (Radon), 118 (Ununoctium, unkown), with a certain repetition of shell numbers, can be seen as a direct consequence of such a full shell structure, if allowed to be repeated when the radius of a shell is not yet large enough to house a full shell of the next dignity. Text book quantum mechanics thus does not explain the periodicity of the periodic table, while the new approach am I pursuing may well do so in a very natural way. Think of that. tisdag 25 augusti 2015 Ulf Danielsson om Klimathot, Hawking och Svarta Hål. Strängfysikern Ulf Danielsson har startat en blogg med Stephen Hawking's besök vid KTH och föreläsning på Stockholm Waterfront som initiellt dragplåster. Ulf skriver gärna om svarta hål, som han verkar tro inneha verklig fysisk existens som "singularitet" till lösningar till Einstein's ekvationer. Ulf verkar även tro på kimatalarmismen som den predikas av IPCC: • När det gäller människogenererad klimatpåverkan är huvudslutsatsen klar: den finns där, och risken att den får betydande följder för den mänskliga civilisationen om inget görs är överhängande. Den senaste IPCC-rapporten gör det omöjligt att dra någon annan generell slutsats. Vi skeptiker som granskat vetenskapen bakom IPCCs klimatalarmism, vet att Ulf i denna fråga blivit helt vilseförd. Frågan är om samma sak gäller för svarta hål? Om det nu är så att man kan hitta singulariteter hos lösningar till Einstein's ekvationer, vilket i sig kan diskuteras eftersom dessa ekvationer är hart när omöjliga att lösa, betyder det att dessa singulariteter också har fysisk realitet? Även om det finns massa i centrum på galaxer som man inte kan se, vilket observationer av galaxers dynamik verkar tyda på, så betyder det väl inte nödvändigtvis att denna osynliga massa utgörs av svart hål? Kan det vara så att IPCCs (farligt tjocka enligt Ulf) rapport utgör ett svart hål ur vilken ingen sann information förmår utstråla? Finite Element Quantum Mechanics 2: Questions without Answers 1. Do isolated quantal systems exist at all? 3. Does quantum mechanics apply to large molecular systems? 4. Is the superposition principle universally valid? 5. Why do so many stationary states not exist? 6. Why are macroscopic bodies localised? 8. Why can approximations be better than the exact solutions? 9. Why is the Born-Oppenheimer picture so successful? 10. Is temperature an observable? Hans Primas gives the following devastating verdict: Finite Element Quantum Mechanics 1: Listening to Bohm David Bohm discusses in the concluding chapter of Quantum Theory the relationship between quantum and classical physics, stating the following charcteristics of classical physics: 1. The world can be analysed into distinct elements. 2. The state of each element can be described in terms of dynamical variables that are specified with arbitrarily high precision. 3. The interrelationship between parts of a system can be described with the aid of exact casual was that define the changes of the above dynamical variables with time in terms of their initial values. The behavior of the system as a whole can be regarded as the result of the interaction by its parts. If we here replace, "arbitrarily high precision" and "exact" with "finite precision", the description 1-3 can be viewed as a description of • the finite element method • as digital physics as digital computation with finite precision • as mathematical simulation of real physics as analog computation with finite precision. My long-term goal is to bring quantum mechanics into a paradigm of classical physics modified by finite precision computation, as a form of computational quantum mechanics, thus bridging the present immense gap between quantum and classical physics. This gap is described by Bohm as follows: • The quantum properties of matter imply the indivisibility unity of all interacting systems. Thus we have contradicted 1 and 2 of the classical theory, since there exist on the quantum level neither well defined elements nor well defined dynamical variables, which describe the behaviour of these elements. My idea is thus to slightly modify classical physics by replacing "arbitrarily high precision" with "finite precision" to encompass quantum mechanics thus opening microscopic quantum mechanics to a machinery which has been so amazingly powerful in the form of finite element methods for macroscopic continuum physics, instead of throwing everything over board and resorting to a game of roulette as in the text book version of quantum mechanics which Bohm refers to. In particular, in this new form of computational quantum mechanics, an electron is viewed as an "element" or a "collection of elements", each element with a distinct non-overlapping spatial presence, with an interacting system of $N$ electrons described by a (complex-valued) wave function $\psi (x,t)$ depending on a 3d space coordinate $x$ and a time coordinate $t$ of the form • $\psi (x,t) = \psi_1(x,t) + \psi_2(x,t)+...+\psi_N(x,t)$, (1) where the electronic wave functions $\psi_j(x,t)$ for $j=1,...,N$, have disjoint supports together filling 3d space, indicating the individual presence of the electrons in space and time. The system wave function $\psi (x,t)$ is required to satisfy a Schrödinger wave equation including a Laplacian asking the composite wave functions $\psi (x,t)$ to be continuous along with derivatives across inter element boundaries. This a is a free boundary problem in 3d space and time and as such readily computable. I have with satisfaction observed that a spherically symmetric shell version of such a finite element model does predict ground state energies in close comparison to observation (within a percent) for all elements in the periodic table, and I will report these results shortly. We may compare the wave function given by (1) with the wave function of text book quantum mechanics as a linear combination of terms of the multiplicative form: • $\psi (x_1,x_2,...x_N,t)=\psi_1(x_1,t)\times\psi_2(x_2,t)\times ...\times\psi_N(x_N,t)$, depending on $N$ 3d space coordinates $x_1,x_2,...,x_N$ and time, where each factor $\psi_j(x_j,t)$ is part of a (statistical) description of the global particle presence of an electron labeled $j$ with $x_j$ ranging over all of 3d space. Such a wave function is uncomputable as the solution to a Schrödinger equation in $3N$ space coordinates, and thus has no scientific value. Nevertheless, this is the text book foundation of quantum mechanics. Text book quantum mechanics is thus based on a model which is uncomputable (and thus useless from scientific point of view), but the model is not dismissed on these grounds. Instead it is claimed that the uncomputable model always is in exact agreement to all observations according to tests of this form: • If a computable approximate version of this model (such as Hartree-Fock with a specific suitably chosen set of electronic orbitals) happens to be in correspondence with observation (due to some unknown happy coincidence), then this is taken as evidence that the exact version is always correct. • If a computable approximate version happens to disagree with observation, which is often the case, then the approximate version is dismissed but the exact model is kept; after all, an approximate model which is wrong (or too approximate) should be possible to view as evidence that an exact model as being less approximate must be more (or fully) correct, right? PS The fact that the finite element method has been such a formidable success for macroscopic problems as systems made up of very many small parts or elements, gives good hope that this method will be at least as useful for microscopic systems viewed to be formed by fewer and possibly simpler (rather than more complex) elements. This fits into a perspective (opposite to the standard view) where microscopics comes out to be more simple than macroscopics, because macroscopics is built from microscopics, and a DNA molecule is more complex than a carbon atom, and a human being more complex than an egg cell. lördag 15 augusti 2015 Popper vs Physics as Finite Precision Computation Today, physics is in a crisis....it is a crisis of understanding...roughly as old as the Copenhagen interpretation of quantum mechanics...(in 1982 Preface to Quantum Theory and the Schism in Physics) Karl Popper's vision expressed in The Postscript to the Logic of Scientific Discovery (with the above book as Vol III), is a science of modern quantum physics which shares the following characteristics of classical physics: 1. Realism 2. Determinism (A) 3. Objectivism. Popper compares this paradigm of rationality with the ruling paradigm of modern physics being the exact opposite as an irrationality characterised by: 1. Idealism 2. Indeterminism (B) 3. Subjectivism. The crisis of modern physics acknowledged by all prominent physicists of today, can be viewed as an effect of (B). It is no wonder that (B) being an irrational opposite to a rational (A), has led to a crisis. The reason the paradigm (A) of classical macroscopic physics was replaced by (B) when Planck-Bohr-Born-Heisenberg shaped the ruling (Copenhagen) paradigm of modern physics, was a perceived impossibility to (i) explain the phenomena of black-body radiation by classical electrodynamics and (ii) to give the standard multi-dimensional (uncomputable) wave function of Schrödinger's equation describing microsopic atomic physics, a physical meaning. I have been led to a version of Popper's paradigm (A) viewing physics as • finite precision computation where (i) and (ii) can be handled in a natural way and the resort to the extreme position (B) can be avoided. This paradigm is outlined as Computational Blackbody Radiation and The World as Computation In particular I have explored a computable three-dimensional alternative version of Schrödinger's equation conforming to (A) and I will present computational results in upcoming posts. In particular, it appears that this (computable) version explains the periodic table more directly than the standard (uncomputable) one. More precisely, an uncomputable mathematical model is useless and cannot be used to explain anything. PS The crisis in physics rooted in the Copenhagen interpretation has deepened after Poppers 1982 analysis, following a well known tactic to handle a pressing problem, which appears to be unsolvable: make the problem even more severe and unsolvable and thereby relieve the pressure from the original problem. Today we can observe this tactic in extreme form with physicists flooding media with fantasy stories about dark matter, dark energy, parallel worlds and cats in superposition of being both dead and alive, all phenomena of which nothing is known. The Dark Ages appears as enlightened against this background. tisdag 4 augusti 2015 Dystert Resultat av KTH-Gate = Noll: SimuleringsTeknik Läggs Ner KTH-gate är benämningen på på den aktion som KTH riktade mot mitt verk som innebar att min ebok Mathematical Simulation Technology (MST) avsedd att användas inom det nya kandidatprogrammet i Simuleringsteknik och Virtuell Design (STVD), mitt under pågående testkurs under HT10 förbjöds av KTH (för en fullständig redogörelse för detta drama, som saknar motsvarighet inom demokratisk stats akademi, se här, härhär och här). Resultatet av censuringripandet blev att kandidatprogrammet separerades från den grupp av lärare som initierat programmet med avsikt att driva detsamma och för detta fått KTHs stöd. Sålunda startade STVD HT12 på en grund av gamla kurser i numerisk analys under ledning av en annan grupp lärare i numerisk analys, detta utan marknadsföring och resultatet blev därefter: Noll intresse, noll söktryck, noll intagningsbetyg, noll aktualitet = noll resultat. KTH insåg efter två år att det var totalt meningslöst att driva ett sådant program och HT14 fattade så Leif Kari, skolchef på skolan för Teknikvetenskap och huvudansvarig för censureringen av MST, det helt följdriktiga beslutet att lägga ner STVD (eller med omskrivning låta det vara "vilande") enligt denna offentliga handling (som registrator vänligt nog grävt fram då varken Leif Kari eller någon annan inblandad velat svara på mina upprepade frågor om status för STVD). Man kan i denna skakande rapport läsa: • väldigt lågt söktryck • stora svårigheter för studenterna att klara studierna • förkunskaper alldeles för svaga • mindre an 20% klarar uppflyttningskraven. Så har då KTH lyckats med sitt uppsåt att stoppa ett alltför lovande initiativ från en alltför internationellt stark gruppering på KTH, under uppvisande av komplett inkompetens på alla nivåer. KTH har således genom censur förstört ett potentiellt högt värde och ersatt det med noll. Bra jobbat enligt KTHs rektor Peter Gudmundson, som aktivt deltog i bokbränningen 2010; när böcker bränns återstår bara aska. Ironiskt nog har Leif Kari och Skolan för Teknikvetenskap dock inte låtit sig nedslås av detta dystra resultat utan arbetar nu aktivt för att uppgradera det havererade kandidatprogrammet i Simuleringsteknik till ett nytt civilingenjörsprogram i Teknisk Matematik enligt detta tilläggsbeslut. Fakultetsrådet har naturligtvis inte tillstyrkt inrättandet av detta program (se här 13d), då logiken saknas: Om KTH inte är kapabelt att driva en kandidatutbildning inom teknisk matematik/simuleringsteknik, är KTH (som landets främsta tekniska högskola) än mindre kapabelt att driva ett civilingenjörsprogram med samma inriktning. PS Så här beskrevs programmet av KTH när det startade 2012: • Simuleringsteknik och virtuell design är ett nytt program på KTH som utvecklats för att möta det ökande behovet av datorsimulering. • Utbildningen ger dig karriärmöjligheter inom många branscher, från verkstads- och processindustri, miljö- och energisektorn, via dataspel och animering, medicin och bioteknik till finansbranschen. • Du kan exempelvis jobba som beräkningskonsult, expert på visualisering och informationsgrafik eller som programdesigner. • Det nya kandidatprogrammet bygger på en mycket stark forsknings- och utbildningsmiljö inom detta område på KTH och är unikt i Sverige. Ja, det är sannerligen unikt med sådan missskötsel, trots (eller kanske på grund av) KTHs priviligierade position.
c74748e7eeff10ec	Tuesday, April 29, 2014 FQXi essay contest 2014: How Should Humanity Steer the Future? This year’s essay contest of the Foundational Questions Institute “How Should Humanity Steer the Future?” broaches a question that is fundamental indeed, fundamental not for quantum gravity but for the future of mankind. I suspect the topic selection has been influenced by the contest being “presented in partnership with” (which I translate into “sponsored by”) not only the John Templeton foundation and Scientific American, but also a philanthropic organization called the “Gruber Foundation” (which I had never heard of before) and Jaan Tallinn. Tallinn is no unknown, he is one of the developers of Skype and when I type his name into Google the auto completion is “net worth”. I met him at the 2011 FQXi conference where he gave a little speech about his worries that artificial intelligence will turn into a threat to humans. I wrote back then a blogpost explaining that I don’t share this particular worry. However, I recall Tallinn’s speech vividly, not because it was so well delivered (in fact, he seemed to be reading off his phone), but because he was so very sincere about it. Most people’s standard reaction in the face of threats to the future of mankind is cynicism or sarcasm, essentially a vocal shoulder shrug, whereas Tallinn seems to have spent quite some time thinking about this. And well, somebody really should be thinking about this... And so I appreciate the topic of this year’s essay contest has a social dimension, not only because it gets tiresome to always circle the same question of where the next breakthrough in theoretical physics will be and the always same answers (let me guess, it’s what you work on), but also because it gives me an outlet for my interests besides quantum gravity. I have always been fascinated by the complex dynamics of systems that are driven by the individual actions of many humans because this reaches out to the larger question of where life on planet Earth is going and why and what all of this is good for. If somebody asks you how humanity should steer the future, a modest reply isn’t really an option, so I have submitted my five step plan to save the world. Well, at least you can’t blame me for not having a vision. The executive summary is that we will only be able to steer at all if we have a way to collectively react to large scale behavior and long-term trends of global systems, and this can only happen if we are able to make informed decisions intuitively, quickly and without much thinking. A steering wheel like this might not be sufficient to avoid running into obstacles, but it is definitely necessary, so that is what we have to start with. The trends that we need to react to are those of global and multi-leveled systems, including economic, social, ecological and politic systems, as well as various infrastructure networks. Presently, we basically fail to act when problems appear. While the problems arise from the interaction of many people and their environment, it is still the individual that has to make decisions. But the individual presently cannot tell how their own action works towards their goals on long distance or time scales. To enable them to make good decisions, the information about the whole system has to be routed back to the individual. But that feedback loop doesn’t presently exist. In principle it would be possible today, but the process is presently far too difficult. The vast majority of people do not have the time and energy to collect the necessary information and make decisions based on it. It doesn’t help to write essays about what we ‘should’ do. People will only act if it’s really simple to do and of immediate relevance for them. Thus my suggestion is to create individual ‘priority maps’ that chart personal values and provide people with intuitive feedback for how well a decision matches with their priorities. A simple example. Suppose you train some software to tell what kind of images you find aesthetically pleasing and what you dislike. You now have various parameters, say colors, shapes, symmetries, composition and so on. You then fill out a questionnaire about preferences for political values. Now rather than long explanations which candidate says what, you get an image that represents how good the match is by converting the match in political values to parameters in an image. You pick the image you like best and are done. The point is that you are being spared having to look into the information yourself, you only get to see the summary that encodes whether voting for that person would work towards what you regard important. Oh, I hear you say, but that vastly oversimplifies matters. Indeed, that is exactly the point. Oversimplification is the only way we’ll manage to overcome our present inability to act. If mankind is to be successful in the long run, we have to evolve to anticipate and react to interrelated global trends in systems of billions of people. Natural selection might do this, but it would take too much time. The priority maps are a technological shortcut to emulate an advanced species that is ‘fit’ in the Darwinian sense, fit to adapt to its changing environment. I envision this to become a brain extension one day. I had a runner up to this essay contribution, which was an argument that research in quantum gravity will be relevant for quantum computing, interstellar travel and technological progress in general. But it would have been a quite impractical speculation (not to mention a self-advertisement of my work on superdeterminism, superluminal information exchange and antigravity). In my mind of course it’s all related – the laws of physics are what eventually drive the evolution of consciousness and also of our species. But I decided to stick with a proposal that I think is indeed realizable today and that would go a long way to enable humanity to steer the future. I encourage you to check out the essays which cover a large variety of ideas. Some of the contributions seem to be very bent towards the aim of making a philosophical case for some understanding of natural law rather than the other, or to find parallels to unsolved problems in physics, but this seems quite a stretch to me. However, I am sure you will find something of interest there. At the very least it will give you some new things to worry about... Saturday, April 26, 2014 Academia isn’t what I expected The Ivory Tower from The Neverending Story. [Source] Talking to the students at the Sussex school let me realize how straight-forward it is today to get a realistic impression of what research in this field looks like. Blogs are a good source of information about scientist’s daily life and duties, and it has also become so much easier to find and make contact with people in the field, either using social networks or joining dedicated mentoring programs. Before I myself got an office at a physics institute I only had a vague idea of what people did there. Absent the lauded ‘role models’ my mental image of academic research formed mostly by reading biographies of the heroes of General Relativity and Quantum Mechanics, plus a stack of popular science books. The latter didn’t contain much about the average researcher’s daily tasks, and to the extent that the former captured university life, it was life in the first half of the 20nd century. I expected some things to have changed during 50 years, notably in technological advances and the ease of travel, publishing, and communication. I finished high school in ’95, so the biggest changes were yet to come. I also knew that disciplines had drifted apart, that philosophy and physics were mostly going separate ways now, and that the days in which a physicist could also be a chemist could also be an artist were long gone. It was clear that academia had generally grown, become more organized and institutionalized, and closer linked to industrial research and applications. I had heard that applying for money was a big part of the game. Those were the days. But my expectations were wrong in many other ways. 20 years, 9 moves and 6 jobs later, here’s the contrast of what I believed theoretical physics would be like to reality: 1. Specialization While I knew that interdisciplinarity had given in to specialization I thought that theoretical physicists would be in close connection to the experimentalists, that they would frequently discuss experiments that might be interesting to develop, or data that required explanation. I also expected theoretical physicists to work closely together with mathematicians, because in the history of physics the mathematics has often been developed alongside the physics. In both cases the reality is an almost complete disconnect. The exchange takes place mostly through published literature or especially dedicated meetings or initiatives. 2. Disconnect I expected a much larger general intellectual curiosity and social responsibility in academia. Instead I found that most researchers are very focused on their own work and nothing but their own work. Not only do institutes rarely if ever have organized public engagement or events that are not closely related to the local research, it’s also that most individual researchers are not interested. In most cases, they plainly don’t have the time to think about anything than their next paper. That disconnect is the root of complaints like Nicholas Kristof’s recent Op-Ed, where calls upon academics: “[P]rofessors, don’t cloister yourselves like medieval monks — we need you!” 3. The Machinery My biggest reality shock was how much of research has turned into manufacturing, into the production of PhDs and papers, papers that are necessary for the next grant, which is necessary to pay the next students, who will write the next papers, iterate. This unromantic hamster wheel still shocks me. It has its good side too though: The standardization of research procedures limits the risks of the individual. If you know how to play along, and are willing to, you have good chances that you can stay. The disadvantage is though that this can force students and postdocs to work on topics they are not actually interested in, and that turns off many bright and creative people. 4. Nonlocality I did not anticipate just how frequent travel and moves are necessary these days. If I had known about this in advance, I think I would have left academia after my diploma. But so I just slipped into it. Luckily I had a very patient boyfriend who turned husband who turned father of my children. 5. The 2nd family The specialization, the single-mindedness, the pressure and, most of all, the loss of friends due to frequent moves create close ties among those who are together in the same boat. It’s a mutual understanding, the nod of been-there-done-that, the sympathy with your own problems that make your colleagues and officemates, driftwood as they often are, a second family. In all these years I have felt welcome at every single institute that I have visited. The books hadn’t told me about this. Experience, as they say, is what you get when you were expecting something else. By and large, I enjoy my job. Most of the time anyway. My lectures at the Sussex school went well, except that the combination of a recent cold and several hours of speaking stressed my voice box to the point of total failure. Yesterday I could only whisper. Today I get out some freak sounds below C2 but that’s pretty much it. It would be funny if it wasn’t so painful. You can find the slides of my lectures here and the guide to further reading here. I hope they live up to your expectations :) Monday, April 21, 2014 Away note I will be traveling the rest of the week to give a lecture at the Sussex graduate school "From Classical to Quantum GR", so not much will happen on this blog. For the school, we were asked for discussion topics related to our lectures, below are my suggestions. Leave your thoughts in the comments, additional suggestions for topics are also welcome. • Is it socially responsible to spend money on quantum gravity research? Don't we have better things to do? How could mankind possibly benefit from quantum gravity? • Can we make any progress on the theory of quantum gravity without connection to experiment? Should we think at all about theories of quantum gravity that do not produce testable predictions? How much time do we grant researchers to come up with predictions? • What is your favorite approach towards quantum gravity? Why? Should you have a favorite approach at all? • Is our problem maybe not with the quantization of gravity but with the foundations of quantum mechanics and the process of quantization? • How plausible is it that gravity remains classical while all the other forces are quantized? Could gravity be neither classical nor quantized? • How convinced are you that the Planck length is at 10-33cm? Do you think it is plausible that it is lower? Should we continue looking for it? • What do you think is the most promising area to look for quantum gravitational effects and why? • Do you think that gravity can be successfully quantized without paying attention to unification? Lara and Gloria say hello and wish you a happy Easter :o) Thursday, April 17, 2014 The Problem of Now [Image Source] Einstein’s greatest blunder wasn’t the cosmological constant, and neither was it his conviction that god doesn’t throw dice. No, his greatest blunder was to speak to a philosopher named Carnap about the Now, with a capital. “The problem of Now”, Carnap wrote in 1963, “worried Einstein seriously. He explained that the experience of the Now means something special for men, something different from the past and the future, but that this important difference does not and cannot occur within physics” I call it Einstein’s greatest blunder because, unlike the cosmological constant and indeterminism, philosophers, and some physicists too, are still confused about this alleged “Problem of Now”. The problem is often presented like this. Most of us experience a present moment, which is a special moment in time, unlike the past and unlike the future. If you write down the equations governing the motion of some particle through space, then this particle is described, mathematically, by a function. In the simplest case this is a curve in space-time, meaning the function is a map from the real numbers to a four-dimensional manifold. The particle changes its location with time. But regardless of whether you use an external definition of time (some coordinate system) or an internal definition (such as the length of the curve), every single instant on that curve is just some point in space-time. Which one, then, is “now”? You could argue rightfully that as long as there’s just one particle moving on a straight line, nothing is happening, and so it’s not very surprising that no notion of change appears in the mathematical description. If the particle would scatter on some other particle, or take a sudden turn, then these instances can be identified as events in space-time. Alas, that still doesn’t tell you whether they happen to the particle “now” or at some other time. Now what? The cause for this problem is often assigned to the timeless-ness of mathematics itself. Mathematics deals in its core with truth values and the very point of using math to describe nature is that these truths do not change. Lee Smolin has written a whole book about the problem with the timeless math, you can read my review here. It may or may not be that mathematics is able to describe all of our reality, but to solve the problem of now, excuse the heresy, you do not need to abandon a mathematical description of physical law. All you have to do is realize that the human experience of now is subjective. It can perfectly well be described by math, it’s just that humans are not elementary particles. The decisive ability that allows us to experience the present moment as being unlike other moments is that we have a memory. We have a memory of events in the past, an imperfect one, and we do not have memory of events in the future. Memory is not in and by itself tied to consciousness, it is tied to the increase of entropy, or the arrow of time if you wish. Many materials show memory; every system with a path dependence like eg hysteresis does. If you get a perm the molecule chains in your hair remember the bonds, not your brain. Memory has nothing to do with consciousness in particular which is good because it makes it much easier to find the flaw in the argument leading to the problem of now. If we want to describe systems with memory we need at the very least two time parameters: t to parameterize the location of the particle and τ to parameterize the strength of memory of other times depending on its present location. This means there is a function f(t,τ) that encodes how strong is the memory of time τ at moment t. You need, in other words, at the very least a two-point function, a plain particle trajectory will not do. That we experience a “now” means that the strength of memory peaks when both time parameters are identical, ie t-τ = 0. That we do not have any memory of the future means that the function vanishes when τ > t. For the past it must decay somehow, but the details don’t matter. This construction is already sufficient to explain why we have the subjective experience of the present moment being special. And it wasn’t that difficult, was it? The origin of the problem is not in the mathematics, but in the failure to distinguish subjective experience of physical existence from objective truth. Einstein spoke about “the experience of the Now [that] means something special for men”. Yes, it means something special for men. This does not mean however, and does not necessitate, that there is a present moment which is objectively special in the mathematical description. In the above construction all moments are special in the same way, but in every moment that very moment is perceived as special. This is perfectly compatible with both our experience and the block universe of general relativity. So Einstein should not have worried. I have a more detailed explanation of this argument – including a cartoon! – in a post from 2008. I was reminded of this now because Mermin had a comment in the recent issue of Nature magazine about the problem of now. In his piece, Mermin elaborates on qbism, a subjective interpretation of quantum mechanics. I was destined to dislike this just because it’s a waste of time and paper to write about non-existent problems. Amazingly however, Mermin uses the subjectiveness of qbism to arrive at the right conclusion, namely that the problem of the now does not exist because our experiences are by its very nature subjective. However, he fails to point out that you don’t need to buy into fancy interpretations of quantum mechanics for this. All you have to do is watch your hair recall sulphur bonds. The summary, please forgive me, is that Einstein was wrong and Mermin is right, but for the wrong reaons. It is possible to describe the human experience of the present moment with the “timeless” mathematics that we presently use for physical laws, it isn’t even difficult and you don’t have to give up the standard interpretation of quantum mechanics for this. There is no problem of Now and there is no problem with Tegmark’s mathematical universe either. And Lee Smolin, well, he is neither wrong nor right, he just has a shaky motivation for his cosmological philosophy. It is correct, as he argues, that mathematics doesn’t objectively describe a present moment. However, it’s a non sequitur that the current approach to physics has reached its limits because this timeless math doesn’t constitute a conflict with our experience. observation. Most people get a general feeling of uneasiness when they first realize that the block universe implies all the past and all the future is equally real as the present moment, that even though we experience the present moment as special, it is only subjectively so. But if you can combat your uneasiness for long enough, you might come to see the beauty in eternal mathematical truths that transcend the passage of time. We always have been, and always will be, children of the universe. Saturday, April 12, 2014 Book review: “The Theoretical Minimum – Quantum Mechanics” By Susskind and Friedman Quantum Mechanics: The Theoretical Minimum What You Need to Know to Start Doing Physics By Leonard Susskind, Art Friedman Basic Books (February 25, 2014) This book is the second volume in a series that we can expect to be continued. The first part covered Classical Mechanics. You can read my review here. The volume on quantum mechanics seems to have come into being much like the first, Leonard Susskind teamed up with Art Friedman, a data consultant whose role I envision being to say “Wait, wait, wait” whenever the professor’s pace gets too fast. The result is an introduction to quantum mechanics like I haven’t seen before. The ‘Theoretical Minimum’ focuses, as its name promises, on the absolute minimum and aims at being accessible with no previous knowledge other than the first volume. The necessary math is provided along the way in separate interludes that can be skipped. The book begins with explaining state vectors and operators, the bra-ket notation, then moves on to measurements, entanglement and time-evolution. It uses the concrete example of spin-states and works its way up to Bell’s theorem, which however isn’t explicitly derived, just captured verbally. However, everybody who has made it through Susskind’s book should be able to then understand Bell’s theorem. It is only in the last chapters that the general wave-function for particles and the Schrödinger equation make an appearance. The uncertainty principle is derived and path integrals are very briefly introduced. The book ends with a discussion of the harmonic oscillator, clearly building up towards quantum field theory there. I find the approach to quantum mechanics in this book valuable for several reasons. First, it gives a prominent role to entanglement and density matrices, pure and mixed states, Alice and Bob and traces over subspaces. The book thus provides you with the ‘minimal’ equipment you need to understand what all the fuzz with quantum optics, quantum computing, and black hole evaporation is about. Second, it doesn’t dismiss philosophical questions about the interpretation of quantum mechanics but also doesn’t give these very prominent space. They are acknowledged, but then it gets back to the physics. Third, the book is very careful in pointing out common misunderstandings or alternative notations, thus preventing much potential confusion. The decision to go from classical mechanics straight to quantum mechanics has its disadvantages though. Normally the student encounters Electrodynamics and Special Relativity in between, but if you want to read Susskind’s lectures as self-contained introductions, the author now doesn’t have much to work with. This time-ordering problem means that every once in a while a reference to Electrodynamics or Special Relativity is bound to confuse the reader who really doesn’t know anything besides this lecture series. It also must be said that the book, due to its emphasis on minimalism, will strike some readers as entirely disconnected from history and experiment. Not even the double-slit, the ultraviolet catastrophe, the hydrogen atom or the photoelectric effect made it into the book. This might not be for everybody. Again however, if you’ve made it through the book you are then in a good position to read up on these topics elsewhere. My only real complaint is that Ehrenfest’s name doesn’t appear together with his theorem. The book isn’t written like your typical textbook. It has fairly long passages that offer a lot of explanation around the equations, and the chapters are introduced with brief dialogues between fictitious characters. I don’t find these dialogues particularly witty, but at least the humor isn’t as nauseating as that in Goldberg’s book. All together, the “Theoretical Minimum” achieves what it promises. If you want to make the step from popular science literature to textbooks and the general scientific literature, then this book series is a must-read. If you can’t make your way through abstract mathematical discussions and prefer a close connection to example and history, you might however find it hard to get through this book. I am certainly looking forward to the next volume. (Disclaimer: Free review copy.) Monday, April 07, 2014 Will the social sciences ever become hard sciences? The term “hard science” as opposed to “soft science” has no clear definition. But roughly speaking, the less the predictive power and the smaller the statistical significance, the softer the science. Physics, without doubt, is the hard core of the sciences, followed by the other natural sciences and the life sciences. The higher the complexity of the systems a research area is dealing with, the softer it tends to be. The social sciences are at the soft end of the spectrum. To me the very purpose of research is making science increasingly harder. If you don’t want to improve on predictive power, what’s the point of science to begin with? The social sciences are soft mainly because data that quantifies the behavior of social, political, and economic systems is hard to come by: it’s huge amounts, difficult to obtain and even more difficult to handle. Historically, these research areas therefore worked with narratives relating plausible causal relations. Needless to say, as computing power skyrockets, increasingly larger data sets can be handled. So the social sciences are finally on the track to become useful. Or so you’d think if you’re a physicist. But interestingly, there is a large opposition to this trend of hardening the social sciences, and this opposition is particularly pronounced towards physicists who take their knowledge to work on data about social systems. You can see this opposition in the comment section to every popular science article on the topic. “Social engineering!” they will yell accusingly. It isn’t so surprising that social scientists themselves are unhappy because the boat of inadequate skills is sinking in the data sea and physics envy won’t keep it afloat. More interesting than the paddling social scientists is the public opposition to the idea that the behavior of social systems can be modeled, understood, and predicted. This opposition is an echo of the desperate belief in free will that ignores all evidence to the contrary. The desperation in both cases is based on unfounded fears, but unfortunately it results in a forward defense. And so the world is full with people who argue that they must have free will because they believe they have free will, the ultimate confirmation bias. And when it comes to social systems they’ll snort at the physicists “People are not elementary particles”. That worries me, worries me more than their clinging to the belief in free will, because the only way we can solve the problems that mankind faces today – the global problems in highly connected and multi-layered political, social, economic and ecological networks – is to better understand and learn how to improve the systems that govern our lives. That people are not elementary particles is not a particularly deep insight, but it collects several valid points of criticism: 1. People are too difficult. You can’t predict them. Humans are made of a many elementary particles and even though you don’t have to know the exact motion of every single one of these particles, a person still has an awful lot of degrees of freedom and needs to be described by a lot of parameters. That’s a complicated way of saying people can do more things than electrons, and it isn’t always clear exactly why they do what they do. That is correct of course, but this objection fails to take into account that not all possible courses of action are always relevant. If it was true that people have too many possible ways to act to gather any useful knowledge about their behavior our world would be entirely dysfunctional. Our societies work only because people are to a large degree predictable. If you go shopping you expect certain behaviors of other people. You expect them to be dressed, you expect them to walk forwards, you expect them to read labels and put things into a cart. There, I’ve made a prediction about human behavior! Yawn, you say, I could have told you that. Sure you could, because making predictions about other people’s behavior is pretty much what we do all day. Modeling social systems is just a scientific version of this. This objection that people are just too complicated is also weak because, as a matter of fact, humans can and have been modeled with quite simple systems. This is particularly effective in situations when intuitive reaction trumps conscious deliberation. Existing examples are traffic flows or the density of crowds when they have to pass through narrow passages. So, yes, people are difficult and they can do strange things, more things than any model can presently capture. But modeling a system is always an oversimplification. The only way to find out whether that simplification works is to actually test it with data. 2. People have free will. You cannot predict what they will do. To begin with it is highly questionable that people have free will. But leaving this aside for a moment, this objection confuses the predictability of individual behavior with the statistical trend of large numbers of people. Maybe you don’t feel like going to work tomorrow, but most people will go. Maybe you like to take walks in the pouring rain, but most people don’t. The existence of free will is in no conflict with discovering correlations between certain types of behavior or preferences in groups. It’s the same difference that doesn’t allow you to tell when your children will speak the first word or make the first step, but that almost certainly by the age of three they’ll have mastered it. 3. People can understand the models and this knowledge makes predictions useless. This objection always stuns me. If that was true, why then isn’t obesity cured by telling people it will remain a problem? Why are the highways still clogged at 5pm if I predict they will be clogged? Why will people drink more beer if it’s free even though they know it’s free to make them drink more? Because the fact that a prediction exists in most cases doesn’t constitute any good reason to change behavior. I can predict that you will almost certainly still be alive when you finish reading this blogpost because I know this prediction is exceedingly unlikely to make you want to prove it wrong. Yes, there are cases when people’s knowledge of a prediction changes their behavior – self-fulfilling prophecies are the best-known examples of this. But this is the exception rather than the rule. In an earlier blogpost, I referred to this as societal fixed points. These are configurations in which the backreaction of the model into the system does not change the prediction. The simplest example is a model whose predictions few people know or care about. 4. Effects don’t scale and don’t transfer. This objection is the most subtle one. It posits that the social sciences aren’t really sciences until you can do and reproduce the outcome of “experiments”, which may be designed or naturally occurring. The typical social experiment that lends itself to analysis will be in relatively small and well-controlled communities (say, testing the implementation of a new policy). But then you have to extrapolate from this how the results will be in larger and potentially very different communities. Increasing the size of the system might bring in entirely new effects that you didn’t even know of (doesn’t scale), and there are a lot of cultural variables that your experimental outcome might have depended on that you didn’t know of and thus cannot adjust for (doesn’t transfer). As a consequence, repeating the experiment elsewhere will not reproduce the outcome. Indeed, this is likely to happen and I think it is the major challenge in this type of research. For complex relations it will take a long time to identify the relevant environmental parameters and to learn how to account for their variation. The more parameters there are and the more relevant they are, the less the predictive value of a model will be. If there are too many parameters that have to be accounted for it basically means doing experiments is the only thing we can ever do. It seems plausible to me, even likely, that there are types of social behavior that fall into this category, and that will leave us with questions that we just cannot answer. However, whether or not a certain trend can or cannot be modeled we will only know by trying. We know that there are cases where it can be done. Geoffry West’s city theory I find a beautiful example where quite simple laws can be found in the midst of all these cultural and contextual differences. In summary. The social sciences will never be as “hard” as the natural sciences because there is much more variation among people than among particles and among cities than among molecules. But the social sciences have become harder already and there is no reason why this trend shouldn’t continue. I certainly hope it will continue because we need this knowledge to collectively solve the problems we have collectively created. Tuesday, April 01, 2014 Do we live in a hologram? Really?? Physicists fly high on the idea that our three-dimensional world is actually two-dimensional, that we live in a hologram, and that we’re all projections on the boundary of space. Or something like this you’ve probably read somewhere. It’s been all over the pop science news ever since string theorists sang the Maldacena. Two weeks ago Scientific American produced this “Instant Egghead” video which is a condensed mashup of all the articles I’ve endured on the topic: The second most confusing thing about this video is the hook “Many physicist now believe that reality is not, in fact, 3-dimensional.” To begin with, physicists haven’t believed this since Minkowski doomed space and time to “fade away into mere shadows”. Moyer in his video apparently refers only to space when he says “reality.” That’s forgiveable. I am more disturbed by the word “reality” that always creeps up in this context. Last year I was at a workshop that mixed physicists with philosophers. Inevitably, upon mentioning the gauge-gravity duality, some philosopher would ask, well, how many dimensions then do we really live in? Really? I have some explanations for you about what this really means. Q: Do we really live in a hologram? A: What is “real” anyway? Q: Having a bad day, yes? A: Yes. How am I supposed to answer a question when I don’t know what it means? Q: Let me be more precise then. Do we live in a hologram as really as, say, we live on planet Earth? A: Thank you, much better. The holographic principle is a conjecture. It has zero experimental evidence. String theorists believe in it because their theory supports a specific version of holography, and in some interpretations black hole thermodynamics hints at it too. Be that as it may, we don’t know whether it is the correct description of nature. Q: So if the holographic principle was the correct description of nature, would we live in a hologram as really as we live on planet Earth? A: The holographic principle is a mathematical statement about the theories that describe nature. There’s a several thousand years long debate about whether or not math is as real as that apple tree in your back yard. This isn’t a question about holography in particular, you could also ask that question also in general relativity: Do we really live in a metric manifold of dimension four and Lorentzian signature? Q: Well, do we? A: On most days I think of the math of our theories as machinery that allows us to describe nature but is not itself nature. On the remaining days I’m not sure what reality is and have a lot of sympathy for Platonism. Make your pick. Q: So if the holographic principle was true, would we live in a hologram as really as we previously thought we live in the space-time of Einstein’s theory of General Relativity? A: A hologram is an image on a 2-dimensional surface that allows one to reconstruct a 3-dimensional image. One shouldn’t take the nomenclature “holographic principle” too seriously. To begin with actual holograms are never 2-dimensional in the mathematical sense; they have a finite width. After all they’re made of atoms and stuff. They also do not perfectly recreate the 3-dimensional image because they have a resolution limit which comes from the wavelength of the light used to take (and reconstruct) the image. A hologram is basically a Fourier transformation. If that doesn’t tell you anything, suffices to say this isn’t the same mathematics as that behind the holographic principle. Q: I keep hearing that the holographic principle says the information of a volume can be encoded on the boundary. What’s the big deal with that? If I get a parcel with a customs declaration, information about the volume is also encoded on the boundary. A: That statement about the encoding of information is sloppy wording. You have to take into account the resolution that you want to achieve. You are right of course in that there’s no problem in writing down the information about some volume and printing it on some surface (or a string for that matter). The point is that the larger the volume the smaller you’ll have to print. Here’s an example. Take a square made out of N2 smaller squares and think of each of them as one bit. They’re either black or white. There are 2N2 different patterns of black and white. In analogy, the square is a box full of matter in our universe and the colors are information about the particles in the inside. Now you want to encode the information about the pattern of that square on the boundary using pieces of the same length as the sidelength of the smaller squares. See image below for N=3. On the left is the division of the square and the boundary, on the right is one way these could encode information. There’s 4N of these boundary pieces and 24N different patterns for them. If N is larger than 4, there are more ways the square can be colored than you have different patterns for the boundary. This means you cannot uniquely encode the information about the volume on the boundary. The holographic principle says that this isn’t so. It says yes, you can always encode the volume on the boundary. Now this means, basically, that some of the patterns for the squares can’t happen. Q: That’s pretty disturbing. Does this mean I can’t pack a parcel in as many ways as I want to? A: In principle, yes. In practice the things we deal with, even the smallest ones we can presently handle in laboratories, are still far above the resolution limit. They are very large chunks compared to the little squares I have drawn above. There is thus no problem encoding all that we can do to them on the boundary. Q: What then is the typical size of these pieces? A: They’re thought to be at the Planck scale, that’s about 10-33 cm. You should not however take the example with the box too seriously. That is just an illustration to explain the scaling of the number of different configurations with the system size. The theory on the surface looks entirely different than the theory in the volume. Q: Can you reach this resolution limit with an actual hologram? A: No you can’t. If you’d use photons with a sufficiently high energy, you’d just blast away the sample of whatever image you wanted to take. However, if you loosely interpret the result of such a high energy blast as a hologram, albeit one that’s very difficult to reconstruct, you would eventually notice these limitations and be able to test the underlying theory. Q: Let me come back to my question then, do we live in the volume or on the boundary? A: Well, the holographic principle is quite a vague idea. It has a concrete realization in the gauge-gravity correspondence that was discovered in string theory. In this case one knows very well how the volume is related to the boundary and has theories that describe each. These both descriptions are identical. They are said to be “dual” and both equally “real” if you wish. They are just different ways of describing the same thing. In fact, depending on what system you describe, we are living on the boundary of a higher-dimensional space rather than in a volume with a lower dimensional surface. Q: If they’re the same why then do we think we live in 3 dimensions and not in 2? Or 4? A: Depends on what you mean with dimension. One way to measure the dimensionality is, roughly speaking, to count the number of ways a particle can get lost if it moves randomly away from a point. The result then depends on what particle you use for the measurement. The particles we deal with will move in 3 dimensions, at least on the distance scales that we typically measure. That’s why we think, feel, and move like we live in 3 dimensions, and nothing wrong with that. The type of particles (or fields) you would have in the dual theories do not correspond to the ones we are used to. And if you ask a string theorist, we live in 11 dimensions one way or the other. Q: I can see then why it is confusing to vaguely ask what dimension “reality” has. But what is the most confusing thing about Moyer’s video? A: The reflection on his glasses. Q: Still having a bad day? A: It’s this time of the month. Q: Okay, then let me summarize what I think I learned here. The holographic principle is an unproved conjecture supported by string theory and black hole physics. It has a concrete theoretical formalization in the gauge-gravity correspondence. There, it identifies a theory in a volume with a theory on the boundary of that volume in a mathematically rigorous way. These theories are both equally real. How “real” that is depends on how real you believe math to be to begin with. It is only surprising that information can always be encoded on the boundary of a volum if you request to maintain the resolution, but then it is quite a mindboggling idea indeed. If one defines the number of dimensions in a suitable way that matches our intuition, we live in 3 spatial dimensions as we always thought we do, though experimental tests in extreme regimes may one day reveal that fundamentally our theories can be rewritten to spaces with different numbers of dimensions. Did I get that right? A: You’re so awesomely attentive. Q: Any plans on getting a dog? A: No, I have interesting conversations with my plants.
8d71a8fda66796b2	Representations of Continuous and Point Groups • Laurent-Patrick Lévy This appendix gives only the minimal requirements for using representations of symmetry groups in solid state physics. Many text books deal with this subject in greater detail and some are given in the Bibliography. Physical systems are often invariant under certain symmetry operations forming a group G. For example, electrons in an atom are subject to a central potential which is invariant under the group of rotations. Likewise, electrons in a solid are in a periodic potential which has translational invariance, but is also invariant under the point group of the lattice. There are 32 point symmetry groups in three dimensions, compatible with translational invariance of a crystal lattice. Wave functions corresponding to stationary solutions of the Schrödinger equation form a vector space, or state space. Hexagonal Mirror Symmetry Unable to display preview. Download preview PDF. Unable to display preview. Download preview PDF. 1. 572. Hammermesh, M. (1964) Group Theory and its Application to Physical Problems. Addison-WesleyGoogle Scholar 2. 573. Serre, J.P. (1967) Representation lineaire des groupes finis. Hermann, ParisGoogle Scholar 3. 574. Wigner, E.P. (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic PressMATHGoogle Scholar 4. 575. Pikus, G.E., Bir, G.L. (1974) Symmetry and Strain Effects in Semiconductors. Wiley, New YorkGoogle Scholar 5. 576. Schläfer, H.L. (1967) Einfürung in die Ligandenfeldtheorie. Akademische Verlaganstalt, FrankfurtGoogle Scholar Copyright information © Springer-Verlag Berlin Heidelberg 2000 Authors and Affiliations • Laurent-Patrick Lévy • 1 1. 1.MPI für Festkörperforschung, Laboratoire des Champs Magnétiques IntensesCNRSGrenoble Cedex 9France Personalised recommendations
12178a0b8eebd38c	Tuesday, November 30, 2010 Quotes of the day I'd rather get punched in the face 10 times than study for those [FINRA] tests again.--Wayne Chrebet [A player lockout next season is a] near certainty. The magnitude of the loss would be at the very least about $160 million to $170 million per team-city.--DeMaurice Smith The big elephant in the room is not Portugal but, of course, it’s Spain. There is not enough official money to bail out Spain if trouble occurs.--Nouriel Roubini Many people seem truly astonished by my decision not to comply with the Federal Bureau of Investigation’s request to wear a wiretap to record conversations with a client. I have even been asked, “Why not just agree to wear the wire to show that no wrongdoing had occurred?” Unfortunately, that requires assuming that I was asked nicely to cooperate. That was not the nature of the proposal I was offered. (And had I agreed, I have no doubt I would have been asked to record conversations with others as well.) A surprise visit by the F.B.I. to your home — especially when your wife and two young children are due to arrive from school at any moment — is a shocking, terrifying experience. It makes me wonder whether they deliberately chose this time of maximum vulnerability. F.B.I. agents are backed by the full force of the federal government, and the ones who arrived at my home made it abundantly clear that they believe I am guilty, and therefore so are all of my clients, and they threatened to arrest me on the spot. ... The S.E.C.’s proper role is to provide guidance and rules to the investment community on what is considered appropriate behavior. If the agency is doing its job, it will take note of untoward activities and issue a cease-and-desist order. But here, the rule-making appears to be decided by the Justice Department. My personal belief is that much of this activity is politically motivated, and will ultimately only delay the return of the confidence of Main Street and Wall Street in our country. Our economy won’t fully heal and return to solid growth so long as the political class maintains its vendetta against business interests in our country. This, along with my firm belief that my clients and I have done nothing wrong, is why I have chosen to take a stand. It’s about fighting for what I believe should be fair dealings between individual citizens and their government.--John Kinnucan How surfing is like investing I've never surfed, but I really liked this: * Know exactly how to exit before you enter. * Being in the right spot is more important than the size of the wave. * The waves you let go are more important than the wave you catch. * If you miss a wave be sure that another will come behind it. * Stay out of the water when wild amateurs are present. * Avoid crowded take offs. * Don't panic in a wipeout and don't fight the current. * Success is achieved by taking risk, not avoiding risk. * Push too hard and you get hurt - don't push hard enough and you can get hurt even worse. * Get in the water yourself to evaluate the current by instinct - the height and direction of the swell, the wind, the tide, the temperature, the coral reef bottoms, the peak, the trough, the time between sets and the position - data without experience will deaden your senses. * Know where the "impact zone" is and how to stay out of it. * Humility trumps arrogance. * You need to be in condition for the worst circumstance in the worst situation not the best circumstance of the best situation. * A successful outing is being able to surf another day. Photo link here. Monday, November 29, 2010 Quotes of the day ... the principle of noncontradiction has been high orthodoxy in Western philosophy since Aristotle mounted a spirited defense of it in his “Metaphysics” — so orthodox that no one seems to have felt the need to mount a sustained defense of it ever since. So the paradox must be of the second kind: there must be something wrong with the argument. Or must there? Not according to a contentious new theory that’s currently doing the rounds. According to this theory, some contradictions are actually true, and the conclusion of the Liar Paradox is a paradigm example of one such contradiction. The theory calls a true contradiction a dialetheia (Greek: “di” = two (way); “aletheia” = truth), and the view itself is called dialetheism. ... Revolutions in logic (of various kinds) have certainly occurred in the past. Arguably, the greatest of these was around the turn of the 20th century, when traditional Aristotelian logic was overthrown, and the mathematical techniques of contemporary logic were ushered in. Perhaps we are on the brink of another.--Graham Priest Predominantly Catholic countries in Europe and elsewhere, such as Ireland, Spain, or Mexico, had larger families than did predominantly Protestant countries, like Sweden or Norway, even after adjusting for the effects on fertility of differences among countries in their average incomes, education, importance of cities, and other variables. Again, the explanation given for this result was that Catholic families were more reluctant to use contraception to reduce the number of children they had. Demographers even used the situation in Ireland to define a class of behavior called “Irish family patterns”, which meant that men and women married late-in their late twenties and early thirties- and that after marriage women gave birth at frequent intervals because couples made little effort to control their births once married. These findings of a strong Catholic “effect” on fertility changed radically during the past 40 years or so. Studies for the United States now show that Catholic families have, if anything, fewer rather than a greater, number of children compared to Protestant families who have similar incomes and education. A similar reversal has occurred in international comparisons. Catholic countries like Spain, Italy, and Poland now have total fertility rates-the number of children born to the average woman over her lifetime- of only 1.4, 1.4, and 1.2, respectively, far below these rates in the predominantly Protestant countries of Northern Europe. Even “Irish” family patterns no longer hold in Ireland, where the typical woman has a little less than two children over her lifetime instead of four or five, even though she is not marrying any later than the typical Irish woman did in the past.--Gary Becker For the most part, however, what we see here [with the latest Wikileaks dispatches] is diplomats doing their proper job: finding out what is happening in the places to which they are posted, working to advance their nation's interests and their government's policies. In fact, my personal opinion of the state department has gone up several notches. In recent years, I have found the American foreign service to be somewhat underwhelming, reach-me-down, dandruffy, especially when compared with other, more confident arms of US government, such as the Pentagon and the treasury. But what we find here is often first rate.--Timothy Garton Ash Some experts, however, questioned WHO's calls for more health donations. "How do you make an impassioned plea for spending more money when we're wasting so much?" asked William Easterly, a foreign aid expert at New York University. He said much of the problem in developing countries is that while donors have spent billions on things like drugs, vaccines and malaria bednets, little has been spent on the health workers needed to distribute them. "Medicines and vaccines don't administer themselves," Easterly said. He also criticized U.N. agencies and major donors like the Bill & Melinda Gates Foundation, who have mostly avoided investing in health systems, preferring instead to build separate programs for illnesses like malaria, polio and AIDS. "That is like doing aerial bombing at 35,000 feet without knowing what you're hitting on the ground," Easterly said. "But investing in medicines for AIDS and malaria makes for much better publicity than investing in health systems."--Maria Cheng So here goes for what Aid Watch is sincerely thankful for: For the largest reduction in world poverty in human history, which has already happened in our generation. For the largest improvement in health and life expectancy in human history, which has already happened in our generation.--Aid Watch Most people operate in an environment of such low risk that both action and inaction have very few consequences. So although we may be attracted to the tiger on the terrace, we prefer to live with the kitty in the kitchen. The tiger on the terrace is operating in a high-risk environment so the consequences of his actions are significant. He is keeping everything stirred up and everyone around him on their toes. He is tireless, always in perpetual motion, relentless, obsessed and fills every room with energy the moment he enters. He has a twinkle in both eyes and all in his path are standing on their toes. While he is on the terrace no one is whining, complaining or sick. They are all running for their lives. That is entertaining for most people to watch, but not to live. The tiger is all things to excess and nothing to moderation. As a consequence, the masses choose to trade the tiger for a kitty in the kitchen and lower risk tolerance. The daily risk appetite of the tiger is just too exhausting even though the thrill quotient is significantly higher. Most people want mediocrity, security, safety and repetition. The kitchens are then filled with whining, sickness, complaints and boredom. That is why there are few tigers left in the world and millions of kitties. Leaders are tigers!--Tom Barrack Sunday, November 28, 2010 The shadow of greed, that is Just when does an artist decide to step away from a piece of work and say, in no uncertain terms, “It’s done?” The “Star Wars” faithful erupted in anger, and with good reason. A healthy chunk of the documentary details the outrage over the “Star Wars” scene involving Han Solo and Greedo, which underwent an ill-advised tweak, to showcase Lucas’ bald overreach. What isn’t addressed, but implied throughout the film, is that Lucas the artist is clearly in decline. When he decided it was time to create the prequel films his filmmaking skills had started to ebb, but his ego had only just begun to grow. Rather than hire a director to help the prequels, as he did with Episodes Five and Six, he took on the task himself. Lucas’ journey toward the Dark Side began when the “Star Wars” toy brigade started hitting the market. The franchise’s merchandise potential caught his eye, turning an iconoclastic filmmaker into a parody of the corporations he once mocked. And where are those small, personal films outside a galaxy far, far away he promised us in interview after interview? Lucas comes off as arrogant here, and blindingly hypocritical for massaging the original trilogy. Years earlier he had testified before Congress about the harm adding color to great black and white films would do to the cultural fabric. Yet here he was performing a similar act to his own trilogy with little regard for the consequences. Image link here. Wednesday, November 24, 2010 I thought Paul Krugman only contradicted himself depending on which party was dominant in Washington, D.C. or around 4 years. Apparently, he can contradict himself in 4 weeks. Quotes of the day It turns out that Bloomberg ran a headline on its terminals at 1:48 EST. The headline read: “Janus Says It Received Inquiry Calling For General Information.” That information wasn’t publicly available to anyone who does not have a Bloomberg terminal. It wasn’t posted on Janus’s website. In fact, it took several minutes before it appeared in any place the general public has access to. So is Bloomberg aiding insider trading? Of course not. It’s a journalistic enterprise whose right to report information is protected by the First Amendment. That’s true even if the information is ‘non-public’ and has obviously been leaked by an insider. And, although it’s less clearly spelled out in the constitution, the right to gate the information so that it is only available to subscribers is probably protected as well. The question is: what’s the difference between what Bloomberg did on Janus today and what the expert networks and consultants are doing?--John Carney Here is what [Stevie Cohen] is not: he is not Warren Buffett, arguably the best investor in history. Buffett is a fundamental investor, a man who takes deep, long-term views on the prospects of one company or another, and then tries to pick his entry point when he's got a large "margin of safety" as protection. It's not difficult to understand why Buffett's early investments in the likes of Coca-Cola (KO), Wells Fargo (WFC), and American Express (AXP) have paid off over the long haul. Cohen is also not Marc Lasry of Avenue Capital, an investor who specializes in so-called distressed investments. When you're a distressed investor, you not only have to have a well-thought opinion on the prospects of a company, you also need to know your way around the capital structure. Size and experience go a long way in the distressed realm, and the big can get bigger by virtue of that edge. When Lasry tries to take control of Donald Trump's bankrupt Atlantic City casinos, we can understand where he's coming from: the assumption that a serious human being could run Trump's casinos better than the Donald himself. Finally, Cohen is not James Simons of Renaissance Technologies, the king of the quantitative investors. While SAC Capital may use some trading algorithms in its day-to-day, Cohen's is not a black-box operation, with software designed and constantly tweaked by a squadron of PhDs like those that work for Simons. We may never learn what Simons' algorithms actually are, but we do accept that computers can outperform humans at quantitative tasks. Cohen, in short, is a trader. According to one knowledgeable investor, Cohen has called what SAC does "information arbitrage." He and his staff are simply whipping stocks around on a day-to-day basis, trying to get an information edge over the guy on the other side of the trade. If that sounds fundamentally useless, that's because it is.--Duff MacDonald Every damn day, if you tune in to any of the 24-hour news outlets, the same pundits retread through the same stuff--they all say the same thing. I spend a great deal of time trying to figure out how the whole DC opinion apparatus remains employed. If there were any justice, their ranks would swell the unemployment rate beyond 10 percent. And still, some moron known as a news executive who hasn't registered a thought beyond mediocre in my lifetime approves of this, and Americans, educated to believe 2+2=5, will put up with anything. Into this horror walks Sarah Palin, who is kind of a sexy librarian, kind of a MILF, kind of just crazy, and altogether does what she wants to do. This, actually, is normal behavior. But we are so used to watching other female politicians compromise in so many ways that there is not enough Vaseline in all of CVS to make the situation comfortable--so Sarah Palin seems completely strange. Unfortunately, Sarah Palin is not very bright, not very thoughtful and not very qualified to run a country. Or a state. But really, are any of the other idiots who want the job so much better?--Elizabeth Wuertzel You can either be an unhelpful and indecisive wimp, or you can be a frickin' idiot. There are no other options. I recommend the frickin' idiot path because it's more masculine.--Scott Adams ... Gore explains why he voted for ethanol subsidies despite believing that they are poor public policy. Note that Mr. Gore refers to his knowingly selling his principles for votes as a “mistake.” Unless he means that his effort didn’t pay off with the top job in the White House, his soul-selling was no “mistake”; he knew exactly what he was doing and why he was doing it. His soul-selling was an instance, pure and simply, of the hypocrisy and lying that is the stuff of too many political campaigns.--Don Boudreaux Palin Derangement Syndromers collectively breathe a sigh of relief Strangely, I do, too: It seems that the Tea Baggers (really, someone should tell those narrow-minded chumps the alternative meaning for that) have been voting in droves to perhaps soften the image of the gobbledegook nattering, gun wielding simpleton, Sarah Palin, who is probably going to be the next US president. However, there is room for hope. That’s because, despite the efforts of the slackjawed xenophobes, Bristol Palin didn’t win Dancing With The Stars. Jennifer Grey and Derek Hough did. While Mom is not my first (second, or third) choice for President in 2012 or 2016, I'm also not worried that she would do much worse than Obama, Bush, Kerry or Gore. She'd probably do better, without the Harvard/Yale elitist brainwash. Truman avoided this, and he was pretty good; then again, LBJ missed as well, and he was terrible. Photo link here. Chart here. Tuesday, November 23, 2010 Economists with trade settlement uncertainty Check out the comments. Quotes of the day Employer callbacks to attractive men are significantly higher than to men with no picture and to plain-looking men, nearly doubling the latter group. Strikingly, attractive women do not enjoy the same beauty premium.--Ben-Gurion University of the Negev, Department of Economics If your child is habitually rude or even violent, then changing behavior has to be your priority; it's a precondition for any deeper progress. You can't discuss your child's feelings while he's screaming or kicking you. And no matter how mellow your child is, you've got to tailor your explanations to his age.--Bryan Caplan The key to understanding why market economies have outperformed planned societies is not recognition of the ubiquity of greed, but understanding of the power of disciplined pluralism. The primary strength of the market economies – and certainly the strength of their economic performance – results from the way in which the market provides freedom to experiment and opportunity to imitate successful innovation, yet is ruthless in cutting off successful experiment. Planned societies, in contrast, have been typically slow to innovate and, when they have done, have been quietly slow to acknowledge when innovation has failed – as most innovations do.--John Kay The Fed isn’t really trying to create inflation.--Scott Sumner Here in the United States, one thing that strikes me about my most liberal friends is how conservative their thinking is at a personal level. For their own children, and in talking about specific other people, they passionately stress individual responsibility. It is only when discussing public policy that they favor collectivism. The tension between their personal views and their political opinions is fascinating to observe. I would not be surprised to find that my friends' attachment to liberal politics is tenuous, and that some major event could cause a rapid, widespread shift toward a more conservative position.--Arnold Kling The main danger to liberty here is not the TSA but rather a set of American attitudes which, at the same time, take our current "war" both far too seriously and also not nearly seriously enough. Overall, I'd like to see less posturing in these debates and more Thucydides.--Tyler Cowen "All of us?" Really, Mr. President?--David Henderson Unbeknownst to the 12 of us [jurors], we were playing out a script to the letter. Our inability to reach an accord, while not the stuff of Law & Order, was not a failure per se but actually set in motion a chain of events that allowed all parties to get the something if not good then at least reasonable for all involved from an otherwise terrible situation. All I could think as I walked to my car after being excused was this: from chaos comes order. This system that we look at and think that it’s in disrepair, that nobody can possibly fix it or in which you have “activist judges” on one side and uncaring, throw-the-book-at-them judges on the other side just isn’t a fair characterization. What you truly have is a proverbial sausage factory: it’s incredibly messy, nothing seems to make sense, nothing looks good or reasonable or even real, but at the end of the line there is something like justice. It doesn’t always look right. It doesn’t always feel right. It doesn’t even always taste right. But it’s at least palatable.--Tux Life The religion of peace further distinguishes itself.--Tony Woodlief Monday, November 22, 2010 From making Subway sandwiches to receiving the Medal of Honor Thank you Sergeant Giunta. The Sal Giunta Story from SebastianJunger/TimHetherington on Vimeo. Quotes of the day You’ve got to understand, guys just want to play the game. Guys don’t really want to practice. We play games like this, what, two in five days? Something like that. One day of practice, so you don’t have to worry about coach Belichick cursing us out all the time. That’ll be great.--Deion Branch Food prices are marching up again mainly because the world economy is recovering from the crisis.--Gary Becker Erskine B. Bowles and Alan K. Simpson, the chairmen of President Obama’s deficit reduction commission, have taken at hard look at these tax expenditures — and they don’t like what they see. In their draft proposal, released earlier this month, they proposed doing away with tax expenditures, which together cost the Treasury over $1 trillion a year. Such a drastic step would allow Mr. Bowles and Mr. Simpson to move the budget toward fiscal sustainability, while simultaneously reducing all income tax rates. Under their plan, the top tax rate would fall to 23 percent from the 35 percent in today’s law (and the 39.6 percent currently advocated by Democratic leadership). This approach has long been the basic recipe for tax reform. By broadening the tax base and lowering tax rates, we can increase government revenue and distort incentives less. That should command widespread applause across the ideological spectrum. Unfortunately, the reaction has been less enthusiastic. ... THERE are certain tax expenditures that I like. My personal favorite is the deduction for charitable giving. It encourages philanthropy and, thus, private rather than governmental solutions to society’s problems. But I know that solving the long-term fiscal problem won’t be easy. Everyone will have to give a little, and perhaps even more than a little. I am willing to give up my favorite tax expenditure if everyone else is willing to give up theirs. The Bowles-Simpson proposal is not perfect, but it is far better than the status quo. The question ahead is whether we can get Senator Porkbelly and Congressman Blowhard to agree.--Greg Mankiw So I like the mortgage interest deduction because it effectively repeals a bad tax.--Steve Landsburg There is no question of 'resisting change'. The only question is what can and should be salvaged from 'devouring time.' Conservation is a labor, not indolence, and it takes discrimination to identify and save a few strands of tradition in the incessant flow of mutability.--Joe Sobran This is not a thought, it’s safe to say, that has ever crossed Bud Selig’s mind.--Ross Douthat Friday, November 19, 2010 Quantitative easing explanation, further explained One of my church elders asked me if the QE video here is true. My explanation to him after the embed. Yes it is 99% correct. The Ben Bernank is probably the pre-eminent expert on the Great Depression. QE is not a good thing, but monetary shocks are worse, and the more money that leaks out for now the better. I am glad he is there, but his policy for the US today sort of contradicts his policy criticism of Japan a decade ago. I am not sure I know anyone better than The Ben Bernank to chair the Fed, with the possible exceptions of Glenn Hubbard, Greg Mankiw, Lars Svensson, or John Taylor. Do you know somebody else more qualified--smart, not a priest of the Church of Unlimited Government, and even keeled in this area? The Fed's dual mandate is price stability (2-3% inflation) and maximum employment (94-96%). The problem is that unemployment is still high, and the only real dial the Fed can turn is money supply and short term interest rates. If it contracts the money supply, standard theory says that unemployment will rise, but inflation will ease. The interesting thing is that there is record low inflation. I speculate it is because people are incentivized to not spend, so it doesn't matter how much money the govt prints, its not being spent. So inflation stays low, and unemployment stays high. There are plenty of competitors to The Goldman Sachs, so even though there is some transaction cost and economic friction in transacting with the US Treasury, it's better than it has ever been in history. If central banks, including the Fed, act irrationally, then smart investment banks and/or hedge funds will make a ton of money (like Soros did against the British pound). If the Fed said that the world should spend $1,000 on every Bible, then The Goldman Sachs would borrow as many Bibles as people would lend them (say, for $50), sell them to government agencies and anyone else who would take them for $1,000, and then wait until Bibles fell back to proper valuations, say $30, buy them back and return them to their owners and make a bazillion dollars. Future taxpayers get hosed. Goldman is seeking and speaking the truth in this case, by forcing prices toward reality and away from foolish policy. They are rarely the false teachers speaking the words our itchy ears wish to hear (with some interesting exceptions). So it turns out Lloyd Blankfein is doing God's work, at least more than irrational central banks. Gold is interesting to people because of primitive conditions In an interview with Sanat Kumar, we learn that platinum and gold were the only stable and rare metals available to civilizations before 1800, and the melting point of platinum is 600 Celsius (1300 Farenheit) degrees hotter. So why is everyone long again? Via AR. Disclosure: I am short GLD. Chart of the day: Historic SPX earnings Source here. Forget Nostradamus John Templeton is quite the predictor. Here is a letter he drafted in 2005, 3 years before his death. An excerpt: Quotes of the day It’s a lot easier to go public than be public.--Frank Quattrone How bad can things get for munis? Very, very bad. During the 1873 Depression more than 24 percent of the outstanding municipal debt defaulted.--John Carney When it comes to warmth, negative behaviors are weighted more heavily. If you act like a jerk once, it’s very hard to redeem yourself, because the intuition people have is that you can’t accidentally be a jerk but you can fake being nice. On competence, it’s flipped. Positive competence is weighted more heavily. People reason that you can’t accidentally get a high SAT score, so it’s okay if you can’t sail a boat.--Amy Cuddy Thursday, November 18, 2010 Wednesday, November 17, 2010 Quotes of the day I wonder if US government agencies have something to hide the conspiracy theorist that I am: U.S. Ends Inquiry of UBS Over Tax Evasion Must be all that Ludlum I read as a youth. Tuesday, November 16, 2010 I'm definitely left-brained Test here. Attributes include: uses logic _detail oriented facts rule words and language present and past math and science can comprehend order/pattern perception knows object name reality based forms strategies except I'm not detailed oriented. I think this generally matches the INTJ Myers Briggs profile assigned to me. From my favorite broker. Monday, November 15, 2010 Chart of the day: Shiller's cyclical adjusted price earnings ratio Source here. Forget about poker, bridge, and backgammon for correlating with success on Wall Street The operative game is now fantasy football: [Stanley Druckenmiller] and [Raj] Rajaratnam, founder of hedge-fund firm Galleon Group, spoke frequently because they were in a fantasy football league together, which also included high-profile hedge-fund names like Paul Tudor Jones and Jim Pallotta [of Raptor Capital]. In somewhat related news, my fantasy teams are doing well. I could be taking first place in the 14-team league after tonight's game (3rd in points), and first in points in the 12-team league (3rd in rankings). A lot of this had to do with picking up Peyton Hillis after Week 1 (and taking a lot of good natured ribbing on allocating a ball carrier who is white). Hey, I've only had 4 down months in the last 24. (And no, my benchmark ain't zero). Quotes of the day People on all levels of income are better off than they were in 1979. The hon. Gentleman [Simon Hughes (Southwark and Bermondsey)] is saying that he would rather that the poor were poorer, provided that the rich were less rich. That way one will never create the wealth for better social services, as we have. What a policy. Yes, he would rather have the poor poorer, provided that the rich were less rich. That is the Liberal policy.--Margaret Thatcher Someone who begins a sentence with “Confidentially” is nearly always betraying a confidence; someone who starts out “Frankly,” or “Honestly,” “To be (completely) honest with you,” or “Let me give it to you straight” brings to mind Ralph Waldo Emerson’s quip: “The louder he talked of his honor, the faster we counted our spoons.”--Erin McKean The right question to ask of the Bowles-Simpson plan is not whether the chairmen’s advice will be followed in full. It would be good if it were, but it will not be. The right question is what Mr Obama does next. Will he see the plan as a way for him to take charge and make the US think hard about ends and means – which he, the rest of the political class and the country as a whole have avoided up to now? Or will he see the plan, for which he himself asked, as political poison and hide? If it is the latter, as seems likely, we can conclude that the US has become ungovernable. The Bowles-Simpson plan matters not just for its merits but even more for what it says about the government’s capacity to act. The stakes are that high. ... That vast empty space in the middle [between the extreme partisan divide of Congress] is the one [Obama] promised to bridge in 2008. I recall, in addition, a good deal of talk about confronting hard questions and leading the country from partisan paralysis to workable solutions. The talk has continued and there has been action as well: nobody could call this an idle administration. But the hardest questions have been shirked and Mr Obama has failed, abjectly, to lead. Mr President, here is your chance. You appointed two excellent chairman to lead your commission. Here is their advice. Accept it. Take ownership of it. Infuriate the zealots on Capitol Hill and, for heaven’s sake, do something with it.--Clive Crook House Speaker Nancy Pelosi called it, “simply unacceptable.” Richard Trumka of the AFL-CIO thinks it is a death sentence for “working Americans.” The conservative Americans for Tax Reform said it was “merely an excuse to raise net taxes on the American people.” And yesterday on TV, Senator Kent Conrad described the recommendations of the Bowles-Simpson deficit cutting commission as “shock therapy.” Please. There’s nothing radical or earth shattering about the Bowles-Simpson proposals. You want something shocking? Try this. One generation of a once-great nation borrows $14 trillion from future generations so that it can fund a lifestyle it can’t afford – and then refuses to entertain reasonable proposals to curtail its lifestyle. Now, that’s shocking. So selfish. But predictably so. After decades of handouts, of LBJ’s Great Society, George Dubya’s Medicare D and now Obama’s “healthcare for all”, we have a fully entitled nation. And it’s much worse than “me, me, me.” American society is “me, me, me – and make sure you use other people’s money.”--Evan Newmark Fed Chairman Bernanke wrote in an article in the Washington Post on November 4th that "The Federal Reserve cannot solve all the economy's problems on its own." The slowdown in the recovery of the American economy is not the result of Fed policy, and cannot be cured by yet another bout of open market operations. This is why the Fed should curtail, and better yet, eliminate its plans for QE2.--Gary Becker [Dubya] Bush, please don’t use the Ivy League, a group to which you clearly belong, as cannon fodder to shape your presidential legacy.--Constance Boozer Today you give away your privacy for nothing, in dribs and drabs. Your credit card company knows some things about you, your phone company knows others, and FaceBook knows a lot. One thing that all of those companies have in common is that the private information they possess involves mostly your past, and not so much your future. When you post pictures on FaceBook, it is a record of where you were, not a prediction of where you will be. Likewise, your credit card company and the phone company have records of what you did, as opposed to what you plan to do next. Privacy about your past is so cheap that you literally give it away. Privacy about your future plans is another matter. That has real value.--Scott Adams Columbia gets its head out of its, er, gets its head into the real world Breaking from a 42-year ban on military activities on campus, students raised the American flag over Low Plaza Thursday morning in a traditional ceremony in honor of Veterans Day. Via IvyGate. Friday, November 12, 2010 Quotes of the day In short [the federal government's Troubled Asset Relief Program, i.e. TARP] may have been the most successful United States program ever. Moreover, it was bipartisan in the sense that both the Republican Administration that created it and the Democratic Administration that fostered it were unified in their beliefs as to what should have been done—and they did it. ... [The media and politicians] continue to excoriate the program as having been harmful to ‘main street’ America. ... The program was never understood by … either group and they acted on their misperceptions by selling the American people an enormous set of falsehoods concerning what was being done and the effect of these programs.--Dick Bove I don’t want to be too hard on [Jim] Webb: Health care isn’t his bailiwick, he’s a freshman Senator, and his work on prison reform is a model of the kind of task that lawmakers should be willing to take up. But in this case, he’s yet another exhibit in the case against the Senate’s bloc of centrists, center-right and center-left alike. There was a period of months where Webb basically wielded a kind of veto power over a bill that he apparently viewed as a political disaster in the making. And by his own account, he let that power go to waste.--Ross Douthat You can get as angry as you want, but you cannot assume away the half of the political spectrum that does not want a massive increase in government spending and income redistribution. If you do, the voters will . . . well, do what they just did, and elect more of those people. ... A top tax rate of 28% on a much broader base is not a giveaway to the rich; it's more than double what Theresa and John Kerry paid in 2003.--Megan McArdle If you accept Chait’s vision of a close-minded, Ayn Randian right and a pragmatic, non-ideological left, you would expect conservatives to be furious over the means-testing and loophole-closing, and liberals to be delighted to have a more redistributionist welfare state. Yet conservative reaction has been muted and respectful (with a notable exception, admittedly) while liberals have been flatly dismissive. Which suggests that maybe, just maybe, American liberalism has more of an ideological commitment to ever-rising government spending than Chait wants to admit.--Ross Douthat I don't find it surprising that people who are not self-supporting would disproportionately break in favor of higher taxes to support a paternalistic program. But the proportion in the general population which supports tax increases is well under 50%, and unsurprisingly, is lowest among the age groups that are going to pay most of the taxes. ... Like a struggling company, we have a big budget gap, and a limited reservoir of tax increases and spending cuts that we can "spend" on fixing it. Every time we raise taxes or cut spending to fund one program, we leave less that can be used for other programs. Taxes cannot go to 100%. Spending will not go to zero. The question, then, is not simply, "Should we raise taxes or the retirement age to fix Social Security?" The question is, "What are the best spending cuts and tax increases to bring our budget into balance?" ... People in polls are lunatics on the budget; they consistently oppose tax increases, oppose spending cuts, and strongly support balancing the budget.--Megan McArdle Unsurprising: Hyperinflation in federal government salaries Source here. Why aren't there more gold shorts out there? I'm the only one that I've talked to that is short. Everyone else that I've talked to is either long, or neutral and trying to talk me out of the short. Although my short is a hesistant one: I'm longer PALL than I am short GLD. Don't want to be short metal right now. If I'm going to be long a metal, than I guess it should have some industrial use and intrinsic value (which gold does not). Megan says Wednesday, November 10, 2010 Superbowl Futures, part 3 Matchbook latest odds (based on midprices): NY Giants 14.7% Pittsburgh 13.4% Baltimore 11.7% NY Jets 10.4% Green Bay 9.8% New Orleans 8.9% New England 8.5% Atlanta 7.8% Indianapolis 7.4% Philadelphia 4.4% San Diego 4.3% Tennessee 4.1% Kansas City 2.4% I am buying IND, GB, NO, and TEN; selling BAL and flipping my long NE back to short. New England is killing me, along with my shorting the NY Giants. P/L Team Position Paid Market Risk $ 2.44 ATL 1 5.6% 8% $ 8.00 $ (5.80) BAL -3 10.1% 12% $(36.00) $ - GB 1 10.0% 10% $ 10.00 $ (0.69) IND 1 7.7% 7% $ 7.00 $ (0.72) KC 2 2.4% 2% $ 4.00 $ (7.63) NE -1 1.4% 9% $ (9.00) $ (0.70) NO -1 8.3% 9% $ (9.00) $ (9.12) NYG -1 5.9% 15% $(15.00) $ 0.91 NYJ 1 9.1% 10% $ 10.00 $ 3.48 PIT 1 9.5% 13% $ 13.00 $ 1.56 TEN 3 3.5% 4% $ 12.00 $ (16.28) Total $ (5.00) Part 2 here. Quotes of the day The real lesson from the story told in “The Social Network” is that businesses are not sold too soon or too late, but that they are instead sold when the founders can no longer take them forward.--Steven Davidoff Smart people can be very clueless when they apply too much precision to imprecise problems.--Eric Falkenstein The US discovered nearly half the drugs approved during [1998-2007], and accounts for roughly that amount of the market, for example. But there are two big exceptions: the UK and Switzerland, which both outperform for their size. In case you're wondering, the league tables look like this: the US leads in the discovery of approved drugs, by a wide margin (118 out of the 252 drugs). Then Japan, the UK and Germany are about equal, in the low 20s each. Switzerland is in next at 13, France at 12, and then the rest of Europe put together adds up to 29. Canada and Australia put together add up to nearly 7, and the entire rest of the world (including China and India) is about 6.5, with most of that being Israel. But while the US may be producing the number of drugs you'd expect, a closer look shows that it's still a real outlier in several respects. The biggest one, to my mind, comes when you use that criterion for innovative structures or mechanisms versus extensions of what's already been worked on, as mentioned in the last post. Looking at it that way, almost all the major drug-discovering countries in the world were tilted towards less innovative medicines.--Derek Lowe We believe a significant drag on economic activity has been the result of overly-burdensome regulatory initiatives, our challenged fiscal situation, legal uncertainty created by the erosion of the rule of law, and unpredictable tax policy. ... QE2 is much more likely to be successful in creating inflation and speculation in financial instruments. It is perverse that on the heels of suffering the after-effects of the collapse of the internet bubble and then the real estate bubble (both of which the Fed disclaims responsibility for creating or supporting), the Fed would like to encourage the formation of yet another asset bubble. ... An FOMC member added that he didn't know "what the world 'bubble' means." How about a rule where if you don't know what a bubble is, you can't serve on the Fed?--Greenlight Capital Profitable, high flying startups like Facebook and Zynga don't bother going public anymore. Why put up with the expense, scrutiny, and distraction? Why risk exposure to class action shareholder suits every time a stock blips? Why kowtow to the whims of grandstanding Congressmen each time one of these potentates feels like holding a hearing to beat up on CEOs? These days the IPOs the public has a chance to participate in are mostly companies bleeding money so profusely that they can no longer be supported by their venture capitalists. Going public is their best alternative to going broke. But wait a minute - how do top tier startups achieve liquidity if they don't go public? Isn't getting rich the main objective of entrepreneurship after founders and early employees succeed in changing the world? Can't Uncle Sam force successful entrepreneurs to bend to the rules before they buy those yachts? Nope. The modern way to cash out is by selling stock on a shadowy collection of unregulated private markets. Emerging exchanges like Second Market and SharesPost as well as innumerable bilateral secondary transactions arranged on the old-boy network offer a free-market escape from the tender protections of Congress. Yes, these private markets are opaque, thin, and bereft of both pricing and company performance information. Which is why by law only the rich and well connected can participate. What do the Income Inequality mongerers have to say about that? By trying to eliminate risk from our public markets, Congress is on its way to eliminating reward.--Bill Frezza Tuesday, November 09, 2010 Quotes of the day Domino’s Pizza was hurting early last year. Domestic sales had fallen, and a survey of big pizza chain customers left the company tied for the worst tasting pies. Then help arrived from an organization called Dairy Management. It teamed up with Domino’s to develop a new line of pizzas with 40 percent more cheese, and proceeded to devise and pay for a $12 million marketing campaign. Consumers devoured the cheesier pizza, and sales soared by double digits. “This partnership is clearly working,” Brandon Solano, the Domino’s vice president for brand innovation, said in a statement to The New York Times. But as healthy as this pizza has been for Domino’s, one slice contains as much as two-thirds of a day’s maximum recommended amount of saturated fat, which has been linked to heart disease and is high in calories. And Dairy Management, which has made cheese its cause, is not a private business consultant. It is a marketing creation of the United States Department of Agriculture — the same agency at the center of a federal anti-obesity drive that discourages over-consumption of some of the very foods Dairy Management is vigorously promoting. Urged on by government warnings about saturated fat, Americans have been moving toward low-fat milk for decades, leaving a surplus of whole milk and milk fat. Yet the government, through Dairy Management, is engaged in an effort to find ways to get dairy back into Americans’ diets, primarily through cheese.--Michael Moss The Great Persuader was going to carry the country into a new age of progressive politics, creating a new consensus behind the power of a reinvigorated state. The earth would start to cool, the waters to recede, the rainbows would appear in the sky. All pundits, including yours truly, get it wrong sometimes, and normally there would be little point in dwelling on past blunders. But it this case, it is worth exhuming these vaporous and embarrassing stupidities for a few moments. Many of our nation’s intellectual leaders wonder why the rest of the country isn’t more respectful of their claims to be guided by and speak for the cool voice of celestial reason. That so many of them gushed over Barack Obama with all of the profundity of reflection and intellectual distance of tweeners at a Justin Bieber concert should help them understand why their claims of superior wisdom are sometimes met with caustic cynicism. A significant chunk of the American liberal intelligentsia completely lost its head over Barack Obama. They mistook hopes and fantasies for reality. Worse, the disease spread to at least some members of the White House team. An administration elected with a mandate to stabilize the country misread the political situation and came to the belief that the country wanted the kinds of serious and deep changes that liberals have wanted for decades. It was 1933, and President Obama was the new FDR. They did not perceive just how wrong they were; nor did they understand how the error undermined the logical case they wanted to make in favor of a bigger role for government guided by smart, well-credentialed liberal wonks. ... Another factor in the President’s political trouble comes from a failure of rhetoric and communication. Musing over the electoral setback, President Obama has spoken of a ‘failure of communication.’ It’s a strange failure for a President so enthusiastically hailed by the mainstream media as the greatest orator of the era.--Walter Russell Mead Monday, November 08, 2010 Proof of public choice theory? Tough standards for slot machines; not so much voting machines. Via Abraham Piper. I'm getting grayer, too Quotes of the day When Ken stands up to Lotso at the end of [Toy Story 3], Lotso yells that there’s a hundred million Barbies just like the one he’s fallen for. Ken affirms that he loves Barbie — that she’s special and unique to him; that she’s not replaceable to him. To be pretentious for a second, that’s a gesture Shakespeare uses a lot — putting wisdom in the mouth of a fool.--Michael Arndt [Istvan] Hargittai is a Hungarian historian who has written extensively on the Nobel Prize itself, Nobel laureates, and some of the greats who did not get the biggest prize in science. His 2003 book The Road to Stockholm includes eleven chapters on the prize and its winners, then a final (and best) chapter: "Who Did Not Win." I have heard Hargittai speak about the subject of this chapter. He is direct and devastating in his judgments on why some of the greats in science were passed over for the big prize. Hargittai puts a human face on science (very much warts and all), which is missing from so much scientific biography. Of course there has been a trend recently in scientific biographies to talk about lust in the lives of their subjects. We all know now that Einstein would not be named husband of the century. Erwin Schrödinger, known for the thought experiment "Schrödinger's Cat," created the Schrödinger equation, central to quantum mechanics, on a winter semester break. At the time he was having an affair with twin young women in one of his classes. He took one twin to the Alps and came back with the equation. But while sins of the flesh haunt the lives of the rich and powerful, Coffey leaves these sins in the background to concentrate on the sins of the spirit central to their success or failure. It is Pride, Envy, and Greed that often make the difference between who gets the big prize and who doesn't.--Neil Gussman San Francisco officials strike a blow for “food justice” by barring the Happy Meal. I think the word “justice” has now jumped the shark. We already have economic justice, social justice, global justice, environmental justice, climate justice, housing justice, transportation justice, even — no kidding — judicial justice. The game seems to be that when you want to force other people to adjust their lives to better suit your preferences, you slap the word “justice” on the end of your slogan and it’s transformed into a golden ticket on the rail car running straight to the tippy top of Moral Mountain. Well, I’m agitating here for definitional justice, and my manifesto is simply that every person who associates the word “justice” with some economically ill-informed utopian fantasy ought to be required to wear a t-shirt that says: I’M GETTING IN TOUCH WITH MY INNER IRON-FISTED AUTOCRAT.--Tony Woodlief The most important step in raising the growth rate is not to increase but rather to lower taxes on capital and entrepreneurship. This implies maintaining essentially all the Bush tax cuts, including those on capital gains and dividends, and those on incomes at all levels, including quite high incomes. The estate tax on very high levels of wealth could be reinstated if politically necessary, but it will only bring in a very small amount of tax revenue, and will be more costly than it is worth. Tax reform also implies a reduction in the corporate income tax, and especially reductions in taxes on incomes of small businesses. Successful small businesses that grow to become large companies, such as Wal-Mart, Starbucks, Microsoft, and Apple, form the foundation of the American economy. They should be strongly encouraged. One goal of such tax reform is to eliminate as much as possible taxes on capital since economic theory basically implies that economic efficiency requires that capital not be taxed in the long run. For the supply of capital in the long run is highly responsive to after-tax rates of return on capital.--Gary Becker Clearly the US needs to move aggressively to shift global demand in its favor because most countries are doing just that – that is what beggar-they-neighbor means, and in a beggar-thy-neighbor world whoever does not respond will bear most of the brunt of the pain. There is no point pretending that we are not in a world in which nearly every country is cheating on trade, and will continue cheating as long as it is able. But the wrong aggressive moves can easily make things worse. I guess this means that the whole trade problem really will best be resolved by an intelligent multilateral agreement, with no cheating and no free-riding, that involves a set of determined and coordinated policies that bring the imbalances down gradually over the next several years. But that is asking for a lot. My guess? Trade disputes will be resolved through more tariffs and currency interventions. No one out there, it seems, is willing to do more than wish the imbalances away. Actually to take the painful rebalancing steps is still not on anyone’s agenda.--Michael Pettis Third-party apps and services can’t pull data from Google without allowing Google to do the same with their data. Think of it as a declaration of data reciprocity. ... To me, the contact info of my friends is my social graph — not Facebook’s social graph. I should be able to take it wherever I wish. My only criticism of Google’s move is that it has taken way too long.--Mathew Ingram At the end of the day, the peculiar story of the downfall of Mark Hurd at the hands of Jodie Fisher remains a drama about what happens when only two people know all the facts, yet many others act on them—with momentous consequences.--Adam Lashinsky But there were other matters weighing on H-P's board. An investigation by The Wall Street Journal into Mr. Hurd's sudden ouster reveals that the letter contained an explosive allegation: that in early 2008, Mr. Hurd told Ms. Fisher of a still-secret H-P plan to buy Electronic Data Systems Corp. Though directors had little reason to doubt Mr. Hurd's assurance this allegation was false, some fretted about it. That is because of what the Journal found was the ultimate reason directors ousted Mr. Hurd: They had lost confidence he was being honest with them about his relationship with Ms. Fisher.--ROBERT A. GUTH, BEN WORTHEN And JUSTIN SCHECK Women assault each other twice as much as men do, and they fight one and half times as much as men do ...--Diego Gambetta Chart of the day: US GDP growth Source here, via Abnormal Returns. Friday, November 05, 2010 Quotes of the day GOP Seizes Control of House I don’t believe for a second that the majority of people who argue for health reform really give a damn about the poor or infirm – it just makes them feel good to say it, or be more acceptable in the company of others. Why do I say this so strongly? Because providing even a very generous level of support to the poor and the infirm is so easily within our reach that it is laughable to suggest otherwise. Instead, “we” use the poor and infirm as pawns in a corporatist game, in a middle-class entitlement game, special-interest game, political-nanny-statism game, and no one is willing to admit it. Look at it this way. You want to argue that the world’s richest economy could easily sacrifice a little income to make sure people do not die (early, unnaturally, due to the poverty that used to kill people) because of income or for the seriously infirm to go without care. I agree. If you add up all of the annual expenditures by local governments, state governments, and the federal government in the United States, you would get a figure around $6 trillion. That would make the US government the world’s single largest economy, and by a factor of 20%.--Michael Rizzo It is hard to exaggerate the emotions invested in the “hopey-changey thing” around the world. Just think of the cheering crowds at Mr Obama’s open-air speech in Berlin in the summer of 2008; the rave reviews given to the newly-elected president’s Cairo speech on Islam and the west; the Nobel Peace Prize awarded to Mr Obama, when he had barely had time to arrange his pens on the Oval Office desk. Now a nasty thought is occurring to the foreigners who invested so much hope in the new president. Perhaps Mr Obama represented not a new beginning in American relations with the rest of the world, but a temporary aberration? Maybe, after a brief stab at internationalism and engagement with the rest of the world, the US will revert to a more unilateralist and nationalist foreign policy?--Gideon Rachman Video of the day: We've figured out a surefire way around the First Amendment Incumbents, that is. Via Don Boudreaux. Thursday, November 04, 2010 Quotes of the day So even the regulator with the best intentions comes to see issues in much the same way as the corporate officers he deals with every day. You require both an abrasive personality and considerable intellectual curiosity to do the job in any other way. And these are not the qualities often sought, or found, in regulators.--John Kay ... the damage done to the Air Force’s reputation, and indeed to the entire military procurement system, may be longstanding. The people who know this [refueling tanker] contract well believe that military officials are being influenced by members of Congress, who have taken sides in the deal along party lines. Politics has always factored into Pentagon programs, but never on a contract of this size that was already stained by corruption and criminality.--Shane Harris This brilliant line* revealed many enduring truths about political humor: that the true challenge is to determine the worst thing an opponent might say about and then to find a way to say it about yourself. That in politics, things are only as bad as things you can’t joke about. That jokes that concede the obvious cost little and earn back something valuable in terms of likeability and credibility. And that if these rules are adhered to, that the resulting joke can be as Machiavellian as anything Machiavelli might have schemed up.--Mark Katz *Jack – Don’t spend one dime more than is necessary. I’ll be damned if I am going to pay for a landslide.--Joe Kennedy, in a fictional telegram to his son
af594fcb7bfdd3f5	Atomic units From Wikipedia, the free encyclopedia - View original article Jump to: navigation, search Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, Hartree atomic units[1] and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In atomic units, the numerical values of the following four fundamental physical constants are all unity by definition: Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in different contexts. Use and notation[edit] Atomic units, like SI units, have a unit of mass, a unit of length, and so on. However, the use and notation is somewhat different from SI. Suppose a particle with a mass of m has 3.4 times the mass of electron. The value of m can be written in three ways: Fundamental atomic units[edit] These four fundamental constants form the basis of the atomic units (see above). Therefore, their numerical values in the atomic units are unity by definition. Fundamental atomic units DimensionNameSymbol/DefinitionValue in SI units[5] masselectron rest mass\!m_\mathrm{e}9.10938291(40)×10−31 kg chargeelementary charge\!e1.602176565(35)×10−19 C actionreduced Planck's constant\hbar = h/(2 \pi)1.054571726(47)×10−34 J·s electric constant-1Coulomb force constant1/(4 \pi \epsilon_0)8.9875517873681×109 kg·m3·s-2·C-2 Related physical constants[edit] Dimensionless physical constants retain their values in any system of units. Of particular importance is the fine-structure constant \alpha = \frac{e^2}{(4 \pi \epsilon_0)\hbar c} \approx 1/137. This immediately gives the value of the speed of light, expressed in atomic units. Some physical constants expressed in atomic units NameSymbol/DefinitionValue in atomic units speed of light\!c\!1/\alpha \approx 137 classical electron radiusr_\mathrm{e}=\frac{1}{4\pi\epsilon_0}\frac{e^2}{m_\mathrm{e} c^2}\!\alpha^2 \approx 5.32\times10^{-5} proton massm_\mathrm{p}m_\mathrm{p}/m_\mathrm{e} \approx 1836 Derived atomic units[edit] Below are given a few derived units. Some of them have proper names and symbols assigned, as indicated in the table. kB is Boltzmann constant. Derived atomic units DimensionNameSymbolExpressionValue in SI unitsValue in more common units lengthbohr\!a_04\pi \epsilon_0 \hbar^2 / (m_\mathrm{e} e^2) = \hbar / (m_\mathrm{e} c \alpha) 5.2917721092(17)×10−11 m[5]0.052917721092(17) nm=0.52917721092(17) Å energyhartree\!E_\mathrm{h}m_\mathrm{e} e^4/(4\pi\epsilon_0\hbar)^2 = \alpha^2 m_\mathrm{e} c^2 4.35974417(75)×10−18 J27.211 eV=627.509 kcal·mol−1 time\hbar / E_\mathrm{h}2.418884326505(16)×10−17 s velocity a_0 E_\mathrm{h} / \hbar = \alpha c2.1876912633(73)×106 m·s−1 force\! E_\mathrm{h} / a_0 8.2387225(14)×10−8 N82.387 nN=51.421 eV·Å−1 temperature\! E_\mathrm{h} / k_\mathrm{B} 3.1577464(55)×105 K pressure E_\mathrm{h} / {a_0}^3 2.9421912(19)×1013 Pa electric field\!E_\mathrm{h} / (ea_0) 5.14220652(11)×1011 V·m−15.14220652(11) GV·cm−1=51.4220652(11) V·Å−1 electric dipole moment e a_0 8.47835326(19)×10−30 C·m2.541746 D SI and Gaussian-CGS variants, and magnetism-related units[edit] There are two common variants of atomic units, one where they are used in conjunction with SI units for electromagnetism, and one where they are used with Gaussian-CGS units.[6] Although the units written above are the same either way (including the unit for electric field), the units related to magnetism are not. In the SI system, the atomic unit for magnetic field is 1 a.u. = \frac{\hbar}{e a_0^2} = 2.35×105 T = 2.35×109 G, and in the Gaussian-cgs unit system, the atomic unit for magnetic field is 1 a.u. = \frac{e}{a_0^2} = 1.72×103 T = 1.72×107 G. (These differ by a factor of α.) Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,[7] \mu_B = \frac{e \hbar}{2 m_e} = 1/2 a.u. and in Gaussian-based atomic units,[8] \mu_B = \frac{e \hbar}{2 m_e c}=\alpha/2\approx 3.6\times 10^{-3} a.u. Bohr model in atomic units[edit] Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model): Non-relativistic quantum mechanics in atomic units[edit] The Schrödinger equation for an electron in SI units is - \frac{\hbar^2}{2m_e} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi}{\partial t} (\mathbf{r}, t). The same equation in au is - \frac{1}{2} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \frac{\partial \psi}{\partial t} (\mathbf{r}, t). For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is: \hat H = - {{{\hbar^2} \over {2 m_e}}\nabla^2} - {1 \over {4 \pi \epsilon_0}}{{e^2} \over {r}}, while atomic units transform the preceding equation into \hat H = - {{{1} \over {2}}\nabla^2} - {{1} \over {r}}. Comparison with Planck units[edit] Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. It should be kept in mind that au were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for quantum gravity and early-universe cosmology. Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and, as a result, the speed of light in atomic units is a large value, 1/\alpha \approx 137. The orbital velocity of an electron around a small atom is of the order of 1 in atomic units, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms much slower than the speed of light (around 2 orders of magnitude slower). There are much larger discrepancies in some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the Planck mass, a mass so large that if a single particle had that much mass it might collapse into a black hole. Indeed, the Planck unit of mass is 22 orders of magnitude larger than the au unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units. See also[edit] Notes and references[edit] 1. ^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society 24 (1) (Cambridge University Press). pp. 89–110. doi:10.1017/S0305004100011919. 2. ^ a b Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5. 3. ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4. 4. ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2. 5. ^ a b "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. Retrieved 1 April 2012. 6. ^ "A note on Units". Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes. 7. ^ Chis, Vasile. "Atomic Units; Molecular Hamiltonian; Born-Oppenheimer Approximation". Molecular Structure and Properties Calculations. Babes-Bolyai University lecture notes. ) 8. ^ Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. p. 380. ISBN 978-0-19-850950-9. External links[edit]
a8ece554b63a4fe4	From Wikipedia, the free encyclopedia (Redirected from Collective excitation) Jump to: navigation, search In physics, quasiparticles and collective excitations (which are closely related) are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with all of the other electrons and nuclei; however it approximately behaves like an electron with a different mass traveling unperturbed through free space. This "electron" with a different mass is called an "electron quasiparticle".[1] In another example, the aggregate motion of electrons in the valence band of a semiconductor is the same as if the semiconductor contained instead positively charged quasiparticles called holes. Other quasiparticles or collective excitations include phonons (particles derived from the vibrations of atoms in a solid), plasmons (particles derived from plasma oscillations), and many others. These fictitious particles are typically called "quasiparticles" if they are related to fermions (like electrons and holes), and called "collective excitations" if they are related to bosons (like phonons and plasmons),[1] although the precise distinction is not universally agreed.[2] Quasiparticles are most important in condensed matter physics, as it is one of the few known ways of simplifying the quantum mechanical many-body problem. General introduction[edit] Solids are made of only three kinds of particles: Electrons, protons, and neutrons. Quasiparticles are none of these; instead they are an emergent phenomenon that occurs inside the solid. Therefore, while it is quite possible to have a single particle (electron or proton or neutron) floating in space, a quasiparticle can instead only exist inside the solid. Motion in a solid is extremely complicated: Each electron and proton gets pushed and pulled (by Coulomb's law) by all the other electrons and protons in the solid (which may themselves be in motion). It is these strong interactions that make it very difficult to predict and understand the behavior of solids (see many-body problem). On the other hand, the motion of a non-interacting particle is quite simple: In classical mechanics, it would move in a straight line, and in quantum mechanics, it would move in a superposition of plane waves. This is the motivation for the concept of quasiparticles: The complicated motion of the actual particles in a solid can be mathematically transformed into the much simpler motion of imagined quasiparticles, which behave more like non-interacting particles. In summary, quasiparticles are a mathematical tool for simplifying the description of solids. They are not "real" particles inside the solid. Instead, saying "A quasiparticle is present" or "A quasiparticle is moving" is shorthand for saying "A large number of electrons and nuclei are moving in a specific coordinated way." Relation to many-body quantum mechanics[edit] Any system, no matter how complicated, has a ground state along with an infinite series of higher-energy excited states. The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 1017 atoms and 1018 electrons. Each of these attracts or repels every other by Coulomb's law. In quantum mechanics, a system is described by a wavefunction, which, if the particles are interacting (as they are in our case), depends on the position of every particle in the system. So, each particle adds three independent variables to the wavefunction, one for each coordinate needed to describe the position of that particle. Because of this, directly approaching the many-body problem of 1018 interacting electrons by straightforwardly trying to solve the appropriate Schrödinger equation is impossible in practice, since it amounts to solving a partial differential equation not just in three dimensions, but in 3x1018 dimensions – one for each component of the position of each particle. One simplifying factor is that the system as a whole, like any quantum system, has a ground state and various excited states with higher and higher energy above the ground state. In many contexts, only the "low-lying" excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the Boltzmann distribution, which implies that very-high-energy thermal fluctuations are unlikely to occur at any given temperature. Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at absolute zero is in the ground state, but if one phonon is added to the crystal (in other words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an elementary excitation. More generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations).[3] When the material is characterized as having "several elementary excitations", this statement presupposes that the different excitations can be combined together. In other words, it presupposes that the excitations can coexist simultaneously and independently. This is never exactly true. For example, a solid with two identical phonons does not have exactly twice the excitation energy of a solid with just one phonon, because the crystal vibration is slightly anharmonic. However, in many materials, the elementary excitations are very close to being independent. Therefore, as a starting point, they are treated as free, independent entities, and then corrections are included via interactions between the elementary excitations, such as "phonon-phonon scattering". Therefore, using quasiparticles / collective excitations, instead of analyzing 1018 particles, one needs only to deal with only a handful of somewhat-independent elementary excitations. It is therefore a very effective approach to simplify the many-body problem in quantum mechanics. This approach is not useful for all systems however: In strongly correlated materials, the elementary excitations are so far from being independent that it is not even useful as a starting point to treat them as independent. Distinction between quasiparticles and collective excitations[edit] Usually, an elementary excitation is called a "quasiparticle" if it is a fermion and a "collective excitation" if it is a boson.[1] However, the precise distinction is not universally agreed.[2] There is a difference in the way that quasiparticles and collective excitations are intuitively envisioned.[2] A quasiparticle is usually thought of as being like a dressed particle: It is built around a real particle at its "core", but the behavior of the particle is affected by the environment. A standard example is the "electron quasiparticle": A real electron particle, in a crystal, behaves as if it had a different mass. On the other hand, a collective excitation is usually imagined to be a reflection of the aggregate behavior of the system, with no single real particle at its "core". A standard example is the phonon, which characterizes the vibrational motion of every atom in the crystal. However, these two visualizations leave some ambiguity. For example, a magnon in a ferromagnet can be considered in one of two perfectly equivalent ways: (a) as a mobile defect (a misdirected spin) in a perfect alignment of magnetic moments or (b) as a quantum of a collective spin wave that involves the precession of many spins. In the first case, the magnon is envisioned as a quasiparticle, in the second case, as a collective excitation. However, both (a) and (b) are equivalent and correct descriptions. As this example shows, the intuitive distinction between a quasiparticle and a collective excitation is not particularly important or fundamental. The problems arising from the collective nature of quasiparticles have also been discussed within the philosophy of science, notably in relation to the identity conditions of quasiparticles and whether they should be considered "real" by the standards of, for example, entity realism.[4][5] Effect on bulk properties[edit] By investigating the properties of individual quasiparticles, it is possible to obtain a great deal of information about low-energy systems, including the flow properties and heat capacity. In the heat capacity example, a crystal can store energy by forming phonons, and/or forming excitons, and/or forming plasmons, etc. Each of these is a separate contribution to the overall heat capacity. The idea of quasiparticles originated in Lev Landau's theory of Fermi liquids, which was originally invented for studying liquid helium-3. For these systems a strong similarity exists between the notion of quasi-particle and dressed particles in quantum field theory. The dynamics of Landau's theory is defined by a kinetic equation of the mean-field type. A similar equation, the Vlasov equation, is valid for a plasma in the so-called plasma approximation. In the plasma approximation, charged particles are considered to be moving in the electromagnetic field collectively generated by all other particles, and hard collisions between the charged particles are neglected. When a kinetic equation of the mean-field type is a valid first-order description of a system, second-order corrections determine the entropy production, and generally take the form of a Boltzmann-type collision term, in which figure only "far collisions" between virtual particles. In other words, every type of mean-field kinetic equation, and in fact every mean-field theory, involves a quasi-particle concept. Examples of quasiparticles and collective excitations[edit] This section contains examples of quasiparticles and collective excitations. The first subsection below contains common ones that occur in a wide variety of materials under ordinary conditions; the second subsection contains examples that arise in particular, special contexts. More common examples[edit] • In solids, an electron quasiparticle is an electron as affected by the other forces and interactions in the solid. The electron quasiparticle has the same charge and spin as a "normal" (elementary particle) electron, and like a normal electron, it is a fermion. However, its mass can differ substantially from that of a normal electron; see the article effective mass.[1] Its electric field is also modified, as a result of electric field screening. In many other respects, especially in metals under ordinary conditions, these so-called Landau quasiparticles[citation needed] closely resemble familiar electrons; as Crommie's "quantum corral" showed, an STM can clearly image their interference upon scattering. • A hole is a quasiparticle consisting of the lack of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor.[1] A hole has the opposite charge of an electron. • A phonon is a collective excitation associated with the vibration of atoms in a rigid crystal structure. It is a quantum of a sound wave. • A magnon is a collective excitation[1] associated with the electrons' spin structure in a crystal lattice. It is a quantum of a spin wave. • A roton is a collective excitation associated with the rotation of a fluid (often a superfluid). It is a quantum of a vortex. • In materials, a photon quasiparticle is a photon as affected by its interactions with the material. In particular, the photon quasiparticle has a modified relation between wavelength and energy (dispersion relation), as described by the material's index of refraction. It may also be termed a polariton, especially near a resonance of the material. • A plasmon is a collective excitation, which is the quantum of plasma oscillations (wherein all the electrons simultaneously oscillate with respect to all the ions). • A polaron is a quasiparticle which comes about when an electron interacts with the polarization of its surrounding ions. More specialized examples[edit] • Composite fermions arise in a two-dimensional system subject to a large magnetic field, most famously those systems that exhibit the fractional quantum Hall effect.[6] These quasiparticles are quite unlike normal particles in two ways. First, their charge can be less than the electron charge e. In fact, they have been observed with charges of e/3, e/4, e/5, and e/7.[7] Second, they can be anyons, an exotic type of particle that is neither a fermion nor boson.[8] • Stoner excitations in ferromagnetic metals • Bogoliubov quasiparticles in superconductors. Superconductivity is carried by Cooper pairs—usually described as pairs of electrons—that move through the crystal lattice without resistance. A broken Cooper pair is called a Bogoliubov quasiparticle.[9] It differs from the conventional quasiparticle in metal because it combines the properties of a negatively charged electron and a positively charged hole (an electron void). Physical objects like impurity atoms, from which quasiparticles scatter in an ordinary metal, only weakly affect the energy of a Cooper pair in a conventional superconductor. In conventional superconductors, interference between Bogoliubov quasiparticles is tough for an STM to see. Because of their complex global electronic structures, however, high-Tc cuprate superconductors are another matter. Thus Davis and his colleagues were able to resolve distinctive patterns of quasiparticle interference in Bi-2212.[10] • A Majorana fermion is a particle which equals its own antiparticle, and can emerge as a quasiparticle in certain superconductors. • Magnetic monopoles arise in condensed matter systems such as spin ice and carry an effective magnetic charge as well as being endowed with other typical quasiparticle properties such as an effective mass. They may be formed through spin flips in frustrated pyrochlore ferromagnets and interact through a Coulomb potential. See also[edit] 1. ^ a b c d e f E. Kaxiras, Atomic and Electronic Structure of Solids, ISBN 0-521-52339-7, pages 65–69. 2. ^ a b c A guide to Feynman diagrams in the many-body problem, by Richard D. Mattuck, p10. "As we have seen, the quasi particle consists of the original real, individual particle, plus a cloud of disturbed neighbors. It behaves very much like an individual particle, except that it has an effective mass and a lifetime. But there also exist other kinds of fictitious particles in many-body systems, i.e. 'collective excitations'. These do not center around individual particles, but instead involve collective, wavelike motion of all the particles in the system simultaneously." 3. ^ Principles of Nanophotonics by Motoichi Ohtsu, p205 [books.google.com/books?id=3za2u8FnCgUC&pg=PA205 google books link] 4. ^ A. Gelfert, 'Manipulative Success and the Unreal', International Studies in the Philosophy of Science Vol. 17, 2003, 245–263 5. ^ B. Falkenburg, Particle Metaphysics (The Frontiers Collection), Berlin: Springer 2007, esp. pp. 243–46 6. ^ Physics Today Article 7. ^ Cosmos magazine June 2008 8. ^ Nature article 9. ^ "Josephson Junctions". Science and Technology Review. Lawrence Livermore National Laboratory. 10. ^ J. E. Hoffman et al.; McElroy, K; Lee, DH; Lang, KM; Eisaki, H; Uchida, S; Davis, JC (2002). "Imaging Quasiparticle Interference in Bi2Sr2CaCu2O8+". Science 297 (5584): 1148–51. arXiv:cond-mat/0209276. Bibcode:2002Sci...297.1148H. doi:10.1126/science.1072640. PMID 12142440. Further reading[edit] • L. D. Landau, Soviet Phys. JETP. 3:920 (1957) • L. D. Landau, Soviet Phys. JETP. 5:101 (1957) • A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (1963, 1975). Prentice-Hall, New Jersey; Dover Publications, New York. • D. Pines, and P. Nozières, The Theory of Quantum Liquids (1966). W.A. Benjamin, New York. Volume I: Normal Fermi Liquids (1999). Westview Press, Boulder. • J. W. Negele, and H. Orland, Quantum Many-Particle Systems (1998). Westview Press, Boulder External links[edit]
c98a0341cf9a354b	Take the 2-minute tour × My question is in the title. I do not really understand why water is not a superfluid. Maybe I make a mistake but the fact that water is not suprfluid comes from the fact that the elementary excitations have a parabolic dispersion curve but for me the question remain. An equivalent way to ask it is: why superfluid helium is described by Gross-Pitaevsky equation and it is not the case for water? share\|improve this question Recent work actually suggests that water may have a superfluid liquid phase – user20145 Jan 23 '13 at 15:24 @x you have to substantiate this claim by a reference or link and a quote, at least from the abstract. – anna v Jan 23 '13 at 15:31 2 Answers 2 You refer to the Landau criterion for superfluidity (there is a separate question whether this is really the best way to think about superfluids, and whether the Landau criterion is necessary and/or sufficient). In a superfluid the low energy excitations are phonons, the dispersion relation is linear $E_p\sim c p$, and the critical velocity is non-zero. In water the degrees of freedom are water molecules, the dispersion relation is quadratic, $E_p\sim p^2/(2m)$, and the critical velocity is zero. The Gross-Pitaevskii equation applies (approximately) to Helium, because in the superfluid phase there is a single particle state which is macroscopically occupied. The GP equation describes the time evolution of the corresponding wave function. In water there are no macroscopically occupied states. You can try to solve the full many-body Schroedinger equation, but at least approximately this problem reduces to cassical kinetic theory. I think the best criterion for superfluidity is irrotational flow: The non-classical moment of inertia, quantization of circulation, and persistent flow in a ring. Again, these don't appear in water because there is no spontaneous symmetry breaking, and no macroscopically occupied state. share\|improve this answer So now my question is why there is no macroscopically occupied state for water ans there is one for helium? In general we don't try to solve Schrödinger equation for helium in order to obtain GP equation, isnt'it? And how can I obtain a classical kinetic equation for water starting from Schrödinger ? – PanAkry Sep 24 '12 at 6:56 A rough criterion is the condition for Bose condensation in an ideal gas, $n\lambda^3\sim 1$, where $n$ is the density and $\lambda$ is the thermal wave length. Note that your question is in some sense backwards: Helium is the exception, water is the rule. Most ordinary fluids solidify instead of becoming superfluid at low $T$. – Thomas Sep 24 '12 at 12:38 Because water is liquid at much too high a temperature. Helium is only superfluid near absolute zero. To have a superfluid, you need the quantum wavelength of the atoms given the environmental decoherence to be longer than the separation between the atoms, so they can coherently come together. share\|improve this answer Your Answer
1030729bd7b394cf	Take the 2-minute tour × I haven't understood this thing: Physics is invariant for CPT trasform...But the Heat or diffusive equation $\nabla^2 T=\partial_t T$ is not invariant for time reversal...but it's P invariant..So CPT simmetry would be violed... What I haven't undestood? Thank you share\|improve this question 2 Answers 2 up vote 10 down vote accepted The heat equation is a macroscopic equation. It describes the flow of heat from hot objects to cold ones. Of course it can not be time-reversible, since the opposite movement never happens. Well, I say 'of course' but you actually have stumbled on something important. As you say, the fundamental laws of nature should be CPT invariant, or at least we expect them to be. The reason the heat equation is not CPT invariant is that it is not a fundamental law, but a macroscopic law emerging from the microscopic laws governing the motions of elementary particles. There is however a problem here, how does this time asymmetry arise from microscopic laws that are themselves time reversal invariant? The answer to that is given by statistical mechanics. While the microscopic laws are time-reversible (I'll focus on T, and leave CP aside), not all states are equally likely with respect to certain choices of the macroscopic variables. There are more configurations of particles corresponding to a room filled with air than with a room where all the air would be concentrated in one corner. It is this asymmetry that forms the basis of all explanations in statistical mechanics. I hope that clears things up a bit. share\|improve this answer Ok! A last question now: the Schrodingher equation isn't invariant for time reversal, for the first derivative in t. But is not that a microscopical law? – Boy Simone Dec 2 '10 at 12:54 Actually, the Schrödinger equation is invariant. But you have to take the complex conjugate of $\psi$. Since $\psi^*$ and $\psi$ have the same probability distributions $\|\psi\|^2$, the physics remains the same. – Raskolnikov Dec 2 '10 at 12:58 Great! Thank you :-) – Boy Simone Dec 2 '10 at 13:14 Nice summary. It worth noting, however, that this is a deep enough topic that multi-hundred page books have been written on the matter. – dmckee Dec 2 '10 at 19:33 @dmckee: Of course, I didn't mean to give an exhaustive explanation. In fact, I left my explanation open to many attacks on purpose. I hope that Boy will think further and come to these questions by himself. But a thorough answer would indeed need a thorough course in statistical mechanics. – Raskolnikov Dec 2 '10 at 22:43 CPT theorem is not a theorem for all of physics but only for a quantum field theory (QFT). Also CPT invariance doesn't mean that QFT is necessarily invariant with respect to any of C, P and T (or PT, TC and CP, which is the same by CPT theorem) transform. Indeed, all of these symmetries are violated by weak interaction. Second, even if the macroscopic laws were completely correct it wouldn't mean that they need to preserve microscopic laws. E.g. most of the microscopic physics is time symmetric (except for small violation by the weak interaction) but second law of thermodynamics (which is universally true for any macroscopic system just by means of logic and statistics) tells you that entropy has to increase with time. We can say that the huge number of particles breaks the microscopical time-symmetry. Now, the heat equation essentially captures dynamics of this time asymmetry of the second law. It tells you that temperatures eventually even out and that is an irreversible process that increases entropy. share\|improve this answer thank you for the answer! And why, in your example, huge number of particles breaks the microscopical time-symmetry? Why don't macroscopic effects preserve microscopical invariance CPT of quantum-field theory? – Boy Simone Dec 2 '10 at 12:39 @Boy: that has to do with statistical mechanics. You should really ask this as a separate question because answer is not completely simple. But in short: any given macroscopic state (given e.g. by energy and pressure) of the system can be realized by many microscopic states. Now your answer boils down to basic questions in probability theory: the more microscopic states there are, the more likely the resulting macroscopic state is. So system is more likely to move to move from the less probable state to more probable state and not in the other way. – Marek Dec 2 '10 at 12:46 Your Answer
6e802cd3967f0549	Albert Einstein His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, the explanation of the perihelion precession of Mercury, the prediction of the deflection of light by gravity (gravitational lensing), the first fluctuation dissipation theorem which explained the Brownian motion of molecules, the photon theory and the wave-particle duality, the quantum theory of atomic motion in solids, the zero-point energy concept, the semi-classical version of the Schrödinger equation, and the quantum theory of a monatomic gas which predicted Bose–Einstein condensation. Einstein published more than 300 scientific and over 150 non-scientific works; he additionally wrote and commentated prolifically on various philosophical and political subjects. His great intelligence and originality has made the word "Einstein" synonymous with genius. Born 14 March 1879 - Ulm, Kingdom of Württemberg, German Empire Died 18 April 1955 (aged 76) - Princeton, New Jersey, USA Resting place Grounds of the Institute for Advanced Study, Princeton, New Jersey. Residence Germany, Italy, Switzerland, USA Ethnicity German Jewish Known for General relativity, Special relativity, Photoelectric effect, Brownian motion, Mass-energy, equivalence, Einstein field equations, Unified Field Theory, Bose–Einstein statistics. Spouse(s) Mileva Marić (1903–1919) Elsa Löwenthal, née Einstein, (1919–1936) Awards Nobel Prize in Physics (1921), Copley Medal (1925), Max Planck Medal (1929), Time Person of the Century.
93d235b2a6aa3550	Open main menu Wikipedia β Statistical mechanics (Redirected from Statistical thermodynamics) Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has a large degree of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws. [1][2][3][note 1] It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics. Statistical mechanics shows how the concepts from macroscopic observations (such as temperature and pressure) are related to the description of microscopic state that fluctuates around an average state. It connects thermodynamic quantities (such as heat capacity) to microscopic behaviour, whereas, in classical thermodynamics, the only available option would be to just measure and tabulate such quantities for various materials. [1] Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. Fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles. Principles: mechanics and ensemblesEdit In physics, there are two types of mechanics usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts: 1. The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics). 2. An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics) Using these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,[note 2] and can be compactly summarized as a density matrix. As is usual for probabilities, the ensemble can be interpreted in different ways:[1] • an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or • the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (empirical probability), in the limit of an infinite number of trials. These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.[note 3] The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. Statistical thermodynamicsEdit The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving. Fundamental postulateEdit A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).[1] There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.[1] Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. A common approach found in many textbooks is to take the equal a priori probability postulate.[2] This postulate states that For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge. The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate: • Ergodic hypothesis: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic. • Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation. • Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest Gibbs entropy (information entropy).[4] Other fundamental postulates for statistical mechanics have also been proposed.[5] Three thermodynamic ensemblesEdit There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.[1] These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. Microcanonical ensemble describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition. Canonical ensemble describes a system of fixed composition that is in thermal equilibrium[note 4] with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy. Grand canonical ensemble describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers. For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.[6] The Gibbs theorem about equivalence of ensembles[7] was developed into the theory of concentration of measure phenomenon,[8] which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.[9] Important cases where the thermodynamic ensembles do not give identical results include: • Microscopic systems. • Large systems at a phase transition. • Large systems with long-range interactions. In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.[2] Thermodynamic ensembles[1] Microcanonical Canonical Grand canonical Fixed variables N, E, V N, T, V μ, T, V Microscopic features • Grand partition function Macroscopic function • Grand potential Calculation methodsEdit Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities. There are some cases which allow exact solutions. • For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics). • Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of Maxwell–Boltzmann statistics, Fermi–Dirac statistics, and Bose–Einstein statistics.[2] • A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few toy models.[10] Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model. Monte CarloEdit One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level. • For rarefied non-ideal gases, approaches such as the cluster expansion use perturbation theory to include the effect of weak interactions, leading to a virial expansion.[3] • For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.[3] • Molecular dynamics computer simulations can be used to calculate microcanonical ensemble averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions. • Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful. Non-equilibrium statistical mechanicsEdit There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example: All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.) In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections. Stochastic methodsEdit One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier. • Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem). The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas. A quantum technique related in theme is the random phase approximation. • BBGKY hierarchy: In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions. • Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions): A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations. Near-equilibrium methodsEdit Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.[3]:664 This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. A few of the theoretical tools used to make this connection include: Hybrid methodsEdit An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.[11][12] Applications outside thermodynamicsEdit The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in: In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.[13] This was the first-ever statistical law in physics.[14] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[15] Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell’s paper and spent much of his life developing the subject further. Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.[16] Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem. The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884.[17][note 5] "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched.[18] Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.[1] Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.[2] See alsoEdit 1. ^ The term statistical mechanics is sometimes used to refer to only statistical thermodynamics. This article takes the broader view. By some definitions, statistical physics is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics. 2. ^ The probabilities in quantum statistical mechanics should not be confused with quantum superposition. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble. 3. ^ Statistical equilibrium should not be confused with mechanical equilibrium. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium. 4. ^ The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system. 5. ^ According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist James Clerk Maxwell in 1871. From: J. Clerk Maxwell, Theory of Heat (London, England: Longmans, Green, and Co., 1871), p. 309: "In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus." 1. ^ a b c d e f g h i Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons. 2. ^ a b c d e Tolman, R. C. (1938). The Principles of Statistical Mechanics. Dover Publications. ISBN 9780486638966. 3. ^ a b c d Balescu, Radu (1975). Equilibrium and Non-Equilibrium Statistical Mechanics. John Wiley & Sons. ISBN 9780471046004. 4. ^ Jaynes, E. (1957). "Information Theory and Statistical Mechanics". Physical Review. 106 (4): 620. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620. 5. ^ a b J. Uffink, "Compendium of the foundations of classical statistical physics." (2006) 6. ^ Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. p. 227. ISBN 9780070518001. 7. ^ Touchette, H. Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels. Journal of Statistical Physics, 159(5), 987-1016, 2015. doi:10.1007/s10955-015-1212-2. 8. ^ Ledoux M. 2001 The Concentration of Measure Phenomenon. (Mathematical Surveys & Monographs No. 89). Providence: AMS. doi:10.1090/surv/089. 9. ^ Gorban, AN; Tyukin, IY. Blessing of dimensionality: mathematical foundations of the statistical physics of data. Philosophical Transactions of the Royal Society A 376 (2118), 20170237, 2018. doi:10.1098/rsta.2017.0237. 10. ^ Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc. ISBN 9780120831807. 11. ^ Altshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. (1982). "Effects of electron-electron collisions with small energy transfers on quantum localisation". Journal of Physics C: Solid State Physics. 15 (36): 7367. Bibcode:1982JPhC...15.7367A. doi:10.1088/0022-3719/15/36/018. 12. ^ Aleiner, I.; Blanter, Y. (2002). "Inelastic scattering time for conductance fluctuations". Physical Review B. 65 (11): 115317. arXiv:cond-mat/0105436 . Bibcode:2002PhRvB..65k5317A. doi:10.1103/PhysRevB.65.115317. 13. ^ See: 16. ^ Ebeling, Werner; Sokolov, Igor M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. World Scientific Publishing Co. Pte. Ltd. pp. 3–12. ISBN 978-90-277-1674-3. (section 1.2) 18. ^ Mayants, Lazar (1984). The enigma of probability and physics. Springer. p. 174. ISBN 978-90-277-1674-3. External linksEdit
91a62f1d4d4856b0	Open main menu Force examples.svg Common symbols F, F, F SI unitnewton (N) Other units dyne, pound-force, poundal, kip In SI base unitskg·m/s2 Derivations from other quantities F = m a DimensionM L T−2 Concepts related to force include: thrust, which increases the velocity of an object; drag, which decreases the velocity of an object; and torque, which produces changes in rotational speed of an object. In an extended body, each part usually applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. Such internal mechanical stresses cause no acceleration of that body as the forces balance one another. Pressure, the distribution of many small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of solid materials, or flow in fluids. Development of the concept Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of friction, and a consequently inadequate view of the nature of natural motion.[2] A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved for nearly three hundred years.[3] By the early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light, and also provided insight into the forces produced by gravitation and inertia. Pre-Newtonian concepts Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground and that they will stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force.[7] This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. The place where the archer moves the projectile was at the start of the flight, and while the projectile sailed through the air, no discernible efficient cause acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation demands a continuum like air for change of place in general.[8] Aristotelian physics began facing criticism in medieval science, first by John Philoponus in the 6th century. The shortcomings of Aristotelian physics would not be fully corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.[9] Newtonian mechanics Sir Isaac Newton described the motion of all objects using the concepts of inertia and force, and in doing so he found they obey certain conservation laws. In 1687, Newton published his thesis Philosophiæ Naturalis Principia Mathematica.[3][10] In this work Newton set out three laws of motion that to this day are the way forces are described in physics.[10] First law Newton's First Law of Motion states that objects continue to move in a state of constant velocity unless acted upon by an external net force (resultant force).[10] This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". That is, Newton's empirical First Law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making rest physically indistinguishable from non-zero constant velocity, Newton's First Law directly connects inertia with the concept of relative velocities. Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". The laws of physics are the same in every inertial frame of reference, that is, in all frames related by a Galilean transformation. For instance, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change as a result of its motion. If a person riding within the vehicle throws a ball straight up, that person will observe it rise vertically and fall vertically and not have to apply a force in the direction the vehicle is moving. Another person, observing the moving vehicle pass by, would observe the ball follow a curving parabolic path in the same direction as the motion of the vehicle. It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and everything inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction of the vehicle. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Inertia therefore applies equally well to constant velocity motion as it does to rest. Though Sir Isaac Newton's most famous equation is Second law A modern statement of Newton's Second Law is a vector equation:[Note 1] where is the momentum of the system, and is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time.[10] By the definition of momentum, where m is the mass and is the velocity.[4]:9-1,9-2 If Newton's second law is applied to a system of constant mass,[Note 2] m may be moved outside the derivative operator. The equation then becomes Newton never explicitly stated the formula in the reduced form above.[11] The use of Newton's Second Law as a definition of force has been disparaged in some of the more rigorous textbooks,[4]:12-1[5]:59[12] because it is essentially a mathematical truism. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.[13][14] Third law Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if is the force of body 1 on body 2 and that of body 2 on body 1, then This law is sometimes referred to as the action-reaction law, with called the action and the reaction. Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies,[15][Note 3] and thus that there is no such thing as a unidirectional force or a force that acts on only one body. In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero: More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.[4]:19-1[5] Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved.[16] In a system of two particles, if is the momentum of object 1 and the momentum of object 2, then Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.[4][5] Special theory of relativity remains valid because it is a mathematical definition.[17]:855–876 But for relativistic momentum to be conserved, it must be redefined as: where is the rest mass and the speed of light. The relativistic expression relating force and acceleration for a particle with constant non-zero rest mass moving in the direction is: is called the Lorentz factor.[18] In the early history of relativity, the expressions and were called longitudinal and transverse mass. Relativistic force does not produce a constant acceleration, but an ever-decreasing acceleration as the object approaches the speed of light. Note that approaches asymptotically an infinite value and is undefined for an object with a non-zero rest mass as it approaches the speed of light, and the theory yields no prediction at that speed. If is very small compared to , then is very close to 1 and is a close approximation. Even for use in relativity, however, one can restore the form of through the use of four-vectors. This relation is correct in relativity when is the four-force, is the invariant mass, and is the four-acceleration.[19] Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction.[3] When two forces act on a point particle, the resulting force, the resultant (also called the net force), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram.[4][5] The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. However, if the forces are acting on an extended body, their respective lines of application must also be specified in order to account for their effects on the motion of the body. Static equilibrium was understood well before the invention of classical mechanics. Objects that are at rest have zero net force acting on them.[22] The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.[3] Forces in quantum mechanics The notion "force" keeps its meaning in quantum mechanics, though one is now dealing with operators instead of classical variables and though the physics is now described by the Schrödinger equation instead of Newtonian equations. This has the consequence that the results of a measurement are now sometimes "quantized", i.e. they appear in discrete portions. This is, of course, difficult to imagine in the context of "forces". However, the potentials V(x,y,z) or fields, from which the forces generally can be derived, are treated similarly to classical position variables, i.e., . This becomes different only in the framework of quantum field theory, where these fields are also quantized. However, already in quantum mechanics there is one "caveat", namely the particles acting onto each other do not only possess the spatial variable, but also a discrete intrinsic angular momentum-like variable called the "spin", and there is the Pauli exclusion principle relating the space and the spin variables. Depending on the value of the spin, identical particles split into two different classes, fermions and bosons. If two identical fermions (e.g. electrons) have a symmetric spin function (e.g. parallel spins) the spatial variables must be antisymmetric (i.e. they exclude each other from their places much as if there was a repulsive force), and vice versa, i.e. for antiparallel spins the position variables must be symmetric (i.e. the apparent force must be attractive). Thus in the case of two fermions there is a strictly negative correlation between spatial and spin variables, whereas for two bosons (e.g. quanta of electromagnetic waves, photons) the correlation is strictly positive. Thus the notion "force" loses already part of its meaning. Feynman diagrams Fundamental forces All of the known forces of the universe are classified into four fundamental interactions. The strong and the weak forces are nuclear forces that act only at very short distances, and are responsible for the interactions between subatomic particles, including nucleons and compound nuclei. The electromagnetic force acts between electric charges, and the gravitational force acts between masses. All other forces in nature derive from these four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between atoms of two surfaces, and the Pauli exclusion principle,[24] which does not permit atoms to pass through each other. Similarly, the forces in springs, modeled by Hooke's law, are the result of electromagnetic forces and the Pauli exclusion principle acting together to return an object to its equilibrium position. Centrifugal forces are acceleration forces that arise simply from the acceleration of rotating frames of reference.[4]:12-11[5]:359 The fundamental theories for forces developed from the unification of different ideas. For example, Sir. Isaac Newton unified, with his universal theory of gravitation, the force responsible for objects falling near the surface of the Earth with the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons.[25] This standard model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation. The complete formulation of the standard model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations suggest that the standard model is incomplete. A Grand Unified Theory that allows for the combination of the electroweak interaction with the strong force is held out as a possibility with candidate theories such as supersymmetry proposed to accommodate some of the outstanding unsolved problems in physics. Physicists are still attempting to develop self-consistent unification models that would combine all four fundamental interactions into a theory of everything. Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is string theory.[6]:212–219 The four fundamental forces of nature[26] Property/Interaction Gravitation Weak Electromagnetic Strong (Electroweak) Fundamental Residual Particles mediating: Graviton (not yet observed) W+ W Z0 γ Gluons Mesons to quarks Strength in the scale of 10−36 10−7 1 Not applicable to hadrons Images of a freely falling basketball taken with a stroboscope at 20 flashes per second. The distance units on the right are multiples of about 12 millimeters. The basketball starts at rest. At the time of the first flash (distance zero) it is released, after which the number of units fallen is equal to the square of the number of flashes. For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.[4][5] Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.[28] Combining these ideas gives a formula that relates the mass ( ) and the radius ( ) of the Earth to the gravitational acceleration: where the vector direction is given by , is the unit vector directed outward from the center of the Earth.[10] In this equation, a dimensional constant is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant,[29] though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation states that the force on a spherical object of mass due to the gravitational pull of mass is where is the distance between the two objects' centers of mass and is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.[10] This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of perturbation analysis[30] were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.[31] Instruments like GRAVITY provide a powerful probe for gravity force detection.[32] Mercury's orbit, however, did not match that predicted by Newton's Law of Gravitation. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however no such planet could be found. When Albert Einstein formulated his theory of general relativity (GR) he turned his attention to the problem of Mercury's orbit and found that his theory added a correction, which could account for the discrepancy. This was the first time that Newton's Theory of Gravity had been shown to be inexact.[33] The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two charges.[17]:519 The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the superposition principle. Coulomb's law unifies all these observations into one succinct statement.[34] Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force.[35]:4-6 to 4-8 Thus the electric field anywhere in space is defined as where is the magnitude of the hypothetical test charge. where is the electromagnetic force, is the magnitude of the charge of the particle, is the electric field, is the velocity of the particle that is crossed with the magnetic field ( ). The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs.[36] These "Maxwell Equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.[37] However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave–particles known as photons. In QED, photons are the fundamental exchange particle, which described all interactions relating to electromagnetism including the electromagnetic force.[Note 4] Strong nuclear There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force[17]:940 is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force[17]:951 is responsible for the decay of certain nucleons into leptons and other types of hadrons.[4][5] The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD).[38] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The (aptly named) strong interaction is the "strongest" of the four fundamental forces. The strong force only acts directly upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between nucleons in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluons, which form part of the virtual pi and rho mesons, which classically transmit the nuclear force (see this topic for more). The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement. Weak nuclear The weak force is due to the exchange of the heavy W and Z bosons. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately 1015 kelvins. Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang. Non-fundamental forces Normal force FN represents the normal force exerted on the object. The normal force is due to repulsive forces of interaction between atoms at close contact. When their electron clouds overlap, Pauli repulsion (due to fermionic nature of electrons) follows resulting in the force that acts in a direction normal to the surface interface between two objects.[17]:93 The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.[4][5] Friction is a surface force that opposes relative motion. The frictional force is directly related to the normal force that acts to keep two solid objects separated at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction. The static friction force ( ) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction ( ) multiplied by the normal force ( ). In other words, the magnitude of the static friction force satisfies the inequality: The kinetic friction force ( ) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals: Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and unstretchable. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action-reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.[39] By connecting the same string multiple times to the same object through the use of a set-up that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. However, even though such machines allow for an increase in force, there is a corresponding increase in the length of string that must be displaced in order to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.[4][5][40] Elastic force Fk is the force that responds to the load on the spring An elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position.[41] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If is the displacement, the force exerted by an ideal spring equals: Continuum mechanics When the drag force ( ) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object ( ), the object reaches a state of dynamic equilibrium at terminal velocity. where is the volume of the object in the fluid and is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[4][5] is the velocity of the object.[4][5] More formally, forces in continuum mechanics are fully described by a stresstensor with terms that are roughly defined as where is the relevant cross-sectional area for the volume for which the stress-tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.[3][5]:133–134[35]:38-1–38-11 Fictitious forces There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force.[42] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[4][5] Because these forces are not genuine they are also referred to as "pseudo forces".[4]:12-11 In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry. As an extension, Kaluza–Klein theory and string theory ascribe electromagnetism and the other fundamental forces respectively to the curvature of differently scaled dimensions, which would ultimately imply that all forces are fictitious. Rotations and torque Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system. Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force is defined relative to an arbitrary reference point as the cross-product: is the moment of inertia of the body is the angular acceleration of the body. This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation. [43] where is the angular momentum of the particle. Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,[44] and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques. Centripetal force For an object accelerating in circular motion, the unbalanced force acting on the object equals:[45] where is the mass of the object, is the velocity of the object and is the distance to the center of the circular path and is the unit vector pointing in the radial direction outwards from the center. This means that the unbalanced centripetal force felt by any object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. The unbalanced force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.[4][5] Kinematic integrals which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem). Similarly, integrating with respect to position gives a definition for the work done by a force:[4]:13-3 which is equivalent to changes in kinetic energy (yielding the work energy theorem).[4]:13-3 Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change in a time interval dt:[4]:13-2 with the velocity. Potential energy Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.[4][5] Conservative forces Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector emanating from spherically symmetric potentials.[48] Examples of this follow: For gravity: where is the gravitational constant, and is the mass of object n. For electrostatic forces: where is electric permittivity of free space, and is the electric charge of object n. For spring forces: where is the spring constant.[4][5] Nonconservative forces For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations that yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.[4][5] Units of measurement The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.[49] The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes. The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2.[49] The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force.[49] An alternative unit of force in a different foot-pound-second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.[49] The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law. The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass.[49] The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated; however it still sees use for some purposes as expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. Other arcane units of force include the sthène, which is equivalent to 1000 N, and the kip, which is equivalent to 1000 lbf. Units of force (SI unit) dyne kilogram-force, pound-force poundal See also Ton-force. Force measurement See also 4. ^ For a complete library on quantum mechanics see Quantum mechanics – References 1. ^ Nave, C. R. (2014). "Force". Hyperphysics. Dept. of Physics and Astronomy, Georgia State University. Retrieved 15 August 2014. 3. ^ a b c d e f g h University Physics, Sears, Young & Zemansky, pp.18–38 11. ^ Howland, R. A. (2006). Intermediate dynamics a linear algebraic approach (Online-Ausg. ed.). New York: Springer. pp. 255&ndash, 256. ISBN 9780387280592. 13. ^ Jammer, Max (1999). Concepts of force : a study in the foundations of dynamics (Facsim. ed.). Mineola, N.Y.: Dover Publications. pp. 220–222. ISBN 9780486406893. 17. ^ a b c d e Cutnell & Johnson 2003 19. ^ Wilson, John B. "Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physics". The Science Realm: John's Virtual Sci-Tech Universe. Archived from the original on 26 June 2009. Retrieved 2008-01-04. 20. ^ "Introduction to Free Body Diagrams". Physics Tutorial Menu. University of Guelph. Archived from the original on 2008-01-16. Retrieved 2008-01-02. 21. ^ Henderson, Tom (2004). "The Physics Classroom". The Physics Classroom and Mathsoft Engineering & Education, Inc. Archived from the original on 2008-01-01. Retrieved 2008-01-02. 25. ^ "Fermions & Bosons". The Particle Adventure. Archived from the original on 2007-12-18. Retrieved 2008-01-04. 26. ^ "Standard model of particles and interactions". Contemporary Physics Education Project. 2000. Retrieved 2 January 2017. 27. ^ Cook, A. H. (1965). "A New Absolute Determination of the Acceleration due to Gravity at the National Physical Laboratory". Nature. 208 (5007): 279. Bibcode:1965Natur.208..279C. doi:10.1038/208279a0. 28. ^ a b Young, Hugh; Freedman, Roger; Sears, Francis and Zemansky, Mark (1949) University Physics. Pearson Education. pp. 59–82 30. ^ Watkins, Thayer. "Perturbation Analysis, Regular and Singular". Department of Economics. San José State University. 31. ^ Kollerstrom, Nick (2001). "Neptune's Discovery. The British Case for Co-Prediction". University College London. Archived from the original on 2005-11-11. Retrieved 2007-03-19. 32. ^ "Powerful New Black Hole Probe Arrives at Paranal". Retrieved 13 August 2015. 33. ^ Siegel, Ethan (20 May 2016). "When Did Isaac Newton Finally Fail?". Forbes. Retrieved 3 January 2017. 34. ^ Coulomb, Charles (1784). "Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de metal". Histoire de l'Académie Royale des Sciences: 229–269. 35. ^ a b c Feynman volume 2 37. ^ Duffin, William (1980). Electricity and Magnetism, 3rd Ed. McGraw-Hill. pp. 364–383. ISBN 0-07-084111-X. 38. ^ Stevens, Tab (10 July 2003). "Quantum-Chromodynamics: A Definition – Science Articles". Archived from the original on 2011-10-16. Retrieved 2008-01-04. 42. ^ Mallette, Vincent (1982–2008). "Inwit Publishing, Inc. and Inwit, LLC – Writings, Links and Software Distributions – The Coriolis Force". Publications in Science and Mathematics, Computing and the Humanities. Inwit Publishing, Inc. Retrieved 2008-01-04. Further reading • Corben, H.C.; Philip Stehle (1994). Classical Mechanics. New York: Dover publications. pp. 28–31. ISBN 0-486-68063-0. • Verma, H.C. (2004). Concepts of Physics Vol 1 (2004 Reprint ed.). Bharti Bhavan. ISBN 8177091875. External links
6fdea6501764b39b	Dismiss Notice Dismiss Notice Join Physics Forums Today! Alternatives to Quantum Mechanics? 1. May 28, 2010 #1 I am curious what are other theories of physics at very small scales besides quantum mechanics? Especially those that aren't probabilistic and undeterminitive, (if there are any at all!) 2. jcsd 3. May 28, 2010 #2 There are none. 4. May 29, 2010 #3 Quantum mechanics is, by definition, physics at the smallest scale. Of course there is also nuclear physics, atomic physics, etc., but QM is the absolute smallest scale because QM revolves around quanta (which are elementary). However, there is, of course, a chance that some things we believe to be quanta (i.e. photons) are composed of something even smaller, meaning that they aren't actually quanta. If you are just looking for other theories, there are lots of theories within QM, nuclear physics, etc. (such as string theory and other unification/quantum gravity theories), but if they deal with the smallest scale (quanta) they will be classified as QM. 5. May 29, 2010 #4 User Avatar Staff Emeritus Science Advisor Gold Member This is a difficult question to answer, because what some consider "another theory", others consider it as a "different interpretation of quantum theory". As you might know, although the formalism (that is, the calculations) of quantum mechanics give very good results, many people aren't happy with what one could call the "philosophy" behind it. So people have, over the last 80 years or so, porposed alternative interpretations of what the calculations mean. And in doing so, they sometimes added formal elements which aren't part of the original quantum formalism. Probably the most famous such "version" is Bohmian mechanics. Other interpretations limit themselves to re-assigning different meanings to the formal elements of quantum mechanics "as we know it" (and can hence with less doubt be called "interpretations"). The "philosophical" viewpoint of these different interpretations is vastly different, and sometimes "religious wars" are fought over them. But as far as I know, on all "experimentable" physics, they come out the same observational results. That's why, if not just different interpretations (because added formal elements), they are physical theories that are equivalent to quantum mechanics (at least for all practical purposes). In other words, on the "hard scientific" level, all these theories are empirically indistinguishable, and it is almost a matter of semantics to call them "different theories", although on the philosophical level, they are totally different, for instance on the level of "determinism", or "stochastic". There are deterministic interpretations of quantum mechanics (Bohmian mechanics for instance). There are "strict" interpretations of the existing formalism (many worlds). There are "shut up and calculate" interpretations (what there "is" is unknowable, we can only calculate what we observe)... For everybody's taste, there must be something. All of these viewpoints have some kinky points, that's probably why none of it stands out clearly above the others... but they all talk about "quantum mechanics" in some or other form, as they are observationally equivalent. Apart from crackpot websites, I've never seen totally different theories in the same domain of applicability, without them trying to establish some kind of equivalence to quantum mechanics (at least for all practical purposes). And there's a good reason for that: the kwantitative predictions of quantum mechanics are impressively verified by experiment, so there is very little "wiggle room" without being in contradiction with observation. In other words, no matter all philosophical and even formal difficulties sometimes, quantum mechanics works very well as a scientific theory. 6. May 29, 2010 #5 User Avatar Science Advisor One should distinguish between 1) "quantum mechanics" as the formalism and 2) the application of quantum mechanics in the physics of atoms, nuclei, elementary particles (in the advanced formalism called quantum field theory) and even in string theory and quantum gravity (which is currently not a final theory but a bunch of research programs). Regarding 2) there may be even smaller scales (below electrons and quarks) even if there is no hint why this should be the case. But there is no attempt to use something different than 1) So Regarding 1) there is no other approach at all. What is debated is its interpretation. 7. May 30, 2010 #6 Thank you everyone, I am very satisfied with the responses. I was just worried a little bit when Einstein did not like the theory, (or did he not like the philosophy?) because he was a very intuitive person. I can feel slightly more assured, since there being no other theory TO learn, (and I would be undoing quit a bit if I were to come up with some theory of my own, not to mention the vast amount of time it would take and not take advantage of all the work done already) to continue and learn quantum mechanics. So with this, I have a few more questions. What math does quantum mechanics use? I have currently finished high school Calculus AP, (so I suppose first year college calculus,) and am prepared to learn vector calculus (and of course MUCH more afterwards). Also, which would be better to learn first, QM or General Relativity? (that probably also depends on which is more math intensive.) Thanks again 8. May 30, 2010 #7 QM uses linear algebra, abstract algebra, analysis, groups, tensors, spinors and topology, to name few. GR uses only linear algebra, analysis and tensors. GR was easier for me to understand, than QM. Plus, GR is somewhat complete, as opposed to QM, which is still under construction, a beta version of a kind. 9. May 30, 2010 #8 Your reply is very misleading in my opinion. For elementary QM you don't need any deep understanding of anything else than linear algebra and a little bit of functional analysis (not rigorous, and this is usually covered in introductory books). The rest you need only for more advanced aspects of QM. (More advanced aspects of ) GR uses all the subjects you mention too, plus much more. In my opinion one should start with some linear algebra and then grasp the elementary aspects of QM. At this level QM is mostly conceptually difficult, while for GR there might be some subtleties connected to the math and confusion might arise without any understanding of Riemannian geometry. Not to mention that one should also know SR before studying GR. What makes you say that? If there is any such opinion, it's probably the other way around. QM is much more tested experimentally in various extremes and situations than GR. 10. May 30, 2010 #9 User Avatar Science Advisor Well, there aren't any known instances where quantum mechanics fails, which means we don't need an alternative. We could of course have a simpler theory, but this is very difficult, because quantum mechanics is in fact quite simple. Not simple in terms of mathematics, no. But simple in terms of relying on very few basic assumptions (postulates). It's not simple in the sense that it's easily intuitive to humans, and it's not simple in the sense that the math are simple. But those two latter concerns don't count because they're anthropocentric; they're a matter of our opinion. Nature is not under any obligation to make things easily understandable by humans! QM has in fact been simplified (as in, reduced to fewer and more basic postulates) since it was first formulated. E.g. 'Spin' was originally a postulate, but is now known to be a consequence of special relativity. Of course, then there also are alternatives: QM was not the first theory developed to explain QM phenomena! For instance, there's the Bohr model of the atom - which is not simpler in terms of relying on fewer assumptions, but simpler in mathematical terms. There's also the field of chemistry - and the rules of chemistry are largely simplifications of quantum mechanical results and properties. The structure of the periodic table, the octet rule, orbital hybridisation, etc. Quantum chemistry is an expanding field, but few chemists solve the Schrödinger equation when they do theory - nor do they need to; their approximate models work well enough most of the time. 11. May 30, 2010 #10 I'm intrigued by this statement. Could you perhaps take a moment to explain how or point me to an appropriate reference? 12. May 30, 2010 #11 User Avatar Staff Emeritus Science Advisor Gold Member In fact, GR and QM are totally different subjects. As QM has much more applications (like chemistry!), I would go for QM first, and leave GR for later idle moments :-) However, before being able to grasp somewhat seriously QM, you must first have a solid grasp of advanced aspects of classical, Newtonian mechanics, which you probably haven't seen. Now, if you want to get a conceptual head start of quantum mechanics, go for Feynman's lectures volume 3. It is an a-typical introduction to QM, which might seem strange, but it gives you a very good idea of quantum mechanics without the full machinery and maths. And it IS serious. (and BTW, read the first two volumes also, they are worth their weight in gold). 13. May 30, 2010 #12 The Poincaré group is the group of all symmetries in SR (translations, rotations and Lorentz boosts). Particles can be classified as representations (irreducible for elementary particles) of the Poincaré group. If you know some group theory (Lie groups and representation theory minimum) I can find some references for you. But loosely speaking it turns out that these (irreducible) representations can be classified by their mass and something called a Little group. 1. For particles with [tex]m^2 > 0[/tex] (massive particles) the little group is SU(2), which physically is nothing but spin (s= 0, 1/2, 1, 3/2...). 2. For [tex]m^2 = 0[/tex] (massless particles) the little group describes helicity. 3. The last possibility is [tex]m^2 < 0[/tex] (tachyons), for which I don't remember the Little group. 14. May 30, 2010 #13 User Avatar Science Advisor element4 already responded to that. One can show that the structure group of spacetime is (loosely speaking) SO(3,1) ~ SU(2)*SU(2). And from these two SU(2) factors one can derive spin with integer and half integer values. The integer values already follow from the SO(3) rotation group in three dimensions, but the half-integer values came as a surprise (and Pauli was attacked by his teacher Sommerfeld introducing factors of 1/2 to explain the Hydrogen spectrum; Sommerfeld was afraid that sooner or later Pauli would introduce 1/3 etc. spoiling quantization at all :-) 15. May 30, 2010 #14 User Avatar Science Advisor In addition to 2. there are for [tex]m^2 = 0[/tex] the so-called continuous (or infinite) helicity states w/o any quantization. I still have to figure out why they are unphysical. 16. May 30, 2010 #15 User Avatar Science Advisor I fully agree. QM - especially its application to atoms, nuclei and elementary particles - all based on the same principles and math - have been tested over decades w/o any hint for any physics not compatible with QM. The only open issue is harmonization of quantum field theory with general relativity to quantum gravity.
f974d441f1d4350c	Music and The Nature of Existence In music, notes start out as whole. A whole note is fundamental to music. From there the only possible, probable and necessary disposition is another whole note or half notes, quarter notes etc. Ohm’s Law. The structure is the same but the mass, volume and measure changes. Music, Pascal’s Triangle and Nature are fundamentally about introspection and growth by these means. You learn by jam session. Nature is learning by jam session. Existence is a jam session. Music, math and nature are singing as they progress in mosaic repertory of emphatic explicate of itself; the quarter note in rhythm signature with the Sun and moon: Time. You are whole necessarily because there exists something called a, “whole note.” This is tacit proof of our own whole existence. A whole note is position positive with variations of itself artistically displayed – Pi. The nature of Music is the nature of existence. Conductors of Emphasis in the key of Truth You are emphasis in the key of Truth. Parallel is the G chord is emphasis in the Key of E among the polyphony of open chord harmony. Everything is in the key of Einstein’s E, which his E of energy – which is Truth. Just like notes inside E can’t leave the Key of E, we are notes that cannot leave the Truth. Your body is the screen that’s the extraction method of distinguishing the G chord from the polyphony of flowing sounds; Schrödinger equation is the math of harmonious sounds that are always on and playing. The Heisenberg uncertainty principle, these notes suspended in potential courses of action. The Heisenberg uncertainty principle is Harmony before procedural execution; before you play the song. In order to emphasize the most exciting chord you can imagine, there needs to be a convection of sound. The G chord needs to be isolated with definition of being. So it can expand in the same space isolated. That’s emphasis. So we need a cogent mask to filter out all the expatiating harmony of E11#9. We need to single out one of the strains of sound so we can enjoy it by itself. So all harmony is altruistically giving of it’s potential and going with your designs to isolate a specific chord. Existence is changing its course of action because of you. Why would we want to isolate the G harmony if all harmony is endlessly resplendently expatiating all over the place? Why do you like some songs and not others? Because we have 5 electric strings in our head and are purely just harmonizing with various songs (people) and reflexively choosing the ones that have the best sound. We are guided by our Intuition of Truth. We know exactly what key we’re in and we are excited to phase in to people’s songs and jam with them. We do this with everyone through Empathy with Emphasis of likeness. Emphasis being “Emotional Phasing.” more specifically, we create an Emotional Pathway with them and then Emotionally Phase in with them. It’s Fusion. It could be called Emfusis. Or Emphasis. I love language. We Emotionally Phase in with the emotional stasis of a person whose chord progression that we like. Phasing in, is fusion of music between two or more points. If we don’t like the song being expanded by a person and can’t leave we will refer to the space as hellish. That’s being forced to listen to terrible out of tune music in a closed space for extended duration. Sometimes this is a “job.” In order to phase in from Schrödinger’s “Phase State” of polyphonic totality, we need a body to convect strains of notes into a song that is exciting to our personal emphasized chord progression. We need a procedural partition that gives us the ability to access some existing harmony while ordering the totality around it. That’s what your fingers are doing on the fret board. That’s why what your body and mind is doing with the totality of life harmonics. You are always in the key of Truth. You’ll know truth when you see it because you will immediately harmonize with it. You are an Altruist because you volunteer to harmonize with everything, even if the harmony is weak. Even if the song is weak you solider on tenaciously in spite of bad music. You are a unconditional love outlet with tacit knowledge of universal harmony. The suggestion I make is to figure out when the music goes bad or changes key and the having the strength to modulate to your preferred key or just stop playing bad music with them. The goal is always play music that you like. If you get stuck playing music you don’t like with someone and can’t leave, because of your persistent altruism, then stop playing with them and find a better rock band. We are the conductor of emphasis in design harmonics.
dfa3173e2f6dda2e	Major Concepts Concepts add up and make up much of what John Searle called the “background”. He recognized, like others before him, e.g. Quine or Wittgenstein, that making sense of the world can’t be based just on the actual empiric “impressions” (data, if you like), but rather that we need concepts that in turn are embedded in the Form of Life whenever we are going to interpret something. Bringing in the concept of “concept” here I have to emphasize right at the beginning that concepts can’t be positively, which means: exhaustively, defined. Concepts articulate directly towards meta-physics, they are even part of it. In the last section of this text we will introduce a structural approach that relates the concept of “concept” into a wider perspective. Yet, describing the role of concepts as “background” certainly does not satisfy “completely”. For Deleuze, for instance, concepts are part of an activity that forms the plane of immanence, first, for each series of thought. Secondarily, however, this gives rise to what he calls the absolute plane of immanence, or pure immanence. In his perspective, this plane is not itself a concept. Yet, here is not the place to criticize Deleuze’s braiding of thought, a life, immanence and transcendence. We try to accomplish both a critique and an extension of Deleuze’s conceptualization elsewhere. We’d like just to mention that concepts should not be messed with definitions, as it is usually done in analytical philosophy, or by functionalists of any kind, creating nonsense such as formal (syntactic) semantics or analytic ontology. Here in his text we’d like to provide brief summaries about important concepts that form the background and the “Image of Thought” of the writing on this blog; it may also be helpful for the reader of any of articles on this blook, since I honestly can’t expect that it is easy to navigate those texts due to the fact that most of them are crossing the “borders” of disciplines or domains. Of course, this list does not form an exhaustive glossary. It just marks the most salient topics from a cross-disciplinary perspective, as far as they are constitutive for a particular attitude I try to actualize, or say, a certain Image of Thought. Its cornerstones? Openness and being anti-axiomatic and heading for consistency, which implies an awareness for meta-critique and conditionability: the question about the conditions of the possible, particularly when it comes to novelty, creativity and imagination. I am strongly convinced that this should not be just a concern for philosophy. It is easy to see why: It concerns both strictly human beings and of a-human entities such as an urban context, epistemic machines or the culture at large. Such, I think I have to emphasize even right to the “beginning” that quite in contrast to analytical positivist stances, the concept of “concept” can’t be conceived as something that could be defined apriori to a particular use case (see this for more details about concepts). Concepts are not only the immaterial joints between empirical experience and intentionality, both as plan and the category of the mental. There are good arguments to regard them as indispensable for any thought, not just in the case of human beings. Together they form a virtual network of potential joints that enable the actualization of potential and at the same time provide the site and the tool to harvest those actualizations. Such we find two very different types of articulations “in” the concept of concept, which “decohere” into one of the forms upon our choreostemic moves. This extends our notion of “understanding” as we described it previously. Put into a nutshell, we conceived of “understanding” as the language game which we humans use to indicate the belief that the extension of the background (scaffold) by the topic at hand is shared between individuals in such a way as to indicate the abilty to reproduce the same effect as anyone else could have triggered understanding the same thing. This sharing of references and inducing of capabilities makes up a lot of what we usually call teaching. Saying “I understand” means to indicate the belief into a resemblance, resonance, or even alignment regarding the way concepts are harvested, linked and used. Throughout this blog we are going to draw upon a variety of abstract concepts. Some are more important than others, some we imported from the complexes of concepts of other authors, some more we assimilated and reshaped them in order to fit them to the rest, a few we supposedly invented (among them: orthoregulation, aspections, choreostemic space, delocutionary acts). Together they form the plane of immanence of our writing. They recur frequently, contextualized in different ways. In order to facilitate a more easy access to the essays, we provide a collection of very brief introductions into those concepts, of which we think that they are “more basic” than others. Of course, and to give only a few examples, something like abstract associativity, the abstract model, categories (in the sense of mathematical category theory), renormalization as a contribution to a theory of generalized measurement, generalized contexts, or the notion of generalized evolution written in purely probabilistic terms without any reference to biology are all quite abstract and also quite fundamental concepts. Yet, they could be found and fully comprehended only on the basis of the basic stuff that we are going to introduce (as briefly as possible) here. For instance, the transition from matter and the material, aka the “body,” to the realm of the immaterial where we find information is still considered by many as one of the core problems in philosophy. We consider it firstly as a pseudo-problem, though, and secondly we hold that any of its formulations is based on a particular (and particularly weird) instantiation of the basic concepts described below. A second example would be provided by the concepts of numbers, or more generally, the concept of the “quantum” or “quantitability” as Bühlmann calls it, which denotes the potential and the conditions for rendering something quantifiable. Quantitability is closely related to the conditions for making something separable, identifiable. It has much to do with the abstract (transcendental) conditions that need to be met in order for becoming able to select something. Such, we are also close to the fields of individuation, associativity, the question about the relation between the material and immaterial. All of those, however, and thus also the question about the role of numbers and mathematics and their form are shaped through the space provided by the concepts below. Another example (the last we will give here) is the problem of causality that is haunting philosophy, natural science as well as the humanities for centuries now. Any attitude towards “causality” is configured by the basic concepts below, which allow us to recognize that “causality” is neither a “phenomenon” nor a transcendental category. It would constitute even a shortfall to consider it merely as a Language Game. It is a name for a particular way to construct the “world” and this name can’t be separated from the “image”, the way we perform this construction. We may ask, in turn , well, what then is the world, how to conceive this? We could answer that we conceive it as a fluid, broiling collection of strands of potential relations that we arrange in order to deal with other “things” than “ourselves”. Of course, this “ourselves” starts to get slippery quite soon. Anyway. The important thing is that others see it differently. And it is this difference that lets appear the concepts below. In the following, I first provide the list of those concepts, before briefly describing each item, garnished with links to the articles and external sources, and last but not least also cross-references within this text that will be indicated by an arrow (➔). The list comprises the following items (active links) 1. Differential and Virtuality 2. Lagrangean Abstraction 3. Probabilization 4. Aspectional Space 5. Localized Space 6. Decoherence 7. Mechanism 8. Vanishing Regress 9. Orthoregulation 10. Transcendence 11. Elementarization 12. Language Game 13. Archaeology 14. Processual Indicative 15. Inverted Experimental Plan 16. Diagram 17. Self-referentiality 18. Choreostemic Space 1. Differential and Virtuality Originally, the differential is a mathematical concept. It has been invented almost at the same time by Newton and by Leibniz, albeit in very different contexts. The differential can be conceived as an inverse relation to the mathematical concept of integration. To integrate a function, i.e. a relation between two quantifiable concepts, one has (1) to find a relation that could yield the starting concepts when integrated (mathematically), then (2) to define the side conditions for the operation of integrating. Quantifiable concepts are often called variables, or when mapped to a Cartesian coordinate system, dimensions. Of course, many relations can’t be differentiated or integrate analytically. Another trick in complicated contexts is to build the differential only with regard to certain aspects, or dimensions. This is called partial differential equation. In physics, the differential is used to describe the regularity of change, each of position over time as velocity or, as second-degree differential, as acceleration, or it is used to calculate the cumulative effect of a dynamically changing, yet describable relationship between two quantifiable concepts. The mathematical differential establishes a very peculiar, if not a unique relationship between quantifiable concepts. There are two remarkable issues. Firstly, expressed as the differential quotient dx over dy (dy differentiated for dx) it symbolizes a special case for the “0 divided by 0”. In other words, despite the actual quantities are vanishing, the whole thing remains well-defined (if certain conditions apply). This “whole thing” is, however, not determined and definable in the same way as its constituents are. It resides completely in the realm of the symbolic. Secondly, this quotient provides an abstraction for the relationship between the quantifiable concepts that we denoted as x and y. It represents a clear concept inclusive the symbols to talk about that abstraction. By means of the differential, abstraction (still mathematical here) is not an ill-defined, fluffy concept anymore. Yet, if we start with the differential (or a system of differential equations) to say anything about the quantifiable concepts and their relation again, we have to instantiate it. In other words, we have to put a selection into effect. Deleuze recognized the philosophical potential of this mathematical concept. Quite obviously, we can’t transfer it directly, representatively, from one domain to the other, i.e. from mathematics to philosophy. It is therefore nothing but stupid to read Deleuze as if he would be a mathematician. Deleuze doesn’t talk about the mathematical differential. The differential is transferable only by means of abstraction, i.e. by means of the philosophical notion of the differential. Such, we equally obviously use the structure of the mathematical differential as a model. Its philosophical relevance derives from two major consequence. We already mentioned it, it is the necessity of instantiation. the first of which is that instantiation can’t be avoided, yet, in the end, it also can’t be fully justified. There are always many ways of instantiating a particular differential. The second implication is that there is irreducible freedom, which nevertheless remains related to the embedding milieu, the language, the community of people sharing the same language. This establishes borders in the genesis of the instance, which must be distinguished from irrelevant representational borders. The elementary figure of interpretation consists of a differential and its instantiation. The differential (in the philosophical sense) is closely linked to the concept of virtuality. Understanding the concept of virtuality becomes increasingly important, since it expresses the necessity for conditioning the possibility for mediality (cf. [6]) as well as the potential. Both of these are in turn salient implications of contemporary information technology. Ultimately, neither the role nor the potential can be fully recognized without a proper understanding of the concept of virtuality and its embedding into the dynamics of generic thought. While the idea of the virtual has been introduced already in classic philosophy, Gilles Deleuze renewed its role in contemporary philosophy, mainly by describing the abstract mechanism of actualisation. The concept of virtuality is about the conditions for potential, which itself is not specified. Between the potential and the transcendental () concept of virtuality a space is spanning which we call the virtual. Neither the potential nor the virtual can be enumerated. Strictly speaking, it would be against the grammar of the potential or the virtual to say “potential objects”. As soon as we speak about objects, we imply identifiability, enumerability (implying set theory as a tool), hence we already talk about possibility . We opposed the virtual and the real: although it could not have been more precise before now, this terminology must be corrected. The virtual is opposed not to the real but to the actual. The virtual is fully real in so far as it is virtual. Exactly what Proust said of states of resonance must be said of the virtual: ‘Real without being actual, ideal without being abstract’; ([1] p.208) The mathematical differential quotient expresses the potential of selecting a tangent. The idea of the tangent is fully present and determinate, though it is “not anywhere” (Note that the language game of selecting a place violates the grammar of the potential!). In contrast, if expressed as difference quotient (before the limes transition), we see an infinite set (or population) of possible items. Ideas are, as Deleuze says, strictly positive. We can neither subtract 1 idea from 3 ideas to get 2 ideas nor can we remove anything from “an” idea to get a “smaller” idea. Ideas are not enumerable at all. Removal and opposition become possible only subsequent to the actualization or in the realm of the possible, that is, once an Idea actualised into a “species”. We call the determination of the virtual content of an Idea differentiation; we call the actualisation of that virtuality into species and distinguished parts differenciation. ([1] p.207) If we talk about the possible, for instance in the context of a probability (), we already actualized the virtual into a space of possible selections. The density or intensity of this selection is already symbolized within an analytical framework. This exclusion of the virtual from certain ways to describe empirical observations is the deep reason why statistics alone is not sufficient to produce forecasts. In turn, whenever we speak about the probable we already limited ourselves to the possible. We already applied certain rules to actualize the aspects that point to the virtual (as the ever present condition), avoiding here the symbolization as of “virtual aspects”. Upon the actualization of a potential we select, we limit what we can perceive as actualized species. Hence I could have said: what we can actually perceive. Any enumeration or symbolization into a formal framework includes actualization, precisely because these actions imply a limiting step. This then includes, of course, also modeling, here taken as the instantiation of the abstract model into a particular model. It is not possible to point to a potential, or the virtual. Hence we can’t take the virtual as a property of space, nor can we build a representational space from it. It is thus nothing but a serious misunderstanding to point to the internet, or the WWW, to Facebook or to Google, and calling this the “virtual space” (cf. [2], among many others). It is also a misunderstanding to call the n-dimensional image of a space or a building “virtual space”. Obviously we can point to such imaging, we can immerse ourselves into it, we can color it etc. There is no color of or for a potential, yet. Something is utterly wrong in such usage of the concept of the virtual. The internet and the WWW represents the technical aspects of mediality—another transcendental () condition. In the usage of tools, symbols (abstract tools) and concepts we refer to the virtual, for we always have to perform myriads of selection. The virtual aspects of “space” around and in a building are present as not-yet-actualised potential always in precisely the same way, whether we talk about the Parthenon, Las Vegas, or a rendering of computational procedures into a visible form. Again, not discerning possibility from the virtual results either in pseudo-paradoxes, or in a fully deterministic world without the possibility for any constructive selection. In so-called information technology, the virtual shows up in a very different way. It is the reversibility and thus the almost, or actually, simultaneous presence of different species, in other words, the full potential for differenciation, which creates an endless and dense presence of the necessity to select. Nevertheless, the virtual is not “inside the computer”, nor is it “inside the mind”. While in more material contexts we can select only once before information decoheres into irreversibility, in electronically mediated contexts the possibility and the potential for ongoing selections keeps intact. Yet, we should not overlook that selecting—and interpreting exactly in the same way—something is first an activity that basically comprises performance and further selections. Performance in turn refers directly to the body. For the decoherence into irreversibility induces in turn the need for interpretation, it is ultimately performance that intensifies our reference to the virtual, while this intensification is before any quantification, i.e. enumeration. The condition for quantification, quantitability, appears as a transcendental condition (), which is “almost virtual”, albeit the relations between quantitability and virtuality are difficult to determine (cf. [5]). Let us take a brief look to the difference. Using the concept of integers (“natural” numbers), we can write down a difference as 7-5. What is the role of the operator, the “minus”? Obviously, as its label already tells us, it symbolizes an operation, an action, hence some rules. Yet, these rules have nothing to do with numbers at all. In the case of arithmetic, they take the numbers as a structural argument. The difference implies orthoregulation () and instantiation, hence the differential and transcendence (). This is the major difference between the philosophy of Derrida and Deleuze. Derrida got stuck at the difference. Its condition he called difference, which, however, does not allow any further argument. It is even deliberately excluded by Derrida to ask about the deconstruction of the differance. Derrida never accepted to proceed into abstraction. As a compensation he needed further strange concepts and anti-concepts, like the “original trail”, that can’t be seen nor interpreted. This negation of the orthoregulation and of the differential (as a philosophical concept) makes Derrida’s philosophy, the deconstructivism so suitable for realism. The difference implies the negative. Deleuze argued (we cited him here) that the “Idea knows nothing about negation”. Nevertheless we have different ideas. These can be related to each other only by means of the Differential. A final remark: it would impossible to discuss the virtual comprehensively even in a whole book, so all the more in a brief section. We just can point to the various online encyclopedias for further aspects. The only things we tried to say is that (1) the Idea of the virtual expresses a necessity, (2) it needs to be distinguished from the possible, (3) the only way to actualize is by means of selection, even if it happens implicitly. Together these aspects of the virtual may serve as a foundation for the primacy of interpretation. The resulting self-referentiality can be turned productive—defending such against the infinite regress—by transcendental constructions as it is instantiated in our concept of the choreostemic space. 2. Lagrangean Abstraction If you have some procedure or rule, you will always have some variables, some operators and some constants. Lagrange’s insight was that the constants can be replaced by a procedure plus further constants, which are very different from the constants we met first. Deleuze adopted this figure in his investigation of thought ([1] p.209) that eventually would result in a transcendental empiricism. The constants on the implied and explicated level we usually call “more abstract,” they are more general, more powerful, but at the same time also less salient than constants on less abstract levels. The important issue is, however, the change of reference for the implied rule. The new rule that is more abstract is replacing a constant on the less abstract level. Hence it is directed towards a rule and to the side-condition of its instantiation, but rather not to the modeled entity itself. Such, the Lagrangean abstraction is closely related to orthoregulation (). The Lagrangean abstraction is not only closely related to the treatment of partial differential equations and thus to the deep structure (the grammar) of mathematics itself. Actually, Lagrange got aware of one of the most powerful tools of thought. As such, Lagrangean abstraction is even at the roots of the philosophical notion of the Differential. 3. Probabilization Our empirical relation to the outer world is a problematic field. In philosophy it is still heavily discussed under the label of the so-called “natural kinds”, a concept that we reject. In a moment you will understand why. There are, of course, also a lot of serious practical issues and implications around it. Quite obviously, our relation to the world does not include a “direct” access. Even logical positivists admit that. We never can recognize the world as such, we never can be sure about something that we have to conceive as empirical. Astonishingly, in urbanism it still (in 2007) elicits an emphasizing response if someone calls it “illusion to believe that we can see reality as it really is” (Linda Pollack, quoting Lefebvre [1]). We take it as a sign for hope. Unfortunately for the realist attitude, even words are empirical items. If we can’t be sure about our empirical observation, what is the consequence of that? In everyday parlance we use the grammatical form of the “conditional 1”, “it could be”. In technoscience we apply the framework of probability. The main consequence thus is a further question: How do we manage to proceed from a probabilistic description, an expectation that comprises different cases, sometimes only implied, but never seen before, to the use of propositions, to a propositional structure? Most appropriately, “object” should be conceived first as a language game (). We then may ask about that game, its rules, the necessary symbols and processes, its usage and its context etc. The next step is pretty simple. We conclude that the language game “object” refers to a small set of only two structural properties. These are (1) a binning operation that refers to a model that is organized by (i) a certain minimum degree of reliability in the context of anticipatory modeling, (ii) a commonly acknowledged threshold for performing the transition from the degree to the propositional usage of the concept, and (2) the indication for the propositional usage of the concept itself, excluding thereby the need for complicated and challenging probabilistic reasoning. Even if a particular extended system of models and theories would help us to conclude that a particular conclusion about the relations and conditions in the outer world is pretty sure, we can not be absolutely sure. Models, theories and, of course, Forms of Life can’t be arranged in a seamless manner, they may even partially contradict each other. Such, the embedding, i.e. the conditioning of a particular conclusion is not stable in its prescription how to organize measurement and modeling, namely regarding the selection of “properties” that are used to organize the raw signals. Adding or removing properties from a descriptive setup results in a different “object”, which establishes a different set of relations, hence in this sense also different facts. Here, we can see/resolve a number of issues. “Objects” are not as such in the world. To claim otherwise would result in the further claim that symbols are in the world, notably (1) as immaterial entities, and (2) independently from any mind-based interpretation. This was the position of Frege, and partially of Russell and Whitehead in the Principia. Wittgenstein was the first to proof that this is wrong. Regarding the field of machines with mental capabilities, we can understand now that the so-called “symbol-grounding problem” is not a problem at. Second, we can not talk reasonably about “learning” with regard to “machines”, as long as we represent the outer world in a structured manner like tables. Tables already presuppose an act of symbolification; even worse, this symbolification implements a direct representation in the space of the possible, that is the already actualised. Ultimately, symbolification is a process or act that expels potentiality. The step from a probabilistic representation of signals from the outer world to the propositional usage of concepts—lets call it PP-transition—can be interpreted as a speciation, hence also as an individuation. Yet, the appearance of those is just a consequence of the necessity for an empirical rooting of thought. The said PP-transition seems to induce a regress, as we would need symbols even for setting up a probabilistic representation of the signals. Yet, the probabilistic representation of these signals is not only not representative, it is generic. The remaining self-referentiality can be dissolved into orthoregulation () and a differential space (philosophical notion of d. here). Renormalization [2] is the practical tool for representing empirical observations that pays most respect to this structure. 4. Aspectional Space The mathematical notion of space comprises two elements: a mapping between enumerable—though not necessarily quantifiable—entities and a structure that describes the allowed operators. Such a structure is, for instance, a group. As a map, a space represents the condition of the possibility to relate entities to each other. In turn, any entity may be conceived as a compound of elements, where ta particular compound then appears at a particular location in the space. A space thus defines the expressibility. This space is an apriori for any further expression. Yet, a particular concept of space should not be taken as a “constant” (see above about Lagrangean Abstraction). It is a tribute to the grammar (and the structure) of mathematics as a societal institution for inventing signs and symbols as well as the relations between them that the dimensions of mathematical spaces are independent from each other. Of course, the notion of independence is a quite problematic one, as we have argued in our investigation of modernism and its presuppositions, and it is by no means a necessary condition for the possibility of a space. If we drop the apriori set determination of independence we get an aspectional space (see this for more details). If the structure of an aspectional space is global and representational, such as for instance the Euclidean or the hyperbolic space is, the 3d-aspectional space can be drawn in 2 dimensions as a ternary diagram. In such a space, the mapped entity is not crushed into elements that appear as independent elements. In the aspectional space, the wholeness, the integrity of the object remains fully intact, even if we would express it by means of the respective formalism. It is clear, that the mere formulation of the aspectional space as a contrast to the Cartesian dimensional space implies a differential, and thus also a particular actualization, which results either in the aspectional space or in the Cartesian space. As we have seen, the difference between them is due to a different stance towards independence. As Wittgenstein elaborated in the context of his image theory—the double aspect images—we may ask what is it that shows up in the possibility for the duality itself? Obviously, the two spaces point towards the virtual of independence/dependence. 5. Local Space The idea is quite simple. Instead of setting up a single space that serves as a global container into which we map different entities, we assign a local instance of space for each entity in the “would-be” global space. This, of course, is very close to the idea of the analysis situs by Leibniz, hence also to his concept of the monad, and also to the concept of topology. Actually, in a flat Euclidean space (and full symmetry of the group defining the operations) it doesn’t make any difference whether we follow the route of the global space or that of the local space. By means of elementary geometric operations such as turning or mirroring we can translate the global and the local spaces into each other without any loss or consequence, except perhaps a simplification regarding some analytical aspects. Yet, this remains true only if the space is indeed flat and fully symmetric regarding its group. For the relations in a hyperbolical space it makes a huge difference whether we use a global space or a local space. If we replace the “standard” symmetric algebra e.g. by a Lie-algebra, there could also be a drastic difference. Since in self-referential contexts space is actively and locally deformed—Mandelbrot calls it “warping”—we should not expect that the assumption of a globally homogeneous space is a reasonable assumption. Instead, we should renormalize the space in terms of the intensities of local qualities. In other words, the reference system is not treated as a constant condition any more, instead, the reference system is conceived as a procedural dynamisation (). An example for the cost of the assumption of global homogeneity may be provided by the Navier-Stokes equation. This equation is used to describe the behavior of fluids. Yet, astonishingly it is still not known (in 2012) whether and under which conditions the NSE has a solution. As far as I know, engineering (and physics) does not apply the trick of the local space to the problem of turbulent flows so far. The method of “energy cascade” across different scales, however, points to the same direction. Yet, whatever approach engineers select, they apply it in global space. Note that local space is fundamentally different from approaches like fractals or wavelets; even as they point into the same direction, they are usually applied globally in the same way. Hence, they should try to localize the problem more radically. In other words, the Navier-Stokes equation and the current approaches may be too representational (not properly abstract) as for allowing for a general solution. 6. Decoherence Decoherence is a language game that has been introduced first in quantum physics. Despite we can measure quants, for instance in the Quantum-Hall-effect, the microscopic description of a quant does not imply an enumerable entity such as a particle. Even the descriptive level of the field is not abstract enough. The Schrödinger equation expresses waves of probability. The problem is that we can make experiments where we do not find fields, i.e. continuums, but rather particles. This is even true for electromagnetic waves. Obviously, there is a conceptual gap between the purely informational description, which actually refers to a potential, and the observable particle. In simple words (hopefully not too simplistic), this gap is closed by the concept of decoherence. Decoherence is thus closely related to interpretation and measurement, hence also to the concepts (and language games) of reversibility vs. irreversibility, and as well to that of information and causality. This allows us to recognize that the effect of decoherence is not limited to the quantum world. It appears as soon as we deal with information and interpretation, particularly however, in context where we are forced to start with an almost purely informational approach. Such contexts are given by self-referential systems as far as they show emergence, in associative networks and certainly in any context where language or symbols are involved. 7. Mechanism The concept of mechanism refers to three elements. (1) a more or less microscopic perspective, (2) a more or less deterministic principle that is implemented on the microscopic level, (3) a population of instances of that principle, making up the macroscopic entity. Mechanism are not limited to the description of material “systems”. They can also be applied to purely conceptual arrangements, albeit this implies the treatment of symbols as quasi-materials”. Machines can be conceived as a particular, “denaturalized” instance of the concept of mechanism. There is usually no population and the principles work on the macroscopic, i.e. global level. Machines reside in the Cartesian space. They are fully deterministic and thus fully actualized, at least such are the hopes of the engineer. Literally, they ought to represent a set of formulas. Mechanisms, on the other hand, result in an entity that we can not describe as a fully actualized ensemble of embodied mechanisms any more. Unfortunately, the engineering of informational machines follows still the route of fully deterministic machines. There are only two concepts I am aware of that transcend determinism: Douglas Hofstadter’s proposal of the Copycat, and our proposal of the extended Self-Organizing Map. Similarly, the concept of mechanism is not yet a sufficiently explored topic in the philosophy of science. It is clear that the concept of mechanism is a necessary element of any practice that s interested to defy determinism. 8. Vanishing Regress The infinite regress is an infamous figure in philosophy and philosophical logic. The label describes a situation, where by means of logic you find, as a conclusion, that you have to apply the same logical argument again (logic taken here as n-valent, with n=2 or larger, but finite.) Hence, it seems that nothing had been gained. Despite its abundance, the infinite regress (IR) is nothing but a mistake known as petitio principii, though on a structural level. It is a classical pseudo-problem. For the basic reason being that there is no such thing like “pure logic” in the empirical world. Logic is a transcendental condition. Any actual instance of logic is incubated with unknown amounts of semantic references. One of the problems is that this usually does not get recognized, hence, important questions remain unspoken. Yet, these semantic references require presuppositions of any kind, abstractly spoken, logic becomes applicable only through the reference to rules outside of logic, to orthoregulation. Hence, there is no infinite regress. Any regress is vanishing as soon as we are going to take it serious. The petitio principii consists in the assumption of the universality and the actuality of logic. The regress shows up only under these presuppositions. It is the presupposition that creates the problem, which is conceived as a logical problem. 9. Orthoregulation Obviously, we necessarily maintain empirical relations to the outer world. We play the anticipation game wherever we can afford, either by modeling or by institutional structures. We expect at least a weak stability of these efforts. Orthoregulations are not directed to the most basic observations, but to the construction of observations, or to the generation and usage of models. Since there is—quite obviously so, because it is intended—much less diversity in models than in basic observations, the generation of rules as orthoregulation is faced with an empirical basis that is much smaller that that we have at our disposal for building even a single model. The same holds for the second level of orthoregulation. Despite we still talk about rule-following, the empirical reference changes. 10. Transcendence Any condition is transcendent to the conditioned. From the perspective of the conditioned, from within the system, if you like, the conditions are not even visible. Thus, transcendence has nothing to with mysticism. Yet, it may well be conceived as a kind of metaphysics: it is a concepts that goes beyond physics (as a science), its claims and its possibilities. Else note that it is very different from the side condition. In order to make conditions visible one has to change them, yet almost in an unintended manner. It will turn out only upon the interpretation of the result if one really has found new conditions, or just a variation of known parameters. Thus, experimentation is necessarily an “embodied” activity, which even includes the laboratory itself. There are important implications. First, we can see conditions only upon the implied differential (). More precisely, conditions are always given as differential, or in a different language game, as orthoregulation (). This points to the second issue: Conditions imply a symbolic description. Even if they don’t get symbolized, e.g. in animals, or on the level of physical descriptions, implied conditions are already possible and thus actualized. Of course, any visibly symbolized condition refers itself to concepts, which in turn are transcendental. There is a particularly interesting issue about conditions and transcendence with regard to emergence. It is not possible to use the language suited for the description of the microscopic level as well for the description of the emergent macroscopic level. The same impossibility we experience in the opposite direction. Despite we see one system, perhaps even in a petri dish, or as the rendering of a computational procedure, we need two different descriptions that are even incommensurable to each other. In complex systems, such as organisms, the emergent levels act back onto the lower levels. Such, the organism is full of irreducible transcendence. Yet, we do not need organisms to observe the implication of transcendence. We may experience it upon the interpretation of any arbitrary series. There are two reasons for this. First, a series changes its appearance if it is chunked in different ways. Second, as an informational entity the series itself provokes the necessity of decoherence, of duality, and thus for simultaneously present, yet incommensurable individuation [2]. 11. Elementarization Whenever we need to render something accessible to measurement, interpretation, or quantification, we need a starting point. Once set, this starting point can’t be revoked anymore. It conditions the possibility of measurement and interpretation. On the level of concepts, such starting points appear as elements. Elements mediate between quantitability, which is close to the virtual, and quantification, that is, the establishment of a suitable space for rendering relations into maps. It is, for instance, impossible to talk about complexity without a preceding elementarization. Without elementarization, we get stuck in binary concepts. This in turn makes any measurement impossible, and hence also interpretation and modeling. Binary (dual) concepts thus are not suitable for any attempt to clarify things by a (partial) transposition into the sayable. We just can assign names and values, both of which are completely arbitrary as species in a any given case. Usually, the consequence of which is some kind of quarrels. 12. Language Game When describing the concept of probabilization () we asked about the consequences of uncertainty regarding our empirical observations. The problem starts, of course, even before that, because we have to set what we are going to observe. For we have to do it in a way that could be accepted as a replication of the observing and would allow to “reproduce” the observations, which requires in turn interpretation, we obviously need language already before the empirical observation. This does not mean that primitive animals or machines would not be able to “observe”. Yet, their experimenting takes at least hundreds of generations, if not millions of years. Evolution is an embodied experiment. Interestingly enough, Gerard Edelman proposed an important role of positive and negative selection processes in both long-term and short-term dynamics in the brain. So, what is a “Language Game”, and why is it called like that? Wittgenstein has been the first who recognized the transcendental () role of language for the domain of the “human”. Of course, it would be nonsense to claim that we can think only by means of language. But any kind of culture is necessarily based on the externalization of thoughts, regardless the material shape of the thinking entities, and this externalization requires some kind of open language. Above we provided a possible argument for that in a very brief form. Second, Wittgenstein calls it a game for two reasons. (1) The use of language is basic, that is, there is neither a formalization nor a meta-language possible. (2) Language is open, locally as well as globally, both in space and time, which means that the rules can not be written down completely, neither apriori nor aposteriori to the act. Else, the rules are in constant evolution, or if you like, negotiation. We invent new rules and new words, drop others, import and assimilate rules and concepts from other languages. Just like in a very complicated game. Language reflects the Form of Life. Without an open language we would not be able to build new concepts as informational entities, nor to use them, for this requires interpretation. Again, animals certainly use concepts as well, but in most these concepts remain closely tied to the respective type of body as a material arrangement, capable only for certain associations. As soon as concepts and bodies disengage, we observe culture, even in animals. As any other game, or set of rules, language games can be challenged and perverted. This fact is the trivial consequence of the need for interpretation. Interpretation is a primary condition for any reference (which ultimately results in an transcendental empiricism). In the vast case of cases language games work quite well. For words are not just pointers, they carry a set of implied rules that indicate how to process them. One particular way to challenge a language game leads to (pseudo-) paradoxes. Paradoxes are downright “language game colliders” (LGC). A LGC is a concept or an argument that relates two (or more) incommensurable language games into a close mutual dependence. Language games are incommensurable if they follow a different grammar, such as for instance countability of instances of a particular type (entity) and the sign for non-countability. This leads for instance to the Sorites-paradox, the paradox of the heap. It goes like this: Given a heap of sand, we remove one grain of sand after the other. First, the heap remains a heap. At some point, however, we can’t consider it a heap anymore. This situation then is commonly called “paradoxical”. Yet, by no means this establishes a paradox. It is just an indication for a LGC. The concept of “heap” excludes the operation of counting, as the “mechanism” of the paradoxical argument emphasis itself. The argument, however, also employs counting, thus a grammatical incommensurability is established. This, of course, induces difficulties. Nevertheless, these difficulties are neither bound to a “heap object”, nor to logic. It is more like the 2-year old boy who gets terrified by the incommensurability of his own wish to do opposite things at once. In the same way, many variants can be constructed. Most of them collide countability with the sign of non-countability, that is the structural declaration that something should not be counted. On the one part of the “paradox” we then find object, numbers, entities, and on the other we find symbols or concepts like “infinity” or “all”. It is very important to understand that difference, because it introduces operations. In logic however, there are no “operations”, because any term has to be reducible to a=a. For the same reason logic is not applicable to emergence, to the use of the differential, or to orthoregulation (). 13. Archaeology An archaeologist digs. He tries to reconstruct forces in the past by using modeling upon the fund artefacts. She takes the sediments as they can be found… well, as they can be constructed. The important thing is that there is no necessity for the sequence of layers of sediments, nor for their naming. That means, that in archaeology we do not deal with causality. Yet, the sediments didn’t get there to build a mountain by chance. There is piecewise coherence, spanning certain periods of time, until an outer event (“comet”) or an inner bifurcation leads to radically different conditions. A small remark is indicated here: Of course, coherence is itself concept whose usage requires rules for its instantiation. It has been Michel Foucault who transferred this structure into the investigation of cultural issues. In culture, the coherence is not given by the (mainly disentangling) geological forces and the (mainly associating) biological forces. Instead we find language, institutions, traditions, images of thought, concepts and morality. These form a meta-fluid assemblage that sometimes is more coherent, and sometimes not, sometimes stable as glass (which physically is fluid), sometimes stable as ice (which is a crystal), sometime as a non-Newtonian fluid where fluidity or viscosity is dependent on local contexts. This multidimensional “carpet” he called the “field of proposals”. At any point in time, we are immersed in that field. The cultural archaeologist does not deal with causality either, but also not with calendar dates. S/He is interested in the emergent patterns of differential fluidity. 14. Processual Indicative The processual indicative (PI) is a concept that is related to language games and words. Which role(s) play words in language? While it is clearly based on an illusion to claim that words refer to objects, it also seems inappropriate to think that they refer directly to concepts or to symbols. Unger once noted that “the cloud doesn’t exist”. Yet, the symbolic value associated to the word “cloud” is absolutely clear—it is part of the propositional world. Following Wittgenstein we tend to propose that vagueness is not a problem of language that could or that should be cured, it is a design principle. Quite in contrast, it is linguistics that must be “cured”. In this situation inferentialism as proposed by Robert Brandom may help. The label “inferentialism” refers to a set of arguments proposing that in the use of language we necessarily ascribe roles to each other in order to provide some of the necessary conditions that allow us to interpret any “released utterance”, which of course need not follow the official school-grammar for the respective language. The language game “cloud” contains a marker that is both linked to the structure and the semantics of the word. This marker indicates two very different things. (1) There is a object without sharp borders, and (2) no precise measurement should be performed. Wittgenstein remarked I know how to use language, or how to use a word. The interesting thing in language and words is not that they are used to reference a putative “object”. Upon applying Lagrangean abstraction we can conceive the “reference” as a constant compound, built from a rule and other constant. The interesting thing about words is the rules that they carry along together with the reference to another sign, or word in more simple cases. It is thus not just the “object” that is indicated by the word “cloud,” but rather a particular procedure, or class of procedures, which I as the primary speaker suggest to use by saying “Oh, there is a cloud.” By means of such procedures a particular style of modeling will be “induced” in my discourse partner, a particular way to actualize an operationalization, leading to such a representation of the external phenomenon that both partners are able to increase their mutual “understanding.” This scheme transparently describes the inner structure of what Charles S. Peirce called a “sign situation.” Words are thus not just primitive symbols, or relations to other words, as they are treated by the community of computer linguists. They are to be conceived as a compound made from a pointer to other (Peircean) signs and a bag of rules how to establish a model about the respective reference to the external world, or to the world of concepts. This second part comes as an invitation, or even as a demand. Hence we call it the processual indicative of words. 15. Inverted Experimental Plan Kasparov once has been asked how he could manage to find the most appropriate move from the billions over billions of possible moves, through which the computer sifts. His answer was: “I take the right one.” In empirical measurement we are faced with the problem to set up the measurement scheme for the most important variables. Yet, we could measure almost an infinite number of different things, of course, even in case of trivial machines. If the experimenter is very experienced, she might know where to look. In other words, she can use already available concepts to organize the measurement. Or, in turn we may say that the measurements reflects the concepts. But how to know? Anyway, if one would know then one could set up a so-called statistical trial plan in order to minimize efforts and to maximize the probability to observe clear differences. Unfortunately, these plans work only for at most a few, usually less than 5, variables. Any real world system is influenced by much more factors. In this situation we can take a fresh look to classification. If we organize according to the scheme shown here, we will find a model that identifies both the most relevant variables as well as the best segmentation of the data to derive the actual proposal about that relevancy. Usually, these two aspects are deeply related to each other, except in the case of random systems, where each “particle” is moving completely independent of the others. Such a model that accomplishes segmentation and selection “simultaneously” can be interpreted as if it would be a result of a statistical trial plan. Thus, we can call the model also an “inverted plan”. From this we can derive a further proposal, namely that as long as a putative model does not behave as an inverted plan it is unsuitable for serving as a basis to proceed from the probabilistic properties of a model to the propositional usage of a model. 16. Diagram A diagram relates a small set of abstract elements in such a way as to demonstrate the irreducible quality of the whole. Such, not any (mathematical) graph is also a diagram, but every diagram is a graph. A diagram is the differential for a given structure, while itself it is of course again a structure. Yet, obviously the inverse does not hold, not every structure is also a diagram. A good example is given by category theory, or by the Peircean sign. Here, on our blog we used diagrams to investigate the structure of modeling (L-scheme) as well as in the case of comparison. Figure 1a,b: (a) Diagrams showing the loop structure of extended modeling (L-scheme) and (b) the basic diagram depicting a category as it is conceived in mathematical category theory [8]. Diagrams work like abstract machines, as Deleuze noted. Usually, they do not resemble neither to the input nor to the output in any obvious way. Of course, they also refer to a purpose or a produce (an implied purpose), even if the produce would be a probabilistic outcome. In an important sense, diagrams are instances of the abstract model. Yet, there is a slight difference. Diagrams show us the relations between the elements of the abstract machine, which are established through the operators in modeling. 17. Self-referentiality Self-referentiality (SR) is a property of an operational arrangement, whether this arrangement is of a more material or more immaterial nature. In a SR arrangement, the result of the operations establish the structural conditions of further operations. This implies two distinctions: First, between a microscopic level and a macroscopic level of description. Second, between the principle (working on the microscopic level) and the entirety. These properties make them different from other conceptual figures like recursion or the infinite regress () (which anyway is impossible). The infinite regress is a logical figure, hence there are no “operations”. The recursion is not defining the conditions in a circular manner, because any recursion can be linearized. It is just a particularly condensed form to express a strictly serial arrangement of repetitive operations. Self-referential arrangements (SRA) can’t be linearized. SRA are also (strongly) different from the cybernetic perspective. A cybernetic arrangement organizes the throughput of a machine-like system without any changes occurring on the structural or conditional level of itself. This is called feedback. In SRA, there is nothing like a feedback, instead the conditions for the mechanism(s) () changes and leads to different operations or behavior. In biological systems we rarely find cybernetic arrangements, probably only in simple wiring patterns in the vegatative nervous system. Even the innumerably cited (and taught) prototype for cybernetics, the “regulation” of blood sugar in an animal organism, can’t be conceived as a cybernetic cycle, as research has demonstrated in the recent decade. An example for a material SRA is the class of Reaction-Diffusion-Systems (RDS), regardless its instantiation. We know for instance of the Turing-RDS, the Turing-McCabe system, the Belousov-Zhabotinsky-system (BZ), the Meinhard-Gierer-System, or the Gray-Scott system. These systems differ with regard to the kinetics of the mechanisms on the microscopic level and the kind of involved mechanisms. While the BZ-system or the McCabe-system do not introduce or remove anything, the Gray-Scott-system models a throughput system, where some produce is removed from the reactor. Even in the case of blood sugar it is much more appropriate to conceive it as an RDS (some parts spatially fixed, others not) than to think about it as a cybernetic control system. An example for a conceptual SRA is the choreostemic space () for which we have described its structure and mechanism in an earlier essay (see also below for a summary). 18. Choreostemic Space The choreostemic space (CS) is a concept whose target is most easily introduced by the following observation. In order to organize our empirical observations we need models. We even need a “system of models” for understanding language. In order to set up models we need symbols and rules that have nothing to do with the model. These rules in turn refer to concepts and to a language. Such, they are implying mediality in a twofold manner. First, through language, second, through the fact, that models never appear as a singularized instance. They always imply further models, directed to the same or to other purposes, but anyway forming a population, which can serve itself as a medium. Rule-following, on the other hand, implies an action, it is a label for the decoherence () into irreversibility, i.e. for interpretation and selection. Rule-following implies virtuality. Such, we arrive at four concepts: model, mediality [4], concept and virtuality (). The CS describes a particular possibility of how to relate these four concepts to each other. This arrangement is, of course, a kind of space (). The particularity of the CS derives from its properties. 1. It is defined in a self-referential manner, it is an operational space that does not allow for representations. 2. The four concepts are conceived as transcendental “directions”, or “headings” (like in the navigation of airplanes) for anything happening in the CS. The transcendental headings are called “aspects”. 3. The structure of the space is hyperbolic and local (). 4. The choreostemic space is an aspectional space (). 5. The containment of the space is the 2nd-order differential . This structure results in some quite remarkable consequences. Since its formal setup is self-referential it is not a definition. More precisely, it does not refer to any other concept. It does not need any axiom as its preceding condition other than the possibility for interpretation. The CS does not contain “locations”, because its containment is the generalized differential, that is a 2nd-order differential. Pointing to a putative location, even the claim to do so, “transports” the entity trying to do so to “elsewhere”. More precisely, any explication is heading thoughts towards the model while “still” keeping a relation to the other aspects, namely concepts. Everything that could be thought starts in the CS and leaves a trace in it. Of course, also talking about the CS, or setting up its structural properties, or the use of the concept “properties”. The same holds for “pointing” or “showing”. Such, it extends, clarifies and provides a general foundation for the concept of thought. Yet, the CS does not “contain” anything that could be thought or said. Through the process of instantiation of the transcendental aspects we arrive at interpretable entities, such as language, images, or intended material arrangements that usually are called design. It is a procedural differential (), the inner structure of interpretation itself. Since any usage of concepts leaves a trace in the CS, the CS can be used as a tool to map different systems of thought, even different Forms of Life. Take for instance an cleric and a scientist, or within philosophy, an idealist (e.g. Fichte) and a pragmatist (e.g. Peirce), or within science a biologist and a physicist, or within religion a bishop or sufi and a derwish. Any of them builds a more or less typical attractor that describes the particular dynamics in thinking. Comparing these attractors allows to investigate resemblances beyond any representation. Hence, it is also suitable get rid of values and binary concepts without falling into relativism or skepticism. We can conceive it as the possibility to construct a kind of measurement device for thought that comprises its own renormalization procedure. The fact that the CS is about the general notion of thought, or, in other words, the generic thought, sometimes gives rise to two misunderstandings. First, the CS can’t be conceived as a concept that would map or represent the episteme. The CS is not about epistemology. Second, saying that it is about generic thought does not mean that it is about “pure” thought. Quite to the opposite, the CS excludes any “purity”. Idealism shows itself just as a particular figure in the CS. All together, the CS is the last outpost of the sayable and the demonstrable. It is possible only through two properties. First, its self-referentiality, second, its differential quality, which makes the instantiation a necessary operation and condition for itself. The final remark thus states that the CS dissolves the idea of philosophical justification. Justification appears as a very small attractor near the attractor of logic, both of which are “virtwhere” on a pre-specific path that leads from the (abstract) model to the (abstract) concept. • [1] Gilles Deleuze, Difference and Repetition. • [2] Marcos Novak, The Transphysical City, 1996. available online. • [3] Linda Pollack, “Constructed Ground: of Scale”, in: Charles Waldheim (ed.), The Landscape Urbanism Reader, 2007. • [4] B. Delamotte (2003). A hint of renormalization. Am.J.Phys. 72 (2004) 170-184, available online: arXiv:hep-th/0212049v3. • [5] Vera Bühlmann, “Serialization, Linearization, Modelling.” The First International Deleuze Studies Conference. Cardiff University, School of English, Communication and Philosophy. Cardiff, Wales UK, August 11-14 2008. • [6] Vera Bühlmann, inhabiting media. Annäherungen an Herkünfte und Topoi medialer Architektonik. PhD thesis, University of Basel (CH) 2009; • [7] Vera Bühlmann, “Articulating quantities, if things depend on whatever can be the case”, forthcoming in the proceedings to The Art of Concept, 3rd Conference: CONJUNCTURE — A Series of Symposia on 21st Century Philosophy, Politics, and Aesthetics, ed. by Nathan Brown and Petar Milat, Multimedia Institut MAMA in Zagreb Kroatia. Leave a Reply You are commenting using your account. Log Out / Change ) Google+ photo Twitter picture Facebook photo Connecting to %s %d bloggers like this:
30b4564c5b06ff9c	By Martin Schottenloher Half I provides an in depth, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in dimensions. The conformal teams are made up our minds and the appearence of the Virasoro algebra within the context of the quantization of two-dimensional conformal symmetry is defined through the class of relevant extensions of Lie algebras and teams. half II surveys extra complex subject matters of conformal box concept equivalent to the illustration concept of the Virasoro algebra, conformal symmetry inside string concept, an axiomatic method of Euclidean conformally covariant quantum box conception and a mathematical interpretation of the Verlinde formulation within the context of moduli areas of holomorphic vector bundles on a Riemann floor. Show description Read or Download A mathematical introduction to conformal field theory PDF Best quantum theory books This publication reexamines the delivery of quantum mechanics, specifically interpreting the advance of an important and unique insights of Bohr. specifically, it supplies a close learn of the advance and the translation given to Bohr's precept of Correspondence. It additionally describes the function that this precept performed in guiding Bohr's study over the serious interval from 1920 to 1927. Lehrbuch der Mathematischen Physik: 4 Quantenmechanik großer Systeme In der Quantentheorie werden Observable durch Operatoren im Hilbert-Raum dargestellt. Der dafA1/4r geeignete mathematische Rahmen sind die Cx - Algebren, welche Matrizen und komplexe Funktionen verallgemeinern. Allerdings benAtigt guy in der Physik auch unbeschrAnkte Operatoren, deren Problematik eigens untersucht werden muA. Introduction to quantum graphs A "quantum graph" is a graph regarded as a one-dimensional complicated and built with a differential operator ("Hamiltonian"). Quantum graphs come up clearly as simplified versions in arithmetic, physics, chemistry, and engineering whilst one considers propagation of waves of varied nature via a quasi-one-dimensional (e. Introduction to quantum mechanics: Schroedinger equation and path integral After a attention of uncomplicated quantum mechanics, this advent goals at an aspect by means of facet therapy of primary functions of the Schrödinger equation at the one hand and the functions of the trail crucial at the different. various from conventional texts and utilizing a scientific perturbation technique, the answer of Schrödinger equations contains additionally people with anharmonic oscillator potentials, periodic potentials, screened Coulomb potentials and a standard singular power, in addition to the research of the big order habit of the perturbation sequence. Extra info for A mathematical introduction to conformal field theory Example text For p _ 1 we have Pk "= (~0 . ~1 + £k " ~2 . . . ~n . 4_ ~k) C= N ~''q Moreover, ~o + ~n+l _[_ (~k -- ¢~k # 0 implies Pk E z(RP'q). Finally, since Pk --* (~o. ~+1) for k --, co it follows that (~o . e. N p'q C ~(Rv,q). m We therefore choose N p'q as the underlying manifold of the conformal compactification. N p'q is a regular quadric in IP,+I(R). Hence it is an n-dimensional compact submanifold of IP~+I(R). N p'q contains ,(R v,q) as a dense subset. We get another description of N p,q using the quotient map "y. E. e. A ~ ~ A~. Altogether, ¢ : N p'q --, N p'q is well-defined. Because of the fact that the metric on R p÷l'q÷l is invariant with respect to A, CA turns out to be conformal. if P is . represented by E Sp x S q. ) Obviously, CA = ¢-A and ¢~1 = Ch-~. In the case CA = CA' for A,A' e O ( p + 1, q+ 1) we have ~/(h~) = ~(A'~) for all ~ e R ~+2 with (~) = 0. Hence, h = r h ' with r e R\{0}. Now A, A' e O(p+l, q+l) implies r - 1 or r - - 1 . 5 Let ~ : M --, ~P,q be a conformal transformation on a connected open subset M C R p'q. A topological group is a group G equipped with a topology, such that the group operation G × G --, G, (g, h) ~ gh, and the inversion map G --, G, g ~ g-i, are continuous. The first three examples are finite-dimensional Lie groups, while the last two examples are infinite-dimensional Lie groups (modeled on Fr~chet spaces). The topology of Diff+(S) will be discussed shortly at the beginning of Sect. 5. ) remains, which is continuous for the strong topology on U(P) (see below for the definition of the strong topology). Download PDF sample Rated 4.74 of 5 – based on 48 votes
8ed514f6eb2f28de	Further disclaimer (thank to ariels for the information): The 'snapshot' mentioned above is a well defined object in dynamics (its mathematical form containing firm proofs and a specific ontology). However, I think the point below still stands. Though the 'snapshot' as defined mathematically may be vastly different than the 'snapshot' the lay person is familiar with, I think Feyerabend would still argue that the very choosing of the term 'snapshot' is a metaphorical/rhetorical one, that cannot be encompassed by an easy rationality... Applying Philosophy of Science Feyerabendian and Lakatosian analyses of Quantum Chaos I will be discussing an article entitled “Chaos on the Quantum Scale” by Mason A. Porter and Richard L. Liboff from the November-December 2001 issue of American Scientist. The article discusses recent advances in recent attempts to model systems that behave chaotically on the quantum (sub-atomic) scale. It will be helpful to briefly summarize the main points of the article: The first few introductory paragraphs relate quantum mechanics and chaos theory by placing emphasis on their respective uses of uncertainty. From this common point of uncertainty, the authors state that because scientists seem to ‘find’ chaotic phenomena at all scales, they cannot rule out the possibility of chaos at the sub-atomic level. The next section of the article is a brief history of chaos theory that describes the early work of Henri Poincaré and mentions the later work in the 1960’s by meteorologist Edward Lorenz. They then explain that chaos has been found in so many disparate disciplines of science, and once again reiterate that they cannot rule it out at the quantum level. Here they also mention possible applications of such quantum level chaos in nanotechnology. From here they move into the largest section of the article, the billiard-themed thought experiment/model. They move from a simple two dimensional billiard table to increasingly more chaotic and quantum-like billiard tables. There is a two-dimensional table with a circular rail, a spherical ‘table’, a spherical table with wave-particles as ‘balls’, and finally, a spherical table with an oscillating boundary and with wave-particles of different frequencies. Within this section, they also explain the more technical aspects of their attempt to model quantum chaos. They explain their plotting methods (the Poincaré section) as well as their mathematical methods as well (the Schrödinger equation and Hamiltonians). With the final few examples they show us that they cannot as yet model true quantum chaos, but only semi-quantum chaos (which requires mathematics from the realm of classical physics as well as quantum mechanics). After this admission, they go on to describe in detail future applications that successful quantum chaotic modeling will have in nanotechnology, from superconducting quantum-interference devices (SQUIDs) to carbon nanotubes. The final sentence of the article sums up the general attitude of the authors: “As we have shown… this theory possesses beautiful mathematical structure and the potential to aid progress in several areas of physics both in theory and in practice” (Porter 537). I shall now attempt to analyze the article in light of two very different ‘theories’ (though one can certainly not firmly be called a ‘theory’): namely, those of Paul Feyerabend and Imre Lakatos. I will begin my discussion with Feyerabend’s thought, and then move on to Lakatos. After these analyses, I will engage both authors with each other, and attempt to bring out certain problems in each of their ‘theories’ that I see myself. Paul Feyerabend introduces the Chinese Edition to his book Against Method by stating his thesis that: the events, procedures and results that constitute the sciences have no common structure; there are no elements that occur in every scientific investigation but are missing elsewhere. Concrete developments… have distinct features and we can often explain why and how these features led to success. But not ever discovery can be accounted for in the same manner, and procedures that paid off in the past may create havoc when imposed on the future. Successful research does not obey general standards; it relies now on one trick, now on another…(AM 1). So, we can (and do) explain why certain scientific developments/revolutions do occur, but we should not expect these explanations to bud into theories, and we should definitely not expect that our explanations should apply in all cases. This inability for universally applicable theories to be universally applied, is not a result of our inability to hit upon the correct theory, but is a result of the non-uniform character of what we call ‘science’. Science is not a homogenous enterprise. It comprises everything from sociology to quantum mechanics. Before we can expect to have an absolute theory (which Feyerabend thinks is neither possible nor desirable) we would have to have an absolute definition of what ‘science’ is. (Here we can see the influence of Wittgenstein’s idea of language games on Feyerabend’s thought). Perhaps science isn’t something we can have a theory about. So, it being understood that Feyerabend believes that ‘science’ is not homogenous, and that we can only explain individual cases with individual criteria, what processes would he think applicable in the article at hand? Obviously this is a difficult question to answer. I think a fruitful way of approaching the task is through a very un-Feyerabendian process. By seeing what he has done in the past (e.g. in his previous analyses of scientific ‘developments’) we may be able to surmise what he would be likely to note in our particular example. In Feyerabend’s analysis of Galileo (specifically in chapter 7 of Against Method) he emphasizes the role of rhetoric, and ‘propaganda’ in scientific change. He states that:>/p> Galileo replaces one natural interpretation by a very different and as yet (1630) at least partly unnatural interpretation. How does he proceed? How does he manage to introduce absurd and counterinductive assertions, such as the assertion that the earth moves, and yet get them a just and attentive hearing? One anticipates that arguments will not suffice - an interesting and highly important limitation of rationalism – and Galileo’s utterances are indeed arguments in appearance only. For Galileo uses propaganda (AM 67). So, as it seems that an analysis of non-argumentative (rhetorical) uses of language aided Feyerabend in his discussion of Galileo. Thus, one possibly fruitful method of analysis may be to search out similar uses of language in our article. Which is precisely what I will do. Here is a good example of the use of non-rational, non-argumentative means of convincing someone of your point: The trail of evidence towards a commingling of quantum mechanics and chaos started late in the 19th century, when … Henri Poincaré started working on equations to predict the positions of the planets as they rotated around the sun (Porter 532). Here we are led to believe by Porter/Liboff that Poincaré’s work is part of a ‘trail of evidence’ that provides support for their work (‘the commingling of quantum mechanics and chaos’). By the appeal to an accepted authority (it is generally accepted in the chaos community that Poincaré is the ‘father of chaos theory’) we are supposed to lend further credence to their own work (though, as we are told in the last portion of the article, this work has not provided a true connection between the two theories). But, is there, in Poincaré’s work any evidence of this commingling of chaos and quantum mechanics? Hardly. The ‘evidence’ they refer to is simply the birth of chaos theory. If we accept their claim, one might analogously state that my birth contains ‘evidence’ for whom I will marry in the future. (Putting aside genetic predisposition toward certain possible mates, this is absurd.) We cannot (rationally) justify the claim that the birth of chaos theory provides evidence for the future ‘commingling’ of that theory with quantum mechanics. It does, however, provide a nice segue for the authors into a historical summary of the birth of chaos theory. Rather than an argument, it is a literary device (like exaggeration, alliteration, etc.) that aids both the achievement of the authors’ goal (describing quantum chaos) and making the text itself more fluid. Staunch rationalists would argue (Feyerabend might say) that this example mistakes a literary device for a scientific argument, and that if we simply separated the two, the problem would dissolve. Feyerabend’s position however, is that we are unable to separate the two. He states in Against Method That interests, forces, propaganda and brainwashing techniques play a much greater role than is commonly believed in …the growth of science, can also be seen from an analysis of the relation between idea and action. It is often taken for granted that a clear and distinct understanding of new ideas precedes, and should precede, their formulation and institutional expression. (An investigation starts with a problem, says Popper.) First, we have an idea, or a problem, then we act, i.e. either speak, or build, or destroy. Yet this is certainly not the way in which small children develop. They use words … they play with them, until they grasp a meaning that has so far been beyond their reach… There is no reason why this mechanism should cease to function in the adult. We must expect, for example, that the idea of liberty could be made clear only by means of the very same actions, which were supposed to create liberty (AM 17). Putting aside the theory of language acquisition proposed here, we see that Feyerabend believes that the form of our investigation is just as important as the content or result of it. Thus, we cannot understand an argument separately from the language it is phrased in, language that often contains suggestive (propagandistic) phrases. In other words what you say is often inseparable from how you say it Analogies to real world objects are also used by Porter/Liboff. For example: “A buckyball has a soccer-ball shape…” (Porter 536); “Nanotubes can also vibrate like a plucked guitar string…” (Porter 537); and, “Such a plot represents a series of snapshots of the system under investigation” (Porter 534). These analogies appear to be used simply to enhance the more abstract qualities of the quantum-chaotic world the authors are describing, and make them more understandable. But, it seems there is more going on here. If we view the article in the Feyerabendian sense that I have been developing above, the choice of metaphor can also affect the readers’ conception of the ‘ideas’ that the authors are attempting to put across. In particular, the ‘snapshot’ analogy seems suggestive to me. What the authors describe as ‘snapshots’ are Poincaré sections taken from higher-than-three dimensional systems. In effect, two-dimensional plots that are, by a mathematical process, abstracted from ‘multi-dimensional masses.’ These are possibly some of the most theoretical objects ever created yet the authors describe them as ‘snapshots’. Obviously there are qualities of the Poincaré section that lend it to the comparison: both a snapshot and a Poincaré section are thought to be reports of a particular time and space. But, other aspects of the comparison may (hopefully, for the Porter/Liboff) lead the reader into accepting highly theoretical concepts as real objects, more so than they would have without the analogy. Obviously the creation of a photographic snapshot is itself based on theory, but it is one that we use (and accept) in everyday life, one that we accept without reservations. Not only that, but the real-life snapshot (as opposed to the Poincaré section snapshot) represents things which we already accept as existing in the real world. In comparing the Poincaré section to a snapshot, the authors attempt to further solidify the reality of the objects that the section represents. Rather than seeing the n-dimensional objects of the Poincaré section as abstract objects, we are now more suggested to picture them as objects like our vacation slides, or wedding photos. Imre Lakatos’ great contribution to the history and philosophy of science (and the historiography of science) is the concept of the research programme. As a general illustration of the role of a research programme, the following quote may be helpful: the great scientific achievements are research programmes which can be evaluated in terms of progressive and degenerating problemshifts; and scientific revolutions consist of one research programme superseding (overtaking in progress) another (Lakatos 115). How can we apply such a methodology to the emergence of quantum-chaos? Well, to start with, we might ask just what research programme, or programmes we are working with. Are quantum mechanics, chaos theory and quantum-chaos all individual research programmes, and, if so, how do we explain the emergence of quantum-chaos (a theory that contains elements of both quantum mechanics and chaos theory) in relation to the other two? I shall attempt to answer these two questions in order. To answer the first, we should define more firmly what Lakatos means by the term ‘research programme’. He states that: The basic unit of appraisal must be not an isolated theory or conjunction of theories but rather a ‘research programme’, with a conventionally accepted (and thus by provisional decision ‘irrefutable’) ‘hard core’ and with a ‘positive heuristic’ which defines problems, outlines the construction of a belt of auxiliary hypotheses, foresees anomalies and turns them victoriously into examples, all according to a preconceived plan. The scientist lists anomalies, but as long as his research programme sustains its momentum, he may freely put them aside. It is primarily the positive heuristic of his programme, not the anomalies, which dictate the choice of his problems (Lakatos 116). So, in order to determine whether or not our three ‘categories’ can be aptly described as research programmes they must have a ‘hard core’ (which I take to mean principles or examples that one has to accept in order to work within the research programme), and also a ‘positive heuristic’ that determines what problems will be addressed (and how to address them). For brevity’s sake I shall limit my discussion to the ‘hard core’ and the problem-determining function of the positive heuristic while ignoring the role of anomalies in negative determination of problems (a role that Lakatos, unlike Popper, believes is secondary to that of the positive heuristic). Quantum mechanics definitely seems to have a ‘hard core’ that its adherents agree is irrefutable, and essential to its elaboration. Historical examples of such an irrefutable core can be found in papers (from the late 19th century to the first quarter of the 20th) by Planck, Bohr, Einstein and others. These papers contain principles that form the unshakeable core of quantum mechanics even now. Here is just one example, which should suffice to illustrate the point: Today we know that no approach which is founded on classical mechanics and electrodynamics can yield a useful radiation formula. … Planck in his fundamental investigation based his radiation formula…on the assumption of discrete portions of energy quanta from which quantum theory developed rapidly (Einstein 63). So, quantum mechanical theory develops directly from Planck’s assumption of quanta. Although this is an oversimplification, it does illustrate that there are basic assumptions which quantum theorists are unwilling to sacrifice. We have our ‘hard core’, now the question is: does quantum mechanics have its own ‘positive heuristic’? I think the easiest way to answer this is to rephrase the question slightly: has quantum mechanics generally determined its own problems positively (i.e. set out to solve them) before they are negatively determined by emergent anomalies? Obviously searching out the ‘general’ answer to this question is well beyond the scope of this essay, but finding a few examples can at least allow us to provisionally classify quantum mechanics as a research programme. One example is the full, and accurate, derivation of Planck’s law. Planck proposed the idea of quanta (discrete units of energy) in 1900 and the perfection of a law describing this idea was worked on until 1926. The idea of quanta was proposed as a basic tenet of quantum mechanics (it was ‘anomalous’ only for the then degenerating research programme of classical mechanics), though it could not be perfectly derived. So, setting it up as a problem, quantum mechanics attempted to ‘solve’ it (and eventually did). The problem of splitting the atom, though it may have been motivated by outside political factors, was internally posed to quantum mechanics as well, and consequently solved as ‘predicted’ by theory. Undoubtedly, then, Lakatos would define quantum mechanics as a research programme, and not merely a theory contained within a larger research programme. Can the same be said of chaos theory? Well, chaos theory seems to have its own ‘hard core’. This much we can see from the Porter/Liboff. The theory’s basic assumption is that some phenomena… depend intimately on a system’s initial conditions, so that an imperceptible change in the beginning value of a variable can make the outcome of a process impossible to predict (Porter 532). All applications of chaos theory work outward from this core principle, which is also historically situated (in the article) through the work of Poincaré: Poincaré started working on equations to predict the positions of the planets as they rotated around the sun… Note the starting positions and velocities, feed them into a set of equations based on Newton’s laws of motion, and the results should predict future positions. But the outcome turned Poincaré’s expectations upside down. With only two planets under consideration, he found that even tiny differences in the initial conditions… elicited substantial changes in future positions (Porter 532). So, like quantum mechanics, the hard core of chaos is situated historically in a few irrefutable examples and principles. For quantum mechanics some examples of the core principles are the Heisenberg uncertainty principle and Planck’s assumption of discrete quanta. The individuals most often recognized historically as exemplars of quantum mechanical theory are Einstein, Bohr, Born, Ehrenfest, to name a few. These examples are constantly cited and referred to both pedagogically, and in scientists’ description of the birth of their field. Chaos theory’s core principle is that we cannot accurately predict the future state of a dynamical (i.e. chaotic) system. This principle is exemplified in the early work of Poincaré (which is generally seen as proto-chaotic) and the later meteorological studies of Lorenz (who is also mentioned by Porter/Liboff). Now we move on to the question of whether or not chaos theory has a positive heuristic, which determines the problems to be solved. It seems, at least prima facie (which is as far as such a limited study can go) that, unlike quantum mechanics (whose scope is internally limited to the ‘quantum realm’) that chaos theory has the potential to be applied to any system. In this respect, can it be considered a research programme? If it has historically been applied only within other research programmes (meteorology, electrodynamics, planetary motion, to name only a few mentioned in the article itself) it does not seem plausible that it can define its own problems and attempt to solve them in seclusion from other research programmes. Rather than a research programme, I propose that chaos theory is a self-contained theory (a modeling or mathematical tool) that functions within a variety of established and independent research programmes. On this view, it would appear that quantum-chaos, far from being an independent research programme, is the result of a development that is internal to the progressive research programme of quantum mechanics. Quantum-chaos is not an entirely new system of ideas, but a growth of new ideas within the boundaries of the quantum realm. That is, without quantum mechanics, there would be no realm in which to create quantum-chaos, and no ‘rules’ with which to describe it. Critique of Feyerabend and Lakatos Now that we have seen a few of the ideas of Feyerabend and Lakatos in application (albeit forcefully) I shall move on to a critical engagement of the two, playing off their views (as well as my own) against one another. I will start with Lakatos. It seems that though the research programme is a valuable historiographical lens with which to view scientific history, it has obvious limitations. Although it enables the historian of science to encompass more examples than something like (what Lakatos calls) a ‘conventionalist’ historiography, it is by no means all encompassing. The main problem that I see with his methodology is one that Lakatos states himself. The methodology of research programmes – like any other theory of scientific rationality – must be supplemented by empirical-external history. No rationality theory will ever solve the problems like why Mendelian genetics disappeared in Soviet Russia in the 1950’s, or why certain schools of research into genetic racial differences or into the economics of foreign aid came into disrepute in the Anglo-Saxon countries in the 1960’s… (Lakatos 119). So, like most other rationalist reconstructions of the history of science, his attempt must be supplemented by psychological, sociological and other explanations. The difference between a falsificationist like Popper and someone like Lakatos is that Lakatos at least admits that there are other factors in the history of science than rational ones. But, for a rationalist project, whose aim is to explain all scientific change, this fundamental problem simply cannot be overcome. The problem is that the human agents in science (who, despite any talk of a ‘third world’ are key agents in scientific change) are never fully, or exclusively, rational. If we are bound by a purely rational reconstruction of the history of science, then the irrational in science (which Lakatos admits exists) will always elude our methodological understanding. Lakatos denies that any theory of scientific rationality can succeed in this task. The problem of irrationality in science is one that I believe Feyerabend can overcome more easily. To him it seems that if a completely rational reconstruction (based on the rigorous application of a specific ‘system’) is bound to fail, then should we not look at the possibility of an irrational, even non-systematic explanation of the history of science? Obviously such an explanation could not be termed a ‘methodology’ but through something like it, we could attempt to explain any historical stage of science. Such an irrational, anti-methodological approach is precisely what Paul Feyerabend calls for. Feyerabend’s explanations do not rely on the constancy of a specific method or concept, but fluctuate based on the particular situation they are attempting to ‘explain’. When talking about a series of lectures he had given at the London School of Economics, Feyerabend sketches out for us his intent: My aim in the lectures was to show that some very simple and plausible rules and standards which both philosophers and scientists regarded as essential parts of rationality were violated in the course of episodes (Copernican Revolution; triumph of the kinetic theory; rise of quantum theory; and so on) they regarded as equally essential. More specifically I tried to show (a) that the rules (standards) were actually violated and that the more perceptive scientists were aware of the violations; and (b) that they had to be violated. Insistence on the rules would not have improved matters, it would have arrested progress (SFS 13). Feyerabend suggests here that not only are rules not always fruitful in science, but that strict adherence to those rules sometimes hinders its progress. The same can be said about historiography of science. If we insist on strict adherence to specific rules in all cases then not only are going to get it ‘wrong’, but we may make it harder to get it ‘right’ (i.e. more useful, less problematic historical descriptions). So, we have discussed a specific problem with Lakatos’ methodology of research programmes and ended up at the seeming inadequacy of all methodologies. But neither I, nor Feyerabend, believe that there are never times when rules can be applied fruitfully to historical analyses. Indeed, Lakatos’ concept of the research programme seems to provide criteria that are more widely applicable than many others proposed before it. It does not fall prey to the rash assumption that science is strictly rational, thought it admits science’s rationality is all that it can explain. This is precisely what Feyerabend wants the rationalists (and particularly the other LSE rationalists) to admit: that we cannot always fit history into the box of rationality (regardless of whether the box is that of falsificationism or the methodology of research programmes). So, on the one hand, Lakatosian research programmes explain more than any other rationalist reconstruction can, but on the other hand, Lakatos admits that (unlike Feyerabend) he cannot explain irrationality in science. How can I criticize Feyerabend? If I accused him of incoherence, or self-contradiction, he would take it as a complement. If one can accept any standard at any time, depending upon the circumstances, then of course one can seem to be contradictory, he would say. I tend to agree with Feyerabend that no rules can be applied absolutely, for all time. But, one might criticize him in his specific historical analyses. For instance, his emphasis on the rhetorical (non-rational) use of language and irrational ‘methods’ of Galileo and Copernicus may ignore some of the important rational features in their work. Though this problem may be inherent to an attack on rationalist reconstructions of science, I think that Feyerabend often ignores salient features of history simply because they are instances of rationality. That being said, I believe that Feyerabend’s philosophy of science provides us with the mindset to build a number of very unique perspectives on the history of science. He tells us that no method can work absolutely, but some methods can work sometimes. Our task is to think for ourselves and create our own interpretations of science, and not to rely on the grandiose systems of our predecessors. Bennett, Jesse. The Cosmic Perspective (1st edition), Addison Wesley Longman, 1999, New York. “Chaos on the Quantum Scale”, Mason A. Porter and Richard L. Liboff, pp.532-537 in American Scientist Volume 89, No. 6 November-December 2001. “Chaos Theory and Fractals”, Jonathan Mendelson and Elana Blumenthal 2000-2001, URL: http://www.mathjmendl.org/chaos/index.html “Early Quantum Mechanics”, J J O'Connor and E F Robertson 1996, URL: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html Einstein, Albert. “On the Quantum Theory of Radiation” pp63-77 in Sources of Quantum Mechanics. Ed. B.L. Van der Waerden, 1968, Dover Publications, New York. Feyerabend, Paul. Against Method, Verso, 1988 1975 New York. (Referred to in the text as AM) Feyerabend, Paul. Science in a Free Society, New Left Books, 1978, London (referred to in the text as SFS). Lakatos, Imre. “History of Science and its Rational Reconstructions” pp.107-127 in Scientific Revolutions, ed. Ian Hacking, 1981, Oxford University Press, New York. Wittgenstein, Ludwig. Philosophical Investigations. Translated by G.E.M. 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6c8ddb61ba43cdc4	Nobel Prizes and Laureates Nobel Prizes and Laureates The Nobel Prize in Physics 1965 Sin-Itiro Tomonaga, Julian Schwinger, Richard P. Feynman Share this: Nobel Lecture Nobel Lecture, December 11, 1965 The Development of the Space-Time View of Quantum Electrodynamics We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize. I realize that a truly scientific paper would be of greater value, but such a paper I could publish in regular journals. So, I shall use this Nobel Lecture as an opportunity to do something of less value, but which I cannot do elsewhere. I ask your indulgence in another manner. I shall include details of anecdotes which are of no value either scientifically, nor for understanding the development of ideas. They are included only to make the lecture more entertaining. I worked on this problem about eight years until the final publication in 1947. The beginning of the thing was at the Massachusetts Institute of Technology, when I was an undergraduate student reading about the known physics, learning slowly about all these things that people were worrying about, and realizing ultimately that the fundamental problem of the day was that the quantum theory of electricity and magnetism was not completely satisfactory. This I gathered from books like those of Heitler and Dirac. I was inspired by the remarks in these books; not by the parts in which everything was proved and demonstrated carefully and calculated, because I couldn't understand those very well. At the young age what I could understand were the remarks about the fact that this doesn't make any sense, and the last sentence of the book of Dirac I can still remember, "It seems that some essentially new physical ideas are here needed." So, I had this as a challenge and an inspiration. I also had a personal feeling, that since they didn't get a satisfactory answer to the problem I wanted to solve, I don't have to pay a lot of attention to what they did do. I did gather from my readings, however, that two things were the source of the difficulties with the quantum electrodynamical theories. The first was an infinite energy of interaction of the electron with itself. And this difficulty existed even in the classical theory. The other difficulty came from some infinites which had to do with the infinite numbers of degrees of freedom in the field. As I understood it at the time (as nearly as I can remember) this was simply the difficulty that if you quantized the harmonic oscillators of the field (say in a box) each oscillator has a ground state energy of (½) and there is an infinite number of modes in a box of every increasing frequency w, and therefore there is an infinite energy in the box. I now realize that that wasn't a completely correct statement of the central problem; it can be removed simply by changing the zero from which energy is measured. At any rate, I believed that the difficulty arose somehow from a combination of the electron acting on itself and the infinite number of degrees of freedom of the field. Well, it seemed to me quite evident that the idea that a particle acts on itself, that the electrical force acts on the same particle that generates it, is not a necessary one - it is a sort of a silly one, as a matter of fact. And, so I suggested to myself, that electrons cannot act on themselves, they can only act on other electrons. That means there is no field at all. You see, if all charges contribute to making a single common field, and if that common field acts back on all the charges, then each charge must act back on itself. Well, that was where the mistake was, there was no field. It was just that when you shook one charge, another would shake later. There was a direct interaction between charges, albeit with a delay. The law of force connecting the motion of one charge with another would just involve a delay. Shake this one, that one shakes later. The sun atom shakes; my eye electron shakes eight minutes later, because of a direct interaction across. Now, this has the attractive feature that it solves both problems at once. First, I can say immediately, I don't let the electron act on itself, I just let this act on that, hence, no self-energy! Secondly, there is not an infinite number of degrees of freedom in the field. There is no field at all; or if you insist on thinking in terms of ideas like that of a field, this field is always completely determined by the action of the particles which produce it. You shake this particle, it shakes that one, but if you want to think in a field way, the field, if it's there, would be entirely determined by the matter which generates it, and therefore, the field does not have any independent degrees of freedom and the infinities from the degrees of freedom would then be removed. As a matter of fact, when we look out anywhere and see light, we can always "see" some matter as the source of the light. We don't just see light (except recently some radio reception has been found with no apparent material source). You see then that my general plan was to first solve the classical problem, to get rid of the infinite self-energies in the classical theory, and to hope that when I made a quantum theory of it, everything would just be fine. That was the beginning, and the idea seemed so obvious to me and so elegant that I fell deeply in love with it. And, like falling in love with a woman, it is only possible if you do not know much about her, so you cannot see her faults. The faults will become apparent later, but after the love is strong enough to hold you to her. So, I was held to this theory, in spite of all difficulties, by my youthful enthusiasm. Then I went to graduate school and somewhere along the line I learned what was wrong with the idea that an electron does not act on itself. When you accelerate an electron it radiates energy and you have to do extra work to account for that energy. The extra force against which this work is done is called the force of radiation resistance. The origin of this extra force was identified in those days, following Lorentz, as the action of the electron itself. The first term of this action, of the electron on itself, gave a kind of inertia (not quite relativistically satisfactory). But that inertia-like term was infinite for a point-charge. Yet the next term in the sequence gave an energy loss rate, which for a point-charge agrees exactly with the rate you get by calculating how much energy is radiated. So, the force of radiation resistance, which is absolutely necessary for the conservation of energy would disappear if I said that a charge could not act on itself. So, I learned in the interim when I went to graduate school the glaringly obvious fault of my own theory. But, I was still in love with the original theory, and was still thinking that with it lay the solution to the difficulties of quantum electrodynamics. So, I continued to try on and off to save it somehow. I must have some action develop on a given electron when I accelerate it to account for radiation resistance. But, if I let electrons only act on other electrons the only possible source for this action is another electron in the world. So, one day, when I was working for Professor Wheeler and could no longer solve the problem that he had given me, I thought about this again and I calculated the following. Suppose I have two charges - I shake the first charge, which I think of as a source and this makes the second one shake, but the second one shaking produces an effect back on the source. And so, I calculated how much that effect back on the first charge was, hoping it might add up the force of radiation resistance. It didn't come out right, of course, but I went to Professor Wheeler and told him my ideas. He said, - yes, but the answer you get for the problem with the two charges that you just mentioned will, unfortunately, depend upon the charge and the mass of the second charge and will vary inversely as the square of the distance R, between the charges, while the force of radiation resistance depends on none of these things. I thought, surely, he had computed it himself, but now having become a professor, I know that one can be wise enough to see immediately what some graduate student takes several weeks to develop. He also pointed out something that also bothered me, that if we had a situation with many charges all around the original source at roughly uniform density and if we added the effect of all the surrounding charges the inverse R square would be compensated by the R2 in the volume element and we would get a result proportional to the thickness of the layer, which would go to infinity. That is, one would have an infinite total effect back at the source. And, finally he said to me, and you forgot something else, when you accelerate the first charge, the second acts later, and then the reaction back here at the source would be still later. In other words, the action occurs at the wrong time. I suddenly realized what a stupid fellow I am, for what I had described and calculated was just ordinary reflected light, not radiation reaction. But, as I was stupid, so was Professor Wheeler that much more clever. For he then went on to give a lecture as though he had worked this all out before and was completely prepared, but he had not, he worked it out as he went along. First, he said, let us suppose that the return action by the charges in the absorber reaches the source by advanced waves as well as by the ordinary retarded waves of reflected light; so that the law of interaction acts backward in time, as well as forward in time. I was enough of a physicist at that time not to say, "Oh, no, how could that be?" For today all physicists know from studying Einstein and Bohr, that sometimes an idea which looks completely paradoxical at first, if analyzed to completion in all detail and in experimental situations, may, in fact, not be paradoxical. So, it did not bother me any more than it bothered Professor Wheeler to use advance waves for the back reaction - a solution of Maxwell's equations, which previously had not been physically used. Professor Wheeler used advanced waves to get the reaction back at the right time and then he suggested this: If there were lots of electrons in the absorber, there would be an index of refraction n, so, the retarded waves coming from the source would have their wave lengths slightly modified in going through the absorber. Now, if we shall assume that the advanced waves come back from the absorber without an index - why? I don't know, let's assume they come back without an index - then, there will be a gradual shifting in phase between the return and the original signal so that we would only have to figure that the contributions act as if they come from only a finite thickness, that of the first wave zone. (More specifically, up to that depth where the phase in the medium is shifted appreciably from what it would be in vacuum, a thickness proportional to l/(n-1). ) Now, the less the number of electrons in here, the less each contributes, but the thicker will be the layer that effectively contributes because with less electrons, the index differs less from 1. The higher the charges of these electrons, the more each contribute, but the thinner the effective layer, because the index would be higher. And when we estimated it, (calculated without being careful to keep the correct numerical factor) sure enough, it came out that the action back at the source was completely independent of the properties of the charges that were in the surrounding absorber. Further, it was of just the right character to represent radiation resistance, but we were unable to see if it was just exactly the right size. He sent me home with orders to figure out exactly how much advanced and how much retarded wave we need to get the thing to come out numerically right, and after that, figure out what happens to the advanced effects that you would expect if you put a test charge here close to the source? For if all charges generate advanced, as well as retarded effects, why would that test not be affected by the advanced waves from the source? I found that you get the right answer if you use half-advanced and half-retarded as the field generated by each charge. That is, one is to use the solution of Maxwell's equation which is symmetrical in time and that the reason we got no advanced effects at a point close to the source in spite of the fact that the source was producing an advanced field is this. Suppose the source s surrounded by a spherical absorbing wall ten light seconds away, and that the test charge is one second to the right of the source. Then the source is as much as eleven seconds away from some parts of the wall and only nine seconds away from other parts. The source acting at time t=0 induces motions in the wall at time +10. Advanced effects from this can act on the test charge as early as eleven seconds earlier, or at t= -1. This is just at the time that the direct advanced waves from the source should reach the test charge, and it turns out the two effects are exactly equal and opposite and cancel out! At the later time +1 effects on the test charge from the source and from the walls are again equal, but this time are of the same sign and add to convert the half-retarded wave of the source to full retarded strength. Thus, it became clear that there was the possibility that if we assume all actions are via half-advanced and half-retarded solutions of Maxwell's equations and assume that all sources are surrounded by material absorbing all the the light which is emitted, then we could account for radiation resistance as a direct action of the charges of the absorber acting back by advanced waves on the source. Many months were devoted to checking all these points. I worked to show that everything is independent of the shape of the container, and so on, that the laws are exactly right, and that the advanced effects really cancel in every case. We always tried to increase the efficiency of our demonstrations, and to see with more and more clarity why it works. I won't bore you by going through the details of this. Because of our using advanced waves, we also had many apparent paradoxes, which we gradually reduced one by one, and saw that there was in fact no logical difficulty with the theory. It was perfectly satisfactory. We also found that we could reformulate this thing in another way, and that is by a principle of least action. Since my original plan was to describe everything directly in terms of particle motions, it was my desire to represent this new theory without saying anything about fields. It turned out that we found a form for an action directly involving the motions of the charges only, which upon variation would give the equations of motion of these charges. The expression for this action A is where is the four-vector position of the ith particle as a function of some parameter . The first term is the integral of proper time, the ordinary action of relativistic mechanics of free particles of mass mi. (We sum in the usual way on the repeated index m.) The second term represents the electrical interaction of the charges. It is summed over each pair of charges (the factor ½ is to count each pair once, the term i=j is omitted to avoid self-action) .The interaction is a double integral over a delta function of the square of space-time interval I2 between two points on the paths. Thus, interaction occurs only when this interval vanishes, that is, along light cones. The fact that the interaction is exactly one-half advanced and half-retarded meant that we could write such a principle of least action, whereas interaction via retarded waves alone cannot be written in such a way. So, all of classical electrodynamics was contained in this very simple form. It looked good, and therefore, it was undoubtedly true, at least to the beginner. It automatically gave half-advanced and half-retarded effects and it was without fields. By omitting the term in the sum when i=j, I omit self-interaction and no longer have any infinite self-energy. This then was the hoped-for solution to the problem of ridding classical electrodynamics of the infinities. It turns out, of course, that you can reinstate fields if you wish to, but you have to keep track of the field produced by each particle separately. This is because to find the right field to act on a given particle, you must exclude the field that it creates itself. A single universal field to which all contribute will not do. This idea had been suggested earlier by Frenkel and so we called these Frenkel fields. This theory which allowed only particles to act on each other was equivalent to Frenkel's fields using half-advanced and half-retarded solutions. There were several suggestions for interesting modifications of electrodynamics. We discussed lots of them, but I shall report on only one. It was to replace this delta function in the interaction by another function, say, f(I2ij), which is not infinitely sharp. Instead of having the action occur only when the interval between the two charges is exactly zero, we would replace the delta function of I2 by a narrow peaked thing. Let's say that f(Z) is large only near Z=0 width of order a2. Interactions will now occur when T2-R2 is of order a2 roughly where T is the time difference and R is the separation of the charges. This might look like it disagrees with experience, but if a is some small distance, like 10-13 cm, it says that the time delay T in action is roughly or approximately, - if R is much larger than a, T=R±a2/2R. This means that the deviation of time T from the ideal theoretical time R of Maxwell, gets smaller and smaller, the further the pieces are apart. Therefore, all theories involving in analyzing generators, motors, etc., in fact, all of the tests of electrodynamics that were available in Maxwell's time, would be adequately satisfied if were 10-13 cm. If R is of the order of a centimeter this deviation in T is only 10-26 parts. So, it was possible, also, to change the theory in a simple manner and to still agree with all observations of classical electrodynamics. You have no clue of precisely what function to put in for f, but it was an interesting possibility to keep in mind when developing quantum electrodynamics. It also occurred to us that if we did that (replace d by f) we could not reinstate the term i=j in the sum because this would now represent in a relativistically invariant fashion a finite action of a charge on itself. In fact, it was possible to prove that if we did do such a thing, the main effect of the self-action (for not too rapid accelerations) would be to produce a modification of the mass. In fact, there need be no mass mi, term, all the mechanical mass could be electromagnetic self-action. So, if you would like, we could also have another theory with a still simpler expression for the action A. In expression (1) only the second term is kept, the sum extended over all i and j, and some function replaces d. Such a simple form could represent all of classical electrodynamics, which aside from gravitation is essentially all of classical physics. Although it may sound confusing, I am describing several different alternative theories at once. The important thing to note is that at this time we had all these in mind as different possibilities. There were several possible solutions of the difficulty of classical electrodynamics, any one of which might serve as a good starting point to the solution of the difficulties of quantum electrodynamics. I would also like to emphasize that by this time I was becoming used to a physical point of view different from the more customary point of view. In the customary view, things are discussed as a function of time in very great detail. For example, you have the field at this moment, a differential equation gives you the field at the next moment and so on; a method, which I shall call the Hamilton method, the time differential method. We have, instead (in (1) say) a thing that describes the character of the path throughout all of space and time. The behavior of nature is determined by saying her whole spacetime path has a certain character. For an action like (1) the equations obtained by variation (of Xim (ai)) are no longer at all easy to get back into Hamiltonian form. If you wish to use as variables only the coordinates of particles, then you can talk about the property of the paths - but the path of one particle at a given time is affected by the path of another at a different time. If you try to describe, therefore, things differentially, telling what the present conditions of the particles are, and how these present conditions will affect the future you see, it is impossible with particles alone, because something the particle did in the past is going to affect the future. Therefore, you need a lot of bookkeeping variables to keep track of what the particle did in the past. These are called field variables. You will, also, have to tell what the field is at this present moment, if you are to be able to see later what is going to happen. From the overall space-time view of the least action principle, the field disappears as nothing but bookkeeping variables insisted on by the Hamiltonian method. As a by-product of this same view, I received a telephone call one day at the graduate college at Princeton from Professor Wheeler, in which he said, "Feynman, I know why all electrons have the same charge and the same mass" "Why?" "Because, they are all the same electron!" And, then he explained on the telephone, "suppose that the world lines which we were ordinarily considering before in time and space - instead of only going up in time were a tremendous knot, and then, when we cut through the knot, by the plane corresponding to a fixed time, we would see many, many world lines and that would represent many electrons, except for one thing. If in one section this is an ordinary electron world line, in the section in which it reversed itself and is coming back from the future we have the wrong sign to the proper time - to the proper four velocities - and that's equivalent to changing the sign of the charge, and, therefore, that part of a path would act like a positron." "But, Professor", I said, "there aren't as many positrons as electrons." "Well, maybe they are hidden in the protons or something", he said. I did not take the idea that all the electrons were the same one from him as seriously as I took the observation that positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole! To summarize, when I was done with this, as a physicist I had gained two things. One, I knew many different ways of formulating classical electrodynamics, with many different mathematical forms. I got to know how to express the subject every which way. Second, I had a point of view - the overall space-time point of view - and a disrespect for the Hamiltonian method of describing physics. I would like to interrupt here to make a remark. The fact that electrodynamics can be written in so many ways - the differential equations of Maxwell, various minimum principles with fields, minimum principles without fields, all different kinds of ways, was something I knew, but I have never understood. It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but, with a little mathematical fiddling you can show the relationship. An example of that is the Schrödinger equation and the Heisenberg formulation of quantum mechanics. I don't know why this is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature. A thing like the inverse square law is just right to be represented by the solution of Poisson's equation, which, therefore, is a very different way to say the same thing that doesn't look at all like the way you said it before. I don't know what it means, that nature chooses these curious forms, but maybe that is a way of defining simplicity. Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing. I was now convinced that since we had solved the problem of classical electrodynamics (and completely in accordance with my program from M.I.T., only direct interaction between particles, in a way that made fields unnecessary) that everything was definitely going to be all right. I was convinced that all I had to do was make a quantum theory analogous to the classical one and everything would be solved. So, the problem is only to make a quantum theory, which has as its classical analog, this expression (1). Now, there is no unique way to make a quantum theory from classical mechanics, although all the textbooks make believe there is. What they would tell you to do, was find the momentum variables and replace them by , but I couldn't find a momentum variable, as there wasn't any. The character of quantum mechanics of the day was to write things in the famous Hamiltonian way - in the form of a differential equation, which described how the wave function changes from instant to instant, and in terms of an operator, H. If the classical physics could be reduced to a Hamiltonian form, everything was all right. Now, least action does not imply a Hamiltonian form if the action is a function of anything more than positions and velocities at the same moment. If the action is of the form of the integral of a function, (usually called the Lagrangian) of the velocities and positions at the same time then you can start with the Lagrangian and then create a Hamiltonian and work out the quantum mechanics, more or less uniquely. But this thing (1) involves the key variables, positions, at two different times and therefore, it was not obvious what to do to make the quantum-mechanical analogue. I tried - I would struggle in various ways. One of them was this; if I had harmonic oscillators interacting with a delay in time, I could work out what the normal modes were and guess that the quantum theory of the normal modes was the same as for simple oscillators and kind of work my way back in terms of the original variables. I succeeded in doing that, but I hoped then to generalize to other than a harmonic oscillator, but I learned to my regret something, which many people have learned. The harmonic oscillator is too simple; very often you can work out what it should do in quantum theory without getting much of a clue as to how to generalize your results to other systems. So that didn't help me very much, but when I was struggling with this problem, I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think that a good place to discuss intellectual matters is a beer party. So, he sat by me and asked, "what are you doing" and so on, and I said, "I'm drinking beer." Then I realized that he wanted to know what work I was doing and I told him I was struggling with this problem, and I simply turned to him and said, "listen, do you know any way of doing quantum mechanics, starting with action - where the action integral comes into the quantum mechanics?" "No", he said, "but Dirac has a paper in which the Lagrangian, at least, comes into quantum mechanics. I will show it to you tomorrow." Next day we went to the Princeton Library, they have little rooms on the side to discuss things, and he showed me this paper. What Dirac said was the following: There is in quantum mechanics a very important quantity which carries the wave function from one time to another, besides the differential equation but equivalent to it, a kind of a kernal, which we might call K(x', x), which carries the wave function j(x) known at time t, to the wave function j(x') at time, t+e Dirac points out that this function K was analogous to the quantity in classical mechanics that you would calculate if you took the exponential of ie, multiplied by the Lagrangian imagining that these two positions x,x' corresponded t and t+e. In other words, Professor Jehle showed me this, I read it, he explained it to me, and I said, "what does he mean, they are analogous; what does that mean, analogous? What is the use of that?" He said, "you Americans! You always want to find a use for everything!" I said, that I thought that Dirac must mean that they were equal. "No", he explained, "he doesn't mean they are equal." "Well", I said, "let's see what happens if we make them equal." So I simply put them equal, taking the simplest example where the Lagrangian is ½Mx2 - V(x) but soon found I had to put a constant of proportionality A in, suitably adjusted. When I substituted for K to get and just calculated things out by Taylor series expansion, out came the Schrödinger equation. So, I turned to Professor Jehle, not really understanding, and said, "well, you see Professor Dirac meant that they were proportional." Professor Jehle's eyes were bugging out - he had taken out a little notebook and was rapidly copying it down from the blackboard, and said, "no, no, this is an important discovery. You Americans are always trying to find out how something can be used. That's a good way to discover things!" So, I thought I was finding out what Dirac meant, but, as a matter of fact, had made the discovery that what Dirac thought was analogous, was, in fact, equal. I had then, at least, the connection between the Lagrangian and quantum mechanics, but still with wave functions and infinitesimal times. It must have been a day or so later when I was lying in bed thinking about these things, that I imagined what would happen if I wanted to calculate the wave function at a finite interval later. I would put one of these factors eieL in here, and that would give me the wave functions the next moment, t+e and then I could substitute that back into (3) to get another factor of eieL and give me the wave function the next moment, t+2e and so on and so on. In that way I found myself thinking of a large number of integrals, one after the other in sequence. In the integrand was the product of the exponentials, which, of course, was the exponential of the sum of terms like eL. Now, L is the Lagrangian and e is like the time interval dt, so that if you took a sum of such terms, that's exactly like an integral. That's like Riemann's formula for the integral Ldt, you just take the value at each point and add them together. We are to take the limit as e-0, of course. Therefore, the connection between the wave function of one instant and the wave function of another instant a finite time later could be obtained by an infinite number of integrals, (because e goes to zero, of course) of exponential where S is the action expression (2). At last, I had succeeded in representing quantum mechanics directly in terms of the action S. This led later on to the idea of the amplitude for a path; that for each possible way that the particle can go from one point to another in space-time, there's an amplitude. That amplitude is e to the times the action for the path. Amplitudes from various paths superpose by addition. This then is another, a third way, of describing quantum mechanics, which looks quite different than that of Schrödinger or Heisenberg, but which is equivalent to them. Now immediately after making a few checks on this thing, what I wanted to do, of course, was to substitute the action (1) for the other (2). The first trouble was that I could not get the thing to work with the relativistic case of spin one-half. However, although I could deal with the matter only nonrelativistically, I could deal with the light or the photon interactions perfectly well by just putting the interaction terms of (1) into any action, replacing the mass terms by the non-relativistic (Mx2/2)dt. When the action has a delay, as it now had, and involved more than one time, I had to lose the idea of a wave function. That is, I could no longer describe the program as; given the amplitude for all positions at a certain time to compute the amplitude at another time. However, that didn't cause very much trouble. It just meant developing a new idea. Instead of wave functions we could talk about this; that if a source of a certain kind emits a particle, and a detector is there to receive it, we can give the amplitude that the source will emit and the detector receive. We do this without specifying the exact instant that the source emits or the exact instant that any detector receives, without trying to specify the state of anything at any particular time in between, but by just finding the amplitude for the complete experiment. And, then we could discuss how that amplitude would change if you had a scattering sample in between, as you rotated and changed angles, and so on, without really having any wave functions. It was also possible to discover what the old concepts of energy and momentum would mean with this generalized action. And, so I believed that I had a quantum theory of classical electrodynamics - or rather of this new classical electrodynamics described by action (1). I made a number of checks. If I took the Frenkel field point of view, which you remember was more differential, I could convert it directly to quantum mechanics in a more conventional way. The only problem was how to specify in quantum mechanics the classical boundary conditions to use only half-advanced and half-retarded solutions. By some ingenuity in defining what that meant, I found that the quantum mechanics with Frenkel fields, plus a special boundary condition, gave me back this action, (1) in the new form of quantum mechanics with a delay. So, various things indicated that there wasn't any doubt I had everything straightened out. It was also easy to guess how to modify the electrodynamics, if anybody ever wanted to modify it. I just changed the delta to an f, just as I would for the classical case. So, it was very easy, a simple thing. To describe the old retarded theory without explicit mention of fields I would have to write probabilities, not just amplitudes. I would have to square my amplitudes and that would involve double path integrals in which there are two S's and so forth. Yet, as I worked out many of these things and studied different forms and different boundary conditions. I got a kind of funny feeling that things weren't exactly right. I could not clearly identify the difficulty and in one of the short periods during which I imagined I had laid it to rest, I published a thesis and received my Ph.D. During the war, I didn't have time to work on these things very extensively, but wandered about on buses and so forth, with little pieces of paper, and struggled to work on it and discovered indeed that there was something wrong, something terribly wrong. I found that if one generalized the action from the nice Langrangian forms (2) to these forms (1) then the quantities which I defined as energy, and so on, would be complex. The energy values of stationary states wouldn't be real and probabilities of events wouldn't add up to 100%. That is, if you took the probability that this would happen and that would happen - everything you could think of would happen, it would not add up to one. Another problem on which I struggled very hard, was to represent relativistic electrons with this new quantum mechanics. I wanted to do a unique and different way - and not just by copying the operators of Dirac into some kind of an expression and using some kind of Dirac algebra instead of ordinary complex numbers. I was very much encouraged by the fact that in one space dimension, I did find a way of giving an amplitude to every path by limiting myself to paths, which only went back and forth at the speed of light. The amplitude was simple (ie) to a power equal to the number of velocity reversals where I have divided the time into steps and I am allowed to reverse velocity only at such a time. This gives (as approaches zero) Dirac's equation in two dimensions - one dimension of space and one of time . Dirac's wave function has four components in four dimensions, but in this case, it has only two components and this rule for the amplitude of a path automatically generates the need for two components. Because if this is the formula for the amplitudes of path, it will not do you any good to know the total amplitude of all paths, which come into a given point to find the amplitude to reach the next point. This is because for the next time, if it came in from the right, there is no new factor ie if it goes out to the right, whereas, if it came in from the left there was a new factor ie. So, to continue this same information forward to the next moment, it was not sufficient information to know the total amplitude to arrive, but you had to know the amplitude to arrive from the right and the amplitude to arrive to the left, independently. If you did, however, you could then compute both of those again independently and thus you had to carry two amplitudes to form a differential equation (first order in time). And, so I dreamed that if I were clever, I would find a formula for the amplitude of a path that was beautiful and simple for three dimensions of space and one of time, which would be equivalent to the Dirac equation, and for which the four components, matrices, and all those other mathematical funny things would come out as a simple consequence - I have never succeeded in that either. But, I did want to mention some of the unsuccessful things on which I spent almost as much effort, as on the things that did work. To summarize the situation a few years after the way, I would say, I had much experience with quantum electrodynamics, at least in the knowledge of many different ways of formulating it, in terms of path integrals of actions and in other forms. One of the important by-products, for example, of much experience in these simple forms, was that it was easy to see how to combine together what was in those days called the longitudinal and transverse fields, and in general, to see clearly the relativistic invariance of the theory. Because of the need to do things differentially there had been, in the standard quantum electrodynamics, a complete split of the field into two parts, one of which is called the longitudinal part and the other mediated by the photons, or transverse waves. The longitudinal part was described by a Coulomb potential acting instantaneously in the Schrödinger equation, while the transverse part had entirely different description in terms of quantization of the transverse waves. This separation depended upon the relativistic tilt of your axes in spacetime. People moving at different velocities would separate the same field into longitudinal and transverse fields in a different way. Furthermore, the entire formulation of quantum mechanics insisting, as it did, on the wave function at a given time, was hard to analyze relativistically. Somebody else in a different coordinate system would calculate the succession of events in terms of wave functions on differently cut slices of space-time, and with a different separation of longitudinal and transverse parts. The Hamiltonian theory did not look relativistically invariant, although, of course, it was. One of the great advantages of the overall point of view, was that you could see the relativistic invariance right away - or as Schwinger would say - the covariance was manifest. I had the advantage, therefore, of having a manifestedly covariant form for quantum electrodynamics with suggestions for modifications and so on. I had the disadvantage that if I took it too seriously - I mean, if I took it seriously at all in this form, - I got into trouble with these complex energies and the failure of adding probabilities to one and so on. I was unsuccessfully struggling with that. Then Lamb did his experiment, measuring the separation of the 2S½ and 2P½ levels of hydrogen, finding it to be about 1000 megacycles of frequency difference. Professor Bethe, with whom I was then associated at Cornell, is a man who has this characteristic: If there's a good experimental number you've got to figure it out from theory. So, he forced the quantum electrodynamics of the day to give him an answer to the separation of these two levels. He pointed out that the self-energy of an electron itself is infinite, so that the calculated energy of a bound electron should also come out infinite. But, when you calculated the separation of the two energy levels in terms of the corrected mass instead of the old mass, it would turn out, he thought, that the theory would give convergent finite answers. He made an estimate of the splitting that way and found out that it was still divergent, but he guessed that was probably due to the fact that he used an unrelativistic theory of the matter. Assuming it would be convergent if relativistically treated, he estimated he would get about a thousand megacycles for the Lamb-shift, and thus, made the most important discovery in the history of the theory of quantum electrodynamics. He worked this out on the train from Ithaca, New York to Schenectady and telephoned me excitedly from Schenectady to tell me the result, which I don't remember fully appreciating at the time. Returning to Cornell, he gave a lecture on the subject, which I attended. He explained that it gets very confusing to figure out exactly which infinite term corresponds to what in trying to make the correction for the infinite change in mass. If there were any modifications whatever, he said, even though not physically correct, (that is not necessarily the way nature actually works) but any modification whatever at high frequencies, which would make this correction finite, then there would be no problem at all to figuring out how to keep track of everything. You just calculate the finite mass correction Dm to the electron mass mo, substitute the numerical values of mo+Dm for m in the results for any other problem and all these ambiguities would be resolved. If, in addition, this method were relativistically invariant, then we would be absolutely sure how to do it without destroying relativistically invariant. After the lecture, I went up to him and told him, "I can do that for you, I'll bring it in for you tomorrow." I guess I knew every way to modify quantum electrodynamics known to man, at the time. So, I went in next day, and explained what would correspond to the modification of the delta-function to f and asked him to explain to me how you calculate the self-energy of an electron, for instance, so we can figure out if it's finite. I want you to see an interesting point. I did not take the advice of Professor Jehle to find out how it was useful. I never used all that machinery which I had cooked up to solve a single relativistic problem. I hadn't even calculated the self-energy of an electron up to that moment, and was studying the difficulties with the conservation of probability, and so on, without actually doing anything, except discussing the general properties of the theory. But now I went to Professor Bethe, who explained to me on the blackboard, as we worked together, how to calculate the self-energy of an electron. Up to that time when you did the integrals they had been logarithmically divergent. I told him how to make the relativistically invariant modifications that I thought would make everything all right. We set up the integral which then diverged at the sixth power of the frequency instead of logarithmically! So, I went back to my room and worried about this thing and went around in circles trying to figure out what was wrong because I was sure physically everything had to come out finite, I couldn't understand how it came out infinite. I became more and more interested and finally realized I had to learn how to make a calculation. So, ultimately, I taught myself how to calculate the self-energy of an electron working my patient way through the terrible confusion of those days of negative energy states and holes and longitudinal contributions and so on. When I finally found out how to do it and did it with the modifications I wanted to suggest, it turned out that it was nicely convergent and finite, just as I had expected. Professor Bethe and I have never been able to discover what we did wrong on that blackboard two months before, but apparently we just went off somewhere and we have never been able to figure out where. It turned out, that what I had proposed, if we had carried it out without making a mistake would have been all right and would have given a finite correction. Anyway, it forced me to go back over all this and to convince myself physically that nothing can go wrong. At any rate, the correction to mass was now finite, proportional to where a is the width of that function f which was substituted for d. If you wanted an unmodified electrodynamics, you would have to take a equal to zero, getting an infinite mass correction. But, that wasn't the point. Keeping a finite, I simply followed the program outlined by Professor Bethe and showed how to calculate all the various things, the scatterings of electrons from atoms without radiation, the shifts of levels and so forth, calculating everything in terms of the experimental mass, and noting that the results as Bethe suggested, were not sensitive to a in this form and even had a definite limit as ag0. The rest of my work was simply to improve the techniques then available for calculations, making diagrams to help analyze perturbation theory quicker. Most of this was first worked out by guessing - you see, I didn't have the relativistic theory of matter. For example, it seemed to me obvious that the velocities in non-relativistic formulas have to be replaced by Dirac's matrix a or in the more relativistic forms by the operators . I just took my guesses from the forms that I had worked out using path integrals for nonrelativistic matter, but relativistic light. It was easy to develop rules of what to substitute to get the relativistic case. I was very surprised to discover that it was not known at that time, that every one of the formulas that had been worked out so patiently by separating longitudinal and transverse waves could be obtained from the formula for the transverse waves alone, if instead of summing over only the two perpendicular polarization directions you would sum over all four possible directions of polarization. It was so obvious from the action (1) that I thought it was general knowledge and would do it all the time. I would get into arguments with people, because I didn't realize they didn't know that; but, it turned out that all their patient work with the longitudinal waves was always equivalent to just extending the sum on the two transverse directions of polarization over all four directions. This was one of the amusing advantages of the method. In addition, I included diagrams for the various terms of the perturbation series, improved notations to be used, worked out easy ways to evaluate integrals, which occurred in these problems, and so on, and made a kind of handbook on how to do quantum electrodynamics. But one step of importance that was physically new was involved with the negative energy sea of Dirac, which caused me so much logical difficulty. I got so confused that I remembered Wheeler's old idea about the positron being, maybe, the electron going backward in time. Therefore, in the time dependent perturbation theory that was usual for getting self-energy, I simply supposed that for a while we could go backward in the time, and looked at what terms I got by running the time variables backward. They were the same as the terms that other people got when they did the problem a more complicated way, using holes in the sea, except, possibly, for some signs. These, I, at first, determined empirically by inventing and trying some rules. I have tried to explain that all the improvements of relativistic theory were at first more or less straightforward, semi-empirical shenanigans. Each time I would discover something, however, I would go back and I would check it so many ways, compare it to every problem that had been done previously in electrodynamics (and later, in weak coupling meson theory) to see if it would always agree, and so on, until I was absolutely convinced of the truth of the various rules and regulations which I concocted to simplify all the work. During this time, people had been developing meson theory, a subject I had not studied in any detail. I became interested in the possible application of my methods to perturbation calculations in meson theory. But, what was meson theory? All I knew was that meson theory was something analogous to electrodynamics, except that particles corresponding to the photon had a mass. It was easy to guess the d-function in (1), which was a solution of d'Alembertian equals zero, was to be changed to the corresponding solution of d'Alembertian equals m2. Next, there were different kind of mesons - the one in closest analogy to photons, coupled via , are called vector mesons - there were also scalar mesons. Well, maybe that corresponds to putting unity in place of the , I would here then speak of "pseudo vector coupling" and I would guess what that probably was. I didn't have the knowledge to understand the way these were defined in the conventional papers because they were expressed at that time in terms of creation and annihilation operators, and so on, which, I had not successfully learned. I remember that when someone had started to teach me about creation and annihilation operators, that this operator creates an electron, I said, "how do you create an electron? It disagrees with the conservation of charge", and in that way, I blocked my mind from learning a very practical scheme of calculation. Therefore, I had to find as many opportunities as possible to test whether I guessed right as to what the various theories were. One day a dispute arose at a Physical Society meeting as to the correctness of a calculation by Slotnick of the interaction of an electron with a neutron using pseudo scalar theory with pseudo vector coupling and also, pseudo scalar theory with pseudo scalar coupling. He had found that the answers were not the same, in fact, by one theory, the result was divergent, although convergent with the other. Some people believed that the two theories must give the same answer for the problem. This was a welcome opportunity to test my guesses as to whether I really did understand what these two couplings were. So, I went home, and during the evening I worked out the electron neutron scattering for the pseudo scalar and pseudo vector coupling, saw they were not equal and subtracted them, and worked out the difference in detail. The next day at the meeting, I saw Slotnick and said, "Slotnick, I worked it out last night, I wanted to see if I got the same answers you do. I got a different answer for each coupling - but, I would like to check in detail with you because I want to make sure of my methods." And, he said, "what do you mean you worked it out last night, it took me six months!" And, when we compared the answers he looked at mine and he asked, "what is that Q in there, that variable Q?" (I had expressions like (tan -1Q) /Q etc.). I said, "that's the momentum transferred by the electron, the electron deflected by different angles." "Oh", he said, "no, I only have the limiting value as Q approaches zero; the forward scattering." Well, it was easy enough to just substitute Q equals zero in my form and I then got the same answers as he did. But, it took him six months to do the case of zero momentum transfer, whereas, during one evening I had done the finite and arbitrary momentum transfer. That was a thrilling moment for me, like receiving the Nobel Prize, because that convinced me, at last, I did have some kind of method and technique and understood how to do something that other people did not know how to do. That was my moment of triumph in which I realized I really had succeeded in working out something worthwhile. At this stage, I was urged to publish this because everybody said it looks like an easy way to make calculations, and wanted to know how to do it. I had to publish it, missing two things; one was proof of every statement in a mathematically conventional sense. Often, even in a physicist's sense, I did not have a demonstration of how to get all of these rules and equations from conventional electrodynamics. But, I did know from experience, from fooling around, that everything was, in fact, equivalent to the regular electrodynamics and had partial proofs of many pieces, although, I never really sat down, like Euclid did for the geometers of Greece, and made sure that you could get it all from a single simple set of axioms. As a result, the work was criticized, I don't know whether favorably or unfavorably, and the "method" was called the "intuitive method". For those who do not realize it, however, I should like to emphasize that there is a lot of work involved in using this "intuitive method" successfully. Because no simple clear proof of the formula or idea presents itself, it is necessary to do an unusually great amount of checking and rechecking for consistency and correctness in terms of what is known, by comparing to other analogous examples, limiting cases, etc. In the face of the lack of direct mathematical demonstration, one must be careful and thorough to make sure of the point, and one should make a perpetual attempt to demonstrate as much of the formula as possible. Nevertheless, a very great deal more truth can become known than can be proven. It must be clearly understood that in all this work, I was representing the conventional electrodynamics with retarded interaction, and not my half-advanced and half-retarded theory corresponding to (1). I merely use (1) to guess at forms. And, one of the forms I guessed at corresponded to changing d to a function f of width a2, so that I could calculate finite results for all of the problems. This brings me to the second thing that was missing when I published the paper, an unresolved difficulty. With d replaced by f the calculations would give results which were not "unitary", that is, for which the sum of the probabilities of all alternatives was not unity. The deviation from unity was very small, in practice, if a was very small. In the limit that I took a very tiny, it might not make any difference. And, so the process of the renormalization could be made, you could calculate everything in terms of the experimental mass and then take the limit and the apparent difficulty that the unitary is violated temporarily seems to disappear. I was unable to demonstrate that, as a matter of fact, it does. It is lucky that I did not wait to straighten out that point, for as far as I know, nobody has yet been able to resolve this question. Experience with meson theories with stronger couplings and with strongly coupled vector photons, although not proving anything, convinces me that if the coupling were stronger, or if you went to a higher order (137th order of perturbation theory for electrodynamics), this difficulty would remain in the limit and there would be real trouble. That is, I believe there is really no satisfactory quantum electrodynamics, but I'm not sure. And, I believe, that one of the reasons for the slowness of present-day progress in understanding the strong interactions is that there isn't any relativistic theoretical model, from which you can really calculate everything. Although, it is usually said, that the difficulty lies in the fact that strong interactions are too hard to calculate, I believe, it is really because strong interactions in field theory have no solution, have no sense they're either infinite, or, if you try to modify them, the modification destroys the unitarity. I don't think we have a completely satisfactory relativistic quantum-mechanical model, even one that doesn't agree with nature, but, at least, agrees with the logic that the sum of probability of all alternatives has to be 100%. Therefore, I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug. I am, of course, not sure of that. This completes the story of the development of the space-time view of quantum electrodynamics. I wonder if anything can be learned from it. I doubt it. It is most striking that most of the ideas developed in the course of this research were not ultimately used in the final result. For example, the half-advanced and half-retarded potential was not finally used, the action expression (1) was not used, the idea that charges do not act on themselves was abandoned. The path-integral formulation of quantum mechanics was useful for guessing at final expressions and at formulating the general theory of electrodynamics in new ways - although, strictly it was not absolutely necessary. The same goes for the idea of the positron being a backward moving electron, it was very convenient, but not strictly necessary for the theory because it is exactly equivalent to the negative energy sea point of view. We are struck by the very large number of different physical viewpoints and widely different mathematical formulations that are all equivalent to one another. The method used here, of reasoning in physical terms, therefore, appears to be extremely inefficient. On looking back over the work, I can only feel a kind of regret for the enormous amount of physical reasoning and mathematically re-expression which ends by merely re-expressing what was previously known, although in a form which is much more efficient for the calculation of specific problems. Would it not have been much easier to simply work entirely in the mathematical framework to elaborate a more efficient expression? This would certainly seem to be the case, but it must be remarked that although the problem actually solved was only such a reformulation, the problem originally tackled was the (possibly still unsolved) problem of avoidance of the infinities of the usual theory. Therefore, a new theory was sought, not just a modification of the old. Although the quest was unsuccessful, we should look at the question of the value of physical ideas in developing a new theory. Many different physical ideas can describe the same physical reality. Thus, classical electrodynamics can be described by a field view, or an action at a distance view, etc. Originally, Maxwell filled space with idler wheels, and Faraday with fields lines, but somehow the Maxwell equations themselves are pristine and independent of the elaboration of words attempting a physical description. The only true physical description is that describing the experimental meaning of the quantities in the equation - or better, the way the equations are to be used in describing experimental observations. This being the case perhaps the best way to proceed is to try to guess equations, and disregard physical models or descriptions. For example, McCullough guessed the correct equations for light propagation in a crystal long before his colleagues using elastic models could make head or tail of the phenomena, or again, Dirac obtained his equation for the description of the electron by an almost purely mathematical proposition. A simple physical view by which all the contents of this equation can be seen is still lacking. Therefore, I think equation guessing might be the best method to proceed to obtain the laws for the part of physics which is presently unknown. Yet, when I was much younger, I tried this equation guessing and I have seen many students try this, but it is very easy to go off in wildly incorrect and impossible directions. I think the problem is not to find the best or most efficient method to proceed to a discovery, but to find any method at all. Physical reasoning does help some people to generate suggestions as to how the unknown may be related to the known. Theories of the known, which are described by different physical ideas may be equivalent in all their predictions and are hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in ones attempt to understand what is not yet understood. I, therefore, think that a good theoretical physicist today might find it useful to have a wide range of physical viewpoints and mathematical expressions of the same theory (for example, of quantum electrodynamics) available to him. This may be asking too much of one man. Then new students should as a class have this. If every individual student follows the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated to understand strong interactions, say, is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction - a direction obvious from an unfashionable view of field theory - who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unusual point of view; one that he may have to invent for himself. I say sacrificed himself because he most likely will get nothing from it, because the truth may lie in another direction, perhaps even the fashionable one. But, if my own experience is any guide, the sacrifice is really not great because if the peculiar viewpoint taken is truly experimentally equivalent to the usual in the realm of the known there is always a range of applications and problems in this realm for which the special viewpoint gives one a special power and clarity of thought, which is valuable in itself. Furthermore, in the search for new laws, you always have the psychological excitement of feeling that possible nobody has yet thought of the crazy possibility you are looking at right now. So what happened to the old theory that I fell in love with as a youth? Well, I would say it's become an old lady, that has very little attractive left in her and the young today will not have their hearts pound anymore when they look at her. But, we can say the best we can for any old woman, that she has been a very good mother and she has given birth to some very good children. And, I thank the Swedish Academy of Sciences for complimenting one of them. Thank you. Copyright © The Nobel Foundation 1965 Share this: To cite this page MLA style: "Richard P. Feynman - Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics". Nobel Media AB 2014. Web. 21 Oct 2014. <> Your mission is to arrange an amazing laser party! Read more about the Nobel Prize in Physics. All you want to know about the Nobel Prize in Physics!
6d547dac174bd531	Chandrasekhar limit Chandrasekhar limit The Chandrasekhar limit limits the mass of bodies made from electron-degenerate matter, a dense form of matter which consists of nuclei immersed in a gas of electrons. The limit is the maximum nonrotating mass which can be supported against gravitational collapse by electron degeneracy pressure. It is named after the astrophysicist Subrahmanyan Chandrasekhar, and is commonly given as being about 1.4 solar masses. As white dwarfs are composed of electron-degenerate matter, no nonrotating white dwarf can be heavier than the Chandrasekhar limit. Computed values for the limit will vary depending on the approximations used, the nuclear composition of the mass, and the temperature. Chandrasekhar, eq. (36),, eq. (58),, eq. (43) gives a value of frac{omega_3^0 sqrt{3pi}}{2}left (frac{hbar c}{G}right )^{3/2}frac{1}{(mu_e m_H)^2}. Here, μe is the average molecular weight per electron, mH is the mass of the hydrogen atom, and ω30≈2.018236 is a constant connected with the solution to the Lane-Emden equation. Numerically, this value is approximately (2/μe)2 · 2.85 · 1030 kg, or 1.43 (2/μe)2 M, where M=1.989·1030 kg is the standard solar mass. As sqrt{hbar c/G} is the Planck mass, MPl≈2.176·10−8 kg, the limit is of the order of MPl3/mH2. For a fully relativistic treatment, the equation of state used will interpolate between the equations P=K1ρ5/3 for small ρ and P=K2ρ4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at Mlimit. This is the Chandrasekhar limit. The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. μe has been set equal to 2. Radius is measured in standard solar radii or kilometers, and mass in standard solar masses. A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature. Lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation. In 1926, the British physicist Ralph H. Fowler observed that the relationship between the density, energy and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei which obeyed Fermi-Dirac statistics. This Fermi gas model was then used by the British physicist E. C. Stoner in 1929 to calculate the relationship between the mass, radius, and density of white dwarfs, assuming them to be homogenous spheres. Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37 kg. In 1930, Stoner derived the internal energy-density equation of state for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately (for μe=2.5) 2.19 · 1030 kg. Stoner went on to derive the pressure-density equation of state, which he published in 1932. These equations of state were also previously published by the Russian physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter. Frenkel's work, however, was ignored by the astronomical and astrophysical community. A series of papers published between 1931 and 1935 had its beginning on a trip from India to England in 1930, where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas. In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state, and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above. Chandrasekhar reviews this work in his Nobel Prize lecture. This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau, who, however, did not apply it to white dwarfs. Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P=K1ρ5/3 universally applicable, even for large ρ. Although Bohr, Fowler, Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar., pp. 110–111 Through the rest of his life, Eddington held to his position in his writings, including his work on his fundamental theory. The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar. In Miller's view: Strong indications of the reliability of Chandrasekhar's formula are: A type Ia supernova apparently from a supra-limit white dwarf Main article: Champagne Supernova. Further reading See also Search another word or see chandrasekhar limiton Dictionary \| Thesaurus \|Spanish Copyright © 2014, LLC. All rights reserved. • Please Login or Sign Up to use the Recent Searches feature
4fffe4741635a696	Archive for the ‘Metaphysical Spouting’ Category “Could a Quantum Computer Have Subjective Experience?” Monday, August 25th, 2014 Author’s Note: Below is the prepared version of a talk that I gave two weeks ago at the workshop Quantum Foundations of a Classical Universe, which was held at IBM’s TJ Watson Research Center in Yorktown Heights, NY. My talk is for entertainment purposes only; it should not be taken seriously by anyone. If you reply in a way that makes clear you did take it seriously (“I’m shocked and outraged that someone who dares to call himself a scientist would … [blah blah]“), I will log your IP address, hunt you down at night, and force you to put forward an account of consciousness and decoherence that deals with all the paradoxes discussed below—and then reply at length to all criticisms of your account. If you’d like to see titles, abstracts, and slides for all the talks from the workshop—including by Charles Bennett, Sean Carroll, James Hartle, Adrian Kent, Stefan Leichenauer, Ken Olum, Don Page, Jason Pollack, Jess Riedel, Mark Srednicki, Wojciech Zurek, and Michael Zwolak—click here. You’re also welcome to discuss these other nice talks in the comments section, though I might or might not be able to answer questions about them. Apparently videos of all the talks will be available before long (Jess Riedel has announced that videos are now available). (Note that, as is probably true for other talks as well, the video of my talk differs substantially from the prepared version—it mostly just consists of interruptions and my responses to them! On the other hand, I did try to work some of the more salient points from the discussion into the text below.) Thanks so much to Charles Bennett and Jess Riedel for organizing the workshop, and to all the participants for great discussions. I didn’t prepare slides for this talk—given the topic, what slides would I use exactly? “Spoiler alert”: I don’t have any rigorous results about the possibility of sentient quantum computers, to state and prove on slides. I thought of giving a technical talk on quantum computing theory, but then I realized that I don’t really have technical results that bear directly on the subject of the workshop, which is how the classical world we experience emerges from the quantum laws of physics. So, given the choice between a technical talk that doesn’t really address the questions we’re supposed to be discussing, or a handwavy philosophical talk that at least tries to address them, I opted for the latter, so help me God. Let me start with a story that John Preskill told me years ago. In the far future, humans have solved not only the problem of building scalable quantum computers, but also the problem of human-level AI. They’ve built a Turing-Test-passing quantum computer. The first thing they do, to make sure this is actually a quantum computer, is ask it to use Shor’s algorithm to factor a 10,000-digit number. So the quantum computer factors the number. Then they ask it, “while you were factoring that number, what did it feel like? did you feel yourself branching into lots of parallel copies, which then recohered? or did you remain a single consciousness—a ‘unitary’ consciousness, as it were? can you tell us from introspection which interpretation of quantum mechanics is the true one?” The quantum computer ponders this for a while and then finally says, “you know, I might’ve known before, but now I just … can’t remember.” I like to tell this story when people ask me whether the interpretation of quantum mechanics has any empirical consequences. Look, I understand the impulse to say “let’s discuss the measure problem, or the measurement problem, or derivations of the Born rule, or Boltzmann brains, or observer-counting, or whatever, but let’s take consciousness off the table.” (Compare: “let’s debate this state law in Nebraska that says that, before getting an abortion, a woman has to be shown pictures of cute babies. But let’s take the question of whether or not fetuses have human consciousness—i.e., the actual thing that’s driving our disagreement about that and every other subsidiary question—off the table, since that one is too hard.”) The problem, of course, is that even after you’ve taken the elephant off the table (to mix metaphors), it keeps climbing back onto the table, often in disguises. So, for better or worse, my impulse tends to be the opposite: to confront the elephant directly. Having said that, I still need to defend the claim that (a) the questions we’re discussing, centered around quantum mechanics, Many Worlds, and decoherence, and (b) the question of which physical systems should be considered “conscious,” have anything to do with each other. Many people would say that the connection doesn’t go any deeper than: “quantum mechanics is mysterious, consciousness is also mysterious, ergo maybe they’re related somehow.” But I’m not sure that’s entirely true. One thing that crystallized my thinking about this was a remark made in a lecture by Peter Byrne, who wrote a biography of Hugh Everett. Byrne was discussing the question, why did it take so many decades for Everett’s Many-Worlds Interpretation to become popular? Of course, there are people who deny quantum mechanics itself, or who have basic misunderstandings about it, but let’s leave those people aside. Why did people like Bohr and Heisenberg dismiss Everett? More broadly: why wasn’t it just obvious to physicists from the beginning that “branching worlds” is a picture that the math militates toward, probably the simplest, easiest story one can tell around the Schrödinger equation? Even if early quantum physicists rejected the Many-Worlds picture, why didn’t they at least discuss and debate it? Here was Byrne’s answer: he said, before you can really be on board with Everett, you first need to be on board with Daniel Dennett (the philosopher). He meant: you first need to accept that a “mind” is just some particular computational process. At the bottom of everything is the physical state of the universe, evolving via the equations of physics, and if you want to know where consciousness is, you need to go into that state, and look for where computations are taking place that are sufficiently complicated, or globally-integrated, or self-referential, or … something, and that’s where the consciousness resides. And crucially, if following the equations tells you that after a decoherence event, one computation splits up into two computations, in different branches of the wavefunction, that thereafter don’t interact—congratulations! You’ve now got two consciousnesses. And if everything above strikes you as so obvious as not to be worth stating … well, that’s a sign of how much things changed in the latter half of the 20th century. Before then, many thinkers would’ve been more likely to say, with Descartes: no, my starting point is not the physical world. I don’t even know a priori that there is a physical world. My starting point is my own consciousness, which is the one thing besides math that I can be certain about. And the point of a scientific theory is to explain features of my experience—ultimately, if you like, to predict the probability that I’m going to see X or Y if I do A or B. (If I don’t have prescientific knowledge of myself, as a single, unified entity that persists in time, makes choices, and later observes their consequences, then I can’t even get started doing science.) I’m happy to postulate a world external to myself, filled with unseen entities like electrons behaving in arbitrarily unfamiliar ways, if it will help me understand my experience—but postulating other versions of me is, at best, irrelevant metaphysics. This is a viewpoint that could lead you Copenhagenism, or to its newer variants like quantum Bayesianism. Of course, there are already tremendous difficulties here, even if we ignore quantum mechanics entirely. Ken Olum was over much of this ground in his talk yesterday (see here for a relevant paper by Davenport and Olum). You’ve all heard the ones about, would you agree to be painlessly euthanized, provided that a complete description of your brain would be sent to Mars as an email attachment, and a “perfect copy” of you would be reconstituted there? Would you demand that the copy on Mars be up and running before the original was euthanized? But what do we mean by “before”—in whose frame of reference? Some people say: sure, none of this is a problem! If I’d been brought up since childhood taking family vacations where we all emailed ourselves to Mars and had our original bodies euthanized, I wouldn’t think anything of it. But the philosophers of mind are barely getting started. You might say, sure, maybe these questions are puzzling, but what’s the alternative? Either we have to say that consciousness is a byproduct of any computation of the right complexity, or integration, or recursiveness (or something) happening anywhere in the wavefunction of the universe, or else we’re back to saying that beings like us are conscious, and all these other things aren’t, because God gave the souls to us, so na-na-na. Or I suppose we could say, like the philosopher John Searle, that we’re conscious, and the lookup table and homomorphically-encrypted brain and Vaidman brain and all these other apparitions aren’t, because we alone have “biological causal powers.” And what do those causal powers consist of? Hey, you’re not supposed to ask that! Just accept that we have them. Or we could say, like Roger Penrose, that we’re conscious and the other things aren’t because we alone have microtubules that are sensitive to uncomputable effects from quantum gravity. But neither of those two options ever struck me as much of an improvement. Yet I submit to you that, between these extremes, there’s another position we can stake out—one that I certainly don’t know to be correct, but that would solve so many different puzzles if it were correct that, for that reason alone, it seems to me to merit more attention than it usually receives. (In an effort to give the view that attention, a couple years ago I wrote an 85-page essay called The Ghost in the Quantum Turing Machine, which one or two people told me they actually read all the way through.) If, after a lifetime of worrying (on weekends) about stuff like whether a giant lookup table would be conscious, I now seem to be arguing for this particular view, it’s less out of conviction in its truth than out of a sense of intellectual obligation: to whatever extent people care about these slippery questions at all, to whatever extent they think various alternative views deserve a hearing, I believe this one does as well. The intermediate position that I’d like to explore says the following. Yes, consciousness is a property of any suitably-organized chunk of matter. But, in addition to performing complex computations, or passing the Turing Test, or other information-theoretic conditions that I don’t know (and don’t claim to know), there’s at least one crucial further thing that a chunk of matter has to do before we should consider it conscious. Namely, it has to participate fully in the Arrow of Time. More specifically, it has to produce irreversible decoherence as an intrinsic part of its operation. It has to be continually taking microscopic fluctuations, and irreversibly amplifying them into stable, copyable, macroscopic classical records. Before I go further, let me be extremely clear about what this view is not saying. Firstly, it’s not saying that the brain is a quantum computer, in any interesting sense—let alone a quantum-gravitational computer, like Roger Penrose wants! Indeed, I see no evidence, from neuroscience or any other field, that the cognitive information processing done by the brain is anything but classical. The view I’m discussing doesn’t challenge conventional neuroscience on that account. Secondly, this view doesn’t say that consciousness is in any sense necessary for decoherence, or for the emergence of a classical world. I’ve never understood how one could hold such a belief, while still being a scientific realist. After all, there are trillions of decoherence events happening every second in stars and asteroids and uninhabited planets. Do those events not “count as real” until a human registers them? (Or at least a frog, or an AI?) The view I’m discussing only asserts the converse: that decoherence is necessary for consciousness. (By analogy, presumably everyone agrees that some amount of computation is necessary for an interesting consciousness, but that doesn’t mean consciousness is necessary for computation.) Thirdly, the view I’m discussing doesn’t say that “quantum magic” is the explanation for consciousness. It’s silent on the explanation for consciousness (to whatever extent that question makes sense); it seeks only to draw a defensible line between the systems we want to regard as conscious and the systems we don’t—to address what I recently called the Pretty-Hard Problem. And the (partial) answer it suggests doesn’t seem any more “magical” to me than any other proposed answer to the same question. For example, if one said that consciousness arises from any computation that’s sufficiently “integrated” (or something), I could reply: what’s the “magical force” that imbues those particular computations with consciousness, and not other computations I can specify? Or if one said (like Searle) that consciousness arises from the biology of the brain, I could reply: so what’s the “magic” of carbon-based biology, that could never be replicated in silicon? Or even if one threw up one’s hands and said everything was conscious, I could reply: what’s the magical power that imbues my stapler with a mind? Each of these views, along with the view that stresses the importance of decoherence and the arrow of time, is worth considering. In my opinion, each should be judged according to how well it holds up under the most grueling battery of paradigm-cases, thought experiments, and reductios ad absurdum we can devise. So, why might one conjecture that decoherence, and participation in the arrow of time, were necessary conditions for consciousness? I suppose I could offer some argument about our subjective experience of the passage of time being a crucial component of our consciousness, and the passage of time being bound up with the Second Law. Truthfully, though, I don’t have any a-priori argument that I find convincing. All I can do is show you how many apparent paradoxes get resolved if you make this one speculative leap. For starters, if you think about exactly how our chunk of matter is going to amplify microscopic fluctuations, it could depend on details like the precise spin orientations of various subatomic particles in the chunk. But that has an interesting consequence: if you’re an outside observer who doesn’t know the chunk’s quantum state, it might be difficult or impossible for you to predict what the chunk is going to do next—even just to give decent statistical predictions, like you can for a hydrogen atom. And of course, you can’t in general perform a measurement that will tell you the chunk’s quantum state, without violating the No-Cloning Theorem. For the same reason, there’s in general no physical procedure that you can apply to the chunk to duplicate it exactly: that is, to produce a second chunk that you can be confident will behave identically (or almost identically) to the first, even just in a statistical sense. (Again, this isn’t assuming any long-range quantum coherence in the chunk: only microscopic coherence that then gets amplified.) It might be objected that there are all sorts of physical systems that “amplify microscopic fluctuations,” but that aren’t anything like what I described, at least not in any interesting sense: for example, a Geiger counter, or a photodetector, or any sort of quantum-mechanical random-number generator. You can make, if not an exact copy of a Geiger counter, surely one that’s close enough for practical purposes. And, even though the two counters will record different sequences of clicks when pointed at identical sources, the statistical distribution of clicks will be the same (and precisely calculable), and surely that’s all that matters. So, what separates these examples from the sorts of examples I want to discuss? What separates them is the undisputed existence of what I’ll call a clean digital abstraction layer. By that, I mean a macroscopic approximation to a physical system that an external observer can produce, in principle, without destroying the system; that can be used to predict what the system will do to excellent accuracy (given knowledge of the environment); and that “sees” quantum-mechanical uncertainty—to whatever extent it does—as just a well-characterized source of random noise. If a system has such an abstraction layer, then we can regard any quantum noise as simply part of the “environment” that the system observes, rather than part of the system itself. I’ll take it as clear that such clean abstraction layers exist for a Geiger counter, a photodetector, or a computer with a quantum random number generator. By contrast, for (say) an animal brain, I regard it as currently an open question whether such an abstraction layer exists or not. If, someday, it becomes routine for nanobots to swarm through people’s brains and make exact copies of them—after which the “original” brains can be superbly predicted in all circumstances, except for some niggling differences that are traceable back to different quantum-mechanical dice rolls—at that point, perhaps educated opinion will have shifted to the point where we all agree the brain does have a clean digital abstraction layer. But from where we stand today, it seems entirely possible to agree that the brain is a physical system obeying the laws of physics, while doubting that the nanobots would work as advertised. It seems possible that—as speculated by Bohr, Compton, Eddington, and even Alan Turing—if you want to get it right you’ll need more than just the neural wiring graph, the synaptic strengths, and the approximate neurotransmitter levels. Maybe you also need (e.g.) the internal states of the neurons, the configurations of sodium-ion channels, or other data that you simply can’t get without irreparably damaging the original brain—not only as a contingent matter of technology but as a fundamental matter of physics. (As a side note, I should stress that obviously, even without invasive nanobots, our brains are constantly changing, but we normally don’t say as a result that we become completely different people at each instant! To my way of thinking, though, this transtemporal identity is fundamentally different from a hypothetical identity between different “copies” of you, in the sense we’re talking about. For one thing, all your transtemporal doppelgängers are connected by a single, linear chain of causation. For another, outside movies like Bill and Ted’s Excellent Adventure, you can’t meet your transtemporal doppelgängers and have a conversation with them, nor can scientists do experiments on some of them, then apply what they learned to others that remained unaffected by their experiments.) So, on this view, a conscious chunk of matter would be one that not only acts irreversibly, but that might well be unclonable for fundamental physical reasons. If so, that would neatly resolve many of the puzzles that I discussed before. So for example, there’s now a straightforward reason why you shouldn’t consent to being killed, while your copy gets recreated on Mars from an email attachment. Namely, that copy will have a microstate with no direct causal link to your “original” microstate—so while it might behave similarly to you in many ways, you shouldn’t expect that your consciousness will “transfer” to it. If you wanted to get your exact microstate to Mars, you could do that in principle using quantum teleportation—but as we all know, quantum teleportation inherently destroys the original copy, so there’s no longer any philosophical problem! (Or, of course, you could just get on a spaceship bound for Mars: from a philosophical standpoint, it amounts to the same thing.) Similarly, in the case where the simulation of your brain was run three times for error-correcting purposes: that could bring about three consciousnesses if, and only if, the three simulations were tied to different sets of decoherence events. The giant lookup table and the Earth-sized brain simulation wouldn’t bring about any consciousness, unless they were implemented in such a way that they no longer had a clean digital abstraction layer. What about the homomorphically-encrypted brain simulation? That might no longer work, simply because we can’t assume that the microscopic fluctuations that get amplified are homomorphically encrypted. Those are “in the clear,” which inevitably leaks information. As for the quantum computer that simulates your thought processes and then perfectly reverses the simulation, or that queries you like a Vaidman bomb—in order to implement such things, we’d of course need to use quantum fault-tolerance, so that the simulation of you stayed in an encoded subspace and didn’t decohere. But under our assumption, that would mean the simulation wasn’t conscious. Now, it might seem to some of you like I’m suggesting something deeply immoral. After all, the view I’m considering implies that, even if a system passed the Turing Test, and behaved identically to a human, even if it eloquently pleaded for its life, if it wasn’t irreversibly decohering microscopic events then it wouldn’t be conscious, so it would be fine to kill it, torture it, whatever you want. But wait a minute: if a system isn’t doing anything irreversible, then what exactly does it mean to “kill” it? If it’s a classical computation, then at least in principle, you could always just restore from backup. You could even rewind and not only erase the memories of, but “uncompute” (“untorture”?) whatever tortures you had performed. If it’s a quantum computation, you could always invert the unitary transformation U that corresponded to killing the thing (then reapply U and invert it again for good measure, if you wanted). Only for irreversible systems are there moral acts with irreversible consequences. This is related to something that’s bothered me for years in quantum foundations. When people discuss Schrödinger’s cat, they always—always—insert some joke about, “obviously, this experiment wouldn’t pass the Ethical Review Board. Nowadays, we try to avoid animal cruelty in our quantum gedankenexperiments.” But actually, I claim that there’s no animal cruelty at all in the Schrödinger’s cat experiment. And here’s why: in order to prove that the cat was ever in a coherent superposition of \|Alive〉 and \|Dead〉, you need to be able to measure it in a basis like {\|Alive〉+\|Dead〉,\|Alive〉-\|Dead〉}. But if you can do that, you must have such precise control over all the cat’s degrees of freedom that you can also rotate unitarily between the \|Alive〉 and \|Dead〉 states. (To see this, let U be the unitary that you applied to the \|Alive〉 branch, and V the unitary that you applied to the \|Dead〉 branch, to bring them into coherence with each other; then consider applying U-1V.) But if you can do that, then in what sense should we say that the cat in the \|Dead〉 state was ever “dead” at all? Normally, when we speak of “killing,” we mean doing something irreversible—not rotating to some point in a Hilbert space that we could just as easily rotate away from. (There followed discussion among some audience members about the question of whether, if you destroyed all records of some terrible atrocity, like the Holocaust, everywhere in the physical world, you would thereby cause the atrocity “never to have happened.” Many people seemed surprised by my willingness to accept that implication of what I was saying. By way of explaining, I tried to stress just how far our everyday, intuitive notion of “destroying all records of something” falls short of what would actually be involved here: when we think of “destroying records,” we think about burning books, destroying the artifacts in museums, silencing witnesses, etc. But even if all those things were done and many others, still the exact configurations of the air, the soil, and photons heading away from the earth at the speed of light would retain their silent testimony to the Holocaust’s reality. “Erasing all records” in the physics sense would be something almost unimaginably more extreme: it would mean inverting the entire physical evolution in the vicinity of the earth, stopping time’s arrow and running history itself backwards. Such ‘unhappening’ of what’s happened is something that we lack any experience of, at least outside of certain quantum interference experiments—though in the case of the Holocaust, one could be forgiven for wishing it were possible.) OK, so much for philosophy of mind and morality; what about the interpretation of quantum mechanics? If we think about consciousness in the way I’ve suggested, then who’s right: the Copenhagenists or the Many-Worlders? You could make a case for either. The Many-Worlders would be right that we could always, if we chose, think of decoherence events as “splitting” our universe into multiple branches, each with different versions of ourselves, that thereafter don’t interact. On the other hand, the Copenhagenists would be right that, even in principle, we could never do any experiment where this “splitting” of our minds would have any empirical consequence. On this view, if you can control a system well enough that you can actually observe interference between the different branches, then it follows that you shouldn’t regard the system as conscious, because it’s not doing anything irreversible. In my essay, the implication that concerned me the most was the one for “free will.” If being conscious entails amplifying microscopic events in an irreversible and unclonable way, then someone looking at a conscious system from the outside might not, in general, be able to predict what it’s going to do next, not even probabilistically. In other words, its decisions might be subject to at least some “Knightian uncertainty”: uncertainty that we can’t even quantify in a mutually-agreed way using probabilities, in the same sense that we can quantify our uncertainty about (say) the time of a radioactive decay. And personally, this is actually the sort of “freedom” that interests me the most. I don’t really care if my choices are predictable by God, or by a hypothetical Laplace demon: that is, if they would be predictable (at least probabilistically), given complete knowledge of the microstate of the universe. By definition, there’s essentially no way for my choices not to be predictable in that weak and unempirical sense! On the other hand, I’d prefer that my choices not be completely predictable by other people. If someone could put some sheets of paper into a sealed envelope, then I spoke extemporaneously for an hour, and then the person opened the envelope to reveal an exact transcript of everything I said, that’s the sort of thing that really would cause me to doubt in what sense “I” existed as a locus of thought. But you’d have to actually do the experiment (or convince me that it could be done): it doesn’t count just to talk about it, or to extrapolate from fMRI experiments that predict which of two buttons a subject is going to press with 60% accuracy a few seconds in advance. But since we’ve got some cosmologists in the house, let me now turn to discussing the implications of this view for Boltzmann brains. (For those tuning in from home: a Boltzmann brain is a hypothetical chance fluctuation in the late universe, which would include a conscious observer with all the perceptions that a human being—say, you—is having right now, right down to false memories and false beliefs of having arisen via Darwinian evolution. On statistical grounds, the overwhelming majority of Boltzmann brains last just long enough to have a single thought—like, say, the one you’re having right now—before they encounter the vacuum and freeze to death. If you measured some part of the vacuum state toward which our universe seems to be heading, asking “is there a Boltzmann brain here?,” quantum mechanics predicts that the probability would be ridiculously astronomically small, but nonzero. But, so the argument goes, if the vacuum lasts for infinite time, then as long as the probability is nonzero, it doesn’t matter how tiny it is: you’ll still get infinitely many Boltzmann brains indistinguishable from any given observer; and for that reason, any observer should consider herself infinitely likelier to be a Boltzmann brain than to be the “real,” original version. For the record, even among the strange people at the IBM workshop, no one actually worried about being a Boltzmann brain. The question, rather, is whether, if a cosmological model predicts Boltzmann brains, then that’s reason enough to reject the model, or whether we can live with such a prediction, since we have independent grounds for knowing that we can’t be Boltzmann brains.) At this point, you can probably guess where this is going. If decoherence, entropy production, full participation in the arrow of time are necessary conditions for consciousness, then it would follow, in particular, that a Boltzmann brain is not conscious. So we certainly wouldn’t be Boltzmann brains, even under a cosmological model that predicts infinitely more of them than of us. We can wipe our hands; the problem is solved! I find it extremely interesting that, in their recent work, Kim Boddy, Sean Carroll, and Jason Pollack reached a similar conclusion, but from a completely different starting point. They said: look, under reasonable assumptions, the late universe is just going to stay forever in an energy eigenstate—just sitting there doing nothing. It’s true that, if someone came along and measured the energy eigenstate, asking “is there a Boltzmann brain here?,” then with a tiny but nonzero probability the answer would be yes. But since no one is there measuring, what licenses us to interpret the nonzero overlap in amplitude with the Boltzmann brain state, as a nonzero probability of there being a Boltzmann brain? I think they, too, are implicitly suggesting: if there’s no decoherence, no arrow of time, then we’re not authorized to say that anything is happening that “counts” for anthropic purposes. Let me now mention an obvious objection. (In fact, when I gave the talk, this objection was raised much earlier.) You might say, “look, if you really think irreversible decoherence is a necessary condition for consciousness, then you might find yourself forced to say that there’s no consciousness, because there might not be any such thing as irreversible decoherence! Imagine that our entire solar system were enclosed in an anti de Sitter (AdS) boundary, like in Greg Egan’s science-fiction novel Quarantine. Inside the box, there would just be unitary evolution in some Hilbert space: maybe even a finite-dimensional Hilbert space. In which case, all these ‘irreversible amplifications’ that you lay so much stress on wouldn’t be irreversible at all: eventually all the Everett branches would recohere; in fact they’d decohere and recohere infinitely many times. So by your lights, how could anything be conscious inside the box?” My response to this involves one last speculation. I speculate that the fact that we don’t appear to live in AdS space—that we appear to live in (something evolving toward) a de Sitter space, with a positive cosmological constant—might be deep and important and relevant. I speculate that, in our universe, “irreversible decoherence” means: the records of what you did are now heading toward our de Sitter horizon at the speed of light, and for that reason alone—even if for no others—you can’t put Humpty Dumpty back together again. (Here I should point out, as several workshop attendees did to me, that Bousso and Susskind explored something similar in their paper The Multiverse Interpretation of Quantum Mechanics.) Does this mean that, if cosmologists discover tomorrow that the cosmological constant is negative, or will become negative, then it will turn out that none of us were ever conscious? No, that’s stupid. What it would suggest is that the attempt I’m now making on the Pretty-Hard Problem had smacked into a wall (an AdS wall?), so that I, and anyone else who stressed in-principle irreversibility, should go back to the drawing board. (By analogy, if some prescription for getting rid of Boltzmann brains fails, that doesn’t mean we are Boltzmann brains; it just means we need a new prescription. Tempting as it is to skewer our opponents’ positions with these sorts of strawman inferences, I hope we can give each other the courtesy of presuming a bare minimum of sense.) Another question: am I saying that, in order to be absolutely certain of whether some entity satisfied the postulated precondition for consciousness, one might, in general, need to look billions of years into the future, to see whether the “decoherence” produced by the entity was really irreversible? Yes (pause to gulp bullet). I am saying that. On the other hand, I don’t think it’s nearly as bad as it sounds. After all, the category of “consciousness” might be morally relevant, or relevant for anthropic reasoning, but presumably we all agree that it’s unlikely to play any causal role in the fundamental laws of physics. So it’s not as if we’ve introduced any teleology into the laws of physics by this move. Let me end by pointing out what I’ll call the “Tegmarkian slippery slope.” It feels scientific and rational—from the perspective of many of us, even banal—to say that, if we’re conscious, then any sufficiently-accurate computer simulation of us would also be. But I tried to convince you that this view depends, for its aura of obviousness, on our agreeing not to probe too closely exactly what would count as a “sufficiently-accurate” simulation. E.g., does it count if the simulation is done in heavily-encrypted form, or encoded as a giant lookup table? Does it matter if anyone actually runs the simulation, or consults the lookup table? Now, all the way at the bottom of the slope is Max Tegmark, who asks: to produce consciousness, what does it matter if the simulation is physically instantiated at all? Why isn’t it enough for the simulation to “exist” mathematically? Or, better yet: if you’re worried about your infinitely-many Boltzmann brain copies, then why not worry equally about the infinitely many descriptions of your life history that are presumably encoded in the decimal expansion of π? Why not hold workshops about how to avoid the prediction that we’re infinitely likelier to be “living in π” than to be our “real” selves? From this extreme, even most scientific rationalists recoil. They say, no, even if we don’t yet know exactly what’s meant by “physical instantiation,” we agree that you only get consciousness if the computer program is physically instantiated somehow. But now I have the opening I want. I can say: once we agree that physical existence is a prerequisite for consciousness, why not participation in the Arrow of Time? After all, our ordinary ways of talking about sentient beings—outside of quantum mechanics, cosmology, and maybe theology—don’t even distinguish between the concepts “exists” and “exists and participates in the Arrow of Time.” And to say we have no experience of reversible, clonable, coherently-executable, atemporal consciousnesses is a massive understatement. Of course, we should avoid the sort of arbitrary prejudice that Turing warned against in Computing Machinery and Intelligence. Just because we lack experience with extraterrestrial consciousnesses, doesn’t mean it would be OK to murder an intelligent extraterrestrial if we met one tomorrow. In just the same way, just because we lack experience with clonable, atemporal consciousnesses, doesn’t mean it would be OK to … wait! As we said before, clonability, and aloofness from time’s arrow, call severely into question what it even means to “murder” something. So maybe this case isn’t as straightforward as the extraterrestrials after all. At this point, I’ve probably laid out enough craziness, so let me stop and open things up for discussion. Integrated Information Theory: Virgil Griffith opines Wednesday, June 25th, 2014 Remember the two discussions about Integrated Information Theory that we had a month ago on this blog? You know, the ones where I argued that IIT fails because “the brain might be an expander, but not every expander is a brain”; where IIT inventor Giulio Tononi wrote a 14-page response biting the bullet with mustard; and where famous philosopher of mind David Chalmers, and leading consciousness researcher (and IIT supporter) Christof Koch, also got involved in the comments section? OK, so one more thing about that. Virgil Griffith recently completed his PhD under Christof Koch at Caltech—as he puts it, “immersing [him]self in the nitty-gritty of IIT for the past 6.5 years.” This morning, Virgil sent me two striking letters about his thoughts on the recent IIT exchanges on this blog. He asked me to share them here, something that I’m more than happy to do: Reading these letters, what jumped out at me—given Virgil’s long apprenticeship in the heart of IIT-land—was the amount of agreement between my views and his. In particular, Virgil agrees with my central contention that Φ, as it stands, can at most be a necessary condition for consciousness, not a sufficient condition, and remarks that “[t]o move IIT from talked about to accepted among hard scientists, it may be necessary for [Tononi] to wash his hands of sufficiency claims.” He agrees that a lack of mathematical clarity in the definition of Φ is a “major problem in the IIT literature,” commenting that “IIT needs more mathematically inclined people at its helm.” He also says he agrees “110%” that the lack of a derivation of the form of Φ from IIT’s axioms is “a pothole in the theory,” and further agrees 110% that the current prescriptions for computing Φ contain many unjustified idiosyncrasies. Indeed, given the level of agreement here, there’s not all that much for me to rebut, defend, or clarify! I suppose there are a few things. 1. Just as a clarifying remark, in a few places where it looks from the formatting like Virgil is responding to something I said (for example, “The conceptual structure is unified—it cannot be decomposed into independent components” and “Clearly, a theory of consciousness must be able to provide an adequate account for such seemingly disparate but largely uncontroversial facts”), he’s actually responding to something Giulio said (and that I, at most, quoted). 2. Virgil says, correctly, that Giulio would respond to my central objection against IIT by challenging my “intuition for things being unconscious.” (Indeed, because Giulio did respond, there’s no need to speculate about how he would respond!) However, Virgil then goes on to explicate Giulio’s response using the analogy of temperature (interestingly, the same analogy I used for a different purpose). He points out how counterintuitive it would be for Kelvin’s contemporaries to accept that “even the coldest thing you’ve touched actually has substantial heat in it,” and remarks: “I find this ‘Kelvin scale for C’ analogy makes the panpsychism much more palatable.” The trouble is that I never objected to IIT’s panpsychism per se: I only objected to its seemingly arbitrary and selective panpsychism. It’s one thing for a theory to ascribe some amount of consciousness to a 2D grid or an expander graph. It’s quite another for a theory to ascribe vastly more consciousness to those things than it ascribes to a human brain—even while denying consciousness to things that are intuitively similar but organized a little differently (say, a 1D grid). A better analogy here would be if Kelvin’s theory of temperature had predicted, not merely that all ordinary things had some heat in them, but that an ice cube was hotter than the Sun, even though a popsicle was, of course, colder than the Sun. (The ice cube, you see, “integrates heat” in a way that the popsicle doesn’t…) 3. Virgil imagines two ways that an IIT proponent could respond to my argument involving the cerebellum—the argument that accuses IIT proponents of changing the rules of the game according to convenience (a 2D grid has a large Φ? suck it up and accept it; your intuitions about a grid’s lack of consciousness are irrelevant. the human cerebellum has a small Φ? ah, that’s a victory for IIT, since the cerebellum is intuitively unconscious). The trouble is that both of Virgil’s imagined responses are by reference to the IIT axioms. But I wasn’t talking about the axioms themselves, but about whether we’re allowed to validate the axioms, by checking their consequences against earlier, pre-theoretic intuitions. And I was pointing out that Giulio seemed happy to do so when the results “went in IIT’s favor” (in the cerebellum example), even though he lectured me against doing so in the cases of the expander and the 2D grid (cases where IIT does less well, to put it mildly, at capturing our intuitions). 4. Virgil chastises me for ridiculing Giulio’s phenomenological argument for the consciousness of a 2D grid by way of nursery rhymes: “Just because it feels like something to see a wall, doesn’t mean it feels like something to be a wall. You can smell a rose, and the rose can smell good, but that doesn’t mean the rose can smell you.” Virgil amusingly comments: “Even when both are inebriated, I’ve never heard [Giulio] nor [Christof] separately or collectively imply anything like this. Moreover, they’re each far too clueful to fall for something so trivial.” For my part, I agree that neither Giulio nor Christof would ever advocate something as transparently silly as, “if you have a rich inner experience when thinking about X, then that’s evidence X itself is conscious.” And I apologize if I seemed to suggest they would. To clarify, my point was not that Giulio was making such an absurd statement, but rather that, assuming he wasn’t, I didn’t know what he was trying to say in the passages of his that I’d just quoted at length. The silly thing seemed like the “obvious” reading of his words, and my hermeneutic powers were unequal to the task of figuring out the non-silly, non-obvious reading that he surely intended. Anyway, there’s much more to Virgil’s letters than the above—including answers to some of my subsidiary questions about the details of IIT (e.g., how to handle unbalanced partitions, and the mathematical meanings of terms like “mechanism” and “system of mechanisms”). Also, in parts of the letters, Virgil’s main concern is neither to agree with me nor to agree with Giulio, but rather to offer his own ideas, developed in the course of his PhD work, for how to move forward and fix some of the problems with IIT. All in all, these are recommended reads for anyone who’s been following this debate. Giulio Tononi and Me: A Phi-nal Exchange Friday, May 30th, 2014 You might recall that last week I wrote a post criticizing Integrated Information Theory (IIT), and its apparent implication that a simple Reed-Solomon decoding circuit would, if scaled to a large enough size, bring into being a consciousness vastly exceeding our own. On Wednesday Giulio Tononi, the creator of IIT, was kind enough to send me a fascinating 14-page rebuttal, and to give me permission to share it here: If you’re interested in this subject at all, then I strongly recommend reading Giulio’s response before continuing further. But for those who want the tl;dr: Giulio, not one to battle strawmen, first restates my own argument against IIT with crystal clarity. And while he has some minor quibbles (e.g., apparently my calculations of Φ didn’t use the most recent, “3.0” version of IIT), he wisely sets those aside in order to focus on the core question: according to IIT, are all sorts of simple expander graphs conscious? There, he doesn’t “bite the bullet” so much as devour a bullet hoagie with mustard. He affirms that, yes, according to IIT, a large network of XOR gates arranged in a simple expander graph is conscious. Indeed, he goes further, and says that the “expander” part is superfluous: even a network of XOR gates arranged in a 2D square grid is conscious. In my language, Giulio is simply pointing out here that a √n×√n square grid has decent expansion: good enough to produce a Φ-value of about √n, if not the information-theoretic maximum of n (or n/2, etc.) that an expander graph could achieve. And apparently, by Giulio’s lights, Φ=√n is sufficient for consciousness! While Giulio never mentions this, it’s interesting to observe that logic gates arranged in a 1-dimensional line would produce a tiny Φ-value (Φ=O(1)). So even by IIT standards, such a linear array would not be conscious. Yet the jump from a line to a two-dimensional grid is enough to light the spark of Mind. Yet even as we admire Giulio’s honesty and consistency, his stance might also prompt us, gently, to take another look at this peanut-butter-moon theory, and at what grounds we had for believing it in the first place. In his response essay, Giulio offers four arguments (by my count) for accepting IIT despite, or even because of, its conscious-grid prediction: one “negative” argument and three “positive” ones. Alas, while your Φ-lage may vary, I didn’t find any of the four arguments persuasive. In the rest of this post, I’ll go through them one by one and explain why. I. The Copernicus-of-Consciousness Argument Like many commenters on my last post, Giulio heavily criticizes my appeal to “common sense” in rejecting IIT. Sure, he says, I might find it “obvious” that a huge Vandermonde matrix, or its physical instantiation, isn’t conscious. But didn’t people also find it “obvious” for millennia that the Sun orbits the Earth? Isn’t the entire point of science to challenge common sense? Clearly, then, the test of a theory of consciousness is not how well it upholds “common sense,” but how well it fits the facts. The above position sounds pretty convincing: who could dispute that observable facts trump personal intuitions? The trouble is, what are the observable facts when it comes to consciousness? The anti-common-sense view gets all its force by pretending that we’re in a relatively late stage of research—namely, the stage of taking an agreed-upon scientific definition of consciousness, and applying it to test our intuitions—rather than in an extremely early stage, of agreeing on what the word “consciousness” is even supposed to mean. Since I think this point is extremely important—and of general interest, beyond just IIT—I’ll expand on it with some analogies. Suppose I told you that, in my opinion, the ε-δ definition of continuous functions—the one you learn in calculus class—failed to capture the true meaning of continuity. Suppose I told you that I had a new, better definition of continuity—and amazingly, when I tried out my definition on some examples, it turned out that ⌊x⌋ (the floor function) was continuous, whereas x2 had discontinuities, though only at 17.5 and 42. You would probably ask what I was smoking, and whether you could have some. But why? Why shouldn’t the study of continuity produce counterintuitive results? After all, even the standard definition of continuity leads to some famously weird results, like that x sin(1/x) is a continuous function, even though sin(1/x) is discontinuous. And it’s not as if the standard definition is God-given: people had been using words like “continuous” for centuries before Bolzano, Weierstrass, et al. formalized the ε-δ definition, a definition that millions of calculus students still find far from intuitive. So why shouldn’t there be a different, better definition of “continuous,” and why shouldn’t it reveal that a step function is continuous while a parabola is not? In my view, the way out of this conceptual jungle is to realize that, before any formal definitions, any ε’s and δ’s, we start with an intuition for we’re trying to capture by the word “continuous.” And if we press hard enough on what that intuition involves, we’ll find that it largely consists of various “paradigm-cases.” A continuous function, we’d say, is a function like 3x, or x2, or sin(x), while a discontinuity is the kind of thing that the function 1/x has at x=0, or that ⌊x⌋ has at every integer point. Crucially, we use the paradigm-cases to guide our choice of a formal definition—not vice versa! It’s true that, once we have a formal definition, we can then apply it to “exotic” cases like x sin(1/x), and we might be surprised by the results. But the paradigm-cases are different. If, for example, our definition told us that x2 was discontinuous, that wouldn’t be a “surprise”; it would just be evidence that we’d picked a bad definition. The definition failed at the only task for which it could have succeeded: namely, that of capturing what we meant. Some people might say that this is all well and good in pure math, but empirical science has no need for squishy intuitions and paradigm-cases. Nothing could be further from the truth. Suppose, again, that I told you that physicists since Kelvin had gotten the definition of temperature all wrong, and that I had a new, better definition. And, when I built a Scott-thermometer that measures true temperatures, it delivered the shocking result that boiling water is actually colder than ice. You’d probably tell me where to shove my Scott-thermometer. But wait: how do you know that I’m not the Copernicus of heat, and that future generations won’t celebrate my breakthrough while scoffing at your small-mindedness? I’d say there’s an excellent answer: because what we mean by heat is “whatever it is that boiling water has more of than ice” (along with dozens of other paradigm-cases). And because, if you use a thermometer to check whether boiling water is hotter than ice, then the term for what you’re doing is calibrating your thermometer. When the clock strikes 13, it’s time to fix the clock, and when the thermometer says boiling water’s colder than ice, it’s time to replace the thermometer—or if needed, even the entire theory on which the thermometer is based. Ah, you say, but doesn’t modern physics define heat in a completely different, non-intuitive way, in terms of molecular motion? Yes, and that turned out to be a superb definition—not only because it was precise, explanatory, and applicable to cases far beyond our everyday experience, but crucially, because it matched common sense on the paradigm-cases. If it hadn’t given sensible results for boiling water and ice, then the only possible conclusion would be that, whatever new quantity physicists had defined, they shouldn’t call it “temperature,” or claim that their quantity measured the amount of “heat.” They should call their new thing something else. The implications for the consciousness debate are obvious. When we consider whether to accept IIT’s equation of integrated information with consciousness, we don’t start with any agreed-upon, independent notion of consciousness against which the new notion can be compared. The main things we start with, in my view, are certain paradigm-cases that gesture toward what we mean: • You are conscious (though not when anesthetized). • (Most) other people appear to be conscious, judging from their behavior. • Many animals appear to be conscious, though probably to a lesser degree than humans (and the degree of consciousness in each particular species is far from obvious). • A rock is not conscious. A wall is not conscious. A Reed-Solomon code is not conscious. Microsoft Word is not conscious (though a Word macro that passed the Turing test conceivably would be). Fetuses, coma patients, fish, and hypothetical AIs are the x sin(1/x)’s of consciousness: they’re the tougher cases, the ones where we might actually need a formal definition to adjudicate the truth. Now, given a proposed formal definition for an intuitive concept, how can we check whether the definition is talking about same thing we were trying to get at before? Well, we can check whether the definition at least agrees that parabolas are continuous while step functions are not, that boiling water is hot while ice is cold, and that we’re conscious while Reed-Solomon decoders are not. If so, then the definition might be picking out the same thing that we meant, or were trying to mean, pre-theoretically (though we still can’t be certain). If not, then the definition is certainly talking about something else. What else can we do? II. The Axiom Argument According to Giulio, there is something else we can do, besides relying on paradigm-cases. That something else, in his words, is to lay down “postulates about how the physical world should be organized to support the essential properties of experience,” then use those postulates to derive a consciousness-measuring quantity. OK, so what are IIT’s postulates? Here’s how Giulio states the five postulates leading to Φ in his response essay (he “derives” these from earlier “phenomenological axioms,” which you can find in the essay): 1. A system of mechanisms exists intrinsically if it can make a difference to itself, by affecting the probability of its past and future states, i.e. it has causal power (existence). 2. It is composed of submechanisms each with their own causal power (composition). 3. It generates a conceptual structure that is the specific way it is, as specified by each mechanism’s concept — this is how each mechanism affects the probability of the system’s past and future states (information). From my standpoint, these postulates have three problems. First, I don’t really understand them. Second, insofar as I do understand them, I don’t necessarily accept their truth. And third, insofar as I do accept their truth, I don’t see how they lead to Φ. To elaborate a bit: I don’t really understand the postulates. I realize that the postulates are explicated further in the many papers on IIT. Unfortunately, while it’s possible that I missed something, in all of the papers that I read, the definitions never seemed to “bottom out” in mathematical notions that I understood, like functions mapping finite sets to other finite sets. What, for example, is a “mechanism”? What’s a “system of mechanisms”? What’s “causal power”? What’s a “conceptual structure,” and what does it mean for it to be “unified”? Alas, it doesn’t help to define these notions in terms of other notions that I also don’t understand. And yes, I agree that all these notions can be given fully rigorous definitions, but there could be many different ways to do so, and the devil could lie in the details. In any case, because (as I said) it’s entirely possible that the failure is mine, I place much less weight on this point than I do on the two points to follow. I don’t necessarily accept the postulates’ truth. Is consciousness a “unified conceptual structure”? Is it “singular”? Maybe. I don’t know. It sounds plausible. But at any rate, I’m far less confident about any these postulates—whatever one means by them!—than I am about my own “postulate,” which is that you and I are conscious while my toaster is not. Note that my postulate, though not phenomenological, does have the merit of constraining candidate theories of consciousness in an unambiguous way. I don’t see how the postulates lead to Φ. Even if one accepts the postulates, how does one deduce that the “amount of consciousness” should be measured by Φ, rather than by some other quantity? None of the papers I read—including the ones Giulio linked to in his response essay—contained anything that looked to me like a derivation of Φ. Instead, there was general discussion of the postulates, and then Φ just sort of appeared at some point. Furthermore, given the many idiosyncrasies of Φ—the minimization over all bipartite (why just bipartite? why not tripartite?) decompositions of the system, the need for normalization (or something else in version 3.0) to deal with highly-unbalanced partitions—it would be quite a surprise were it possible to derive its specific form from postulates of such generality. I was going to argue for that conclusion in more detail, when I realized that Giulio had kindly done the work for me already. Recall that Giulio chided me for not using the “latest, 2014, version 3.0″ edition of Φ in my previous post. Well, if the postulates uniquely determined the form of Φ, then what’s with all these upgrades? Or has Φ’s definition been changing from year to year because the postulates themselves have been changing? If the latter, then maybe one should wait for the situation to stabilize before trying to form an opinion of the postulates’ meaningfulness, truth, and completeness? III. The Ironic Empirical Argument Or maybe not. Despite all the problems noted above with the IIT postulates, Giulio argues in his essay that there’s a good a reason to accept them: namely, they explain various empirical facts from neuroscience, and lead to confirmed predictions. In his words: [A] theory’s postulates must be able to explain, in a principled and parsimonious way, at least those many facts about consciousness and the brain that are reasonably established and non-controversial. For example, we know that our own consciousness depends on certain brain structures (the cortex) and not others (the cerebellum), that it vanishes during certain periods of sleep (dreamless sleep) and reappears during others (dreams), that it vanishes during certain epileptic seizures, and so on. Clearly, a theory of consciousness must be able to provide an adequate account for such seemingly disparate but largely uncontroversial facts. Such empirical facts, and not intuitions, should be its primary test… [I]n some cases we already have some suggestive evidence [of the truth of the IIT postulates' predictions]. One example is the cerebellum, which has 69 billion neurons or so — more than four times the 16 billion neurons of the cerebral cortex — and is as complicated a piece of biological machinery as any. Though we do not understand exactly how it works (perhaps even less than we understand the cerebral cortex), its connectivity definitely suggests that the cerebellum is ill suited to information integration, since it lacks lateral connections among its basic modules. And indeed, though the cerebellum is heavily connected to the cerebral cortex, removing it hardly affects our consciousness, whereas removing the cortex eliminates it. I hope I’m not alone in noticing the irony of this move. But just in case, let me spell it out: Giulio has stated, as “largely uncontroversial facts,” that certain brain regions (the cerebellum) and certain states (dreamless sleep) are not associated with our consciousness. He then views it as a victory for IIT, if those regions and states turn out to have lower information integration than the regions and states that he does take to be associated with our consciousness. But how does Giulio know that the cerebellum isn’t conscious? Even if it doesn’t produce “our” consciousness, maybe the cerebellum has its own consciousness, just as rich as the cortex’s but separate from it. Maybe removing the cerebellum destroys that other consciousness, unbeknownst to “us.” Likewise, maybe “dreamless” sleep brings about its own form of consciousness, one that (unlike dreams) we never, ever remember in the morning. Giulio might take the implausibility of those ideas as obvious, or at least as “largely uncontroversial” among neuroscientists. But here’s the problem with that: he just told us that a 2D square grid is conscious! He told us that we must not rely on “commonsense intuition,” or on any popular consensus, to say that if a square mesh of wires is just sitting there XORing some input bits, doing nothing at all that we’d want to call intelligent, then it’s probably safe to conclude that the mesh isn’t conscious. So then why shouldn’t he say the same for the cerebellum, or for the brain in dreamless sleep? By Giulio’s own rules (the ones he used for the mesh), we have no a-priori clue whether those systems are conscious or not—so even if IIT predicts that they’re not conscious, that can’t be counted as any sort of success for IIT. For me, the point is even stronger: I, personally, would be a million times more inclined to ascribe consciousness to the human cerebellum, or to dreamless sleep, than I would to the mesh of XOR gates. For it’s not hard to imagine neuroscientists of the future discovering “hidden forms of intelligence” in the cerebellum, and all but impossible to imagine them doing the same for the mesh. But even if you put those examples on the same footing, still the take-home message seems clear: you can’t count it as a “success” for IIT if it predicts that the cerebellum in unconscious, while at the same time denying that it’s a “failure” for IIT if it predicts that a square mesh of XOR gates is conscious. If the unconsciousness of the cerebellum can be considered an “empirical fact,” safe enough for theories of consciousness to be judged against it, then surely the unconsciousness of the mesh can also be considered such a fact. IV. The Phenomenology Argument I now come to, for me, the strangest and most surprising part of Giulio’s response. Despite his earlier claim that IIT need not dovetail with “commonsense intuition” about which systems are conscious—that it can defy intuition—at some point, Giulio valiantly tries to reprogram our intuition, to make us feel why a 2D grid could be conscious. As best I can understand, the argument seems to be that, when we stare at a blank 2D screen, we form a rich experience in our heads, and that richness must be mirrored by a corresponding “intrinsic” richness in 2D space itself: [I]f one thinks a bit about it, the experience of empty 2D visual space is not at all empty, but contains a remarkable amount of structure. In fact, when we stare at the blank screen, quite a lot is immediately available to us without any effort whatsoever. Thus, we are aware of all the possible locations in space (“points”): the various locations are right “there”, in front of us. We are aware of their relative positions: a point may be left or right of another, above or below, and so on, for every position, without us having to order them. And we are aware of the relative distances among points: quite clearly, two points may be close or far, and this is the case for every position. Because we are aware of all of this immediately, without any need to calculate anything, and quite regularly, since 2D space pervades most of our experiences, we tend to take for granted the vast set of relationship[s] that make up 2D space. And yet, says IIT, given that our experience of the blank screen definitely exists, and it is precisely the way it is — it is 2D visual space, with all its relational properties — there must be physical mechanisms that specify such phenomenological relationships through their causal power … One may also see that the causal relationships that make up 2D space obtain whether the elements are on or off. And finally, one may see that such a 2D grid is necessary not so much to represent space from the extrinsic perspective of an observer, but to create it, from its own intrinsic perspective. Now, it would be child’s-play to criticize the above line of argument for conflating our consciousness of the screen with the alleged consciousness of the screen itself. To wit: Just because it feels like something to see a wall, doesn’t mean it feels like something to be a wall. You can smell a rose, and the rose can smell good, but that doesn’t mean the rose can smell you. However, I actually prefer a different tack in criticizing Giulio’s “wall argument.” Suppose I accepted that my mental image of the relationships between certain entities was relevant to assessing whether those entities had their own mental life, independent of me or any other observer. For example, suppose I believed that, if my experience of 2D space is rich and structured, then that’s evidence that 2D space is rich and structured enough to be conscious. Then my question is this: why shouldn’t the same be true of 1D space? After all, my experience of staring at a rope is also rich and structured, no less than my experience of staring at a wall. I perceive some points on the rope as being toward the left, others as being toward the right, and some points as being between two other points. In fact, the rope even has a structure—namely, a natural total ordering on its points—that the wall lacks. So why does IIT cruelly deny subjective experience to a row of logic gates strung along a rope, reserving it only for a mesh of logic gates pasted to a wall? And yes, I know the answer: because the logic gates on the rope aren’t “integrated” enough. But who’s to say that the gates in the 2D mesh are integrated enough? As I mentioned before, their Φ-value grows only as the square root of the number of gates, so that the ratio of integrated information to total information tends to 0 as the number of gates increases. And besides, aren’t what Giulio calls “the facts of phenomenology” the real arbiters here, and isn’t my perception of the rope’s structure a phenomenological fact? When you cut a rope, does it not split? When you prick it, does it not fray? At this point, I fear we’re at a philosophical impasse. Having learned that, according to IIT, 1. a square grid of XOR gates is conscious, and your experience of staring at a blank wall provides evidence for that, 2. by contrast, a linear array of XOR gates is not conscious, your experience of staring at a rope notwithstanding, 3. the human cerebellum is also not conscious (even though a grid of XOR gates is), and 4. unlike with the XOR gates, we don’t need a theory to tell us the cerebellum is unconscious, but can simply accept it as “reasonably established” and “largely uncontroversial,” I personally feel completely safe in saying that this is not the theory of consciousness for me. But I’ve also learned that other people, even after understanding the above, still don’t reject IIT. And you know what? Bully for them. On reflection, I firmly believe that a two-state solution is possible, in which we simply adopt different words for the different things that we mean by “consciousness”—like, say, consciousnessReal for my kind and consciousnessWTF for the IIT kind. OK, OK, just kidding! How about “paradigm-case consciousness” for the one and “IIT consciousness” for the other. Completely unrelated announcement: Some of you might enjoy this Nature News piece by Amanda Gefter, about black holes and computational complexity. Wednesday, May 21st, 2014 Happy birthday to me! Recently, lots of people have been asking me what I think about IIT—no, not the Indian Institutes of Technology, but Integrated Information Theory, a widely-discussed “mathematical theory of consciousness” developed over the past decade by the neuroscientist Giulio Tononi. One of the askers was Max Tegmark, who’s enthusiastically adopted IIT as a plank in his radical mathematizing platform (see his paper “Consciousness as a State of Matter”). When, in the comment thread about Max’s Mathematical Universe Hypothesis, I expressed doubts about IIT, Max challenged me to back up my doubts with a quantitative calculation. So, this is the post that I promised to Max and all the others, about why I don’t believe IIT. And yes, it will contain that quantitative calculation. But first, what is IIT? The central ideas of IIT, as I understand them, are: (1) to propose a quantitative measure, called Φ, of the amount of “integrated information” in a physical system (i.e. information that can’t be localized in the system’s individual parts), and then (2) to hypothesize that a physical system is “conscious” if and only if it has a large value of Φ—and indeed, that a system is more conscious the larger its Φ value. I’ll return later to the precise definition of Φ—but basically, it’s obtained by minimizing, over all subdivisions of your physical system into two parts A and B, some measure of the mutual information between A’s outputs and B’s inputs and vice versa. Now, one immediate consequence of any definition like this is that all sorts of simple physical systems (a thermostat, a photodiode, etc.) will turn out to have small but nonzero Φ values. To his credit, Tononi cheerfully accepts the panpsychist implication: yes, he says, it really does mean that thermostats and photodiodes have small but nonzero levels of consciousness. On the other hand, for the theory to work, it had better be the case that Φ is small for “intuitively unconscious” systems, and only large for “intuitively conscious” systems. As I’ll explain later, this strikes me as a crucial point on which IIT fails. The literature on IIT is too big to do it justice in a blog post. Strikingly, in addition to the “primary” literature, there’s now even a “secondary” literature, which treats IIT as a sort of established base on which to build further speculations about consciousness. Besides the Tegmark paper linked to above, see for example this paper by Maguire et al., and associated popular article. (Ironically, Maguire et al. use IIT to argue for the Penrose-like view that consciousness might have uncomputable aspects—a use diametrically opposed to Tegmark’s.) Anyway, if you want to read a popular article about IIT, there are loads of them: see here for the New York Times’s, here for Scientific American‘s, here for IEEE Spectrum‘s, and here for the New Yorker‘s. Unfortunately, none of those articles will tell you the meat (i.e., the definition of integrated information); for that you need technical papers, like this or this by Tononi, or this by Seth et al. IIT is also described in Christof Koch’s memoir Consciousness: Confessions of a Romantic Reductionist, which I read and enjoyed; as well as Tononi’s Phi: A Voyage from the Brain to the Soul, which I haven’t yet read. (Koch, one of the world’s best-known thinkers and writers about consciousness, has also become an evangelist for IIT.) So, I want to explain why I don’t think IIT solves even the problem that it “plausibly could have” solved. But before I can do that, I need to do some philosophical ground-clearing. Broadly speaking, what is it that a “mathematical theory of consciousness” is supposed to do? What questions should it answer, and how should we judge whether it’s succeeded? The most obvious thing a consciousness theory could do is to explain why consciousness exists: that is, to solve what David Chalmers calls the “Hard Problem,” by telling us how a clump of neurons is able to give rise to the taste of strawberries, the redness of red … you know, all that ineffable first-persony stuff. Alas, there’s a strong argument—one that I, personally, find completely convincing—why that’s too much to ask of any scientific theory. Namely, no matter what the third-person facts were, one could always imagine a universe consistent with those facts in which no one “really” experienced anything. So for example, if someone claims that integrated information “explains” why consciousness exists—nope, sorry! I’ve just conjured into my imagination beings whose Φ-values are a thousand, nay a trillion times larger than humans’, yet who are also philosophical zombies: entities that there’s nothing that it’s like to be. Granted, maybe such zombies can’t exist in the actual world: maybe, if you tried to create one, God would notice its large Φ-value and generously bequeath it a soul. But if so, then that’s a further fact about our world, a fact that manifestly couldn’t be deduced from the properties of Φ alone. Notice that the details of Φ are completely irrelevant to the argument. Faced with this point, many scientifically-minded people start yelling and throwing things. They say that “zombies” and so forth are empty metaphysics, and that our only hope of learning about consciousness is to engage with actual facts about the brain. And that’s a perfectly reasonable position! As far as I’m concerned, you absolutely have the option of dismissing Chalmers’ Hard Problem as a navel-gazing distraction from the real work of neuroscience. The one thing you can’t do is have it both ways: that is, you can’t say both that the Hard Problem is meaningless, and that progress in neuroscience will soon solve the problem if it hasn’t already. You can’t maintain simultaneously that (a) once you account for someone’s observed behavior and the details of their brain organization, there’s nothing further about consciousness to be explained, and (b) remarkably, the XYZ theory of consciousness can explain the “nothing further” (e.g., by reducing it to integrated information processing), or might be on the verge of doing so. As obvious as this sounds, it seems to me that large swaths of consciousness-theorizing can just be summarily rejected for trying to have their brain and eat it in precisely the above way. Fortunately, I think IIT survives the above observations. For we can easily interpret IIT as trying to do something more “modest” than solve the Hard Problem, although still staggeringly audacious. Namely, we can say that IIT “merely” aims to tell us which physical systems are associated with consciousness and which aren’t, purely in terms of the systems’ physical organization. The test of such a theory is whether it can produce results agreeing with “commonsense intuition”: for example, whether it can affirm, from first principles, that (most) humans are conscious; that dogs and horses are also conscious but less so; that rocks, livers, bacteria colonies, and existing digital computers are not conscious (or are hardly conscious); and that a room full of people has no “mega-consciousness” over and above the consciousnesses of the individuals. The reason it’s so important that the theory uphold “common sense” on these test cases is that, given the experimental inaccessibility of consciousness, this is basically the only test available to us. If the theory gets the test cases “wrong” (i.e., gives results diverging from common sense), it’s not clear that there’s anything else for the theory to get “right.” Of course, supposing we had a theory that got the test cases right, we could then have a field day with the less-obvious cases, programming our computers to tell us exactly how much consciousness is present in octopi, fetuses, brain-damaged patients, and hypothetical AI bots. In my opinion, how to construct a theory that tells us which physical systems are conscious and which aren’t—giving answers that agree with “common sense” whenever the latter renders a verdict—is one of the deepest, most fascinating problems in all of science. Since I don’t know a standard name for the problem, I hereby call it the Pretty-Hard Problem of Consciousness. Unlike with the Hard Hard Problem, I don’t know of any philosophical reason why the Pretty-Hard Problem should be inherently unsolvable; but on the other hand, humans seem nowhere close to solving it (if we had solved it, then we could reduce the abortion, animal rights, and strong AI debates to “gentlemen, let us calculate!”). Now, I regard IIT as a serious, honorable attempt to grapple with the Pretty-Hard Problem of Consciousness: something concrete enough to move the discussion forward. But I also regard IIT as a failed attempt on the problem. And I wish people would recognize its failure, learn from it, and move on. In my view, IIT fails to solve the Pretty-Hard Problem because it unavoidably predicts vast amounts of consciousness in physical systems that no sane person would regard as particularly “conscious” at all: indeed, systems that do nothing but apply a low-density parity-check code, or other simple transformations of their input data. Moreover, IIT predicts not merely that these systems are “slightly” conscious (which would be fine), but that they can be unboundedly more conscious than humans are. To justify that claim, I first need to define Φ. Strikingly, despite the large literature about Φ, I had a hard time finding a clear mathematical definition of it—one that not only listed formulas but fully defined the structures that the formulas were talking about. Complicating matters further, there are several competing definitions of Φ in the literature, including ΦDM (discrete memoryless), ΦE (empirical), and ΦAR (autoregressive), which apply in different contexts (e.g., some take time evolution into account and others don’t). Nevertheless, I think I can define Φ in a way that will make sense to theoretical computer scientists. And crucially, the broad point I want to make about Φ won’t depend much on the details of its formalization anyway. We consider a discrete system in a state x=(x1,…,xn)∈Sn, where S is a finite alphabet (the simplest case is S={0,1}). We imagine that the system evolves via an “updating function” f:Sn→Sn. Then the question that interests us is whether the xi‘s can be partitioned into two sets A and B, of roughly comparable size, such that the updates to the variables in A don’t depend very much on the variables in B and vice versa. If such a partition exists, then we say that the computation of f does not involve “global integration of information,” which on Tononi’s theory is a defining aspect of consciousness. More formally, given a partition (A,B) of {1,…,n}, let us write an input y=(y1,…,yn)∈Sn to f in the form (yA,yB), where yA consists of the y variables in A and yB consists of the y variables in B. Then we can think of f as mapping an input pair (yA,yB) to an output pair (zA,zB). Now, we define the “effective information” EI(A→B) as H(zB \| A random, yB=xB). Or in words, EI(A→B) is the Shannon entropy of the output variables in B, if the input variables in A are drawn uniformly at random, while the input variables in B are fixed to their values in x. It’s a measure of the dependence of B on A in the computation of f(x). Similarly, we define EI(B→A) := H(zA \| B random, yA=xA). We then consider the sum Φ(A,B) := EI(A→B) + EI(B→A). Intuitively, we’d like the integrated information Φ=Φ(f,x) be the minimum of Φ(A,B), over all 2n-2 possible partitions of {1,…,n} into nonempty sets A and B. The idea is that Φ should be large, if and only if it’s not possible to partition the variables into two sets A and B, in such a way that not much information flows from A to B or vice versa when f(x) is computed. However, no sooner do we propose this than we notice a technical problem. What if A is much larger than B, or vice versa? As an extreme case, what if A={1,…,n-1} and B={n}? In that case, we’ll have Φ(A,B)≤2log2\|S\|, but only for the boring reason that there’s hardly any entropy in B as a whole, to either influence A or be influenced by it. For this reason, Tononi proposes a fix where we normalize each Φ(A,B) by dividing it by min{\|A\|,\|B\|}. He then defines the integrated information Φ to be Φ(A,B), for whichever partition (A,B) minimizes the ratio Φ(A,B) / min{\|A\|,\|B\|}. (Unless I missed it, Tononi never specifies what we should do if there are multiple (A,B)’s that all achieve the same minimum of Φ(A,B) / min{\|A\|,\|B\|}. I’ll return to that point later, along with other idiosyncrasies of the normalization procedure.) Tononi gives some simple examples of the computation of Φ, showing that it is indeed larger for systems that are more “richly interconnected” in an intuitive sense. He speculates, plausibly, that Φ is quite large for (some reasonable model of) the interconnection network of the human brain—and probably larger for the brain than for typical electronic devices (which tend to be highly modular in design, thereby decreasing their Φ), or, let’s say, than for other organs like the pancreas. Ambitiously, he even speculates at length about how a large value of Φ might be connected to the phenomenology of consciousness. To be sure, empirical work in integrated information theory has been hampered by three difficulties. The first difficulty is that we don’t know the detailed interconnection network of the human brain. The second difficulty is that it’s not even clear what we should define that network to be: for example, as a crude first attempt, should we assign a Boolean variable to each neuron, which equals 1 if the neuron is currently firing and 0 if it’s not firing, and let f be the function that updates those variables over a timescale of, say, a millisecond? What other variables do we need—firing rates, internal states of the neurons, neurotransmitter levels? Is choosing many of these variables uniformly at random (for the purpose of calculating Φ) really a reasonable way to “randomize” the variables, and if not, what other prescription should we use? The third and final difficulty is that, even if we knew exactly what we meant by “the f and x corresponding to the human brain,” and even if we had complete knowledge of that f and x, computing Φ(f,x) could still be computationally intractable. For recall that the definition of Φ involved minimizing a quantity over all the exponentially-many possible bipartitions of {1,…,n}. While it’s not directly relevant to my arguments in this post, I leave it as a challenge for interested readers to pin down the computational complexity of approximating Φ to some reasonable precision, assuming that f is specified by a polynomial-size Boolean circuit, or alternatively, by an NC0 function (i.e., a function each of whose outputs depends on only a constant number of the inputs). (Presumably Φ will be #P-hard to calculate exactly, but only because calculating entropy exactly is a #P-hard problem—that’s not interesting.) I conjecture that approximating Φ is an NP-hard problem, even for restricted families of f’s like NC0 circuits—which invites the amusing thought that God, or Nature, would need to solve an NP-hard problem just to decide whether or not to imbue a given physical system with consciousness! (Alas, if you wanted to exploit this as a practical approach for solving NP-complete problems such as 3SAT, you’d need to do a rather drastic experiment on your own brain—an experiment whose result would be to render you unconscious if your 3SAT instance was satisfiable, or conscious if it was unsatisfiable! In neither case would you be able to communicate the outcome of the experiment to anyone else, nor would you have any recollection of the outcome after the experiment was finished.) In the other direction, it would also be interesting to upper-bound the complexity of approximating Φ. Because of the need to estimate the entropies of distributions (even given a bipartition (A,B)), I don’t know that this problem is in NP—the best I can observe is that it’s in AM. In any case, my own reason for rejecting IIT has nothing to do with any of the “merely practical” issues above: neither the difficulty of defining f and x, nor the difficulty of learning them, nor the difficulty of calculating Φ(f,x). My reason is much more basic, striking directly at the hypothesized link between “integrated information” and consciousness. Specifically, I claim the following: Yes, it might be a decent rule of thumb that, if you want to know which brain regions (for example) are associated with consciousness, you should start by looking for regions with lots of information integration. And yes, it’s even possible, for all I know, that having a large Φ-value is one necessary condition among many for a physical system to be conscious. However, having a large Φ-value is certainly not a sufficient condition for consciousness, or even for the appearance of consciousness. As a consequence, Φ can’t possibly capture the essence of what makes a physical system conscious, or even of what makes a system look conscious to external observers. The demonstration of this claim is embarrassingly simple. Let S=Fp, where p is some prime sufficiently larger than n, and let V be an n×n Vandermonde matrix over Fp—that is, a matrix whose (i,j) entry equals ij-1 (mod p). Then let f:Sn→Sn be the update function defined by f(x)=Vx. Now, for p large enough, the Vandermonde matrix is well-known to have the property that every submatrix is full-rank (i.e., “every submatrix preserves all the information that it’s possible to preserve about the part of x that it acts on”). And this implies that, regardless of which bipartition (A,B) of {1,…,n} we choose, we’ll get EI(A→B) = EI(B→A) = min{\|A\|,\|B\|} log2p, and hence Φ(A,B) = EI(A→B) + EI(B→A) = 2 min{\|A\|,\|B\|} log2p, or after normalizing, Φ(A,B) / min{\|A\|,\|B\|} = 2 log2p. Or in words: the normalized information integration has the same value—namely, the maximum value!—for every possible bipartition. Now, I’d like to proceed from here to a determination of Φ itself, but I’m prevented from doing so by the ambiguity in the definition of Φ that I noted earlier. Namely, since every bipartition (A,B) minimizes the normalized value Φ(A,B) / min{\|A\|,\|B\|}, in theory I ought to be able to pick any of them for the purpose of calculating Φ. But the unnormalized value Φ(A,B), which gives the final Φ, can vary greatly, across bipartitions: from 2 log2p (if min{\|A\|,\|B\|}=1) all the way up to n log2p (if min{\|A\|,\|B\|}=n/2). So at this point, Φ is simply undefined. On the other hand, I can solve this problem, and make Φ well-defined, by an ironic little hack. The hack is to replace the Vandermonde matrix V by an n×n matrix W, which consists of the first n/2 rows of the Vandermonde matrix each repeated twice (assume for simplicity that n is a multiple of 4). As before, we let f(x)=Wx. Then if we set A={1,…,n/2} and B={n/2+1,…,n}, we can achieve EI(A→B) = EI(B→A) = (n/4) log2p, Φ(A,B) = EI(A→B) + EI(B→A) = (n/2) log2p, and hence Φ(A,B) / min{\|A\|,\|B\|} = log2p. In this case, I claim that the above is the unique bipartition that minimizes the normalized integrated information Φ(A,B) / min{\|A\|,\|B\|}, up to trivial reorderings of the rows. To prove this claim: if \|A\|=\|B\|=n/2, then clearly we minimize Φ(A,B) by maximizing the number of repeated rows in A and the number of repeated rows in B, exactly as we did above. Thus, assume \|A\|≤\|B\| (the case \|B\|≤\|A\| is analogous). Then clearly EI(B→A) ≥ \|A\|/2, EI(A→B) ≥ min{\|A\|, \|B\|/2}. So if we let \|A\|=cn and \|B\|=(1-c)n for some c∈(0,1/2], then Φ(A,B) ≥ [c/2 + min{c, (1-c)/2}] n, Φ(A,B) / min{\|A\|,\|B\|} = Φ(A,B) / \|A\| = 1/2 + min{1, 1/(2c) – 1/2}. But the above expression is uniquely minimized when c=1/2. Hence the normalized integrated information is minimized essentially uniquely by setting A={1,…,n/2} and B={n/2+1,…,n}, and we get Φ = Φ(A,B) = (n/2) log2p, which is quite a large value (only a factor of 2 less than the trivial upper bound of n log2p). Now, why did I call the switch from V to W an “ironic little hack”? Because, in order to ensure a large value of Φ, I decreased—by a factor of 2, in fact—the amount of “information integration” that was intuitively happening in my system! I did that in order to decrease the normalized value Φ(A,B) / min{\|A\|,\|B\|} for the particular bipartition (A,B) that I cared about, thereby ensuring that that (A,B) would be chosen over all the other bipartitions, thereby increasing the final, unnormalized value Φ(A,B) that Tononi’s prescription tells me to return. I hope I’m not alone in fearing that this illustrates a disturbing non-robustness in the definition of Φ. But let’s leave that issue aside; maybe it can be ameliorated by fiddling with the definition. The broader point is this: I’ve shown that my system—the system that simply applies the matrix W to an input vector x—has an enormous amount of integrated information Φ. Indeed, this system’s Φ equals half of its entire information content. So for example, if n were 1014 or so—something that wouldn’t be hard to arrange with existing computers—then this system’s Φ would exceed any plausible upper bound on the integrated information content of the human brain. And yet this Vandermonde system doesn’t even come close to doing anything that we’d want to call intelligent, let alone conscious! When you apply the Vandermonde matrix to a vector, all you’re really doing is mapping the list of coefficients of a degree-(n-1) polynomial over Fp, to the values of the polynomial on the n points 0,1,…,n-1. Now, evaluating a polynomial on a set of points turns out to be an excellent way to achieve “integrated information,” with every subset of outputs as correlated with every subset of inputs as it could possibly be. In fact, that’s precisely why polynomials are used so heavily in error-correcting codes, such as the Reed-Solomon code, employed (among many other places) in CD’s and DVD’s. But that doesn’t imply that every time you start up your DVD player you’re lighting the fire of consciousness. It doesn’t even hint at such a thing. All it tells us is that you can have integrated information without consciousness (or even intelligence)—just like you can have computation without consciousness, and unpredictability without consciousness, and electricity without consciousness. It might be objected that, in defining my “Vandermonde system,” I was too abstract and mathematical. I said that the system maps the input vector x to the output vector Wx, but I didn’t say anything about how it did so. To perform a computation—even a computation as simple as a matrix-vector multiply—won’t we need a physical network of wires, logic gates, and so forth? And in any realistic such network, won’t each logic gate be directly connected to at most a few other gates, rather than to billions of them? And if we define the integrated information Φ, not directly in terms of the inputs and outputs of the function f(x)=Wx, but in terms of all the actual logic gates involved in computing f, isn’t it possible or even likely that Φ will go back down? This is a good objection, but I don’t think it can rescue IIT. For we can achieve the same qualitative effect that I illustrated with the Vandermonde matrix—the same “global information integration,” in which every large set of outputs depends heavily on every large set of inputs—even using much “sparser” computations, ones where each individual output depends on only a few of the inputs. This is precisely the idea behind low-density parity check (LDPC) codes, which have had a major impact on coding theory over the past two decades. Of course, one would need to muck around a bit to construct a physical system based on LDPC codes whose integrated information Φ was provably large, and for which there were no wildly-unbalanced bipartitions that achieved lower Φ(A,B)/min{\|A\|,\|B\|} values than the balanced bipartitions one cared about. But I feel safe in asserting that this could be done, similarly to how I did it with the Vandermonde matrix. More generally, we can achieve pretty good information integration by hooking together logic gates according to any bipartite expander graph: that is, any graph with n vertices on each side, such that every k vertices on the left side are connected to at least min{(1+ε)k,n} vertices on the right side, for some constant ε>0. And it’s well-known how to create expander graphs whose degree (i.e., the number of edges incident to each vertex, or the number of wires coming out of each logic gate) is a constant, such as 3. One can do so either by plunking down edges at random, or (less trivially) by explicit constructions from algebra or combinatorics. And as indicated in the title of this post, I feel 100% confident in saying that the so-constructed expander graphs are not conscious! The brain might be an expander, but not every expander is a brain. Before winding down this post, I can’t resist telling you that the concept of integrated information (though it wasn’t called that) played an interesting role in computational complexity in the 1970s. As I understand the history, Leslie Valiant conjectured that Boolean functions f:{0,1}n→{0,1}n with a high degree of “information integration” (such as discrete analogues of the Fourier transform) might be good candidates for proving circuit lower bounds, which in turn might be baby steps toward P≠NP. More strongly, Valiant conjectured that the property of information integration, all by itself, implied that such functions had to be at least somewhat computationally complex—i.e., that they couldn’t be computed by circuits of size O(n), or even required circuits of size Ω(n log n). Alas, that hope was refuted by Valiant’s later discovery of linear-size superconcentrators. Just as information integration doesn’t suffice for intelligence or consciousness, so Valiant learned that information integration doesn’t suffice for circuit lower bounds either. As humans, we seem to have the intuition that global integration of information is such a powerful property that no “simple” or “mundane” computational process could possibly achieve it. But our intuition is wrong. If it were right, then we wouldn’t have linear-size superconcentrators or LDPC codes. I should mention that I had the privilege of briefly speaking with Giulio Tononi (as well as his collaborator, Christof Koch) this winter at an FQXi conference in Puerto Rico. At that time, I challenged Tononi with a much cruder, handwavier version of some of the same points that I made above. Tononi’s response, as best as I can reconstruct it, was that it’s wrong to approach IIT like a mathematician; instead one needs to start “from the inside,” with the phenomenology of consciousness, and only then try to build general theories that can be tested against counterexamples. This response perplexed me: of course you can start from phenomenology, or from anything else you like, when constructing your theory of consciousness. However, once your theory has been constructed, surely it’s then fair game for others to try to refute it with counterexamples? And surely the theory should be judged, like anything else in science or philosophy, by how well it withstands such attacks? [Endnote: See also this related post, by the philosopher Eric Schwetzgebel: Why Tononi Should Think That the United States Is Conscious. While the discussion is much more informal, and the proposed counterexample more debatable, the basic objection to IIT is the same.] Update (5/22): Here are a few clarifications of this post that might be helpful. (1) The stuff about zombies and the Hard Problem was simply meant as motivation and background for what I called the “Pretty-Hard Problem of Consciousness”—the problem that I take IIT to be addressing. You can disagree with the zombie stuff without it having any effect on my arguments about IIT. (2) I wasn’t arguing in this post that dualism is true, or that consciousness is irreducibly mysterious, or that there could never be any convincing theory that told us how much consciousness was present in a physical system. All I was arguing was that, at any rate, IIT is not such a theory. (3) Yes, it’s true that my demonstration of IIT’s falsehood assumes—as an axiom, if you like—that while we might not know exactly what we mean by “consciousness,” at any rate we’re talking about something that humans have to a greater extent than DVD players. If you reject that axiom, then I’d simply want to define a new word for a certain quality that non-anesthetized humans seem to have and that DVD players seem not to, and clarify that that other quality is the one I’m interested in. (4) For my counterexample, the reason I chose the Vandermonde matrix is not merely that it’s invertible, but that all of its submatrices are full-rank. This is the property that’s relevant for producing a large value of the integrated information Φ; by contrast, note that the identity matrix is invertible, but produces a system with Φ=0. (As another note, if we work over a large enough field, then a random matrix will have this same property with high probability—but I wanted an explicit example, and while the Vandermonde is far from the only one, it’s one of the simplest.) (5) The n×n Vandermonde matrix only does what I want if we work over (say) a prime field Fp with p>>n elements. Thus, it’s natural to wonder whether similar examples exist where the basic system variables are bits, rather than elements of Fp. The answer is yes. One way to get such examples is using the low-density parity check codes that I mention in the post. Another common way to get Boolean examples, and which is also used in practice in error-correcting codes, is to start with the Vandermonde matrix (a.k.a. the Reed-Solomon code), and then combine it with an additional component that encodes the elements of Fp as strings of bits in some way. Of course, you then need to check that doing this doesn’t harm the properties of the original Vandermonde matrix that you cared about (e.g., the “information integration”) too much, which causes some additional complication. (6) Finally, it might be objected that my counterexamples ignored the issue of dynamics and “feedback loops”: they all consisted of unidirectional processes, which map inputs to outputs and then halt. However, this can be fixed by the simple expedient of iterating the process over and over! I.e., first map x to Wx, then map Wx to W2x, and so on. The integrated information should then be the same as in the unidirectional case. Update (5/24): See a very interesting comment by David Chalmers. Retiring falsifiability? A storm in Russell’s teacup Friday, January 17th, 2014 My good friend Sean Carroll took a lot of flak recently for answering this year’s Edge question, “What scientific idea is ready for retirement?,” with “Falsifiability”, and for using string theory and the multiverse as examples of why science needs to break out of its narrow Popperian cage. For more, see this blog post of Sean’s, where one commenter after another piles on the beleaguered dude for his abandonment of science and reason themselves. My take, for whatever it’s worth, is that Sean and his critics are both right. Sean is right that “falsifiability” is a crude slogan that fails to capture what science really aims at. As a doofus example, the theory that zebras exist is presumably both “true” and “scientific,” but it’s not “falsifiable”: if zebras didn’t exist, there would be no experiment that proved their nonexistence. (And that’s to say nothing of empirical claims involving multiple nested quantifiers: e.g., “for every physical device that tries to solve the Traveling Salesman Problem in polynomial time, there exists an input on which the device fails.”) Less doofusly, a huge fraction of all scientific progress really consists of mathematical or computational derivations from previously-accepted theories—and, as such, has no “falsifiable content” apart from the theories themselves. So, do workings-out of mathematical consequences count as “science”? In practice, the Nobel committee says sure they do, but only if the final results of the derivations are “directly” confirmed by experiment. Far better, it seems to me, to say that science is a search for explanations that do essential and nontrivial work, within the network of abstract ideas whose ultimate purpose to account for our observations. (On this particular question, I endorse everything David Deutsch has to say in The Beginning of Infinity, which you should read if you haven’t.) On the other side, I think Sean’s critics are right that falsifiability shouldn’t be “retired.” Instead, falsifiability’s portfolio should be expanded, with full-time assistants (like explanatory power) hired to lighten falsifiability’s load. I also, to be honest, don’t see that modern philosophy of science has advanced much beyond Popper in its understanding of these issues. Last year, I did something weird and impulsive: I read Karl Popper. Given all the smack people talk about him these days, I was pleasantly surprised by the amount of nuance, reasonableness, and just general getting-it that I found. Indeed, I found a lot more of those things in Popper than I found in his latter-day overthrowers Kuhn and Feyerabend. For Popper (if not for some of his later admirers), falsifiability was not a crude bludgeon. Rather, it was the centerpiece of a richly-articulated worldview holding that millennia of human philosophical reflection had gotten it backwards: the question isn’t how to arrive at the Truth, but rather how to eliminate error. Which sounds kind of obvious, until I meet yet another person who rails to me about how empirical positivism can’t provide its own ultimate justification, and should therefore be replaced by the person’s favorite brand of cringe-inducing ugh. Oh, I also think Sean might have made a tactical error in choosing string theory and the multiverse as his examples for why falsifiability needs to be retired. For it seems overwhelmingly likely to me that the following two propositions are both true: 1. Falsifiability is too crude of a concept to describe how science works. 2. In the specific cases of string theory and the multiverse, a dearth of novel falsifiable predictions really is a big problem. As usual, the best bet is to use explanatory power as our criterion—in which case, I’d say string theory emerges as a complex and evolving story. On one end, there are insights like holography and AdS/CFT, which seem clearly to do explanatory work, and which I’d guess will stand as permanent contributions to human knowledge, even if the whole foundations on which they currently rest get superseded by something else. On the other end, there’s the idea, championed by a minority of string theorists and widely repeated in the press, that the anthropic principle applied to different patches of multiverse can be invoked as a sort of get-out-of-jail-free card, to rescue a favored theory from earlier hopes of successful empirical predictions that then failed to pan out. I wouldn’t know how to answer a layperson who asked why that wasn’t exactly the sort of thing Sir Karl was worried about, and for good reason. Finally, not that Edge asked me, but I’d say the whole notions of “determinism” and “indeterminism” in physics are past ready for retirement. I can’t think of any work they do, that isn’t better done by predictability and unpredictability. Luke Muehlhauser interviews me about philosophical progress Saturday, December 14th, 2013 I’m shipping out today to sunny Rio de Janeiro, where I’ll be giving a weeklong course about BosonSampling, at the invitation of Ernesto Galvão. Then it’s on to Pennsylvania (where I’ll celebrate Christmas Eve with old family friends), Israel (where I’ll drop off Dana and Lily with Dana’s family in Tel Aviv, then lecture at the Jerusalem Winter School in Theoretical Physics), Puerto Rico (where I’ll speak at the FQXi conference on Physics of Information), back to Israel, and then New York before returning to Boston at the beginning of February. Given this travel schedule, it’s possible that blogging will be even lighter than usual for the next month and a half (or not—we’ll see). In the meantime, however, I’ve got the equivalent of at least five new blog posts to tide over Shtetl-Optimized fans. Luke Muehlhauser, the Executive Director of the Machine Intelligence Research Institute (formerly the Singularity Institute for Artificial Intelligence), did an in-depth interview with me about “philosophical progress,” in which he prodded me to expand on certain comments in Why Philosophers Should Care About Computational Complexity and The Ghost in the Quantum Turing Machine. Here are (abridged versions of) Luke’s five questions: 1. Why are you so interested in philosophy? And what is the social value of philosophy, from your perspective? 2. What are some of your favorite examples of illuminating Q-primes [i.e., scientifically-addressable pieces of big philosophical questions] that were solved within your own field, theoretical computer science? 3. Do you wish philosophy-the-field would be reformed in certain ways? Would you like to see more crosstalk between disciplines about philosophical issues? Do you think that, as Clark Glymour suggested, philosophy departments should be defunded unless they produce work that is directly useful to other fields … ? 4. Suppose a mathematically and analytically skilled student wanted to make progress, in roughly the way you describe, on the Big Questions of philosophy. What would you recommend they study? What should they read to be inspired? What skills should they develop? Where should they go to study? 5. Which object-level thinking tactics … do you use in your own theoretical (especially philosophical) research? Are there tactics you suspect might be helpful, which you haven’t yet used much yourself? For the answers—or at least my answers—click here! PS. In case you missed it before, Quantum Computing Since Democritus was chosen by Scientific American blogger Jennifer Ouellette (via the “Time Lord,” Sean Carroll) as the top physics book of 2013. Woohoo!! The Ghost in the Quantum Turing Machine Saturday, June 15th, 2013 I’ve been traveling this past week (in Israel and the French Riviera), heavily distracted by real life from my blogging career. But by popular request, let me now provide a link to my very first post-tenure publication: The Ghost in the Quantum Turing Machine. Here’s the abstract: In honor of Alan Turing’s hundredth birthday, I unwisely set out some thoughts about one of Turing’s obsessions throughout his life, the question of physics and free will. I focus relatively narrowly on a notion that I call “Knightian freedom”: a certain kind of in-principle physical unpredictability that goes beyond probabilistic unpredictability. Other, more metaphysical aspects of free will I regard as possibly outside the scope of science. I examine a viewpoint, suggested independently by Carl Hoefer, Cristi Stoica, and even Turing himself, that tries to find scope for “freedom” in the universe’s boundary conditions rather than in the dynamical laws. Taking this viewpoint seriously leads to many interesting conceptual problems. I investigate how far one can go toward solving those problems, and along the way, encounter (among other things) the No-Cloning Theorem, the measurement problem, decoherence, chaos, the arrow of time, the holographic principle, Newcomb’s paradox, Boltzmann brains, algorithmic information theory, and the Common Prior Assumption. I also compare the viewpoint explored here to the more radical speculations of Roger Penrose. The result of all this is an unusual perspective on time, quantum mechanics, and causation, of which I myself remain skeptical, but which has several appealing features. Among other things, it suggests interesting empirical questions in neuroscience, physics, and cosmology; and takes a millennia-old philosophical debate into some underexplored territory. See here (and also here) for interesting discussions over on Less Wrong. I welcome further discussion in the comments section of this post, and will jump in myself after a few days to address questions (update: eh, already have). There are three reasons for the self-imposed delay: first, general busyness. Second, inspired by the McGeoch affair, I’m trying out a new experiment, in which I strive not to be on such an emotional hair-trigger about the comments people leave on my blog. And third, based on past experience, I anticipate comments like the following: “Hey Scott, I didn’t have time to read this 85-page essay that you labored over for two years. So, can you please just summarize your argument in the space of a blog comment? Also, based on the other comments here, I have an objection that I’m sure never occurred to you. Oh, wait, just now scanning the table of contents…” So, I decided to leave some time for people to RTFM (Read The Free-Will Manuscript) before I entered the fray. For now, just one remark: some people might wonder whether this essay marks a new “research direction” for me. While it’s difficult to predict the future (even probabilistically :-) ), I can say that my own motivations were exactly the opposite: I wanted to set out my thoughts about various mammoth philosophical issues once and for all, so that then I could get back to complexity, quantum computing, and just general complaining about the state of the world. “Quantum Information and the Brain” Thursday, January 24th, 2013 A month and a half ago, I gave a 45-minute lecture / attempted standup act with the intentionally-nutty title above, for my invited talk at the wonderful NIPS (Neural Information Processing Systems) conference at Lake Tahoe. Video of the talk is now available at VideoLectures net. That site also did a short written interview with me, where they asked about the “message” of my talk (which is unfortunately hard to summarize, though I tried!), as well as the Aaron Swartz case and various other things. If you just want the PowerPoint slides from my talk, you can get those here. Now, I could’ve just given my usual talk on quantum computing and complexity. But besides increasing boredom with that talk, one reason for my unusual topic was that, when I sent in the abstract, I was under the mistaken impression that NIPS was at least half a “neuroscience” conference. So, I felt a responsibility to address how quantum information science might intersect the study of the brain, even if the intersection ultimately turned out to be the empty set! (As I say in the talk, the fact that people have speculated about connections between the two, and have sometimes been wrong but for interesting reasons, could easily give me 45 minutes’ worth of material.) Anyway, it turned out that, while NIPS was founded by people interested in modeling the brain, these days it’s more of a straight machine learning conference. Still, I hope the audience there at least found my talk an amusing appetizer to their hearty meal of kernels, sparsity, and Bayesian nonparametric regression. I certainly learned a lot from them; while this was my first machine learning conference, I’ll try to make sure it isn’t my last. (Incidentally, the full set of NIPS videos is here; it includes great talks by Terry Sejnowski, Stanislas Dehaene, Geoffrey Hinton, and many others. It was a weird honor to be in such distinguished company — I wouldn’t have invited myself!) A causality post, for no particular reason Friday, November 2nd, 2012 The following question emerged from a conversation with the machine learning theorist Pedro Domingos a month ago. Consider a hypothetical race of intelligent beings, the Armchairians, who never take any actions: never intervene in the world, never do controlled experiments, never try to build anything and see if it works. The sole goal of the Armchairians is to observe the world around them and, crucially, to make accurate predictions about what’s going to happen next. Would the Armchairians ever develop the notion of cause and effect? Or would they be satisfied with the notion of statistical correlation? Or is the question kind of silly, the answer depending entirely on what we mean by “developing the notion of cause and effect”? Feel free to opine away in the comments section. Why Many-Worlds is not like Copernicanism Saturday, August 18th, 2012 [Update (8/26): Inspired by the great responses to my last Physics StackExchange question, I just asked a new one---also about the possibilities for gravitational decoherence, but now focused on Gambini et al.'s "Montevideo interpretation" of quantum mechanics. Also, on a completely unrelated topic, my friend Jonah Sinick has created a memorial YouTube video for the great mathematician Bill Thurston, who sadly passed away last week. Maybe I should cave in and set up a Twitter feed for this sort of thing...] [Update (8/26): I've now posted what I see as one of the main physics questions in this discussion on Physics StackExchange: "Reversing gravitational decoherence." Check it out, and help answer if you can!] [Update (8/23): If you like this blog, and haven't yet read the comments on this post, you should probably do so! To those who've complained about not enough meaty quantum debates on this blog lately, the comment section of this post is my answer.] [Update: Argh! For some bizarre reason, comments were turned off for this post. They're on now. Sorry about that.] I’m in Anaheim, CA for a great conference celebrating the 80th birthday of the physicist Yakir Aharonov. I’ll be happy to discuss the conference in the comments if people are interested. In the meantime, though, since my flight here was delayed 4 hours, I decided to (1) pass the time, (2) distract myself from the inanities blaring on CNN at the airport gate, (3) honor Yakir’s half-century of work on the foundations of quantum mechanics, and (4) honor the commenters who wanted me to stop ranting and get back to quantum stuff, by sharing some thoughts about a topic that, unlike gun control or the Olympics, is completely uncontroversial: the Many-Worlds Interpretation of quantum mechanics. Proponents of MWI, such as David Deutsch, often argue that MWI is a lot like Copernican astronomy: an exhilarating expansion in our picture of the universe, which follows straightforwardly from Occam’s Razor applied to certain observed facts (the motions of the planets in one case, the double-slit experiment in the other). Yes, many holdouts stubbornly refuse to accept the new picture, but their skepticism says more about sociology than science. If you want, you can describe all the quantum-mechanical experiments anyone has ever done, or will do for the foreseeable future, by treating “measurement” as an unanalyzed primitive and never invoking parallel universes. But you can also describe all astronomical observations using a reference frame that places the earth is the center of the universe. In both cases, say the MWIers, the problem with your choice is its unmotivated perversity: you mangle the theory’s mathematical simplicity, for no better reason than a narrow parochial urge to place yourself and your own experiences at the center of creation. The observed motions of the planets clearly want a sun-centered model. In the same way, Schrödinger’s equation clearly wants measurement to be just another special case of unitary evolution—one that happens to cause your own brain and measuring apparatus to get entangled with the system you’re measuring, thereby “splitting” the world into decoherent branches that will never again meet. History has never been kind to people who put what they want over what the equations want, and it won’t be kind to the MWI-deniers either. This is an important argument, which demands a response by anyone who isn’t 100% on-board with MWI. Unlike some people, I happily accept this argument’s framing of the issue: no, MWI is not some crazy speculative idea that runs afoul of Occam’s razor. On the contrary, MWI really is just the “obvious, straightforward” reading of quantum mechanics itself, if you take quantum mechanics literally as a description of the whole universe, and assume nothing new will ever be discovered that changes the picture. Nevertheless, I claim that the analogy between MWI and Copernican astronomy fails in two major respects. The first is simply that the inference, from interference experiments to the reality of many-worlds, strikes me as much more “brittle” than the inference from astronomical observations to the Copernican system, and in particular, too brittle to bear the weight that the MWIers place on it. Once you know anything about the dynamics of the solar system, it’s hard to imagine what could possibly be discovered in the future, that would ever again make it reasonable to put the earth at the “center.” By contrast, we do more-or-less know what could be discovered that would make it reasonable to privilege “our” world over the other MWI branches. Namely, any kind of “dynamical collapse” process, any source of fundamentally-irreversible decoherence between the microscopic realm and that of experience, any physical account of the origin of the Born rule, would do the trick. Admittedly, like most quantum folks, I used to dismiss the notion of “dynamical collapse” as so contrived and ugly as not to be worth bothering with. But while I remain unimpressed by the specific models on the table (like the GRW theory), I’m now agnostic about the possibility itself. Yes, the linearity of quantum mechanics does indeed seem incredibly hard to tinker with. But as Roger Penrose never tires of pointing out, there’s at least one phenomenon—gravity—that we understand how to combine with quantum-mechanical linearity only in various special cases (like 2+1 dimensions, or supersymmetric anti-deSitter space), and whose reconciliation with quantum mechanics seems to raise fundamental problems (i.e., what does it even mean to have a superposition over different causal structures, with different Hilbert spaces potentially associated to them?). To make the discussion more concrete, consider the proposed experiment of Bouwmeester et al., which seeks to test (loosely) whether one can have a coherent superposition over two states of the gravitational field that differ by a single Planck length or more. This experiment hasn’t been done yet, but some people think it will become feasible within a decade or two. Most likely it will just confirm quantum mechanics, like every previous attempt to test the theory for the last century. But it’s not a given that it will; quantum mechanics has really, truly never been tested in this regime. So suppose the interference pattern isn’t seen. Then poof! The whole vast ensemble of parallel universes spoken about by the MWI folks would have disappeared with a single experiment. In the case of Copernicanism, I can’t think of any analogous hypothetical discovery with even a shred of plausibility: maybe a vector field that pervades the universe but whose unique source was the earth? So, this is what I mean in saying that the inference from existing QM experiments to parallel worlds seems too “brittle.” As you might remember, I wagered $100,000 that scalable quantum computing will indeed turn out to be compatible with the laws of physics. Some people considered that foolhardy, and they might be right—but I think the evidence seems pretty compelling that quantum mechanics can be extrapolated at least that far. (We can already make condensed-matter states involving entanglement among millions of particles; for that to be possible but not quantum computing would seem to require a nasty conspiracy.) On the other hand, when it comes to extending quantum-mechanical linearity all the way up to the scale of everyday life, or to the gravitational metric of the entire universe—as is needed for MWI—even my nerve falters. Maybe quantum mechanics does go that far up; or maybe, as has happened several times in physics when exploring a new scale, we have something profoundly new to learn. I wouldn’t give much more informative odds than 50/50. The second way I’d say the MWI/Copernicus analogy breaks down arises from a closer examination of one of the MWIers’ favorite notions: that of “parochial-ness.” Why, exactly, do people say that putting the earth at the center of creation is “parochial”—given that relativity assures us that we can put it there, if we want, with perfect mathematical consistency? I think the answer is: because once you understand the Copernican system, it’s obvious that the only thing that could possibly make it natural to place the earth at the center, is the accident of happening to live on the earth. If you could fly a spaceship far above the plane of the solar system, and watch the tiny earth circling the sun alongside Mercury, Venus, and the sun’s other tiny satellites, the geocentric theory would seem as arbitrary to you as holding Cheez-Its to be the sole aim and purpose of human civilization. Now, as a practical matter, you’ll probably never fly that spaceship beyond the solar system. But that’s irrelevant: firstly, because you can very easily imagine flying the spaceship, and secondly, because there’s no in-principle obstacle to your descendants doing it for real. Now let’s compare to the situation with MWI. Consider the belief that “our” universe is more real than all the other MWI branches. If you want to describe that belief as “parochial,” then from which standpoint is it parochial? The standpoint of some hypothetical godlike being who sees the entire wavefunction of the universe? The problem is that, unlike with my solar system story, it’s not at all obvious that such an observer can even exist, or that the concept of such an observer makes sense. You can’t “look in on the multiverse from the outside” in the same way you can look in on the solar system from the outside, without violating the quantum-mechanical linearity on which the multiverse picture depends in the first place. The closest you could come, probably, is to perform a Wigner’s friend experiment, wherein you’d verify via an interference experiment that some other person was placed into a superposition of two different brain states. But I’m not willing to say with confidence that the Wigner’s friend experiment can even be done, in principle, on a conscious being: what if irreversible decoherence is somehow a necessary condition for consciousness? (We know that increase in entropy, of which decoherence is one example, seems intertwined with and possibly responsible for our subjective sense of the passage of time.) In any case, it seems clear that we can’t talk about Wigner’s-friend-type experiments without also talking, at least implicitly, about consciousness and the mind/body problemand that that fact ought to make us exceedingly reluctant to declare that the right answer is obvious and that anyone who doesn’t see it is an idiot. In the case of Copernicanism, the “flying outside the solar system” thought experiment isn’t similarly entangled with any of the mysteries of personal identity. There’s a reason why Nobel Prizes are regularly awarded for confirmations of effects that were predicted decades earlier by theorists, and that therefore surprised almost no one when they were finally found. Were we smart enough, it’s possible that we could deduce almost everything interesting about the world a priori. Alas, history has shown that we’re usually not smart enough: that even in theoretical physics, our tendencies to introduce hidden premises and to handwave across gaps in argument are so overwhelming that we rarely get far without constant sanity checks from nature. I can’t think of any better summary of the empirical attitude than the famous comment by Donald Knuth: “Beware of bugs in the above code. I’ve only proved it correct; I haven’t tried it.” In the same way, I hereby declare myself ready to support MWI, but only with the following disclaimer: “Beware of bugs in my argument for parallel copies of myself. I’ve only proved that they exist; I haven’t heard a thing from them.”
b4f7a1247a067fc4	Symmetry Symmetry 2073-8994 Molecular Diversity Preservation International (MDPI) 10.3390/sym6020396 symmetry-06-00396 Article Invisibility and Symmetry: A Simple Geometrical Viewpoint Sánchez-SotoLuis L.* MonzónJuan J. Departamento de Optica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain; E-Mail: Author Contributions: Both authors contributed equally to the theoretical analysis, numerical calculations, and writing of the paper. Author to whom correspondence should be addressed; E-Mail:; Tel.: +34-91-3944-680; Fax: +34-91-3944-683. 06 2014 22 05 2014 6 2 396 408 24 02 2014 12 05 2014 14 05 2014 © 2014 by the authors; licensee MDPI, Basel, Switzerland. 2014 We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy. PT symmetry SL(2, ℂ) Lorentz group Hyperbolic geometry The work of Bender and coworkers [16] has triggered considerable efforts to understand complex potentials that have neither parity ( ) nor time-reversal symmetry ( ), yet they retain combined invariance. These systems can exhibit real energy eigenvalues, thus suggesting a plausible generalization of quantum mechanics. This speculative concept has motivated an ongoing debate in several forefronts [7,8]. Quite recently, the prospect of realizing -symmetric potentials within the framework of optics has been put forward [9,10] and experimentally tested [11]. The complex refractive index takes on here the role of the potential, so they can be realized through a judicious inclusion of index guiding and gain/loss regions. These -synthetic materials can exhibit several intriguing features [1214], one of which will be the main interest of this paper, namely, unidirectional invisibility [1517]. In all these matters, the time-honored transfer-matrix method is particularly germane [18]. However, a quick look at the literature immediately reveals the different backgrounds and habits in which the transfer matrix is used and the very little cross talk between them. To remedy this flaw, we have been capitalizing on a number of geometrical concepts to gain further insights into the behavior of one-dimensional scattering [1926]. Indeed, when one think in a unifying mathematical scenario, geometry immediately comes to mind. Here, we keep going this program and examine the action of the transfer matrices associated to invisible scatterers. Interestingly enough, when viewed in SO(1, 3), they turn to be nothing but parabolic Lorentz transformations, also called null rotations, which play a crucial role in the determination of the little group of massless particles. Furthermore, borrowing elementary techniques of hyperbolic geometry, we reinterpret these matrices as parallel displacements, which are motions without Euclidean counterpart. We stress that our formulation does not offer any inherent advantage in terms of efficiency in solving practical problems; rather, it furnishes a general and unifying setting to analyze the transfer matrix for complex potentials, which, in our opinion, is more than a curiosity. Basic Concepts on Transfer Matrix To be as self-contained as possible, we first briefly review some basic facts on the quantum scattering of a particle of mass m by a local complex potential V(x) defined on the real line ℝ [2734]. Although much of the renewed interest in this topic has been fuelled by the remarkable case of symmetry, we do not use this extra assumption in this Section. The problem at hand is governed by the time-independent Schrödinger equation H Ψ ( x ) = [ d 2 d x 2 + U ( x ) ] Ψ ( x ) = ɛ Ψ ( x )where ε = 2mE2 and U(x) = 2mV(x)/ћ2, E being the energy of the particle. We assume that U(x) → 0 fast enough as x → ±∞, although the treatment can be adapted, with minor modifications, to cope with potentials for which the limits U± = limx→±∞ U(x) are different. Since U(x) decays rapidly as \|x\| ∞, solutions of (1) have the asymptotic behavior Ψ ( x ) = { A + e + ikx + A e ikx x B + e + ikx + B e ikx x Here, k2 = ε, A± and B± are k-dependent complex coefficients (unspecified, at this stage), and the subscripts + and — distinguish right-moving modes exp(+ikx) from left-moving modes exp(−ikx), respectively. The problem requires to work out the exact solution of (1) and invoke the appropriate boundary conditions, involving not only the continuity of Ψ(x) itself, but also of its derivative. In this way, one has two linear relations among the coefficients A± and B±, which can be solved for any amplitude pair in terms of the other two; the result can be expressed as a matrix equation that translates the linearity of the problem. Frequently, it is more advantageous to specify a linear relation between the wave amplitudes on both sides of the scatterer, namely, ( B + B ) = M ( A + A ) M is the transfer matrix, which depends in a complicated way on the potential U(x). Yet one can extract a good deal of information without explicitly calculating it: let us apply (3) successively to a right-moving [(A+ = 1, B = 0)] and to a left-moving wave [(A+ = 0, B = 1)], both of unit amplitude. The result can be displayed as ( T 0 ) = M ( 1 R ) , ( R r 1 ) = M ( 0 T r ) ,where Tℓ,r and Rℓ,r are the transmission and reflection coefficients for a wave incoming at the potential from the left and from the right, respectively, defined in the standard way as the quotients of the pertinent fluxes [35]. With this in mind, Equation (4) can be thought of as a linear superposition of the two independent solutions Ψ k ( x ) = { e + ikx + R ( x ) e ikx x , T ( k ) e + ikx x , Ψ k r ( x ) = { T r ( k ) e ikx x , e ikx + R r ( k ) e + ikx x ,which is consistent with the fact that, since ε > 0, the spectrum of the Hamiltonian (1) is continuous and there are two linearly independent solutions for a given value of ε. The wave function Ψ k ( x ) represents a wave incident from −∞ [exp(+ikx)] and the interaction with the potential produces a reflected wave [R(k) exp(−ikx)] that escapes to −∞ and a transmitted wave [T(k) exp(+ikx)] that moves off to −∞. The solution Ψ k ( x ) can be interpreted in a similar fashion. Because of the Wronskian of the solutions (5) is independent of x, we can compute W ( Ψ k , Ψ k r ) = Ψ k Ψ k r Ψ k Ψ k r first for x → −∞ and then for x → ∞; this gives i 2 k W ( Ψ k , Ψ k r ) = T r ( k ) = T ( k ) T ( k ) We thus arrive at the important conclusion that, irrespective of the potential, the transmission coefficient is always independent of the input direction. Taking this constraint into account, we go back to the system (4) and write the solution for M as M 11 ( k ) = T ( k ) R ( k ) R r ( k ) T ( k ) , M 12 ( k ) = R r ( k ) T ( k ) , M 21 ( k ) = R ( k ) T ( k ) , M 22 ( k ) = 1 T ( k ) A straightforward check shows that det M = +1, so M ∊ SL(2, ℂ); a result that can be drawn from a number of alternative and more elaborate arguments [36]. One could also relate outgoing amplitudes to the incoming ones (as they are often the magnitudes one can externally control): this is precisely the scattering matrix, which can be concisely formulated as ( B + A ) = S ( A + B )with matrix elements S 11 ( k ) = T ( k ) , S 12 ( k ) = R r ( k ) , S 21 ( k ) = R ( k ) , S 22 ( k ) = T ( k ) Finally, we stress that transfer matrices are very convenient mathematical objects. Suppose that V1 and V2 are potentials with finite support, vanishing outside a pair of adjacent intervals I1 and I2. If M1 and M2 are the corresponding transfer matrices, the total system (with support I1 U I2) is described by M = M 1 M 2 This property is rather helpful: we can connect simple scatterers to create an intricate potential landscape and determine its transfer matrix by simple multiplication. This is a common instance in optics, where one routinely has to treat multilayer stacks. However, this important property does not seem to carry over into the scattering matrix in any simple way [37,38], because the incoming amplitudes for the overall system cannot be obtained in terms of the incoming amplitudes for every subsystem. Spectral Singularities The scattering solutions (5) constitute quite an intuitive way to attack the problem and they are widely employed in physical applications. Nevertheless, it is sometimes advantageous to look at the fundamental solutions of (1) in terms of left- and right-moving modes, as we have already used in (2). Indeed, the two independent solutions of (1) can be formally written down as [39] Ψ k ( + ) ( x ) = e + ikx + x K + ( x , x ) e + i k x d x Ψ k ( ) ( x ) = e ikx + x K ( x , x ) e i k x d x The kernels K±(x,x ′) enjoy a number of interesting properties. What matters for our purposes is that the resulting Ψ k ( ± ) ( x ) are analytic with respect to k in ℂ+ = {z ∊ ℂ\| Imz > 0} and continuous on the real axis. In addition, it is clear that Ψ k ( + ) ( x ) = e + ikx x , Ψ k ( ) ( x ) = e ikx x that is, they are the Jost functions for this problem [31]. Let us look at the Wronskian of the Jost functions W ( Ψ k ( ) , Ψ k ( + ) ), which, as a function of k, is analytical in ℂ+. A spectral singularity is a point k* ∊ ℝ+ of the continuous spectrum of the Hamiltonian (1) such that W ( Ψ k * ( ) , Ψ k * ( + ) ) = 0so Ψ k ( ± ) ( x ) become linearly dependent at k* and the Hamiltonian is not diagonalizable. In fact, the set of zeros of the Wronskian is bounded, has at most a countable number of elements and its limit points can lie in a bounded subinterval of the real axis [40]. There is an extensive theory of spectral singularities for (1) that was started by Naimark [41]; the interested reader is referred to, e.g., Refs. [4246] for further details. The asymptotic behavior of Ψ k ± ( x ) at the opposite extremes of ℝ with respect to those in (12) can be easily worked out by a simple application of the transfer matrix (and its inverse); viz, Ψ k ( ) ( x ) = M 12 e + ikx + M 22 e ikx x Ψ k ( + ) ( x ) = M 22 e + ikx M 21 e ikx x Using Ψ k ± ( x ) in (12) and (14), we can calculate i 2 k W ( Ψ k ( ) , Ψ k ( + ) ) = M 22 ( k )Upon comparing with the definition (13), we can reinterpret the spectral singularities as the real zeros of M22(k) and, as a result, the reflection and transmission coefficients diverge therein. The converse holds because M12(k) and M21(k) are entire functions, lacking singularities. This means that, in an optical scenario, spectral singularities correspond to lasing thresholds [4749]. One could also consider the more general case that the Hamiltonian (1) has, in addition to a continuous spectrum corresponding to k ∊ ℝ+, a possibly complex discrete spectrum. The latter corresponds to the square-integrable solutions of that represent bound states. They are also zeros of M22(k), but unlike the zeros associated with the spectral singularities these must have a positive imaginary part [36]. The eigenvalues of S are s ± = 1 M 22 ( k ) [ 1 ± 1 M 11 ( k ) M 22 ( k ) ]At a spectral singularity, s+ diverges, while s_ → M11(k)/2, which suggests identifying spectral singularities with resonances with a vanishing width. Invisibility and <inline-graphic xlink:href="symmetry-06-00396i1.tif"/> Symmetry As heralded in the Introduction, unidirectional invisibility has been lately predicted in materials. We shall elaborate on the ideas developed by Mostafazadeh [50] in order to shed light into this intriguing question. The potential U(x) is called reflectionless from the left (right), if R(k) = 0 and Rr(k) ≠ 0 [Rr(k) = 0 and R(k) ≠ 0]. From the explicit matrix elements in (7) and (9), we see that unidirectional reflectionlessness implies the non-diagonalizability of both M and S. Therefore, the parameters of the potential for which it becomes reflectionless correspond to exceptional points of M and S [51,52]. The potential is called invisible from the left (right), if it is reflectionless from left (right) and in addition T(k) = 1. We can easily express the conditions for the unidirectional invisibility as M 12 ( k ) 0 , M 11 ( k ) = M 22 ( k ) = 1 ( left invisible ) M 21 ( k ) 0 , M 11 ( k ) = M 22 ( k ) = 1 ( right invisible ) Next, we scrutinize the role of -symmetry in the invisibility. For that purpose, we first briefly recall that the parity transformation “reflects” the system with respect to the coordinate origin, so that x−x and the momentum p ↦ − p. The action on the wave function is Ψ ( x ) ( P Ψ ) ( x ) = Ψ ( x ) On the other hand, the time reversal inverts the sense of time evolution, so that xx, p−p and i ↦ −i. This means that the operator implementing such a transformation is antiunitary and its action reads Ψ ( x ) ( T Ψ ) ( x ) = Ψ * ( x ) Consequently, under a combined transformation, we have Ψ ( x ) ( P T Ψ ) ( x ) = Ψ * ( x ) Let us apply this to a general complex scattering potential. The transfer matrix of the -transformed system, we denote by M( ), fulfils ( A + * A * ) = M ( P T ) ( B + * B * ) Comparing with (3), we come to the result M ( P T ) = ( M 1 ) and, because det M = 1, this means M 11 P T M 22 , M 12 P T M 12 * , M 21 P T M 21 * , M 22 P T M 11 * , When the system is invariant under this transformation [M( ) = M], it must hold M 1 = M a fact already noticed by Longhi [48] and that can be also recast as [53] Re ( R T ) = Re ( R r T ) = 0 This can be equivalently restated in the form ρ τ = ± π / 2 , ρ r τ = ± π / 2with τ = arg(T) and ρℓ,r = arg(Rℓ,r). Hence, if we look at the complex numbers R, Rr, and T as phasors, Equation (26) tell us that R and Rr are always collinear, while T is simultaneously perpendicular to them. We draw the attention to the fact that the same expressions have been derived for lossless symmetric beam splitters [54]: we have shown that they hold true for any -symmetric structure. A direct consequence of (23) is that there are particular instances of -invariant systems that are invisible, although not every invisible potential is invariant. In this respect, it is worth stressing, that even ( -symmetric) potentials do not support unidirectional invisibility and the same holds for real ( -symmetric) potentials. In optics, beam propagation is governed by the paraxial wave equation, which is equivalent to a Schrödinger-like equation, with the role of the potential played here by the refractive index. Therefore, a necessary condition for a complex refractive index to be invariant is that its real part is an even function of x, while the imaginary component (loss and gain profile) is odd. Relativistic Variables To move ahead, let us construct the Hermitian matrices X = ( X + X ) ( X + X * ) = ( \| X + \| 2 X + X * X + * X \| X \| 2 )where X± refers to either A± or B±; i.e., the amplitudes that determine the behavior at each side of the potential. The matrices X are quite reminiscent of the coherence matrix in optics or the density matrix in quantum mechanics. One can verify that M acts on X by conjugation X = M X M The matrix X′ is associated with the amplitudes B± and X with A±. Let us consider the set σμ = ( ,σ), with Greek indices running from 0 to 3. The σμ are the identity and the standard Pauli matrices, which constitute a basis of the linear space of 2 × 2 complex matrices. For the sake of covariance, it is convenient to define σ̃μ = σμ = ( ,σ), so that [55] Tr ( σ ˜ μ σ ν ) = 2 δ ν μand δ ν μ ( M ) is the Kronecker delta. To any Hermitian matrix X we can associate the coordinates x μ = 1 2 Tr ( X σ ˜ μ ) The congruence (28) induces in this way a transformation x μ = Λ ν μ ( M ) x νwhere Λ ν μ ( M ) can be found to be Λ ν μ ( M ) = 1 2 Tr ( σ ˜ μ M σ ν M )This equation can be solved to obtain M from Λ. The matrices M and −M generate the same Λ, so this homomorphism is two-to-one. The variables xμ are coordinates in a Minkovskian (1+3)-dimensional space and the action of the system can be seen as a Lorentz transformation in SO(1, 3). Having set the general scenario, let us have a closer look at the transfer matrix corresponding to right invisibility (the left invisibility can be dealt with in an analogous way); namely, M = ( 1 R 0 1 )where, for simplicity, we have dropped the superscript from Rr, as there is no risk of confusion. Under the homomorphism (32) this matrix generates the Lorentz transformation Λ ( M ) = ( 1 + \| R \| 2 / 2 Re R Im R \| R \| 2 / 2 Re R 1 0 Re R Im R 0 1 Im R \| R \| 2 / 2 Re R Im R 1 \| R \| 2 / 2 )According to Wigner [56], the little group is a subgroup of the Lorentz transformations under which a standard vector sμ remains invariant. When sμ is timelike, the little group is the rotation group SO(3). If sμ is spacelike, the little group are the boosts SO(1, 2). In this context, the matrix (34) is an instance of a null rotation; the little group when sμ is a lightlike or null vector, which is related to E(2), the symmetry group of the two-dimensional Euclidean space [57]. If we write (34) in the form Λ(M) = exp(iN), we can easily work out that N = ( 0 Re R Im R 0 Re R 0 0 Re R Im R 0 0 Im R 0 Re R Im R 0 )This is a nilpotent matrix and the vectors annihilated by N are invariant by Λ(M). In terms of the Lie algebra so(1, 3), N can be expressed as N = Re R ( K 1 + J 2 ) Im R ( K 2 + J 1 )where Ki generate boosts and Ji rotations (i = 1, 2, 3) [58]. Observe that the rapidity of the boost and the angle of the rotation have the same norm. The matrix N define a two-parameter Abelian subgroup. Let us take, for the time being, Re R = 0, as it happens for -invariant invisibility. We can express K2 + J1 as the differential operator K 2 + J 1 ( x 2 0 + x 0 2 ) + ( x 2 3 x 3 2 ) = x 2 ( 0 + 3 ) + ( x 0 x 3 ) 2 As we can appreciate, the combinations x 2 , x 0 x 3 , ( x 0 ) 2 ( x 1 ) 2 ( x 3 ) 2remain invariant. Suppressing the inessential coordinate x2, the flow lines of the Killing vector (37) is the intersection of a null plane, x0x3 = c2 with a hyperboloid (x0)2 − (x1)2 − (x3)2 = c3. The case c3 = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes. Hyperbolic Geometry and Invisibility Although the relativistic hyperboloid in Minkowski space constitute by itself a model of hyperbolic geometry (understood in a broad sense, as the study of spaces with constant negative curvature), it is not the best suited to display some features. Let us consider the customary tridimensional hyperbolic space ℍ3, defined in terms of the upper half-space {(x, y, z) ∊ ℝ3\|z > 0}, equipped with the metric [59] d s 2 = d x 2 + d y 2 + d z 2 z The geodesics are the semicircles in ℍ3 orthogonal to the plane z = 0. We can think of the plane z = 0 in R3 as the complex plane ℂ with the natural identification (x, y, z) ↦ w = x + iy. We need to add the point at infinity, so that ℂ̂ = ℂU ∞, which is usually referred to as the Riemann sphere and identify ℂ̂ as the boundary of ℍ3. Every matrix M in SL(2, ℂ) induces a natural mapping in ℂ via Möbius (or bilinear) transformations [60] w = M 11 w + M 12 M 21 w + M 22Note that any matrix obtained by multiplying M by a complex scalar λ gives the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples. In other words, we need to quotient out SL(2, ℂ) by its center { , − }: the resulting quotient group is known as the projective linear group and is usually denoted PSL(2, ℂ). Observe that we can break down the action (40) into a composition of maps of the form w w + λ , w λ w , w 1 / wwith λ ∊ ℂ. Then we can extend the Mobius transformations to all ℍ3 as follows: ( w , z ) ( w + λ , z ) , ( w , z ) ( λ w , \| λ \| z ) , ( w , z ) ( w * \| w 2 \| + z 2 , z \| w 2 \| + z 2 )The expressions above come from decomposing the action on ℂ̂ of each of the elements of PSL(2, ℂ) in question into two inversions (reflections) in circles in ℂ̂. Each such inversion has a unique extension to ℍ3 as an inversion in the hemisphere spanned by the circle and composing appropriate pairs of inversions gives us these formulas. In fact, one can show that PSL(2, ℂ) preserves the metric on ℍ3. Moreover every isometry of ℍ3 can be seen to be the extension of a conformal map of ℂ̂ to itself, since it must send hemispheres orthogonal to ℂ̂ to hemispheres orthogonal to ℂ̂, hence circles in ℂ̂ to circles in ℂ̂. Thus all orientation-preserving isometries of ℍ3 are given by elements of PSL(2, ℂ) acting as above. In the classification of these isometries the notion of fixed points is of utmost importance. These points are defined by the condition w′ = w in (40), whose solutions are w f = ( M 11 M 22 ) ± [ Tr ( M ) ] 2 4 2 M 21So, they are determined by the trace of M. When the trace is a real number, the induced Mobius transformations are called elliptic, hyperbolic, or parabolic, according [Tr(M)]2 is lesser than, greater than, or equal to 4, respectively. The canonical representatives of those matrices are [61] ( e i θ / 2 0 0 e i θ / 2 ) , elliptic ( e ξ / 2 0 0 e ξ / 2 ) , hyperbolic ( 1 λ 0 1 ) , parabolicwhile the induced geometrical actions are w = w e i θ , w = w e ξ , w = w + λthat is, a rotation of angle θ (so fixes the axis z), a squeezing of parameter ξ (it has two fixed points in ℂ̂, no fixed points in ℍ3, and every hyperplane in ℍ3 that contains the geodesic joining the two fixed points in ℂ̂ is invariant); and a parallel displacement of magnitude λ, respectively. We emphasize that this later action is the only one without Euclidean analogy Indeed, in view of (33), this is precisely the action associated to an invisible scatterer. The far-reaching consequences of this geometrical interpretation will be developed elsewhere. Concluding Remarks We have studied unidirectional invisibility by a complex scattering potential, which is characterized by a set of invariant equations. Consequently, the -symmetric invisible configurations are quite special, for they possess the same symmetry as the equations. We have shown how to cast this phenomenon in term of space-time variables, having in this way a relativistic presentation of invisibility as the set of null rotations. By resorting to elementary notions of hyperbolic geometry, we have interpreted in a natural way the action of the transfer matrix in this case as a parallel displacement. We think that our results are yet another example of the advantages of these geometrical methods: we have devised a geometrical tool to analyze invisibility in quite a concise way that, in addition, can be closely related to other fields of physics. We acknowledge illuminating discussions with Antonio F. Costa, José F. Carineña and José María Montesinos. Financial support from the Spanish Research Agency (Grant FIS2011-26786) is gratefully acknowledged. Conflicts of Interest The authors declare no conflict of interest. References BenderC.M.BoettcherS.Real spectra in non-Hermitian Hamiltonians having symmetryPhys. Rev. Lett.19988052435246 BenderC.M.BoettcherS.MeisingerP.N.-symmetric quantum mechanicsJ. Math. Phys.19994022012229 BenderC.M.BrodyD.C.JonesH.F.Complex extension of quantum mechanicsPhys. Rev. Lett.20028910.1103/PhysRevLett.89.270401 BenderC.M.BrodyD.C.JonesH.F.Must a Hamiltonian be Hermitian?Am. J. Phys.20037110951102 BenderC.M.Making sense of non-Hermitian HamiltoniansRep. Prog. Phys.2007709471018 BenderC.M.MannheimP.D. symmetry and necessary and sufficient conditions for the reality of energy eigenvaluesPhys. Lett. A201037416161620 AssisP.Non-Hermitian Hamiltonians in Field Theory PT-symmetry and ApplicationsVDMSaarbrücken, Germany2010 MoiseyevN.Non-Hermitian Quantum MechanicsCambridge University PressCambridge, UK2011 El-GanainyR.MakrisK.G.ChristodoulidesD.N.MusslimaniZ.H.Theory of coupled optical -symmetric structuresOpt. Lett.20073226322634 BendixO.FleischmannR.KottosT.ShapiroB.Exponentially fragile symmetry in lattices with localized eigenmodesPhys. Rev. Lett.200910310.1103/PhysRevLett.103.030402 RuterC.E.MakrisK.G.El-GanainyR.ChristodoulidesD.N.SegevM.KipD.Observation of parity-time symmetry in opticsNat. Phys.20106192195 MakrisK.G.El-GanainyR.ChristodoulidesD.N.MusslimaniZ.H.Beam dynamics in symmetric optical latticesPhys. Rev. Lett.2008100103904:1103904:4 LonghiS.Bloch oscillations in complex crystals with symmetryPhys. Rev. Lett.2009103123601:1123601:4 SukhorukovA.A.XuZ.KivsharY.S.Nonlinear suppression of time reversals in -symmetric optical couplersPhys. Rev. A20108210.1103/PhysRevA.82.043818 AhmedZ.BenderC.M.BerryM.V.Reflectionless potentials and symmetryJ. Phys. A200538L627L630 LinZ.RamezaniH.EichelkrautT.KottosT.CaoH.ChristodoulidesD.N.Unidirectional invisibility dnduced by -symmetric periodic structuresPhys. Rev. Lett.201110610.1103/PhysRevLett.106.213901 LonghiS.Invisibility in -symmetric complex crystalsJ. Phys. A20114410.1088/1751-8113/44/48/485302 Sánchez-SotoL.L.MonzónJ.J.BarriusoA.G.CariñenaJ.The transfer matrix: A geometrical perspectivePhys. Rep.2012513191227 MonzónJ.J.Sánchez-SotoL.L.Lossles multilayers and Lorentz transformations: More than an analogyOpt. Commun.199916216 MonzónJ.J.Sánchez-SotoL.L.Fullly relativisticlike formulation of multilayer opticsJ. Opt. Soc. Am. A19991620132018 MonzónJ.J.YonteT.Sánchez-SotoL.L.Basic factorization for multilayersOpt. Lett.200126370372 YonteT.MonzónJ.J.Sánchez-SotoL.L.CariñenaJ.F.López-LacastaC.Understanding multilayers from a geometrical viewpointJ. Opt. Soc. Am. A200219603609 MonzónJ.J.YonteT.Sánchez-SotoL.L.CariñenaJ.F.Geometrical setting for the classification of multilayersJ. Opt. Soc. Am. A200219985991 BarriusoA.G.MonzónJ.J.Sánchez-SotoL.L.General unit-disk representation for periodic multilayersOpt. Lett.20032815011503 BarriusoA.G.MonzónJ.J.Sánchez-SotoL.L.CariñenaJ.F.Vectorlike representation of multilayersJ. Opt. Soc. Am. A20042123862391 BarriusoA.G.MonzónJ.J.Sánchez-SotoL.L.CostaA.F.Escher-like quasiperiodic heterostructuresJ. Phys. A200942192002:1192002:9 MugaJ.G.PalaoJ.P.NavarroB.EgusquizaI.L.Complex absorbing potentialsPhys. Rep.2004395357426 LevaiG.ZnojilM.Systematic search for -symmetric potentials with real spectraJ. Phys. A20003371657180 AhmedZ.Schrödinger transmission through one-dimensional complex potentialsPhys. Rev. A200164042716:1042716:4 AhmedZ.Energy band structure due to a complex, periodic, -invariant potentialPhys. Lett. A2001286231235 MostafazadehA.Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energiesPhys. Rev. Lett.2009102220402:1220402:4 CannataF.DedonderJ.P.VenturaA.Scattering in -symmetric quantum mechanicsAnn. Phys.2007322397433 ChongY.D.GeL.StoneA.D.-symmetry breaking and laser-absorber modes in optical scattering systemsPhys. Rev. Lett.201110610.1103/PhysRevLett.106.093902 AhmedZ.New features of scattering from a one-dimensional non-Hermitian (complex) potentialJ. Phys. A20124510.1088/1751-8113/45/3/032004 BoonsermP.VisserM.One dimensional scattering problems: A pedagogical presentation of the relationship between reflection and transmission amplitudesThai J. Math.201088397 MostafazadehA.Mehri-DehnaviH.Spectral singularities, biorthonormal systems and a two-parameter family of complex point interactionsJ. Phys. A20094210.1088/1751-8113/42/12/125303 AktosunT.A factorization of the scattering matrix for the Schrödinger equation and for the wave equation in one dimensionJ. Math. Phys.19923338653869 AktosunT.KlausM.van der MeeC.Factorization of scattering matrices due to partitioning of potentials in one-dimensional Schrödinger-type equationsJ. Math. Phys.19963758975915 MarchenkoV.A.Sturm-Liouville Operators and Their ApplicationsAMS ChelseaProvidence, RI, USA1986 TuncaG.BairamovE.Discrete spectrum and principal functions of non-selfadjoint differential operatorCzech J. Math.199949689700 NaimarkM.A.Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axisAMS Transl.196016103193 PavlovB.S.The nonself-adjoint Schrödinger operatorsTopics Math. Phys.1967187114 NaimarkM.A.Linear Differential Operators: Part IIUngarNew York, NY, USA1968 SamsonovB.F.SUSY transformations between diagonalizable and non-diagonalizable HamiltoniansJ. Phys. A200538L397L403 AndrianovA.A.CannataF.SokolovA.V.Spectral singularities for non-Hermitian one-dimensional Hamiltonians: Puzzles with resolution of identityJ. Math. Phys.201051052104:1052104:22 Chaos-CadorL.García-CalderónG.Resonant states for complex potentials and spectral singularitiesPhys. Rev. A20138710.1103/PhysRevA.87.042114 SchomerusH.Quantum noise and self-sustained radiation of - symmetric systemsPhys. Rev. Lett.201010410.1103/PhysRevLett.104.233601 LonghiS.-symmetric laser absorberPhys. Rev. A20108210.1103/PhysRevA.82.031801 MostafazadehA.Nonlinear spectral singularities of a complex barrier potential and the lasing threshold conditionPhys. Rev. A20138710.1103/PhysRevA.87.063838 MostafazadehA.Invisibility and symmetryPhys. Rev. A20138710.1103/PhysRevA.87.012103 MüllerM.RotterI.Exceptional points in open quantum systemsJ. Phys. A200841244018:1244018:15 Mehri-DehnaviH.MostafazadehA.Geometric phase for non-Hermitian Hamiltonians and its holonomy interpretationJ. Math. Phys.200849082105:1082105:17 MonzónJ.J.BarriusoA.G.Montesinos-AmilibiaJ.M.Sánchez-SotoL.L.Geometrical aspects of PT-invariant transfer matricesPhys. Rev. A20138710.1103/PhysRevA.87.012111 MandelL.WolfE.Optical Coherence and Quantum Optics.Cambridge University PressCambridge, UK1995 BarutA.O.Ra̧czkaR.Theory of Group Representations and ApplicationsPWNWarszaw, Poland1977Section 17.2 WignerE.On unitary representations of the inhomogeneous Lorentz groupAnn. Math.193940149204 KimY.S.NozM.E.Theory and Applications of the Poincare GroupReidelDordrecht, The Netherlands1986 WeinbergS.The Quantum Theory of FieldsCambridge University PressCambridge, UK2005Volume 1 IversenB.Hyperbolic GeometryCambridge University PressCambridge, UK1992Chapter VIII RatcliffeJ.G.Foundations of Hyperbolic ManifoldsSpringerBerlin, Germany2006Section 4.3 AndersonJ.W.Hyperbolic GeometrySpringerNew York, NY, USA1999Chapter 3
4d5e02c73da0f816	I asked a variety of people in different careers about the use of math in their lives. They answered any or all of the following questions: 1. Do you use math in your current profession? What type of math do you use? What is an example of the math you use? 2. What do you wish you had paid better attention to in math class? What has been the most useful information, or skill, taken from the math classes? 3. Do you utilize math skills to maintain your home or personal life? Finances? Budget? 4. Do you use the logic or problem solving skills learned in math class to work through situations in your professional or personal life? Please look over their responses. They are alphabetical by profession, with a list of professions on the right hand side panel. Alicia, Business and Chemistry Math in current profession… I have used math in every field I've worked in. When I worked in a lab, I used everything up to and including calc 2 (integrals). In marketing I actually used math MORE than when I worked in the lab! Lots of algebra, probability, and statistics. Spreadsheet skills are KEY - use will use spreadsheets in just about any career. Skills that were useful or wish I had paid more attention to… The single most important thing I learned in math (all of school really) is CRITICAL THINKING. If you can develop good critical thinking skills you will be MILES ahead of the competition when you are applying for jobs. I wish I still had my stats book. Math to maintain your home/personal life?... All the time! Surface area and volume of simple and complex objects (like say a wall or rectangular room vs molding with rounded parts and/or steps or two level room. I do a lot of math keeping track of our finances. Taking the time to do financial math can save you tens of thousands of dollars over the life of a home loan, make you money with the best possible saving vehicles, and save you from predatory lenders (who will laon you way more than you can afford to pay back, then take your car/house/what have you). Logic or problem solving skills?... Constantly. Everything from trying to repair things in the home to navigating workplace politics to arguing with my husband. Ryan, Engineer The best math classes I ever took were physics 1 and 2, hands down. The math instruction I've had (in real math classes) has mostly sucked. The curriculum is probably to blame more than the instructors, but way too much class time was spent on procedural rather than "connective" work - in other words, that which fosters our ability to connect what we were supposed to be learning with other things we had learned before. If someone asks me why math is important, I tell them one reason it's important is because it's the foundation upon which all science education rests. Anyone who is considering a career that has anything to do with science will have a much easier time of it if their math is solid. Robley, English Instructor Math in current profession… I primarily use very basic math in my job - things like calculating student grades, percentages, and weighing certain projects more heavily than others. Frequently I encounter students who have no idea how to do this basic addition and averaging to determine their current or final grades. This is actually really problematic for students because they are unable to make decisions about their standing in class, their participating in the class, and how to reach the desired grade in the course. Skills that were useful or wish I had paid more attention to… Geometry. I HATED geometry, it was surprisingly hard for me. But I find this is a tool I used frequently when fixing my house. When I'm building a garden in my back yard and need to calculate the supplies I need or what angle to cut the wood at to create a certain shape (I have a six sided feature we created in our yard that took FOREVER to figure out). Logic or problem solving skills?... Many of the courses that are required in college (and high school) can seem pointless while we're in the class. It's often difficult to understand the "use" of a certain piece of history or set of skills. One thing that I'm frequently struck by, however, is how much this knowledge does add value to my life. Even things I don't "use" in my daily life (for example, understanding the history of the Panama canal) add value to my life. I find this knowledge allows me to be better engaged in the world around me. I'm more prepared to understand the things I read (literature, news papers, blogs, FB status updates even); I'm more able to engage in conversations with people that I meet about a variety of subjects; I'm able to understand or question how current events (in my life and in the world) will effect the future; in short I'm more prepared to participate in the world. The knowledge, critical thinking skills and technical skills (like math and writing) that you are learning while you are in college serve not only to prepare you for a specific career, but to prepare you to be engaged participants in the world. This gives you a level of control over the things that are happening around you that you can't gain in any other way. My experience has been that much of this knowledge developed a "purpose" long after I left college and I am continually surprised at the variety of purposes I find for this knowledge. Tim, Finance Math in current profession… We use calculus and linear algebra every day as part of option value calculations and numerical procedures we code up to calculate the fair value of a financial instrument. Skills that were useful or wish I had paid more attention to… Too often, math beyond algebra gets too abstract. For example linear algebra is immensely useful, but most courses I've seen don't drive home the real world applications. So I think I paid attention, but the teaching was so math focused, it was hard to see how I would use it until much later. Math to maintain your home/personal life?... I use math to calculate different budget scenarios at home. For example, if I pay $50 more on my mortgage, how much sooner will I pay off my mortgage? Logic or problem solving skills?... I use stats every day at work, including probability theory to help make decisions. Travis, Chemist Math in current profession… calculate mole to mole ratios for chemical synthesis. Skills that were useful or wish I had paid more attention to… quantitative data for chemical analysis. Angela, Chemistry, B.S. Math in current profession… For chemistry math problems I'll have to use math to calculate concentration/molarity, volumes, and fractions of molecules included in a square area for lattice structures. We use logs to calculate the pH, and we use basic math to calculate total energy/entropy/enthalpy required in a process. Not to mention all the calculus we use to calculate the probability of an electron's location at any given moment, as well as the time-related Schrödinger equation for a wave function! YUCK. Math to maintain your home/personal life?... I use the "counting money back" technique every time I purchase something to be sure I receive the right amount of change. I use the 10% +half of the 10% to calculate tips! Every time I explain to people that's how I do it, they're always like "oh.. that's so easy". Chris, Computational Biologist Math in current profession… I'm a scientist who uses computers to help understand why we get cancer and how we might treat it more effectively. I use math every day, especially statistics. When we look at the DNA of a tumor, we see thousands of mutations. Statistics helps us decide which ones are most likely to be helping that cancer grow, and which ones might be good targets for new drugs. Skills that were useful or wish I had paid more attention to… I was never a great math student in high school and college. Because of that, I've had to go back to basics and teach myself a lot of things later on. If I had really learned the fundamentals of Algebra earlier, it would have made my life easier and really saved me a lot of time! Math to maintain your home/personal life?... We're saving up for a house and planning for our first kid. I need to understand concepts like compound interest to understand where my money goes and how much I need to save. Logic or problem solving skills?... Math and science have a lot in common. Both can be hard sometimes, but by stepping through problems logically and trying different approaches, you can usually figure out the answers. Erin, Geologist Math in current profession… I do use math all the time. The main thing that we try to do is find where we think oil is accumulated below the earth's surface. Once we have found an interesting area, we need to determine the volume of oil that could potentially be in there. So we do volume calculations - There are a lot of factors involved in that - but essentially we need to know the volume of the entire rock body that would have the oil in it, then the porosity of the rock, and how much oil would actually fit in that pore space. All of the factors involved in determining the volumes are imprecise, so we have to assign a range of the possibilities and then simulate the range of volume outcomes for that We interpret seismic data - which is like an ultrasound of the earth - sound waves are sent down into the earth, and recording their reflections back to the surface allow us to see the layers below the earth's surface (up to 8 or so km!) So we have to know about wave theory (maybe that is more physics...) Skills that were useful or wish I had paid more attention to… I wish I had paid more attention to the principles of statistics. Or, more accurately, that I had taken a statistics class... Math to maintain your home/personal life?... Yes! I use math in maintaining a budget and keeping track of where I am financially. I report all my expenses and income and keep that up to date monthly. Logic or problem solving skills?... Right now I live in Denmark and their currency is the Danish crown. When I go shopping, I convert the price to US dollars so I can get a sense of how expensive things are. I use logic in planning my projects and managing my time throughout the day. For example, what time and other resources do I have, what is my due date, what kind of product can I produce in that amount of time? Also, when i read the news, my knowledge of statistics helps me (I do have some....) to recognize sloppy reporting and know which news articles to take seriously. Skye, Greenhouse Manager Math in current profession… I have to calculate man hours for planting times at my greenhouse eg. 8 people planting for 3 hours is 24 hours of work. Marilyn, Health Insurance Agent Math in current profession… I use math all the time in comparing plans and their rates. It is mostly simple math, and some percentages etc. I also use math in figuring tax savings etc. Skills that were useful or wish I had paid more attention to… I wish I had listened to my Personal Finance teacher more. This class was all about math. Now that I know how important it is to begin saving at a young age, I wish I had started earlier. As I mentioned above, the financial calculations have been a great skill to have over the years. Math to maintain your home/personal life?... Math to maintain your home/personal life?.. I use math in my personal life while shopping and figuring out if I should refinance my home etc., but most importantly in budgeting. It is mostly simple math. I use geometry and other math occasionally when I am fixing things around the house. I often use a financial calculator that allows me to see what the future value of my investments will be given the amount and number of payments I make at a given rate of return. (Everyone should know how to do this to inspire them to save money!!!) I use math all the time with my home budgeting and finances. I calculate how much I need to save each month to pay taxes, or insurance premiums that come once a year. I've even set up a very simple budget that allows me to spend wisely and save money for the future. With math I am also able to see how stupid it is to use credit cards/get into debt. I recently used math to show my daughter the advantages of driving a 'paid for' used car rather than leasing a new car. We figured that in just 9 years she would be at least $25,000 ahead if she drove a used car. Logic or problem solving skills?... When I solve problems I love to use math because it usually gives me a clear cut answer. You've heard the saying "the numbers don't lie" and that is the truth. Still, sometimes there is more than just the numbers that need to be considered, for example, do we take the higher paying job if it means more stress and sacrificing time with family? - etc. So you can't rely just on the math, but it does help you weigh your options. I always check the numbers. I am really into SIMPLE budgeting. Budgeting seems to scare a lot of people off, but it really does not have to be that hard and it is so important. Without a money plan or a budget, people can get into major financial problems, which in turn can lead to a lot of stress and pain. Regardless of your career, budgeting and saving money are vital (math intensive) skills to have. If I could give advice to young college students I would encourage them to learn math and start budgeting and saving now!! Emlyn, Homemaker & Part-time curatorial assistant at Museum of Natural History Math in current profession… I use math for both professions. At the museum, we do a significant amount of measuring in taking care of our objects. We use both metric and American measurement types, and so must have a basic understanding of the comparison between the two. We also use computer math functions, such as categorical numbering using our database as well as Excel spreadsheet columns. And yes, we even use math to fill in the correct hours on our timecards! Skills that were useful or wish I had paid more attention to… The most useful information from math classes is basic math (addition, subtraction, multiplication, division) WITHOUT using a calculator. I wish I had paid better attention so I could do all of that in my head quickly! Math to maintain your home/personal life?... At home I use math every day. I am a mom of two, a 2-year-old and a 4-year-old. Meals are often mini fights about fair portions... I use division and reasoning to make sure everyone gets their sandwich cut into 4 pieces. I use measurement and math when making recipes, even basic macaroni and cheese! Budgeting and finances require addition and subtraction all the time. Balancing the checkbook to make sure we haven't overspent on our debit card and matching that against the bank's records is a very important skill. Logic or problem solving skills?... Logic and problem solving get used in my professional life at the museum when we are trying to find the best places to store and display objects. Measurements are crucial as well as logic: two big baskets probably won't fit on the same shelf! At home, logic and problem solving are again crucial, as with the even sandwiches. :) Kyle, Lawyer Math in current profession… I am self employed and I use math with my bookkeeping, profit and loss, cost vs time evaluation, interest calulations, etc. Also, I always have to problem solve and have logical applications. Nadya, Marketing Math in current profession… Marketing is actually full of math. I've used math to compute how much food to order for an event or how many fun give-a-ways to order. How much money we have in the budget and where it can be best used, how many visitors to the website actually looked around, how many people read emails and responded, effectiveness of promotions, how many extra newsletters to order to account for mailing run off (basically the errors the mailhouse might make), how many bags of candy to buy to hand out at Halloween in the branches and much, much more. Marketing involves a lot more than just making things pretty or fun! Math to maintain your home/personal life?... Definitely used in budgeting. Also used in figuring out which can of green beans is the best price or other groceries (figuring per oz prices), estimating how much my grocery bill should be while shopping so I can tell if an error is made when being rung up. Logic or problem solving skills?... I do use logic quite a bit but I'm not sure if I learned it in math class. Occasionally proofs have come in handy though when trying to illustrate to other people that a process will work. Danny, Neuroscience Researcher (Graduate Student) Math in current profession… I use algebra to make solutions needed for my experiments. I must calculate the mass of the drug that I want in my solution for a given volume of saline to reach the concentration that I need for my experiment. I also use algebra/arithmetic to convert different measures of the concentration of a solution to discuss the experiment with other scientists. When administering drugs to live animals, dose of the drug is described as mass of drug per mass of the animal (e.g. 5 mg/kg). However, if we are doing experiments in brain slices, we express the dosage of a drug as its molarity (20 µM). I also do a lot of statistics on the results of the experiments. I write matlab scripts to analyze my data, which could basically be written as long, complicated, algebraic equations. I also use built-in matlab functions to do a statistical multivariate, repeated measure analysis of variance (ANOVA) to determine if the results for different experimental conditions are different enough to be statistically significant, which we can then report in scientific journals. Skills that were useful or wish I had paid more attention to… I paid pretty good attention in my math classes, since I liked math. My problem has mostly been forgetting certain concepts since I haven't been in a math class in over 10 years. A solid understanding of algebra is probably the most important for everyday life. Most problems can be solved with back-of-the-napkin type math using algebra and simple arithmetic. Math to maintain your home/personal life?... Mostly I would say this consists of just some quick mental arithmetic to make sure some company didn't make a mistake and charge me too much or something. Logic or problem solving skills?... Logic and problem solving are hugely helpful in my job as a research scientist. Without them you cannot do science. Janice, Nurse (Medical Intensive Care Unit) Math in current profession… I use math all the time in my job as a nurse: estimate when the IV bag will be finished; basic adding and subtractding for intake and out put totals (24 hr fluid balance); I use algebra to calculate drug administration, setting IV pumps, etc. Everything is in metric system so it is important to understand that. Math to maintain your home/personal life?... In my everyday life, I use it to do my checkbook/bank statement, determine price of something I am buying (if it is a good price or not) and how much I will save using coupons, 20% off, etc. I also calculate if I am getting a good gas mileage with my car (miles/gallon of gas). I use measurements in cooking and adjusting recipes i.e. making half the recipe or doubling, etc. I use it when buying material for crafts/sewing. The list goes on! I am sure I use it even more than that! Jessica, Office Manager & Landscape Designer Math in current profession… I surprisingly use math a lot, which is a surprise for me because it was a real struggle for me through school. I use it as an office manager entering the finances to the theatre I work for and keeping the books up to date. I also use it as a landscape designer; square footage, volume for things like compost or gravel, dimensions, etc. Kevin, Petroleum Engineer Math in current profession… I use algebra for Multivariate statistics, I use geometry for wellbore calculations, and I use calculus for finding the rate of change, inflection points, and completed work (area under the curve). Skills that were useful or wish I had paid more attention to… I wish I paid closer attention to statistics, the most important skill learned is solving algebraic equations. Solutions to real world problems come about by solving equations and have contributed to my project at Shell. Christine, Statistician & Medical Researcher (Graduate Student) Math in current profession… I do research on statistical methods to infer biological networks. For example, inside of each of our cells, there are reactions occurring constantly. In a diseased state, this network of reactions is altered in ways that lead to a loss of cellular function. Magnetic resonance spectroscopy can be used to profile tissue samples, and from this we can identify which molecules are present and estimate their concentrations. We can then use statistical methods to infer the reaction network by looking at the covariance of the concentrations of the molecules. This is a difficult problem since the data is noisy (i.e. there is random variation in the measurement), the networks are complex, and we often have small sample sizes. We are developing statistical methods that allow us to incorporate prior knowledge on the degree of connectivity of the network and known chemical reactions. These techniques can improve the reliability of our inference. The resulting identification of the reactions under diseased conditions can increase scientific understanding of the mechanisms of disease and also guide the development of future treatments. Skills that were useful or wish I had paid more attention to… I still wish that I could explain mathematical concepts better! It's really nice to be able to help out other students, and you can feel sure that you understand something once you are able to explain it someone else. Right now, I work a lot with medical researchers, and I often have to explain statistical methods to them. I am still trying to get better at how I communicate these ideas. Erin, Swimming Coach Math in current profession… add up the amount of yardage that my swimmers are swimming and calculate how much yardage is needed for a mile. Math to maintain your home/personal life?... balance a checkbook. Ben, TV Production Math in current profession… I do base 60 Time math daily (e.g. adding and subtracting the number of minutes in an hour), subtracting one time from another to find the difference between the two. Also I use math in trying to calculate time cues for production, which is tough since I have to do it backwards in real time. Alicia, Veterinarian Math in current profession… Yes, I use math everyday in my profession. I use basic algebra and stoichiometry for fluid rate calculation, drug dosage calculations, basic unit conversions, etc. I love trigonometry and know I will get to use some of it in when I go into my residency for sports med/rehab and have to study/use biomechanics, basic trig is also used a lot in orthopedic surgeries. Skills that were useful or wish I had paid more attention to… I wish I understood physics better which has a lot of math in it because then I would understand ultrasounds better and be able to use my machine better. The most useful skill I learned in math class is basic math, like fractions. I am constantly surprised by how many people can’t understand that ¼ of a tablet is the same as 0.25 of a tablet… Math to maintain your home/personal life?... Yes, making a monthly budget and sticking to it has been important given the amount of educational debt load I have. Also, calculating how much interest accrues in my loans is important to know each year. Andrea, Walmart Customer Service Manager and Supervisor Math in current profession… I use it every day to calculate the ad matching we do. For example say you are buying 10.29 pounds of oranges from rancho market and their ad is 8 pounds for .99 cents you need to calculate how their rates compare with Walmart’s. At Walmart we sell oranges by unit, so we have to figure out the price per orange because not all our produce is by the pound. The reason I am giving you this example is I hear all day long from the younger cashiers that they hate math and I don't know how to figure it out. I show them all how to do this so they not only can handle the ad matches and so they don't feel stupid. I sympathize with a lot of them because math was not my best subject. I also use math in my profession to figure out the difference I need to give customers as a credit if they paid a higher rate of tax than in Utah. Skills that were useful or wish I had paid more attention to… Basic math, and math that you need in the real world, and how to do this in your head without a calculator. And that if you have struggled in the past with math that you can get better, for example, I used to struggle with arithmetic, but now I can do it in my head. Find someone to help you break it down so that you can understand. I am very grateful that I had teachers throughout my schooling who took time to do this for me. Math to maintain your home/personal life?... I use it to figure out how to cut fabric, to double check the machine on prices, and for gas mileage. The desire to learn and gain knowledge (being able to solve the math problems like discounts, or being able to use the computer, etc) has promoted me from a temporary employee to a manager. Jim, Web Developer Math in current profession… I definitely use math in programming. Simple arithmetic is everywhere in programming, but also many of the concepts of programming are based on math. For example, a key concept of programming is understanding variables (assigning labels to data), which is the basis for algebra. But most of the actual work of programming is more like solving proofs in geometry, at least conceptually. The idea of tying together a series of steps to produce a specific result is exactly the work programmers do. Skills that were useful or wish I had paid more attention to… I wish I had taken a statistics class. I think some of the concepts and algorithms would be useful, if for no other reason than to have a clearer idea on how to express what sort of algorithm I am looking for. The most useful thing from math for me was discovering that I enjoyed doing geometry proofs. I think "math" is a huge field and most people actually like certain parts of it, even if they "hate math". You don't have to like everything, but I think everyone should try to expose themselves to as many fields of mathematics as they can. Math to maintain your home/personal life?... I do use math for finances, but not in a straightforward way. I think having a strong understanding of probabilities and percentages can be very helpful for understanding things, such as the fact that if you buy a stock and it loses 50% of its value, the stock will have to gain 100% to be worth its original amount. Logic or problem solving skills?... All the time. I think developing good problem solving skills is useful in almost every aspect of life and business, but the real key is being able to identify and translate real world problems into solvable problems. I always hated solving word problems in math, but I think that's an important skill to master. Business and Chemistry English Instructor Chemistry, B.S. Computational Biologist Greenhouse manager Health Insurance Agent Homemaker, Part-time curatorial assistant at Museum of Natural History Neuroscience Researcher Nurse (Medical Intensive Care Nurse) Office Manager & Landscape Designer Petroleum Engineer Statistician, Medical Researcher Swimming Coach TV production Walmart Customer Service Manager/Supervisor Web Developer This free website was made using Yola. No HTML skills required. Build your website in minutes. Go to www.yola.com and sign up today! Make a free website with Yola
398067b420c8a637	The key to making useful nanoelectronic devices from graphene is to first understand, and then be able to control, the flow of electrons through tiny snippets of the material. The absence of a bandgap in pure graphene means that although its electrical conductivity is the highest of any material bar none, it is nearly impossible to shut it off completely. Researchers at MIT and elsewhere have recently figured out not only how to build precisely defined bandgaps into composites of graphene and boron nitride, but they have also uncovered the deeper electronic structure of the material and found that it contains some of the most fascinating physics known. What the MIT researchers basically did was take single layers of hexagonal graphene and stack them up against single layers of hexagonal boron nitride. The key is to be able to control the degree of alignment between the layers, and therefore the ease with which electrons can hop and slide from one layer to the next. In order to coax the graphene-boron honeycomb into exposing its hidden behaviors, some additional outside influence needs to be imposed. One effective way to do this is to chill everything down to within a fraction of absolute zero and add a massive, out-of-plane magnetic field. When you do that, it becomes possible to see and measure a rarefied theoretical beast known as ‘Hofstadler’s Butterfly’. This fractal pattern emerges when the electronic energy levels of the material are plotted against the applied magnetic field. Originally proposed by Douglass Hofstadler in 1976, his signature butterfly has only recently been experimentally observed in the lab. Another way to state more precisely what these ‘energy levels’ really are is to define them as the observable property of the wave function, or more specifically, as the plotted electron density. In the present case, the researchers used fields up to 45 Tesla that were available at the National High Magnetic Field Laboratory in Tallahassee. For comparison, that’s about five times as large as the most powerful MRI machine in common use. As a graphical representation of the fractal structure of the energy spectrum for electrons in a magnetic field, the butterfly structure has an intrinsic ‘self-similarity’ which can be quantified by a parameter known as the fractal dimension. Intriguingly, just a short time ago, other researchers have found evidence for this exact kind of ‘quantum critical’ semiconductor behavior in many proteins commonly used for various enzymatic functions in cells. They were even able to calculate this fractal dimension for several of them. We won’t delve too much further into how it is figured (there are different ways), other than to say that it is a general measure of complexity we can casually define for our purposes here as the ratio of the change in detail to the change in scale. When the graphene-boron honeycombs are stacked out of alignment, they create something known as a ‘Moire pattern’. When the angle of rotation between the layers was low, the material showed an insulating behavior, and when high, the conducting state was seen. Butterflies and Moire patterns aside, what really has the researchers excited is some of the other physical effects they were able get from graphene. The researchers previously demonstrated something known as a ‘quantum spin Hall state’ when they applied a magnetic field with an in-plane orientation. The field forced electrons at the edge of the material to move in opposite directions, and in separate lanes, according their spin. In contrast to the unidirectional current flow of electrons in a regular metal, a material that behaves as a ‘topological insulator’ would be useful in several spintronic applications. If all that terminology isn’t enough physics for you, there’s more. While the famous Schrödinger equation (which gives the wave functions mentioned above) describes the behavior of electrons in most materials, electron behavior in graphene is ‘ultrarelativistic’ and therefore is better described using the lesser-known Dirac equation. Compared with normal materials, where electron velocity is subrelativistic, electrons in graphene composites configured with just the right alignment can flow at significantly greater speeds, and need to be described with a different formalism. Furthermore, when many layers of graphene are properly stacked together (with associated greater strength), they can still show the high conduction seen in a single layer. There are still many details to be worked out before we can put graphene-based semiconductor devices into mainstream use. However, where before we couldn’t say much more than that electron flow within a layer is expected to be significantly different from that across layers, we can now point to a much richer physics of graphene.
bb1d0b50782218ad	Stochastic differential equation From Wikipedia, the free encyclopedia Jump to: navigation, search A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is itself a stochastic process. SDEs are used to model diverse phenomena such as fluctuating stock prices or physical systems subject to thermal fluctuations. Typically, SDEs incorporate random white noise which can be thought of as the derivative of Brownian motion (or the Wiener process); however, it should be mentioned that other types of random fluctuations are possible, such as jump processes. The earliest work on SDEs was done to describe Brownian motion in Einstein's famous paper, and at the same time by Smoluchowski. However, one of the earlier works related to Brownian motion is credited to Bachelier (1900) in his thesis 'Theory of Speculation'. This work was followed upon by Langevin. Later Itō and Stratonovich put SDEs on more solid mathematical footing. In physical science, SDEs are usually written as Langevin equations. These are sometimes confusingly called "the Langevin equation" even though there are many possible forms. These consist of an ordinary differential equation containing a deterministic part and an additional random white noise term. A second form is the Smoluchowski equation and, more generally, the Fokker-Planck equation. These are partial differential equations that describe the time evolution of probability distribution functions. The third form is the stochastic differential equation that is used most frequently in mathematics and quantitative finance (see below). This is similar to the Langevin form, but it is usually written in differential form. SDEs come in two varieties, corresponding to two versions of stochastic calculus. Stochastic calculus[edit] Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Ito stochastic calculus and the Stratonovich stochastic calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003) and conveniently, one can readily convert an Ito SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down. Numerical solutions[edit] Numerical solution of stochastic differential equations and especially stochastic partial differential equations is a young field relatively speaking. Almost all algorithms that are used for the solution of ordinary differential equations will work very poorly for SDEs, having very poor numerical convergence. A textbook describing many different algorithms is Kloeden & Platen (1995). Methods include the Euler–Maruyama method, Milstein method and Runge–Kutta method (SDE). Use in physics[edit] In physics, SDEs are typically written in the Langevin form and referred to as "the Langevin equation." For example, a general coupled set of first-order SDEs is often written in the form: \dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum_{m=1}^ng_i^m(\mathbf{x})\eta_m(t),\, where \mathbf{x}=\{x_i\|1\le i\le k\} is the set of unknowns, the f_i and g_i are arbitrary functions and the \eta_m are random functions of time, often referred to as "noise terms". This form is usually usable because there are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. If the g_i are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case where : g(x) \propto x. Additive noise is the simpler of the two cases; in that situation the correct solution can often be found using ordinary calculus and in particular the ordinary chain rule of calculus. However, in the case of multiplicative noise, the Langevin equation is not a well-defined entity on its own, and it must be specified whether the Langevin equation should be interpreted as an Ito SDE or a Stratonovich SDE. In physics, the main method of solution is to find the probability distribution function as a function of time using the equivalent Fokker-Planck equation (FPE). The Fokker-Planck equation is a deterministic partial differential equation. It tells how the probability distribution function evolves in time similarly to how the Schrödinger equation gives the time evolution of the quantum wave function or the diffusion equation gives the time evolution of chemical concentration. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques include the path integration that draws on the analogy between statistical physics and quantum mechanics (for example, the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables) or by writing down ordinary differential equations for the statistical moments of the probability distribution function.[citation needed] Use in probability and mathematical finance[edit] The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different. This notation makes the exotic nature of the random function of time \eta_m in the physics formulation more explicit. It is also the notation used in publications on numerical methods for solving stochastic differential equations. In strict mathematical terms, \eta_m cannot be chosen as an ordinary function, but only as a generalized function. The mathematical formulation treats this complication with less ambiguity than the physics formulation. A typical equation is of the form \mathrm{d} X_t = \mu(X_t,t)\, \mathrm{d} t + \sigma(X_t,t)\, \mathrm{d} B_t , where B denotes a Wiener process (Standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding integral equation X_{t+s} - X_{t} = \int_t^{t+s} \mu(X_u,u) \mathrm{d} u + \int_t^{t+s} \sigma(X_u,u)\, \mathrm{d} B_u . The equation above characterizes the behavior of the continuous time stochastic process Xt as the sum of an ordinary Lebesgue integral and an Itō integral. A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length δ the stochastic process Xt changes its value by an amount that is normally distributed with expectation μ(Xttδ and variance σ(Xtt)² δ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function μ is referred to as the drift coefficient, while σ is called the diffusion coefficient. The stochastic process Xt is called a diffusion process, and is usually a Markov process. The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution. Both require the existence of a process Xt that solves the integral equation version of the SDE. The difference between the two lies in the underlying probability space (Ω, F, Pr). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. An important example is the equation for geometric Brownian motion \mathrm{d} X_t = \mu X_t \, \mathrm{d} t + \sigma X_t \, \mathrm{d} B_t. which is the equation for the dynamics of the price of a stock in the Black Scholes options pricing model of financial mathematics. There are also more general stochastic differential equations where the coefficients μ and σ depend not only on the present value of the process Xt, but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process, X, is not a Markov process, and it is called an Itō process and not a diffusion process. When the coefficients depends only on present and past values of X, the defining equation is called a stochastic delay differential equation. Existence and uniqueness of solutions[edit] As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itō SDEs taking values in n-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2). Let T > 0, and let \mu : \mathbb{R}^{n} \times [0, T] \to \mathbb{R}^{n}; \sigma : \mathbb{R}^{n} \times [0, T] \to \mathbb{R}^{n \times m}; be measurable functions for which there exist constants C and D such that \big\| \mu (x, t) \big\| + \big\| \sigma (x, t) \big\| \leq C \big( 1 + \| x \| \big); \big\| \mu (x, t) - \mu (y, t) \big\| + \big\| \sigma (x, t) - \sigma (y, t) \big\| \leq D \| x - y \|; for all t ∈ [0, T] and all x and y ∈ Rn, where \| \sigma \|^{2} = \sum_{i, j = 1}^{n} \| \sigma_{ij} \|^{2}. Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment: \mathbb{E} \big[ \| Z \|^{2} \big] < + \infty. Then the stochastic differential equation/initial value problem \mathrm{d} X_{t} = \mu (X_{t}, t) \, \mathrm{d} t + \sigma (X_{t}, t) \, \mathrm{d} B_{t} \mbox{ for } t \in [0, T]; X_{0} = Z; has a Pr-almost surely unique t-continuous solution (tω) \|→ Xt(ω) such that X is adapted to the filtration FtZ generated by Z and Bs, s ≤ t, and \mathbb{E} \left[ \int_{0}^{T} \| X_{t} \|^{2} \, \mathrm{d} t \right] < + \infty. See also[edit] • Adomian, George (1983). Stochastic systems. Mathematics in Science and Engineering (169). Orlando, FL: Academic Press Inc. • Adomian, George (1986). Nonlinear stochastic operator equations. Orlando, FL: Academic Press Inc. • Adomian, George (1989). Nonlinear stochastic systems theory and applications to physics. Mathematics and its Applications (46). Dordrecht: Kluwer Academic Publishers Group. • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1. • Teugels, J. and Sund B. (eds.) (2004). Encyclopedia of Actuarial Science. Chichester: Wiley. pp. 523–527. • C. W. Gardiner (2004). Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer. p. 415. • Thomas Mikosch (1998). Elementary Stochastic Calculus: with Finance in View. Singapore: World Scientific Publishing. p. 212. ISBN 981-02-3543-7. • Seifedine Kadry, (2007). A Solution of Linear Stochastic Differential Equation. USA: WSEAS TRANSACTIONS on MATHEMATICS, April 2007. p. 618. ISSN 1109-2769. • Bachelier, L., (1900). Théorie de la speculation (in French), PhD Thesis. NUMDAM: In English in 1971 book 'The Random Character of the Stock Market' Eds. P.H. Cootner. • P.E. Kloeden and E. Platen, (1995). Numerical Solution of Stochastic Differential Equations,. Springer,.
92b0aa1bad597320	Take the 2-minute tour × Consider standard quantum mechanics, but forget about the collapse of the wavefunction. Instead, use decoherence through interaction with the environment to bring the evolving quantum state into an eigenstate (rspt arbitrarily close by). Question: Can this theory be fundamentally deterministic? If one takes into account that the variables of the environment are not known, then the evolution is of course 'undetermined' in a probabilistic sense, but that isn't the question. Question is if fundamentally quantum mechanics with environmentally induced decoherence can be deterministic. Note that I'm not saying it has to be. I might be mistaken, but it seems to me decoherence could be followed by an actual non-deterministic process still, so the decoherence alone doesn't settle the question of determinism or non-determinism. Question is if one still needs a non-deterministic ingredient? Update: Please note that I asked whether the evolution can fundamentally be deterministic, or whether it has to be non-deterministic. It is clear to me that for all practical purposes it will appear non-deterministic. Note also that my question does not refer to the prepared state after tracing out the environmental degrees of freedom, but to the full evolution of system and environment. Does one need a non-deterministic ingredient to reproduce quantum mechanics, or can it with the help of decoherence be only apparently non-deterministic yet fundamentally deterministic? share\|improve this question What definition of determinism are you using? – user1708 Jan 21 '11 at 14:45 If you know the state at time t_1, you can in principle calculate everything that's going to happen, or did happen at time t_2. – WIMP Jan 23 '11 at 10:32 You need to rephrase your question if both myself and Matt have misunderstood your intention. Are you now asking if you can deterministically collapse to a particular eigenstate? The answer to that is no, because it violates linearity. – Joe Fitzsimons Jan 23 '11 at 10:56 @ Joe: I just updated the question, hope it's clearer now? You don't need to collapse exactly to a particular eigenstate, just arbitrarily close by (as I already stated in my original question). – WIMP Jan 23 '11 at 11:15 I've posted an updated answer to answer the question as I now understand it. – Joe Fitzsimons Jan 23 '11 at 12:22 5 Answers 5 up vote 3 down vote accepted Short answer to "Can this theory be fundamentally deterministic?": No. Decoherence is the diagonalization of the density matrix in a preferred basis, with the off-diagonals vanishing at late times. Since you can get the same final diagonal matrix from several possible initial pure states of the system under consideration, there's a necessary loss of information and irreversibility. (I'm guessing this is what you meant by non-deterministic) A bit more detail: Decoherence proceeds by the rapid establishment of entanglement-induced correlations between the system and the infinite degrees of freedom of the decohering environment. The second law prevents this process from being reversible (since S has to always increase, and S is zero for the pure state, while it is greater than zero for the decohered mixed state). If you take the second law to be fundamental, then the non-determinism here is fundamental too. UPDATE: The updated question now refers to the full evolution of system + environment, in other words, the entire universe. Since there's nothing else for the universe to entangle with, it will remain in a pure state and evolve deterministically for ever if it always was in a pure state. I however don't know if the universe is in a pure state or a mixed state. Anyone? share\|improve this answer I would answer the same thing, so plus one point. ;-) – Luboš Motl Jan 21 '11 at 19:29 Another way of saying this is the process is not unitary. – Lawrence B. Crowell Jan 22 '11 at 3:34 This isn't technically correct. Decoherence can be the result of an entirely deterministic (and unitary process), since you only care about the reduced density matrix for the system in question. Open quantum systems -do- decohere, even though the entire process is unitary. The larger wavefunction of the entire system is still pure, but the state of the local system becomes mixed. – Joe Fitzsimons Jan 22 '11 at 6:26 My understanding is that the evolution turns non-unitary once you've traced out the environmental degrees of freedom. That's not what I'm asking for. Also, time evolution doesn't need to be unitary to be deterministic. – WIMP Jan 23 '11 at 10:34 I concur with the criticisms of this answer: you are making the mistake of only considering the subsystem in question, not the total system. The subsystem loses information, yes, but it's lost to the total system, which may well be reversible as a whole. – Greg Graviton Jan 23 '11 at 14:19 I don't disagree with the other answers, but I want to try to use different words: Evolution of a quantum state is deterministic in the sense that it is given by the Hamiltonian. Quantum mechanics doesn't need anything beyond unitary evolution. So in that sense, the answer is deterministic. However, decoherence means that eventually a quantum state may evolve into a superposition of very nearly orthogonal states, which for large enough systems will resemble to arbitrarily high precision the answer you might get from assuming that there is a nondeterministic, nonunitary process of "wavefunction collapse." For all practical purposes, since we are large classical observers, we can observe only one such nearly-orthogonal combination, and the interference with other possible outcomes will be unmeasurably small. So, in this sense the outcome is "nondeterministic." If this seems counterintuitive to you, consider something analogous about classical statistical mechanics and thermodynamics. If I start with a collection of gas molecules all bunched up in one corner of the room, it is a very atypical (low-entropy) state under any natural coarse-graining of phase space. Now, by entirely reversible interactions, it can become a typical (high-entropy) state with molecules scattered all over the room. This process appears to have lost information, in the sense that I would have to do many, many difficult measurements to ascertain that a short time before this was a very special state indeed. But really, the underlying physics is deterministic, so in principle the final state remembers where it came from, although for all practical purposes if I tried to evolve it backwards I would never discover the right answer. (To be clear, I'm not claiming a very sharp analogy here. But I'm saying that the notion that microscopically deterministic evolution can be consistent with apparent or "for all practical purposes" loss of determinism in real observations is something that might be more intuitive in this context.) share\|improve this answer Good answer! – Joe Fitzsimons Jan 22 '11 at 6:28 It doesn't seem counterintuitive to me, but that wasn't my question. I'm not asking "for all practical purposes." I know that for all practical purposes it will appear non-deterministic in the sense that we won't be able to predict what's going to happen -- too many variables in the environment. I'm asking if the time evolution of the system is fundamentally 'in principle' deterministic. That of course can only be before you've averaged over the variables of the environment. – WIMP Jan 23 '11 at 10:41 Then I would say yes, it is "in principle" deterministic; the Hamiltonian specifies how everything evolves. Most of my answer was just trying to explain how to reconcile this with the observation that quantum effects look nondeterministic. – Matt Reece Jan 23 '11 at 15:41 Thanks. Yes, that was also my understanding. – WIMP Jan 24 '11 at 7:40 +1 for thermodynamics mention, I think some day decoherence and 2nd law will shake hands – HDE Jan 19 '12 at 0:16 UPDATE: It seems that we have not been answering the question WIMP intended. Here is an updated answer to deal with what I now understand to be the question: Given any unknown quantum state $\|\psi\rangle$, can there be any deterministic process which will make it collapse onto a particular state $\|\phi \rangle$, if $\|\phi \rangle\langle \psi\| \neq 0$? The answer to this question is no, because it violates the linearity of quantum mechanics, allowing us to distinguish between non-orthogonal states. This is trivial, because states orthogonal to $\|\phi \rangle$ will have zero probability of collapsing onto it. This may not seem like a big deal, but it turns out that linearity is fundemantal to quantum mechanics on many levels. If we remove this constraint, then entanglement can be used to signal, and hence create problems with causality. No signalling seems one of the most fundamental features of physics, showing up in many independent theories (electromag, quantum mechanics, relativity, etc.). To see how this can be done, consider an entangled state $\frac{1}{\sqrt{2}}(\|01\rangle - \|10\rangle)$. This is the anti-symmetric state: for any basis $\sigma$ a measurement resulting in outcome $m$ will leave the other qubit in the opposite eigenstate of $\sigma$. Thus, if you could deterministically collapse onto the state $\|0\rangle$ then you can be sure the your half of the EPR pair was not left in state $\|1\rangle$ after the measurement on the other half. So, for Alice to communicate with Bob, she need only choose to measure in the $X$ or $Z$ basis. Measuring in $X$ will mean Bob receives the output $\|0\rangle$ with probability 1, where as measuring in $Z$ will return result $\|1\rangle$ with probability $\frac{1}{2}$. Although this is probabilistic, you can repeat the process arbitrarily many times to get exponentially close to perfect communication. This instantaneous communication breaks causality. If you allow all states to collapse to the target state, then the only solution is a channel which swaps the state with another ancilla system. Systems which can perform such deterministic collapse can always be used to signal, as well as allowing all sorts of additional weirdness like efficient solutions to PSPACE-complete problems in computation and time travel. As a result, this is totally impossible within the current framework of physical theories, and there are very substantial reasons to believe that it is a feature of any physical theory that is valid in our world. The answer is no, if by deterministic you mean possessing a local hidden variable interpretation. This follows directly from the observed violations of Bell's inequality, which ever interpretation of quantum mechanics you choose (what you are referring to is known as the Everett interpretation of quantum mechanics). Bell's inequality works as follows: Given to possible local measurement operators ($A_i$ and $B_i$) at each of two localions $i \in \{1,2\}$, what is the maximum value of the expectation value of $\langle A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2\rangle$. What Bell showed was that this can take on a value of at most 2 for any local hidden variables theory. However quantum mechanics allows it to take on values up to $2\sqrt{2}$, and many experiments have recorded violations of this inequality, showing values in the range $2 < v \leq 2\sqrt{2}$. This essentially rules out a local hidden variable model. If, however, you mean can the unitary interaction of two particles give rise to decoherence, then the answer is yes, as follows: Imagine two particles initially in the state $1/\sqrt{2}(\|0\rangle + \|1\rangle)$. Now imagine they interact via an Ising interaction. After a certain time, they will be in the joint state $1/2(\|00\rangle - \|10\rangle - \|01\rangle + \|11\rangle)$. This is still a pure state, and so no decoherence has occurred. However, imagine one of these particles moves off far away (into the environment). If we only have access to one of these particles, then its reduced density matrix will be $1/2(\|0\rangle \langle 0\|+\|1\rangle \langle 1\|)$, which is simply a classical random distribution over the two orthogonal states, the same as would occur do to a collapse of the wavefunction. share\|improve this answer Thanks for the reply. Even though you've correctly interpreted the meaning of deterministic, you haven't answered my question. You're saying the theory can't be deterministic because that would be in conflict with experimental tests on Bell's inequality. That's not correct. The theory could also be non-local instead, you just need to violate one of the assumptions to prove the theorem. – WIMP Jan 23 '11 at 10:38 @WIMP: The first line of my answer reads: "if by deterministic you mean possessing a local hidden variable interpretation". Certainly global hidden variables can be made to work, but then they always can. – Joe Fitzsimons Jan 23 '11 at 10:42 Yes, I know. But that non-local hidden variable theories are not excluded by Bell's theorem isn't an answer to my question. If you want to pursue that line of thought, the question would then be whether decoherence, rspt the entanglement with the environment is a sort of non-locality that spoils the assumptions to Bell's inequality and thus, isn't excluded by experiment. Also, I was wondering about the possibility of deterministic evolution from a theoretical rather than an experimental point of view. – WIMP Jan 23 '11 at 10:53 @WIMP: Can you please revise the question to make it clear exactly what you are asking? It's currently not clear either from the question or subsequent comments. – Joe Fitzsimons Jan 23 '11 at 10:59 @ Joe: Thanks for the update. Two things: First, you don't need to collapse exactly into \|0>, you just need to get close enough so it would 'for all practical purposes' appear to be an eigenstate, see question & update. Second, that the evolution is fundamentally deterministic doesn't mean that you could in practice deterministically collapse. (Besides this, I didn't ask if it leads to instantaneous messaging as you seem to think. Also, instantaneous messaging doesn't necessarily cause problems with causality, but that's a different point.) – WIMP Jan 24 '11 at 9:13 Not sure if you've ever heard of the De Broglie-Bohm pilot wave interpretation of QM, but it is a fundamentally deterministic interpretation. share\|improve this answer Yes, I've heard of it. But my question wasn't if there is any deterministic interpretation of QM, but if the standard interpretation with decoherence instead of collapse can be deterministic. – WIMP Jan 24 '11 at 7:45 My answer is NO. There is a problem in many QM considerations that we have an isolated system obeying (deterministic!) Schrödinger equation that is a subject of mystical "measurements" that introduce non-deterministic behavior. But one can't make a measurement and leave system isolated (indeed saying that system is isolated is always an approximation in QM, and much heavier than in CM). In fact measurement is an act of introducing interactions with measuring apparatus, measurer, coffee measurer drinks, ... -- so the measurement result can be in theory calculated, but would involve inaccessible and enormous amounts of information. This makes it practically non-deterministic, but fundamentally it is no better than classical chaos. share\|improve this answer Actually, you are arguing that the answer is yes, rather than no. I haven't asked whether it is practically non-deterministic, but whether it can be fundamentally deterministic though appear non-deterministic (much like chaos indeed). – WIMP Jan 24 '11 at 7:48 My answer to "Does decoherence need non-determinism?" is No (-; – mbq Jan 24 '11 at 10:19 Your Answer
4ffa5b715d603f5e	Sunday, January 20, 2008 What's Wrong About Physics Today? All of It! Via my new favorite website,, I've found a treasure trove of physics denial. Typically we'll see relativity denial or abuse of quantum mechanics to serve whatever ideology you choose. But unless one is going for Neal Adams level crankery, one keeps the denial to a few topics. Not this guy. For pretty much any physics concept a layperson would know, and a lot they wouldn't, this guy has a refutation. He covers all the common ones, of course. Relativity, quantum mechanics, the big bang, etc. But we also see a few surprises. For instance, the author attempts to explain away Maxwell's equations, lasers, the Boltzmann distribution, even energy conservation. How does he do? Well, since the site is too long for an exhaustive analysis, I'll pick apart a few examples. Curved Space: The concept of a 'curved space', which is essential for present cosmological models, is logically flawed because space can only be defined by the distance between two objects, which is however by definition always given by a straight line. Ah, circular reasoning at its best. Why can't curved space exist? Because distances between objects are straight lines. Why are distances between objects straight lines? Because space is flat, not curved. The shortest distances between objects—geodesics—are straight lines only in Euclidean geometry. Since we can measure that the shortest path for light is not a straight line when passing a massive object, we know that matter curves space. Gravitation: Modern theories of gravitation assume that the gravitational force between two masses is not an instantaneous interaction but is communicated by field quanta (gravitons) moving with the speed of light. However, this model can be shown to result in different forces in different inertial systems and contradicts therefore the definition of a force. [Emphasis mine] Sorry, guy. You need to show your work. Schrödinger Equation: Present day Quantum Theory has been developed from the original observation that radiation emitted by an atom appears in the form of discrete spectral lines. The Schrodinger Equation could reproduce this theoretically by postulating a wave equation for the atom which yields only certain energy values as a solution. The associated wave functions are continuous functions in space and therewith do not allow to exactly specify the location of atomic electrons. This has led to the interpretation that electrons as such do not exist as localized particles within the atom but only as some diffuse 'cloud' or even only as mathematical objects. This assumption however is an unallowed generalization of the Schrodinger Equation which strictly makes sense only if applied to radiative transitions. The actual (classical) location of the electron is completely unrelated to the wave functions of the radiative states (apart from a statistical connection) and any non-radiative physical effects (e.g. elastic atomic collisions) can therefore be calculated by the principles of classical physics without any logical contradictions. Many of the wider applications of the Schrödinger Equation are therefore completely unfounded and inadequate. Never mind that quantum mechanics as formulated by the Schrödinger equation accurately predicts the results of experiments. Never mind that the classical picture of the electron doesn't, and in fact quantum mechanics solved outstanding problems in the classical model. Never you mind those things, because they're not allowed. In fact, while the Schrödinger equation was written to reproduce the spectral emissions of hydrogen, it can be applied to innumerable systems. And why shouldn't it? It's a differential equation that describes how a quantum system evolves in time and space. One supplies a few parameters of the problem at hand and—ideally—solves for the behavior at all future times. It would be really strange if a completely different differential equation were required for every different situation. It's not inconceivable, though. Know how we tell? Experiments, that's how. And quantum mechanics, through the Schrödinger equation, is great at predicting the outcome of experiments. How well did he do? I'm not impressed. Obviously he fails as a scientist because he substitutes (crappy) argument for evidence. But he does pretty well as a crank; he's very ambitious to say the least. However, I've given you merely a slice of what's contained within. I encourage you to go to the site. Perhaps you'll find something I missed that makes all his arguments hang together. Ben said... Anything in there about bees not being able to fly? Flavin said... I don't think so. But he doesn't believe in the Bernoulli effect, so maybe that's related. Prazzie said... Great, I've nearly completed my collection of "Methods of Torture to Keep in Mind When People Annoy Me". This will go nicely next to my AIG bookmark and the RSS feed link to Dinesh D'Souza's blog. Thanks. *Edit - I missed a """. The horror!
ff28f8afa23fb521	Tuesday, December 22, 2009 Analytic continuation of 3F2, 4F3 and higher functions As of a recent commit, mpmath can evaluate the analytic continuation of the generalized hypergeometric function p+1Fp for any p. Previously 2F1 (the Gaussian hypergeometric function) was supported -- see earlier blog posts -- but not 3F2 and higher. This addition means that the generalized hypergeometric function is finally supported essentially everywhere where it is "well-posed" (in the sense that the series has nonzero radius of convergence), so it is a rather significant improvement. Unfortunately, the implementation is still not perfect, but I decided to commit the existing code since it is quite useful already (and long overdue). As proof of operation, I deliver plots of 3F2 and 4F3, requiring both \|z\| < 1 and \|z\| > 1: from mpmath import * f1 = lambda z: hyp3f2(1,2,3,4,5,z) f2 = lambda z: hyper([1,2,3,4],[5,6,7],z) plot([f1,f2], [-2,2]) A portrait of 3F2 restricted to the unit circle: plot(lambda x: hyp3f2(1,2,3,4,5,exp(2pijx)), [-1,1]) A numerical value of 5F4 with z on the unit circle, with general complex parameters: >>> mp.dps = 50 >>> print hyper([1+j,0.5,-2j,1,0.5+0.5j],[0.5j,0.25,-j,-1-j],-1) (-1.8419705729324526212110109087877199070037836117341 - High precision values of 3F2 at z = 1 and z = 1 + ε: >>> print hyp3f2(1,1,2,3.5,1.5,1) >>> print hyp3f2(1,1,2,3.5,1.5,'1.0001') (2.2626812356987790952469291649495098300894035980837 - A complex plot of 3F2: >>> mp.dps = 5 >>> cplot(lambda z: hyp3f2(2.5,3,4,1,2.25,z), [-2,2], [-2,2], ... points=50000) A bit of theoretical background: the hypergeometric series pFq has an infinite radius of convergence when p ≤ q, so it can in principle be evaluated then by adding sufficiently many terms at sufficiently high precision (although in practice asymptotic expansions must be used for large arguments, as mpmath does). In the balanced case when p = q+1, i.e. for 1F0(a,z), 2F1(a,b,c,z), 3F2(...), 4F3(...), ..., the series converges only when \|z\| < 1 (plus in some special instances). This is due to the fact that the hypergeometric function has a singularity (a pole or branch point, depending on parameters) at z = 1, in turn owing to the fact that the hypergeometric differential equation is singular at z = 1. (The reason that the p = q+1 and not p = q case is "balanced" is that there is an extra, implicit factorial in the hypergeometric series.) The main ingredient of the analytic continuation of p+1Fp is the inversion formula which replaces z with 1/z and thus handles \|z\| > 1. This was easy to implement -- the only complication is that integer parameters result in singular gamma factors, but the mechanisms to handle those automatically were already in place. There was no particular reason why I hadn't added that code already. The tricky part is the unit circle, and the close vicinity thereof, where neither series converges (quickly). I've been looking for good ways handle this, with mixed results. When I posed the problem on this blog several months back, a reader suggested this paper by Wolfgang Bühring which gives a series for the analytic continuation around z = 1. My finding from trying to implement it is that the rate of convergence of the series generally is poor, and therefore it is not immediately effective for computation. However, convergence acceleration with nsum improves the situation considerably in many cases. Some parameter combinations render the convergence acceleration useless, but even then, it can give a few correct digits, so it is better than nothing (although the implementation should probably warn the user when the result probably is inaccurate). I'm unfortunately not aware of any parameter transformations that would substantially improve convergence. The current implementation uses this method for 3F2 when \|z-1\| is small; it should work for 4F3 and higher too, but the series coefficients are much more complicated (involving multiply nested sums), so that's yet to be done. For the rest of the unit circle, I've settled for simply using convergence acceleration directly. This essentially just amounts to passing the hypergeometric series to nsum, which applies Richardson extrapolation and iterated Shanks transformations. The Shanks transformation is actually perfect for this -- it's almost tailor made for convergence acceleration of the balanced hypergeometric series -- and is able to sum the series even outside the circle of convergence, using just a few terms. This covers most of the unit circle -- the catch is just that the acceleration asymptotically deteriorates and ultimately becomes useless close to z = 1, so complementary method (such as the Bühring's is still required there). Mathematica seems to support unit-circle evaluation of hypergeometric functions quite well. Unfortunately, I don't know how it does it. According to Wolfram's Some Notes On Internal Implementation page, The hypergeometric functions use functional equations, stable recurrence relations, series expansions, asymptotic series and Padé approximants. Methods from NSum and NIntegrate are also sometimes used. This looks similar to what I'm doing -- high order Padé approximants and Mathematica's NSum should be equivalent to the nsum-based series acceleration in mpmath. But probably Mathematica employs some more tricks as well. I've also tested direct integration of the hypergeometric differential equation and of Mellin-Barnes integral representations, but these approaches don't seem to work much better (at least not without many improvments), and at best seem to be relatively slow. Thursday, September 10, 2009 Python floats and other unusual things spotted in mpmath New mpf implementation As an example, the following runs at 1.7x the speed: Fixed-precision arithmetic >>> from mp4 import mp, fp >>> mp.pretty = True <class 'mp4.ctx_mp.mpf'> <type 'complex'> Lambert W function >>> mp.lambertw(7.5) >>> mp.lambertw(3+4j) (1.28156180612378 + 0.533095222020971j) >>> fp.lambertw(7.5) >>> fp.lambertw(3+4j) >>> 1/timing(mp.lambertw, 7.5) >>> 1/timing(fp.lambertw, 7.5) Hurwitz zeta function >>> s=0.5+10j >>> mp.hurwitz(s); fp.hurwitz(s) (1.54489522029675 - 0.115336465271273j) >>> s=0.5+1000j >>> mp.hurwitz(s); fp.hurwitz(s) (0.356334367194396 + 0.931997831232994j) >>> s = 0.5+100000j >>> mp.hurwitz(s); fp.hurwitz(s) (1.07303201485775 + 5.7808485443635j) Hypergeometric functions >>> mp.hyp1f1(2.5, 1.2, -4.5) >>> fp.hyp1f1(2.5, 1.2, -4.5) >>> mp.hyp1f1(2.5, 1.2, -30.5) >>> fp.hyp1f1(2.5, 1.2, -30.5) >>> mp.hyp1f1(2.5, 1.2, -300.5) >>> mp.besselk(2.5, 7.5) >>> fp.besselk(2.5, 7.5) Numerical calculus >>> fp.cplot(fp.gamma, points=400000) Thursday, August 13, 2009 Released: mpmath 0.13 I released mpmath 0.13 today. See the announcement for details. Soon coming to a Sage or SymPy near you! I've blogged extensively about the new features that went into this version over the last two months. The short version of the changelog is "lots of special functions". For convenience, here a list of all posts: I also suggest a look at the documentation and particularly the special functions section. Tuesday, August 11, 2009 Torture testing special functions There is more to be done: • Some functions don't yet pass the torture tests General::ovfl: Overflow occurred in computation. General::ovfl: Overflow occurred in computation. General::ovfl: Overflow occurred in computation. General::stop: Further output of General::ovfl will be suppressed during this calculation. (... nothing for 5 minutes) Interrupt> a Out[47]= $Aborted And mpmath (instantaneously): >>> mp.dps = 15; mp.pretty = True Sunday, August 9, 2009 3D visualization of complex functions with matplotlib Principal-branch logarithmic gamma function: Gamma function: Imaginary part of Lambert W function, 0th branch: Riemann zeta function in the critical strip: Surface + wireframe plot: The phase of the Barnes G-function: from mpl_toolkits.mplot3d import Axes3D from matplotlib import cm import pylab import numpy as np import mpmath mpmath.dps = 5 # Use instead of arg for a continuous phase def arg2(x): return mpmath.sin(mpmath.arg(x)) f = lambda z: arg2(mpmath.cos(z)) fig = pylab.figure() ax = Axes3D(fig) X = np.arange(-5, 5, 0.125) Y = np.arange(-5, 5, 0.125) X, Y = np.meshgrid(X, Y) xn, yn = X.shape W = X0 for xk in range(xn): for yk in range(yn): w = float(f(z)) if w != w: raise ValueError W[xk,yk] = w except (ValueError, TypeError, ZeroDivisionError): # can handle special values here print xk, xn # can comment out one of these Tuesday, August 4, 2009 Coulomb wave functions and orthogonal polynomials Work has been a bit slow over the last two weeks, partially due to the fact that I was away during one of them. Nevertheless, I've been able to add a couple of more functions to mpmath, as described in the following post below. With these functions committed, I'm probably going to stop adding features and just do a new release as soon as possible (some documentation and general cleanup will be required first). Coulomb wave functions I have, due to a request, implemented the Coulomb wave functions (commit). The Coulomb wave functions are used in quantum physics; they solve the radial part of the Schrödinger equation for a particle in a 1/r potential. The versions in mpmath are fully general. They work not only for an arbitrarily large or small radius, but for complex values of all parameters. Some example evaluations: >>> mp.dps = 25; mp.pretty = True >>> coulombf(4, 2.5, 1000000) >>> coulombf(2+3j, 1+j, 100j) (2.161557009297068729111076e+37 + 1.217455850269101085622437e+39j) >>> coulombg(-3.5, 2, '1e-100') The definitions of the Coulomb wave functions for complex parameters are based mostly on this paper by N. Michel, which describes a special-purpose C++ implementation. I'm not aware of any other software that implements Coulomb wave functions, except for GSL which only supports limited domains in fixed precision. The Coulomb wave functions are numerically difficult to calculate: the canonical representation requires complex arithmetic, and the terms vary by many orders of magnitude even for small parameter values. Arbitrary-precision arithmetic, therefore, is almost necessary even to obtain low-precision values, unless one uses much more specialized evaluation methods. The current mpmath versions are simple and robust, although they can be fairly slow in worst cases (i.e. when they need to use a high internal precision). Timings for average/good cases are in the millisecond range, >>> timing(coulombf, 3, 2, 10) >>> timing(coulombg, 3, 2, 10) but they can climb up towards a second or so when large cancellations occur. Someone who needs faster Coulomb wave functions, say for a limited range of parameters, could perhaps find mpmath useful for testing a more specialized implementation against. The speed is plenty for basic visualization purposes. Below I've recreated graphs 14.1, 14.2 and 14.5 from Abramowitz & Stegun. mp.dps = 5 # Figure 14.1 F1 = lambda x: coulombf(x,1,10) F2 = lambda x: coulombg(x,1,10) plot([F1,F2], [0,15], [-1.2, 1.7]) # Figure 14.3 F1 = lambda x: coulombf(0,0,x) F2 = lambda x: coulombf(0,1,x) F3 = lambda x: coulombf(0,5,x) F4 = lambda x: coulombf(0,10,x) F5 = lambda x: coulombf(0,x/2,x) plot([F1,F2,F3,F4,F5], [0,25], [-1.2,1.6]) # Figure 14.5 F1 = lambda x: coulombg(0,0,x) F2 = lambda x: coulombg(0,1,x) F3 = lambda x: coulombg(0,5,x) F4 = lambda x: coulombg(0,10,x) F5 = lambda x: coulombg(0,x/2,x) plot([F1,F2,F3,F4,F5], [0,30], [-2,2]) Also, a plot for complex values, cplot(lambda z: coulombg(1+j, -0.5, z)): Orthogonal polynomials I've added the (associated) Legendre functions of the first and second kind, Gegenbauer, (associated) Laguerre and Hermite functions (commits 1, 2, 3, 4). This means that mpmath finally supports all the classical orthogonal polynomials. As usual, they work in generalized form, for complex values of all parameters. A fun thing to do is to verify orthogonality of the orthogonal polynomials using numerical quadrature: >>> mp.dps = 15 >>> chop(quad(lambda t: exp(-t)t2laguerre(3,2,t)laguerre(4,2,t), [0,inf])) >>> chop(quad(lambda t: exp(-t2)hermite(3,t)hermite(4,t), [-inf,inf])) As another basic demonstration, here are some Legendre functions of the second kind, visualized: F1 = lambda x: legenq(0,0,x) F2 = lambda x: legenq(1,0,x) F3 = lambda x: legenq(2,0,x) F4 = lambda x: legenq(3,0,x) F5 = lambda x: legenq(4,0,x) plot([F1,F2,F3,F4,F5], [-1,1], [-1.3,1.3]) Implementing these functions is a lot of work, I've found. The general case is simple (just fall back to generic hypergeometric code), but the special cases (singularities, limits, asymptotically large arguments) are easy to get wrong and there are lots of them to test and document. My general methodology is to implement the special functions as the Wolfram Functions site defines them (if they are listed there), test my implementation against exact formulas, and then test numerical values against Mathematica. Unfortunately, Wolfram Functions often leaves limits undefined, and is not always consistent with Mathematica. In fact, Mathematica is not even consistent with itself. Consider the following: In[277]:= LaguerreL[-1, b, z] // FunctionExpand Out[277]= 0 In[278]:= LaguerreL[-1, -2, z] // FunctionExpand z 2 E z Out[278]= ----- It's tempting to just leave such a case undefined in mpmath and move on (and perhaps fix it later if it turns out to be used). Ideally mpmath should handle all singular cases correctly and consistently, and it should state in the documentation precisely what it will calculate for any given input so that the user doesn't have to guess. Testing and documenting special cases is very time-consuming, however, and although I'm making progress, this work is far from complete. Wednesday, July 15, 2009 Hurwitz zeta function, Dirichlet L-series, Appell F1 I've added three more functions to mpmath since the last blog update: the Hurwitz zeta function, Dirichlet L-series and the Appell hypergeometric function. Hurwitz zeta function The Hurwitz zeta function is available with the syntax hurwitz(s, a=1, derivative=0). It's a separate function from zeta for various reasons (one being to make the plain zeta function as simple as possible). With the optional third argument, it not only computes values, but arbitrarily high derivatives with respect to s (more on which later). It's quite fast at high precision; e.g. the following evaluation takes just 0.25 seconds (0.6 seconds from cold cache) on my laptop: >>> mp.dps = 1000 >>> hurwitz(3, (1,3)) # (1,3) specifies the exact parameter 1/3 With s = π instead, it takes 3.5 seconds. For comparison, Mathematica 6 took 0.5 and 8.3 seconds respectively (on an unloaded remote system which is faster than my laptop, but I'm not going to guess by how much). The function is a bit slow at low precision, relatively speaking, but fast enough (in the 1-10 millisecond range) for plotting and such. Of course, it works for complex s, and in particular on the critical line 1/2+it. Below, I've plotted \|ζ(1/2+it,1)\|2, \|ζ(1/2+it,1/2)\|2, \|ζ(1/2+it,1/3)\|2; the first image shows 0 ≤ t < 30 and the second is with 1000 ≤ t < 1030. To show some more complex values, here are plots of ζ(s, 1), ζ(s, 1/3) and ζ(s, 24/25) for 100 ≤ Im(s) ≤ 110. I picked the range [-2, 3] for the real part to show that the reflection formula for Re(s) < 0 works: In fact, the Hurwitz zeta implementation works for complex values of both s and a. Here is a plot of hurwitz(3+4j, a) with respect to a: The evaluation can be slow and/or inaccurate for nonrational values of a, though (in nonconvergent cases where the functional equation isn't used). Zeta function performance Right now the implementations of the Riemann zeta function and the Hurwitz zeta function in mpmath are entirely separate. In fact, they even use entirely different algorithms (hurwitz uses Euler-Maclaurin summation, while zeta uses the Borwein approximation). This is useful for verifying correctness of either function. Regarding performance, it appears that the Borwein approximation is slightly faster for small \|Im(s)\| while Euler-Maclaurin is massively faster for large \|s\|. The following benchmark might give an idea: >>> from mpmath import >>> mp.dps = 15; mp.pretty = True >>> timing(zeta, 0.5+10j); timing(hurwitz, 0.5+10j) >>> timing(zeta, 0.5+100j); timing(hurwitz, 0.5+100j) >>> timing(zeta, 0.5+1000j); timing(hurwitz, 0.5+1000j) >>> timing(zeta, 0.5+10000j); timing(hurwitz, 0.5+10000j) As the last evaluation shows, the Borwein algorithm is still not doing too poorly at 0.5+10000j from a warm cache, but at this point the cache is 50 MB large. At 0.5+100000j my laptop ran out of RAM and Firefox crashed (fortunately Blogger autosaves!) so it clearly doesn't scale. The Euler-Maclaurin algorithm caches a lot of Bernoulli numbers, but the memory consumption of this appears to be negligible. With hurwitz (and the function value, for those interested), >>> timing(hurwitz, 0.5+100000j) >>> hurwitz(0.5+100000j) (1.07303201485775 + 5.7808485443635j) and the memory usage of the Python interpreter process only climbs to 12 MB. Just by optimizing the pure Python code, I think the Hurwitz zeta function (at low precision) could be improved by about a factor 2 generally and perhaps by a factor 3-4 on the critical line (where square roots can be used instead of logarithms -- zeta uses this optimization but hurwitz doesn't yet); with Cython, perhaps by as much as a factor 10. The real way to performance is not to use arbitrary-precision arithmetic, though. Euler-Maclaurin summation for zeta functions is remarkably stable in fixed-precision arithmetic, so there is no problem using doubles for most applications. As I wrote a while back on sage-devel, a preliminary version of my Hurwitz zeta code for Python complex was 5x faster than Sage's CDF zeta (in a single test, mind you). If there is interest, I could add such a version, perhaps writing it in Cython for Sage (for even greater speed). The Hurwitz zeta function can be differentiated easily like so: >>> mp.dps = 25 >>> hurwitz(2+3j, 1.25, 3) (-0.01266157985314340398338543 - 0.06298953579777517606036907j) >>> diff(lambda s: hurwitz(s, 1.25), 2+3j, 3) (-0.01266157985314340398338543 - 0.06298953579777517606036907j) For simple applications one can just as well use the numerical diff function as above with reasonable performance, but this isn't really feasible at high precision and/or high orders (numerically computing an nth derivative at precision p requires n+1 function evaluations, each at precision (n+1)×p). The specialized code in hurwitz requires no extra precision (although it still needs to perform extra work) and therefore scales up to very high precision and high orders. Here for example is a derivative of order 100 (taking 0.5 seconds) and one of order 200 (taking 4 seconds): >>> hurwitz(2+3j, 1.25, 100) (2.604086240825183469410664e+107 - 1.388710675207202633247271e+107j) >>> hurwitz(2+3j, 1.25, 200) (2.404124816484789309285653e+274 + 6.633220818921777955999455e+273j) It should be possible to calculate a sequence of derivatives much more quickly than with separate calls, although this isn't implemented yet. One use for this is to produce high-degree Taylor series. This could potentially be used in a future version of mpmath, for example to provide very fast moderate-precision (up to 50 digits, say) function for multi-evaluation of the Riemann zeta function on the real line or not too far away from the real line. This in turn would speed up other functions, such as the prime zeta function and perhaps polylogarithms in some cases, which are computed using series over many Riemann zeta values. Dirichlet L-series With the Hurwitz zeta function in place, it was a piece of cake to also supply an evaluation routine for arbitrary Dirichlet L-series. The way it works it that you provide the character explicitly as a list (so it also works for other periodic sequences than Dirichlet characters), and it evaluates it as a sum of Hurwitz zeta values. For example (copypasting from the docstring), the following defines and verifies some values of the Dirichlet beta function, which is defined by a Dirichlet character modulo 4: >>> B = lambda s, d=0: dirichlet(s, [0, 1, 0, -1], d) >>> B(0); 1./2 >>> B(1); pi/4 >>> B(2); +catalan >>> B(2,1); diff(B, 2) >>> B(-1,1); 2catalan/pi >>> B(0,1); log(gamma(0.25)2/(2pisqrt(2))) >>> B(1,1); 0.25pi(euler+2ln2+3ln(pi)-4ln(gamma(0.25))) Appell hypergeometric function The function appellf1 computes the Appell F1 function which is a hypergeometric series in two variables. I made the implementation reasonably fast by rewriting it as a single series in hyp2f1 (with the side effect of supporting the analytic continuation with that for 2F1). A useful feature of the Appell F1 function is that it provides a closed form for some integrals, including those of the form with arbitrary parameter values, for arbitrary endpoints (and that's a quite general integral indeed). Comparing with numerical quadrature: >>> def integral(a,b,p,q,r,x1,x2): ... a,b,p,q,r,x1,x2 = map(mpmathify, [a,b,p,q,r,x1,x2]) ... f = lambda x: x*r (x+a)*p (x+b)q ... def F(x): ... v = x(r+1)/(r+1) * (a+x)*p (b+x)*q ... v = (1+x/a)*(-p) ... v = (1+x/b)*(-q) ... v = appellf1(r+1,-p,-q,2+r,-x/a,-x/b) ... return v ... print "Num. quad:", quad(f, [x1,x2]) ... print "Appell F1:", F(x2)-F(x1) >>> integral('1/5','4/3','-2','3','1/2',0,1) Num. quad: 9.073335358785776206576981 Appell F1: 9.073335358785776206576981 >>> integral('3/2','4/3','-2','3','1/2',0,1) Num. quad: 1.092829171999626454344678 Appell F1: 1.092829171999626454344678 Num. quad: 1106.323225040235116498927 Appell F1: 1106.323225040235116498927 Also incomplete elliptic integrals are covered, so you can for example define one like this: >>> def E(z, m): ... if (pi/2).ae(z): ... return ellipe(m) ... return 2round(re(z)/pi)ellipe(m) + mpf(-1)*round(re(z)/pi)\ ... sin(z)appellf1(0.5,0.5,-0.5,1.5,sin(z)2,msin(z)*2) >>> z, m = 1, 0.5 >>> E(z,m); quad(lambda t: sqrt(1-msin(t)*2), [0,pi/4,3pi/4,z]) >>> z, m = 3, 2 (1.057495752337234229715836 + 1.198140234735592207439922j) (1.057495752337234229715836 + 1.198140234735592207439922j) The Appell series isn't really faster than numerical quadrature, but it's possibly more robust (the singular points need to be given to quad as above to obtain any accuracy, for example). Mpmath doesn't yet have incomplete elliptic integrals; the version above could be a start, although I'll probably want to try a more canonical approach. Sunday, July 12, 2009 Another Mathematica bug Here is mpmath's evaluation for two Struve function values: >>> struvel(1+j, 100j) (0.1745249349140313153158106 + 0.08029354364282519308755724j) >>> struvel(1+j, 700j) (-0.1721150049480079451246076 + 0.1240770953126831093464055j) The same values in Mathematica: In[52]:= N[StruveL[1+I, 100I], 25] Out[52]= 0.1745249349140313153158107 + 0.0802935436428251930875572 I In[53]:= N[StruveL[1+I, 700I], 25] Out[53]= -0.2056171312291138282112197 + 0.0509264284065420772723951 I I'm almost certain that the second value returned by Mathematica is wrong. The value from mpmath agrees with a high-precision direct summation of the series defining the Struve L function, and even Mathematica gives the expected value if one rewrites the L function in terms of the H function: In[59]:= n=1+I; z=700I Out[59]= 700 I In[60]:= N[-I Exp[-n Pi I/2] StruveH[n, I z], 25] Out[60]= -0.1721150049480079451246076 + 0.1240770953126831093464055 I Maple also agrees with mpmath: > evalf(StruveL(1+I, 700I), 25); -0.1721150049480079451246076 + 0.1240770953126831093464055 I So unless Mathematica uses some nonstandard definition of Struve functions, unannounced, this very much looks like a bug in their implementation. Wolfram Alpha reproduces the faulty value, so this still appears to be broken in Mathematica 7. Friday, July 10, 2009 Improved incomplete gamma and exponential integrals; Clausen functions The SVN trunk of mpmath now contains much improved implementations of the incomplete gamma function (gammainc()) as well as the exponential integrals (ei(), e1(), expint()). Although the code is not quite perfect yet, this was a rather tedious undertaking, so I'm probably going to work on something entirely different for a while and give these functions another iteration later. The incomplete gamma function comes in three flavors: the lower incomplete gamma function, the upper incomplete gamma function, and the generalized (two-endpoints) incomplete gamma function. The generalized incomplete gamma function is defined as which reduces to the lower function when a = 0, and the upper version when b = +∞. A huge number of integrals occurring in pure and applied mathematics have this form (even Gaussian integrals, with a change of variables) so a solid incomplete gamma is quite important. It's especially important to ensure both speed in correctness in asymptotic cases. The lower incomplete gamma function is easiest to implement, because it's essentially just a rescaled version of the hypergeometric 1F1 function and the 1F1 implementation already works well. Not much change was made here, so I'm going to write some about the other cases instead. The exponential integrals are essentially the same as the upper incomplete gamma function and mostly share code. Remarks about the upper gamma function therefore also apply to the exponential integrals. Upper gamma performance The upper incomplete gamma function is hard because the two main series representations are an asymptotic series involving 2F0 that doesn't always converge, and an 1F1 series that suffers badly from cancellation for even moderately large arguments. The problem is to decide when to use which series. The 2F0 series is very fast when it converges, while the 1F1 series is quite slow (due to the need for extra precision) just below the point where 2F0 starts to converge. After some experimentation, I decided to change the implementation of 2F0. Instead of performing a heuristic, conservative test to determine whether the series will converge (sometimes claiming falsely that it won't), it now always goes ahead to sum the series and raises an error only when it actually doesn't converge. Thus the asymptotic series will always be used when possible, and although this leads to a slight slowdown for smaller arguments, it avoids worst-case slowness. The most important use for the incomplete gamma function, I believe, is in asymptotics, so I think this is a correct priority. As a result, you can now do this: >>> from mpmath import >>> mp.dps = 25; mp.pretty = True >>> gammainc(10, 100) >>> gammainc(10, 10000000000000000) >>> gammainc(3+4j, 1000000+1000000j) (-1.257913707524362408877881e-434284 + 2.556691003883483531962095e-434284j) The following graph compares old and new performance. The y axis shows the reciprocal time (higher is better) for computing gammainc(3.5, x) as x ranges between 0 and 100, at the standard precision of 15 digits. Red is the old implementation and blue is the new. The code also works with complex numbers, of course; replacing x with jx gives a virtually identical graph (slightly scaled down due to the general overhead of complex arithmetic). It's very visible where the asymptotic series kicks in, and the speed from then on is about 2000 evaluations per second which is relatively good. The new implementation is regrettably up to 3x slower than the old one for smaller x, although the slowdown is a bit misleading since the old version was broken and gave inaccurate results. The big dip in the blue graph at x = 10 is due to the automatic cancellation correction which the old code didn't use. The gap between the asymptotic and non-asymptotic cases could be closed by using specialized series code for the lower incomplete gamma function, or using the Legendre continued fraction for intermediate cases (this comes with some problems however, such as accurately estimating the rate of convergence, and the higher overhead for evaluating a continued fraction than a series). This will certainly be worth doing, but I'm not going to pursue those optimizations right now for reasons already stated. Some good news is that the graph above shows worst-case behavior, where the generic code is used, due the parameter 3.5. I've also implemented fast special-purpose code for the case when the first parameter is a (reasonably small) integer. This also means that the exponential integrals E1(x), Ei(x) as well as En(x) for integer n can be evaluated efficiently. Here is a speed comparison between the old and new implementations of the ei(x) function, again at standard precision. There is actually no change in algorithm here: the old implementation used a Taylor series for small arguments and an asymptotic series for large arguments. The difference is due to using only low-level code; this turned out to buy a factor 2 in the Taylor case and more than an order of magnitude (!) in the asymptotic case. The results are similar for the E1 function and with a complex argument. It is similar (only a bit slower) for gammainc(n,x) and expint(n,x) with a small integer value for n, although so far fast code is only implemented for real x in those cases. Accurate generalized incomplete gamma function The generalized incomplete gamma function can be written either as the difference of two upper gammas, or as the difference of two lower gammas. Which representation is better depends on the arguments. In general, one will work while the other will lead to total cancellation. gammainc is now clever enough to switch representations. This uses a difference of lower gamma functions behind the scenes: >>> gammainc(10000000, 3) - gammainc(10000000, 2) # Bad >>> gammainc(10000000, 2, 3) # Good This uses a difference of upper gamma functions behind the scenes: >>> gammainc(2, 0, 100000001) - gammainc(2, 0, 100000000) # Bad >>> gammainc(2, 100000000, 100000001) # Good Some demo plots Here are two plots of the upper gamma functions and exponential integrals (for various values of the first parameter). A lot of time went into getting the correct branch cuts in the low-level code (and writing tests for them), so please appreciate the view of the imaginary parts. T1 = lambda x: gammainc(-2,x) T2 = lambda x: gammainc(-1,x) T3 = lambda x: gammainc(0,x) T4 = lambda x: gammainc(1,x) T5 = lambda x: gammainc(2,x) T1 = lambda x: expint(-2,x) T2 = lambda x: expint(-1,x) T3 = lambda x: expint(0,x) T4 = lambda x: expint(1,x) T5 = lambda x: expint(2,x) And a complex plot of gammainc(3+4j, 1/z): A plot of gammainc(1/z, -1/z, 1/z); a rather nonsensical function (but that is besides the point): Clausen functions Unrelated to the gamma functions, I've also implemented Clausen functions: These functions are just polylogarithms in disguise, but convenient as standalone functions. With them one can evaluate certain divergent Fourier series for example: >>> clsin(-2, 3) >>> nsum(lambda k: k2 sin(3k), [1,inf]) They also work for complex arguments (and are related to zeta functions): >>> clsin(2+3j, 1+2j) (1.352229437254898401329125 + 1.401881614736751520048876j) >>> clcos(2+3j, pi) (-1.042010539574581174661637 - 0.2070574989958949174656102j) >>> altzeta(2+3j) (1.042010539574581174661637 + 0.2070574989958949174656102j) >>> chop(clcos(zetazero(2), pi/2)) Monday, June 29, 2009 Meijer G, more hypergeometric functions, fractional differentiation Bessel functions, etc. >>> mp.dps = 30 (0.0 + 5.17370851996688078482437428203e+43429448190325182754j) >>> print bessely(1,1020) >>> print besselk(1,1020) Fractional derivatives >>> mp.dps = 30 The Meijer G-function >>> mp.dps = 15 3 1 2 2 3 1 2 2 Out[3]= {8.90265, 0.0017597930166135139087} Out[4]= {12.8231, 3.3409355534341801158987353523397047765918571151576 10 } Out[5]= {59.017, -2.0782671663885270791 10 } Out[6]= {83.3753, -2.8700325450226332558088281915945389986057044454640 10 } Out[7]= {451.365, 2.439257690719956395903324691434088756714300374716395499173\ > 70196218529840153673260714339051464703903148052541923961351654 10 } Complex roots >>> for k in range(5): -12.0 (1.32982316461435 + 0.966173083818997j) -12.0 (-0.507947249855734 + 1.56330088863444j) -12.0 -1.64375182951723 -12.0 (-0.507947249855734 - 1.56330088863444j) -12.0 (1.32982316461435 - 0.966173083818997j) >>> print power(2,1e-100)-1 >>> print powm1(2, 1e-100) That will be all for now. Friday, June 19, 2009 Massive hypergeometric update [Update: as further proof that the asymptotic expansions are working, I've plotted 1/erfi(1/z3), Ai(1/z3) and Bi(1/z3) around 0: End of update.] Today I committed a large patch to mpmath that significantly improves the state of hypergeometric functions. It's the result of about a week of work (plus some earlier research). Perhaps most importantly, I've implemented the asymptotic expansions for 0F1 and 1F1 (the expansions for 2F1 were discussed in the previous post). They should now work for arbitrarily large arguments, as such: >>> from mpmath import >>> mp.dps = 25 >>> print hyp0f1(3,100) >>> print hyp0f1(3,100000000) >>> print hyp0f1(3,1050) >>> print hyp0f1(3,-1050) >>> print hyp0f1(1000,1000+10*8j) (-1.101783528465991973738237e+4700 - 1.520418042892352360143472e+4700j) >>> print hyp1f1(2,3,1010) >>> print hyp1f1(2,3,-1010) >>> print hyp1f1(2,3,10*10j) (-9.750120502003974585202174e-11 - 1.746239245451213207369885e-10j) I also implemented 2F0 and U (Kummer's second function), mostly as a byproduct of the fact that 2F0 is needed for the asymptotic expansions of both 0F1 and 1F1. 2F0 is an interesting function: it is given by a divergent series (it converges only in special cases where it terminates after finitely many steps): However, it can be assigned a finite value for all z by expressing it in terms of U. Hence, mpmath can now compute the regularized sum of 2F0(a,b,z) for any arguments, say these ones: >>> print hyp2f0(5, -1.5, 4) (0.0000005877300438912428637649737 + 89.51091139854661783977495j) >>> print hyp2f0(5, -1.5, -4) (This ought to be a novel feature; SciPy and GSL implement 2F0, but only special cases thereof, and Mathematica doesn't have a direct way to evaluate 2F0.) It's the asymptotic case where z → 0, where a truncation of the 2F0 series can be used, that is used for the expansions at infinity of 0F1 and 1F1. Back to 0F1 and 1F1, the asymptotic expansions of these functions are important because they permit many special functions to be evaluated efficiently for large arguments. So far I've fixed erf, erfc, erfi, airyai and airybi to take advantage of this fact (except for erf and erfc of a real variable, all these functions were previously slow even for a moderately large argument, say \|z\| > 100). Examples that now work (well, some of them possibly theoretically worked before too, but probably required hours or ages of universe to finish): >>> print erf(10000+10000j) (1.000001659143196966967784 - 0.00003985971242709750831972313j) >>> print erfi(1000000000) >>> print erfc(1000-5j) (-1.27316023652348267063187e-434287 - 4.156805871732993710222905e-434288j) >>> print airyai(1010) >>> print airybi(1010) >>> print airyai(-1010) >>> print airybi(-1010) >>> print airyai(10*10 (1+j)) (5.711508683721355528322567e-186339621747698 + 1.867245506962312577848166e-186339621747697j) >>> print airybi(10*10 (1+j)) (-6.559955931096196875845858e+186339621747689 - 6.822462726981357180929024e+186339621747690j) An essential addition in this patch is the function hypercomb which evaluates a linear combination of hypergeometric series, with gamma function and power weights: This is an extremely general function. Here is a partial list of functions that can be represented more or less directly by means of it: • Regularized hypergeometric series • The generalized incomplete gamma and beta functions (and their regularizations) • Bessel functions • Airy, Whittaker, Kelvin, Struve functions, etc • Error functions • Exponential, trigonometric and hyperbolic integrals • Legendre, Chebyshev, Jacobi, Laguerre, Gegenbauer polynomials • The Meijer G-function That's most of Abramowitz & Stegun, and means that the remaining hypergeometric-type functions available in Mathematica or Maxima but absent in mpmath will be easy to implement in the near future. With these additions, mpmath will have the most comprehensive support for numerical hypergeometric-type functions of any open source software, and should be very close to Mathematica. The most important virtue of hypercomb is not that it allows for more concise implementations of various hypergeometric-type functions, although that is a big advantage too. The main idea is that hypercomb can deal with singular subexpressions, and particularly with gamma function poles that cancel against singularities in the hypergeometric series. These cases are almost more common than the nonsingular cases in practice, and hypercomb saves the trouble of handling them in every separate function. Thus, in principle, a numerically correct implementation of hypercomb leads to correct implementations of all the functions in the list above. It's not a silver bullet, of course. For example, if a particular but very common case of some common function triggers an expensive limit evaluation in hypercomb, then it's probably better to handle that case with special-purpose code. There are also most likely some bugs left in hypercomb, although by now I have tested a rather large set of examples; large enough to be confident that it works soundly. To show an example of how it works, an implementation of the Bessel J function might look like this: def besj(n,z): z = mpmathify(z) h = lambda n: [([z/2],[n],[],[n+1],[],[n+1],-(z/2)*2)] return hypercomb(h, [n]) >>> mp.dps = 30 >>> print besselj(3,10.5) >>> print besselj(-3,10.5) >>> print besj(3,10.5) >>> print besj(-3,10.5) It gives the same value as the current Bessel J implementation in mpmath. Note that it works even when n is a negative integer, whereas naively evaluating the equation defining J(n,z) hits a gamma function pole and a division by zero in the hypergeometric series. Thanks to the support for asymptotic expansions, this implementation (unlike the current besselj will also work happily with large arguments): >>> print besj(3,1e9) I'm soon going to fix all the Bessel functions in mpmath along these lines, as I already did with erf, airyai, etc. Here is another, more involved example (quoting directly from the docstring for hypercomb). The following evaluates with a=1, z=3. There is a zero factor, two gamma function poles, and the 1F1 function is singular; all singularities cancel out to give a finite value: >>> from mpmath import >>> mp.dps = 15 >>> print hypercomb(lambda a: [([a-1],[1],[a-3],[a-4],[a],[a-1],3)], [1]) >>> print -9*exp(3) With some tweaks, the same code perhaps with a few tweaks could be used to symbolically evaluate hypergeometric-type functions (in the sense of rewriting them as pure hypergeometric series, and then possibly evaluating those symbolically as a second step). Symbolic support for hypergeometric functions is a very interesting (and hard) problem, and extremely important for computer algebra, but unfortunately I don't have time to work on that at the moment (there is more than enough to do just on the numerical side).
105877bd2e46fe87	SolarStains.ca - Laplace's Demon Home Gallery Classes Articles Testimonials About Contact Laplace's Demon 88 cm W x 135 cm H x 15 cm D / 11 kg (34.5" W x 53" H x 6" D / 24 lb) Available for purchase Laplace’s Demon examines the conflicting propositions of determinism, randomness, and divine intervention. Which universe do we live in: causal, chancy, or arbitrated? Things around us seem deterministic, at least some of them. A ball rolls down a slope in a predictable manner. And it does it here and there, and it does it now and later in the same way, given the same conditions. To let randomness enter our world, which is to say to allow a non-deterministic transgression, one needs to explain the relationship between what is caused and what is not caused. Down there, in the realms of space at the Plank length, the subatomic world goes against our intuitions. Above it is the world of atoms, the world of matter. One layer up is the world of chemistry. Another layer up emerges the biology, from unicellular to complex and to pluricellular, and from vegetal to intelligence and to consciousness. Move up again and find the psychological and the sociological layers, with abstract thinking, grace, and humour. But beauty starts in the laws of nature. An electron can't tell a lie. But again, it doesn't know prestige or reverence either. The “demon” is an athlete of knowledge reaching out in an acrobatic vault for the roulette, as the symbol of randomness, through the dark subatomic world background. The roulette is at the end of a temporal sequence of matter arising from vacuum1, accompanied by the probability waves and other entities. But why such participative desire? I'm following the impulse here to share some of the developments of our understanding of how the Universe works. Although the inquisitive mind who reached my page for the artistic appeal can find excellent scientific literature and programs out there, I want to open the discussion myself with few remarks. While immersing myself into learning about the limits of our scientific understanding of the objective reality I came to realise in the past decade that the meadow of intellectual conundrums, enigmas, mysteries and even dramas encountered in the search for ultimate truth, which is the project of science, offers all the feelings that one can possibly seek when not satisfied with the general understanding of the things around him/her. Learning about them it's perplexing, astonishing, disconcerting, puzzling, unsettling and, at times, demoralizing. But the best part is that one can even get to try to solve them. It's a highly specialised field, indeed, and it's ambitious. Nevertheless, a mystery does not hold there for veneration, but for exploration. And there is a plethora of them, some being a couple of millenia old, intermingling between science, philosophy, and spirituality. That's the bare source of my enthusiasm. The story behind my work presented here begins with Pierre-Simon Laplace2 who, looking into the work of his predecessor, Isaac Newton3, at the beginning of the 19th century, realized that the chapter of celestial mechanics sustained a startling problem of equilibrium. The equations seemed to suggest that the planets of the Solar System were on the course of leaving their orbits. The instability problem was known by the father of the Law of the Universal Gravitation4, which for corrections relied on periodic divine intervention. Not exactly a mechanistic solution, despite that the very laws of motion and universal gravitation of Newton imposed the deterministic5 vocabulary into the scientific discourse. Laplace, however, understanding very well what determinism means to the world, argued that all events must follow the universal rule of "cause-and-effect". Therefore, having known the state of all the particles in the universe at a particular instant, and knowing all the laws of nature, he taught, a "vast intelligence"6 shall be able to predict the state of the universe... a second later, an hour later, or after an eternity. Moreover, calculating the states of the universe backward in time it was to him just matter of applying the equation. The point here being that from the beginning until the end of Time no random occurrence happens, no un-caused causal event can penetrate the system. Laplace did not propose a better solution to the equations, it was only after 100 years when a better approximation of Gravity was put forward by Einstein, compounding time and space in a new sort of material called "the fabric of space-time"7. Even without the math on hand, though, in his wisdom Laplace was convinced that by virtue of determinism the past and future states of the universe are entailed. The year was 1814. This troubling statement implied the jittering realisation that free will8 is merely an illusion. Without free will there is no guilt, no sin, and hence no admonition. Later, with the development of the science of quantum mechanics9, however, the guilt-free universe was about to be shaken. Firstly, Heisenberg’s uncertainty principle10 proposed the idea that one cannot know the state of a particle with infinite precision. The universe was un-knowable. Secondly, the Schrödinger’s probability wave function made things fuzzier and probabilistic, leading to an interpretation called "superposition"11. Again, a description of an un-knowable (before observation) universe. Then quantum entanglement12 showed that locality13 is not a reliable attribute either, because one event at one end of the universe could influence the behaviour of its pair at the other end of the universe with no one knowing it. A concept hard to swallow for Einstein. Furthermore, spontaneity was also proven as inexistent. There is much more, and the list is fascinatingly populated with questions like "why all electrons look the same?", "why time flows one way?", "why the unreasonable effectiveness of mathematics in describing the nature?". Formulations like these seem to deny the idea of perfect knowledge. And modern physics produces new theories and conjectures that surpass by far the technological capability. In the case of gravity waves it took more than 100 years to confirm their existence. A great success. Others are not conceivably testable. In the light of such convoluted enunciations the randomness may have a chance(!) after all. Admittedly, such statement requires a big gap to cover scientifically. Artistically, however, it deserves exploration. But mastering so many fields of the natural science is absolutely meritorious. All the concepts mentioned here are to admire the depths of our understandings. And while we may not be able to know the state of the universe with infinite precision, does our lack of knowledge does make room for randomness, or for non-deterministic, intentional intervention? I suspend my thoughts in front of this dilemma but... I let my affect to guide me. Free will must be saved. 1 Vacuum: the concept has been used in science and philosophy to indicate a medium in which objects exist; it indicates the absence of a being, the nothingness, the empty space, the void, the luminiferous aether, aether - each of these names purports more or less different properties; the best definition of vacuum is probably that of a volume (of space) from which it was removed everything that it can be removed. The energy of empty space is greater than zero; experiments like the Casimir effect have been tested successfully. Virtual particles arrive from vacuum into the real world, even though for a very short time. 2 Pierre-Simon, marquis de Laplace (1749–1827): French scholar and polymath, Laplace believed that all the events in the Universe are produced by agents of causality, and that one can determine the state of a system if one knows the prior state of the system and the rules that governs them. Applying Newtonian's law of gravitation to the entire solar system he observed that its complexity made mathematical solutions impossible to draw. Even though too complicated, yet he thought of it as a deterministic construct. Moreover, despite the discord between his equations and the orbits of Jupiter and Saturn, Laplace asserted that the planetary motion is invariable. His confidence came, perhaps, from later advancement of mathematics and science, and from his unshaken belief in determinism. 3 Isaac Newton (1642–1726): English physicist, mathematician, and astronomer; a key figure in the scientific revolution with many scientific contributions in optics, mechanics and gravity. 4 The Law of Universal Gravitation: was first published in 1687 by Isaac Newton, the theory states that objects attract each other with a force determined by a relation between their masses, the distance between them, and a universal constant. 5 Determinism: the idea of a causal flow of events in the universe is reinforced by the mechanical laws of Isaac Newton published in the Philosophiæ Naturalis Principia Mathematica in 1687. Besides the law of universal attraction, Newton proposed a set of equations describing the principles of motion with constant velocity and with constant acceleration. Therefore, by knowing the initial state of a system, one could make predictions about its state at a later time. 6 Vast intelligence: the term that Pierre-Simon Laplace used to describe the concept of an intelligent entity that has the ability to know the state of all the particles of the Universe; with this data, and knowing all the laws that govern in nature, he/she then can compute the future as well as the past. This entity was later known as “Laplace’s Demon.” 7 The fabric of space-time: Einstein's theory of General Relativity describes gravity as a dynamic influence exerted by two massive objects on each-other in what he called the “fabric of space-time”, and in which the motion of the objects follows the slopes created in this material by their own masses; the trajectories of the two objects result from the continuous deformation of this “fabric of space-time”. Time becomes a function of gravitational magnitude and rate of motion, thus a player, and not a fixed background. 8 Free will: free will is "the canonical designator for a significant kind of control over one’s actions." (by Stanford Encyclopedia of Philosophy). The term "significant" acknowledges a deterministic factor in our control over our actions, implying, at minimum, that determinism cannot be ruled out. To the limit, some argue, our control is null despite our impression of ownership over our decisions. 9 Quantum mechanics: QM is the discipline that explores the behaviours and the properties of the subatomic elements; the term “quantum” indicates that the energy of these elements comes in discrete amounts; it is one of the most successful theories allowing for impressive technological advancement. In the effort to unify gravity with the other fundamental forces, electro-magnetic included, one remarkable approach is to quantify gravity itself; the framework for such approach is called "Quantum Loop Gravity". It is not unconceivable, however, that a new way of thinking would be necessary to solve this problem. As of our current understanding, the gravity and the fundamental forces are not of contradictory natures, they are of totally different natures that have no common ground to begin talking. 10 Uncertainty principle: put forward in 1927 by Werner Heisenberg, the principle puts a limit on the precision with which the product of a particle’s position and its momentum is to be known; the more one learns about one of the two parameters, the greater the error of the other parameter. The very act of observation influences the object's behaviour; ultimately, a photon must bounce off by it in order to give information. The observer becomes part of the environment. This may be an indication that the finesse of our technical investigation is now comparable to the Universe's structural make. 11 Superposition: the superposition is a property of the Schrödinger equation which assembles the notion of "wave" with that of "probability". In quantum mechanics, the Copenhagen interpretation says that a particle occupies not a single point in space, but a region of space, and not one single point at the time, but all of them at once. Its spread-out but indivisible energy collapses into a point when an interaction occurs. Now, historically, an interaction was taught of an act of observation made by a conscious being. A vocabulary inaccuracy, rather, from which was derived that the Universe only exists for the conscious being to see it. This line of thinking was discarded, eventually. Further notes: a) As an alternative to the Copenhagen interpretation, the Everettian multi-verse proposition suggests that whenever two options are available, the Universe splits in two parallel Universes; the two options then become certainties in each Universe. b) Given the wavy characteristic of all that is, the superposition seems justifiable - a wave needs something of a "body" nature, rather than a "pointy" consistency. The ancient Greek atomists then must've start on a delusory footing when stated that the smallest constituent of the universe is particle-like. 12 Quantum entanglement: this is the property of two paired particles to remain in instantaneous correspondence with each-other at any distance (larger systems can be entangled too). Einstein rejected this interpretation on the account of “hidden variables”, and called it “spooky action at the distance”. John S Bell proposed a test in 1935 to verify this, and eventually the results invalidated Einstein’s approach. It is worth mentioning an idea proposed in 2019 by Sean Carroll—that “locality” shall be defined in terms of interactions, rather than distance. Hence two objects are called in each-other's proximity if they interact instantaneously; the space between them is irrelevant. 13 Locality: this paradigm tributary to our macro-world experience means that for an object to have an influence on another object it has to touch it; the only exception known for a while were the magnets. Then the subatomic world revealed that there is no real "touching". However, the objects have to be next to each-other to transmit energy. Except for the entangled particles. Final note: The above foundational concepts in QM&G successfully predict how they operate, but they are not presented with a describing mechanism in the same fashion in which I know that the car goes right when I turn the steering wheel right, but I don't know what gears, pivots, or pulleys are under the hood. Copyright © 2020 Solar Stains
7b934842d0ff86ae	First Week School Second Week School Third Week Discussion Meeting First Week School Angelo Bassi Introduction to collapse models We will introduce the idea of spontaneous wave function collapse models and will present the Ghirardi-Rimini-Weber (GRW) model, the first model of this kind. We will discuss why and how nonlinear modifications to the Schrödinger equation have always to be accompanied by appropriate stochastic terms, in order to avoid superluminal signalling. We will discuss in detail the amplification mechanism, which allows to describe both the quantum properties of microscopic systems and the classical properties of macroscopic objects within a single dynamical framework. We will present the Continuous Spontaneous Localization (CSL) model, which somehow has become the reference model in the literature. We will also introduce gravity-related models, the most famous being the Diosi-Penrose (DP) model. We will review the most promising tests of collapse models, which range from matter-wave interferometry to non-interferometric tests, to cosmological observations. We will show which region of the parameter space of the CSL model has been excluded by experimental data. Useful references: • The GRW model: G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986). • The CSL model: G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A  42, 78 (1990). • The DP model: L. Diosi, Phys. Rev. A 40, 1165 (1989).  P. Penrose, Gen. Rel. Grav. 28, 581 (1996).  • A review of collapse models: Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013). [arXiv:1204.4325] Dustin Lazarovici The measurement problem and some mild solutions We shall introduce two solutions of the measurement problems of quantum mechanics, which do not change the Schrödinger evolution of the wave function: Bohmian Mechanics and Many Worlds. We shall explain on the basis of Bohmian Mechanics how randomness has to be understood in a deterministic theory of nature, since both Bohmian Mechanics and Many Worlds are deterministic. We shall further but shortly explain the emergence of operator observables and Heisenberg's uncertainty relation. We shall also discuss Bell’s inequalities and the nonlocality of Nature, which has been firmly established by now. If time permits, I shall touch upon so called on decoherent histories, which is another attempt to solve the measurement problem, but which is problematical with respect to the various no go theorems, like Kochen and Specker. Useful references: • Detlef Dürr, Stefan Teufel: Bohmian Mechanics, Springer, 2009. • John S. Bell: Speakable and Unspeakable in Quantum Mechanics (2nd Edition), Cambridge University Press, 2004. • Jean Bricmont: Making Sense of Quantum Mechanics, Springer, 2016. • Tim Maudlin: Three measurement problems, Topoi 14(1), 1995. • Sheldon Goldstein: Bohmian Mechanics, in: The Stanford Encyclopedia of Philosophy Andrè Grossardt Quantum mechanics and gravitation – what we know, what we don't know, and what we think we know 1. Mechanics of a point mass – nonrelativistic to relativistic and classical to quantum We will briefly discuss the quantisation of the relativistic point particle and of the Klein-Gordon field. We will further discuss how the limit to nonrelativistic quantum mechanics can be obtained, and what issues arise with this step. 2. Gravitation – from Newton to Einstein and back We will review the principles of general relativity and its Newtonian limit. Then we focus on the problems that arise when we try to couple classical gravity to quantum matter. 3. What we know: nonrelativistic particles in a Newtonian gravitational potential We will discuss the experiments conducted with quantum matter in the gravitational field of the Earth. 4. Equivalence principle in classical and quantum mechanics A brief review will be given on the equivalence principle in classical general relativity. We will discuss how questions arising when generalising the equivalence principle to quantum physics. 5. What we don't know: a very brief history of quantum gravity In a short presentation an overview of the different approaches to quantum gravity will be given. 6. What we think we know (I): perturbative quantum gravity vs. semiclassical gravity We will discuss the perturbative approach to quantum gravity as a quantum field theory, how it compares to a possible semiclassical alternative, and how it could be experimentally tested. 7. What we think we know (II): Quantum mechanics in curved space-time beyond Newton We discuss what is known about the behaviour of quantum systems in a gravitational field beyond the nonrelativistic limit. We will review previously considered effects such as gravity-induced decoherence and discuss open questions and problems. • Wolfgang Rindler, Essential Relativity (second ed.), Springer, 1977 • Thanu Padmanabhan, Gravitation, Cambridge University Press, 2010 • Claus Kiefer, Quantum Gravity (third ed.), Clarendon Press, 2012 • D. Giulini, C. Kiefer, C. Lämmerzahl (eds.), Quantum Gravity, Lecture Notes in Physics, Springer, 2003 • R. Colella, A.W. Overhauser, and S.A. Werner, Observation of Gravitationally Induced Quantum Interference, Physical Review Letters 34 (1975) 1472 • Daniel M. Greenberger, The neutron interferometer as a device for illustrating the strange behavior of quantum systems, Reviews of Modern Physics 55 (1983) 875 • Domenico Giulini, Equivalence principle, quantum mechanics, and atom-interferometric tests. In: F. Finster et al. (eds.), Quantum Field Theory and Gravity, pp. 345-370, Birkhäuser/Springer, 2012. arXiv:1105.0749 [gr-qc] • James Mattingly, Is Quantum Gravity Necessary? In: A.J. Kox and J. Eisenstaedt (eds.), The Universe of General Relativity, Einstein Studies 11, pp. 327-338, Birkhäuser/Springer, 2005. • Roger Penrose, On the Gravitization of Quantum Mechanics 1: Quantum State Reduction, Foundations of Physics 44 (2014) 557. Alex Matzkin Weak measurements What is the value of a given physical property of a quantum system at some intermediate time between the system preparation in an initial state and its final detection ? The answer to this question hinges on the peculiar status of quantum measurements. The standard answer according to standard quantum mechanics would be that this question is meaningless and leads at best to counterfactual paradoxes. Standard quantum mechanics affords nevertheless another answer, in the form of a minimally perturbing and experimentally feasible protocol known as “weak measurements”. The aim of these lectures will be to introduce weak measurements and to appraise the relevance of the weak measurement protocol in order to shed light on the properties of evolving quantum systems. The lectures will be structured according to the following outline: 1. Introduction: Properties and measurements 2. Measurements in Quantum (and also in Classical) Mechanics 3. Weak measurement protocol 4. Weak values and properties of quantum systems Some References: • Y. Aharonov, D. Z. Albert, and L. Vaidman , How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351-1354 (1988) • Y. Aharonov and D. Rohrlich, Quantum Paradoxes, Wiley, 2005 • A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Asking photons where they have been, Phys. Rev. Lett. 111, 240402 (2013) • J. Dressel, Weak values as interference phenomena, Phys. Rev. A 91, 032116 (2015) • A. Matzkin, Observing trajectories with weak measurements in quantum systems in the semiclassical regime, Phys. Rev. Lett. 109, 150407 (2012) • S. Kocsis , B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer, Science 332, 1170 (2011). Tejinder Singh Trace Dynamics: Quantum theory as an emergent phenomenon It has been suggested, for various reasons, that quantum theory maybe an approximation to a deeper theory. In these talks, we will begin by reviewing these reasons. We will then describe one concrete attempt to develop such an underlying theory, namely the theory of Trace Dynamics [TD], proposed by Stephen Adler and collaborators. TD is a classical dynamical theory of matrices, possessing an important global unitary invariance. It is shown how, as a result of coarse-graining, quantum theory emerges as the equilibrium statistical thermodynamics of the underlying microscopic TD. Statistical fluctuations about equilibrium can in principle result in a stochastic nonlinear modification of quantum theory, thus providing a theoretical underpinning for the phenomenological collapse model known as Continuous Spontaneous Localisation [CSL]. We will end by mentioning the outstanding unsolved problems of this program, and attempts to include gravitation.The tutorial will be used to present some ongoing research work in this context, and to have an open-ended discussion with the audience on the subject matter of these talks. References that the students will find useful: • S. L. Adler, Quantum theory as an emergent phenomenon (Cambridge University Press, Cambridge, 2004) Available in part at arXiv:hep-th/0206120 Original Papers: • S. L. Adler, Nucl. Phys. B 415, 195 (1994) [arXiv:hep-th/9306009]: Generalised quantum dynamics • S. L. Adler and A. C. Millard, Nucl. Phys. B 473, 199 (1996) [arXiv:hep-th/9508076]: Generalised quantum dynamics as prequantum mechanics Condensed reviews: • P. Pearle, Stud. Hist. Philos. Mod. Phys. 36, 716 (2005) [arXiv:quant-ph/0602078] • Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013) [arXiv:1204.4325] Rafael Sorkin The quantum measure (and how to measure it) When utilized appropriately, the path-integral offers an alternative to the ordinary quantum formalism of state-vectors, self-adjoint operators, and external observers - an alternative that seems closer to the underlying reality and more in tune with quantum gravity. The basic dynamical relationships are then expressed, not by a propagator, but by the quantum measure, a set-function μ that assigns to every (suitably regular) set E of histories its generalized measure μ(E). (The idea is that μ is to quantum mechanics what the Wiener-measure is to Brownian motion.) Except in the special case where E is an instrument-event, μ(E) cannot be interpreted as a probability, as it is neither additive nor bounded above by unity. Nor, in general, can it be interpreted as the expectation value of a projection operator (or POVM). Nevertheless, I will describe how one can ascertain μ(E) experimentally for any desired E, by means of an arrangement which, in a well-defined sense, filters out the histories that do not belong to E. 1. Alvaro M. Frauca and Rafael D. Sorkin, ``How to Measure the Quantum Measure'', to appear on arXiv 2016 Oct 10, and in IJTP eventually. 2. Sukanya Sinha and Rafael D. Sorkin, ``A Sum-over-histories Account of an EPR(B) Experiment'', Found. of Phys. Lett. 4, 303-335 (1991). 3. Rafael D. Sorkin, ``Quantum Mechanics as Quantum Measure Theory'', Mod. Phys. Lett. A 9, 3119-3127 (1994). ArXiv: gr-qc/9401003. 4. Rafael D. Sorkin, ``Quantum dynamics without the wave function'' J. Phys. A: Math. Theor. 40, 3207-3221 (2007). ArXiv: quant-ph/0610204. Bassano Vacchini Introduction to non-Markovian open quantum systems dynamics The lectures will be devoted to introduce the basic ideas and formalism for the description of open quantum systems, namely quantum systems whose interaction with an external environment cannot be neglected. As a consequence the reduced system dynamics is irreversible and features typical phenomena such as dissipation and decoherence. We will introduce the notion of quantum dynamical map and the basic dynamical equations governing the time evolution of an open quantum system, considering both the fundamental Gorini-Kossakowski-Sudarshan-Lindblad master equation and more general time evolutions including time-convolutionless and memory kernel master equations. We will further consider recent developments pointing to possible definitions of memory effects in a quantum reduced dynamics. In particular we will explore a recently introduced notion of quantum non-Markovianity based on the distinguishability of quantum states as quantified by the trace distance and connecting memory effects with an information flow between system and environment. We will consider the connection of this approach with the divisibility of quantum dynamical maps. Finally we will show how this strategy leads to the actual measurement of non-Markovianity and also allows for the detection of initial system-environment correlations. The structure of the lectures will be as follows: • Lecture 1: Foundations of open quantum system theory. • Lecture 2: Lindblad theory and generalized master equations • Lecture 3: Definitions and measure of non-Markovianity • Lecture 4: Detections of non-Markovianity and of initial correlations Lecture 1+2 General references: [1, 2, 3] Seminal papers: [4, 5] Collisional decoherence: [6] Memory kernels: [7, 8] Lecture 3+4 General references: [9, 10, 11] Seminal papers: [12, 13, 14] Classical-quantum connection: [15] Dynamics with initial conditions: [16] [1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002) [2] A. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction (Springer, 2012) [3] B. Vacchini, Lecture notes on advanced quantum mechanics [4] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976) [5] G. Lindblad, Rep. Math. Phys. 10, 393 (1976) [6] B. Vacchini and K. Hornberger, Phys. Rep. 478, 71 (2009) [7] H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008) [8] B. Vacchini, Phys. Rev. Lett. xx, xxxxxx (2016) [9] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016) [10] A. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014) [11] I. de Vega and D. Alonso, Rev. Mod. Phys. (2016) [12] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009) [13] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010) [14] E.-M. Laine, J. Piilo, and H.-P. Breuer, EPL 92, 60010 (2010) [15] B. Vacchini, A. Smirne, E.-M. Laine, J. Piilo, and H.-P. Breuer, New J. Phys. 13, 093004 (2011) [16] B. Vacchini and G. Amato, Sci. Rep. 6, 37328 (2016) Apoorva Patel Quantum Trajectory formalism for Weak Measurements Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. Investigation of the restrictions needed on the ensemble of quantum trajectories, so as to reproduce projective measurement in the appropriate limit, shows that the Born rule follows when the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation. That implies that the noise and the attraction have a common origin in the measurement interaction between the system and the apparatus. We analyse the quantum trajectory ensemble for the dynamics of quantum diffusion and quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter, which can be tested in weak measurement experiments. We comment on how the specific noise may arise in the measuring apparatus. Second Week School Daniele Faccio Optical Models for Gravity In 1981 Unruh published a paper that initiated the field of Analogue Gravity. Originally just a theoretical endeavour, this is now involves a remarkably broad range of research areas such as Bose-Einstein condensates, optics, nonlinear optics, superfluids and hydrodynamics. More than that, and possibly the main legacy of the field, these different field often come together and find an overlap or common inspiration through the ideas of analogue gravity. In a nutshell, Analogue gravity refers to the attempt to reproduce certain aspects of the quantum field theory in specific curved spacetime geometries. Einstein’s theory of general relativity can be seen to be composed of two parts: the first is essentially a theory of geometry, i.e. the geometry of curved spacetimes. The second part are the Einsteina equations that describe how mass or the energy stress tensor modifies the surrounding spacetime metric and how this in turn modifies the mass distribution. A full quantum gravity theory must necessarily account for the full nonlinear dynamics of the Einstein equations - this level of quantisation is the main challenge of contemporary physics and is even celebrated in Hollywood movies. However, a semi-classical approach is also possible whereby one quantizes the fields, e.g. the electromagnetic field and studies the evolution of these fields on a classical spacetime metric. Such classical spacetime metrics, whilst usually considered to be the result of mass or exotic oject such as balck holes, may actually be encountered in very trivial situations and therefore easily controllable on Earth-based laboratories. The first model investigated by Unruh was simply a flowing body of water. He showed that it is possibly, by controlling the flow, to create a horizon for acoustic waves propagating inside the flow. Even more interestingly, he discovrered that the same mathematical procedures applied by hawking to predict the quantum-vacuum seeded blackbody emission from black holes also applies to these lab systems. The implications of this is that whilst it is not possible to directly verify if a black hole emits Hawking radiation, we may certainly verify in the lab if the mathematical model that predicts this emission is indeed correct. Recent results have provided possible evidence of Hawking emission from analogue black holes or horizons created in a BEC, a flowing body of water and an optical light pulse propagating in a dielectric medium. We will take a closer look at these effects, starting from a basic overview of the main curved spacetimes that are being studies, or could be studies using analogue gravity. We will look at the basic mathematical tools used to predict particle emission from a curved or time dependence spacetime metric and then apply this to a few simple cases. We will then use this to look in detail at how optical models for gravity work with some working examples. Note: this will be an experimentalist’s view of the topic. Equations will be infrequent and incomplete with more weight given to intuitive understanding of the physics at play. Lecture 1: Optical Models for Gravity, part I - optical media that change in time Contents: General overview of analogue gravity, why analogue gravity?, the “one trick pony” approach to QFT in curved spacetimes, Unruh and Hawking radiation, negative frequencies, zero-permittivity media and expanding cosmologies Lecture 2: Optical Models for Gravity, part II - superfluids made of light and rotating spacetimes Contents: Photon fluid basics, spacetime structures in photon fluids, hydrodynamic turbulence in photon fluids, Penrose superradiance from rotating black holes, Zel’dovich superradiance Lecture 3: Optical Models for Gravity, part III - Newton-Schrodinger equation in optics Contents: NSE basics, Boson stars, quantum droplets of light • S. Hawking, Nature (London) 248, 30 (1974). • C. Barcelo, S. Liberati, and M. Visser, Living Rev. Relativity 8, 12 (2005) • Analogue Gravity Phenomenology, D. Faccio, F. Belgiorno, S. Cacciatori, V. Gorini, S. Liberati, U. Moschella eds., Springer (2013) • T. Philbin et al., Science 319, 1367 (2008) • Measurement of Stimulated Hawking Emission in an Analogue System, S. Weinfurtner et al., Phys. Rev. Lett. 106, 021302 (2010) • Laser pulse analogues for gravity and analogue Hawking radiation, D. Faccio, Contemp. Phys., 53, 97 (2012) • Optical Black Hole Lasers, D. Faccio, T. Arane, M. Lamperti, U. Leonhardt, Class. Quant. Gravity 29, 224009 (2012) • Negative frequency resonant radiation, E. Rubino et al., Phys. Rev. Lett., 108, 253901 (2012) • Hawking radiation from ultrashort laser pulse filaments, F. Belgiorno et al., Phys. Rev. Lett., 105, 203901 (2010) • Observation of self-amplifying Hawking radiation in an analogue black-hole laser, J. Steinhauer, Nat. Phys 10, 864–869 (2014) • Observation of quantum Hawking radiation and its entanglement in an analogue black hole, J. Steinhauer, Nat. Phys 12, 959–965 (2016) • Ya. B. Zel'dovich, Pis'ma Zh. Eksp. Teor. Fiz. {14, 270 (1971); Zh. Eksp. Teor. Fiz. 62, 2076 (1972); [JETP Lett. 14}, 180 (1971)][Sov. Phys. JETP 35, 1085 (1972)]. • Ya. B. Zel'dovich, L. V. Rozhanskii, A. A. Starobinskii, Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, 29, I008-I016 (1986). • R. Penrose, General Relativity and Gravitation, 34, 1141 (2002) [reprinted from Rivista del Nuovo Cimento, Numero Speziale I, 257 (1969)]. • ``Superradiance'', R. Brito, V. Cardoso, P. Piani, Springer (2015) • J. D. Beckenstein and M. Schiffer, Phys. Rev. D, 58, 064014 (1998). Nikolai Kiesel Quantum Cavity Optomechanics At first glance, the minute momentum of photons seems an unlikely candidate to provide control over the motion of massive objects. Yet, today on-chip mechanical oscillators can be manipulated with only a few photons and momentum transfer from lasers can manipulate the mechanical properties of gram-scale mirrors. The relevance of this reasearch covers new approaches to answer fundamental questions about the quantum behaviour of massive objects, applications in quantum information science, and novel methods for sensing of forces and acceleration. All of this is part of cavity quantum optomechanics [1, 2, 3], where methods from quantum optics are employed to taylor the interaction between light and mechanical resonators. Theory on the subject has been investigated already over a rather long time and soon after the millenium first ground-breaking experiments have been conducted. However, only the last seven years have witnessed experimens that unambiguously demonstrate optomechanics at the quantum level. A few examples are the demonstration of entanglement between a microwave field and a mechanical resonator [4], the generation of squeezed mechanical states [5], and non-classical correlations between single photons and phonons [6]. In this lecture, we will understand the underlying principles of quantum cavity-optomechanical interaction, some of the common methods to measure its effects and examples of broadly used cavity-optomechanical systems . We will furthermore look exemplarily at a selection of the recent ground-breaking experiments to illustrate solutions and challenges in the field. 1. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, Cavity Optomechanics, Rev. Mod. Phys. 86, 1391 (2014). 2. M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, Quantum optomechanics - throwing a glance, JOSA B 27 A189 (2010). 3. T. J. Kippenberg and K. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172 (2008). 4. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, K. W. Lehnert, Entangling Mechanical Motion with Microwave Fields, Science 342, 710 (2013). 5. E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab, Quantum squeezing of motion in a mechanical resonator, Science 349, 952 (2015). 6. R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, S. Gröblacher, Nonclassical correlations between single photons and phonons from a mechanical oscillator, Nature 530, 313-316 (2016). Markus Arndt These lectures will cover methods to perform matterwave interferometry experiments. The students will gain an insight into the complexity of such experiments, such as how to coherently split matterwaves. The second lecture will give a overview of experiments done on atom and molecule matterwave interferometry, while lecture three is focusing on applications of atom and molecule interferometry for metrology and sensing. 1. Lecture 1: Realizations and ideas around coherent Matter-Wave beam splitters 1. Amplitude beam splitters: 1. Atom optics realizations of the Hadamard Gate using Rabi cycles 2. Atom optics realizations using Raman transitions 2. Wave front beam splitters: 1. nanomechanical gratings, the role of van der Waals forces 2. phase gratings, the role of the dipole force 3. photo-depletion gratings, many ways to knock it out 4. wires, discs and algae: many ways are leading to Rome 3. Far-field diffraction: experiments with 1. Atoms and dimers 2. small clusters 3. large molecules 4. Wide angular momentum beam splitters: 1. High order Bragg diffraction 2. Bloch oscillations 5. Coherent atom optics in the time domain 1. Phase modulation 2. Slits and double slits in the time domain 1. Lecture 2: Concepts and realization of matter-wave interferometers with atoms and molecules 1. Mechanical interferometer for atoms 2. Ramsey-Bordé Interferometer for atoms and molecules 3. Kasevich Chu interferometer for atoms 4. Near-field interferometry with atoms and molecules 1. Talbot-diffraction of light 2. Talbot diffraction of atoms 3. Talbot-Lau interferometry with light 4. Talbot-Lau interferometry with atoms and fullerenes 5. Kapitza-Dirac Talbot-Lau interferometry with complex molecules 6. OTIMA interferometry in the time domain 5. An introduction to theoretical concepts of matter-wave interferometry 1. Lecture 3: Applications of matter-wave interferometers with atoms and molecules 1. Atoms: 1. Gravity sensing and the equivalence principle 2. Measuring the gravitational constant G 3. Rotation sensing 4. measurement of h/m and the fine structure constant 1. Molecules: 1. polarizability and structural conformers 2. dipole moments 3. optical spectroscopy Andrea Vinante Detection of weak forces and quantum foundational problems Detection of small forces is at heart of many fundamental physics experiments. Here, we are primarily interested in understanding how experiments detecting weak classical forces using mechanical systems can be used to test modifications of quantum mechanics, such as spontaneous collapse models. These techniques are often regarded as non-interferometric, as opposed to direct tests making use of quantum superposition states. I will introduce the general features of weak force detection experiment, and illustrate the main sources of noise and dissipation in relevant experiments, including and ultrasensitive force microscopy and gravitational wave detectors. Finally, we will discuss the state of the art in testing collapse models using mechanical systems. Hendrik Ulbricht Testing fundamental physics with table-top experiments We will discuss trapping and cooling experiments of optically levitated nanoparticles [1]. We will report on the cooling of all translational motional degrees of freedom of a single trapped silica particle to 1mK simultaneously at vacuum of 10-5 mbar using a parabolic mirror to form the optical trap. We will further report on the squeezing of a thermal motional state of the trapped particle by rapid switch of the trap frequency [2]. We will further discuss ideas to experimentally test quantum mechanics by means of collapse models [3] by both matter-wave interferometry [4] and non-interferometric methods [5]. While first experimental bounds by non-interferometric tests have been achieved during the last year by a number of different experiments according to non-interferometric experiments [4], we shall also report on different matterwave interferometry experiments to test the quantum superposition principle directly for 1 million atomic mass unit (amu) particles. We will further discuss some ideas to probe the interplay between quantum mechanics and gravitation by (levitated) optomechanics experiments. One idea is to seek first experimental evidence about the fundamentally quantum or classical nature of gravity by using the torsional motion of a non-spherical trapped particle, while a second idea is to test the effect of the gravity related shift of energy levels of the mechanical harmonic oscillator, which is predicted by semi-classical gravity (the so-called Schrdinger-Newton equation) [6]. The idea is to complement topic covered during the first week of this school from the experimental perspective. 1. Vovrosh, J., M. Rashid, D. Hempston, J. Bateman, and H. Ulbricht, Controlling the Motion of a Nanoparticle Trapped in Vacuum, arXiv:1603.02917 (2016). 2. Rashid, M., T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston,M. S. Kim, and H. Ulbricht, Experimental Realisation of a Thermal Squeezed State of Levitated Optomechanics, arXiv:1607.05509 (2016). 3. Bassi, A., K. Lochan, S. Satin, T.P. Singh, and H. Ulbricht, Models of Wave-function Collapse, Underlying Theories, and Experimental Tests, Rev. Mod. Phys. 85, 471 - 527 (2013); 4. Bateman, J., S. Nimmrichter, K. Hornberger, and H. Ulbricht, Near-field interferometry of a free-falling nanoparticle from a point-like source, Nat. Com. 5, 4788 (2014); Wan, C., et al. Free Nano-Object Ramsey Interferometry for Large Quantum Superpositions, Phys. Rev. Lett. 117, 143003 (2016). 5. Bahrami, M., M. Paternostro, A. Bassi, and H. Ulbricht, Non-interferometric Test of Collapse Models in Optomechanical Systems, Phys. Rev. Lett. 112, 210404 (2014); Bera, S., B. Motwani, T.P. Singh, and H. Ulbricht, A proposal for the experimental detection of CSL induced random walk, Sci. Rep. 5, 7664 (2015). 6. Grossardt, A., J. Bateman, H. Ulbricht, and A. Bassi, Optomechanical test of the Schroedinger-Newton equation, Phys. Rev. D 93, 096003 (2016). Saikat Ghosh Feeback Control: taming atoms and nano-drums with electronic feedback Over last few decades, spectacular sophistication has been achieved in experiments involving macroscopic systems that behave quantum mechanically. These include laser-cooled and trapped atoms, ions and condensates, artificially fabricated mesoscopic quantum circuits, micro and nano-mechanical as well as a myriad of hybrid systems with a mixed combination. Experimentally, a skeletal structure that guides almost all these experiments is provided by control theory, which explains how to observe fluctuations (mostly classical) and compensate for it in real time. In this short lecture, I will pick up examples to explain the basics of control theory, as applicable to experiments with quantum systems. In particular, I will talk about how feedback control is used in widely different cases, for experiments pursued in our laboratory, with cold atoms, cavities and graphene resonators. The tutorial problems will mostly consists of understanding few simple op-amp circuits, debugging and interpreting them in the language of feedback control. We will end with a short overview of how these ideas extend to quantum feedback control. Urbasi Sinha Experimental Quantum Measure: Connection with the Superposition principle and the Born Rule These two lectures will deal with the experimental attempts at measuring the Quantum Measure. [1]. There have been several attempts at experimentally bounding the Quantum Measure using photons, atoms, NMR among others [2–6] . Recently, it has been measured to be a definite non-zero for the first time [7]. These lectures will discuss these experiments, the usual difficulties faced in such precision experiments and future experiments which could be attempted in this genre. We will also discuss the implications of such experiments in measuring the deviation from the Superposition principle in interference experiments [8, 9] as well as the Born Rule for probabilities. 1. R. D. Sorkin, Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A. 9, 3119 (1994). 2. U.Sinha, C.Couteau, T.Jennewein, R.Laflamme, G.Weihs, Ruling out multi-order interference in quantum mechanics. Science 329, 418-421 (2010). 3. S¨ollner, I. et al. Testing born’s rule in quantum mechanics for three mutually exclusive events. Found Phys. 42, 742-751 (2012). 4. Park, D. K. Moussa, O. Laflamme, R. Three path interference using nuclear magnetic resonance: a test of the consistency of Born’s rule. New J. Phys. 14, 113025 (2012). 5. T.Kauten, R.Keil, T.Kaufmann, B.Pressl, C.Brukner, G.Weihs, Obtaining tight bounds on higher-order interferences with a 5-path interferometer. arXiv :1508.03253v3 6. Markus Arndt: Private communication 7. G.Rengaraj, U.Prathwiraj, Surya N.Sahoo, R.Somashekhar and U.Sinha, Experimental measure of the correction term in the Superposition Principle, arXiv:1610.09143. 8. R.Sawant, J.Samuel, A.Sinha, S.Sinha, U.Sinha, Non classical paths in quantum interference experiments. Phys.Rev.Lett.113, 120406 (2014). 9. A.Sinha, Aravind H.V., U.Sinha, On the Superposition principle in interference experiments. Scientific Reports 5, 10304 (2015). Gregor Weihs Sources of nonclassical light Applications of nonclassical states of light in quantum optics,foundational tests, metrology, and quantum communication requireappropriate, tailored sources. The typically desired states are singlephotons, entangled photon pairs, multiphoton states, or squeezed statesof light. Various techniques have been used to produce these states. Iwill speak about sources using nonlinear optics in dielectrics andsemiconductors as well as sources based on single semiconductor quantumemitters. Third Week Discussion Meeting Markus Arndt A tale of two limits: Quantum interferometry exploring the limits of high mass and biological complexity "Quantum mechanics and its relativistic brother, quantum field theory, have remained the uncontested winners of theoretical physics throughout the last century, with surprising accuracy and experimental confirmation in the microscopic world. The basic concepts of quantum physics seem however to challenge our philosophical notions that we grew up to like in our everyday world. It is therefore legitimate to ask whether quantum mechanics is a universal theory or rather the limit of something more universal. How will quantum physics behave in a limit where the mass is sufficiently big to feel the warp of space-time? How will quantum physics affect the constituents of life? What fundamental or technological challenges do we face and what progress can we foresee in exploring these two interfaces to the macroscopic world? I will explore these two questions from an experimental and a conceptual viewpoint." Tjerk Oosterkamp A clock containing a massive object in a superposition of states; what makes Penrosian wavefunction collapse tick? Penrose has been advocating the view that the collapse of the wave function is rooted in the incompatibility between general relativity and quantum mechanics. On the basis of conceptual analysis, he arrived at an estimate for the collapse time. To better understand his estimate, in this paper we present a thought experiment, which singles out the role of time-dilations in massive superpositions. First we investigate the behavior of a hypothetical clock containing a component which can be in a superposition of states. The clock contains a massive object, whose only purpose is to introduce a curvature of space time into the problem. We find that a state of this massive object with a smaller radius, but with the same mass, experiences a larger time dilation. Considering a coherent superposition of the large and small object, introduces an ambiguity in the definition of a common time for both states. We assert that this time ambiguity can be thought to affect the time evolution of a state in different ways and that the relative phase difference between these different interpretations can be calculated. We postulate that the wave function collapse will occur when this phase difference becomes of order unity. An absolute energy scale enters this equation and we recover Penrose's estimate for the collapse time by equating the absolute energy scale to the rest mass of the object. Daniele Faccio Optical simulations of problems in quantum cosmology I will give a brief overview of ongoing research endeavours aimed at reproducing basic QFT predictions that are relevant to cosmology. Examples are photon production from expanding spacetimes (e.g. cosmological expansion), Penrose superradiance from a rotating black hole and recent experiments showing an optical analogue of the Newton-Schrodinger equation. Urbasi Sinha Quantum Superposition, Weak measurements and Higher dimensional quantum systems This talk will give a general overview of experiments in different genres that are being performed in the Quantum Information and Computing lab at RRI, Bengaluru. We will discuss an experiment which deals with measuring the deviation from the naive application of the Superposition Principle in interference experiments [1–3]. Our recently concluded experiment [4] reports the first successful measurement of the non-zero Sorkin parameter (which is commonly used in Quantum Measure Theory). Next, we will discuss an ongoing experiment involving higher dimensional quantum systems. Maximally entangled qudits are subjects of interest in many quantum information protocols and fundamental tests of quantum mechanics. Transverse spatial correlation obtained from spontaneous parametric down converted photons is one of the simplest methods that could be readily implemented using slit based interferometric systems. Recently, it was shown that, the angular spectrum of the incident pump can be transferred to the signal-idler bi-photon pair in SPDC process. Tapping on to this, we attempt to harness qutrit- qutrit correlations in spatial degrees of freedom by making the pump have a profile of a triple slit [5]. This experiment could pave the way for using the spatial degree of freedom in experiments based on long distance Quantum Communication. Finally, we will discuss an ongoing experiment which aims to infer the expectation value of Non Hermitian operators using weak measurements [6]. Weak values of Pauli operators acting on polarization state vectors are traditionally measured from the shift of the position pointer states which gets coupled to polarization degree of freedom due to the birefringent crystal, in a pre and post selected ensemble. Here, we present an alternative way to infer weak values that takes advantage of an interferometric setup. 5. Surya N. Sahoo, D.Ghosh, E.Kaur, T.Jennewein, P.Kolenderski and U.Sinha, Measuring Spatial Correlations in qutrits, to be submitted,(2016). 6. A.Pati, U.Singh and U.Sinha, Measuring Non-Hermitian operators via weak values, Physical Review A 92 052120, (2015) Nikolai Kiesel Levitated Cavity Optomechanics Cavity Optomechanics with clamped devices has been tremendously successful in the control of massive mechanical oscillators at the quantum level. However, isolating clamped microfabricated devices from their environment is an extremely hard task. Levitating mesoscopic objects has been suggested as a route to enable orders of magnitude improvement in thermal isolation compared to clamped devices. Experimentally, optically levitated resonators are already en par with the best clamped mechanical resonators in that respect. Levitated cavity optomechanics combines ultimate isolation from the environment with the full toolbox of cavity optomechanics available to control the quantum state of motion. The approach thus provides a route to combine the quantum state preparation techniques offered by quantum optics with the option for free-fall experiments that enable matter-wave interferometry. I will present the state of the art in the field and results from our ongoing experiments. The latter include studies of nanoparticles in hollow-core photonic crystal fibers and cavity-optomechanical control of nanoparticles. Finally, I will discuss some of the ideas how to exploit levitated cavity optomechanics in the context of stochastic thermodynamics and for fundamental tests of quantum physics Tom van der Reep Smoothly breaking unitarity One of the remaining issues in quantum mechanics is the apparent discrepancy between this theory and our classical world. To probe the boundary between the two realms, we propose to build a microwave interferometer that contains a travelling wave parametric amplifier (TWPA) in each of its arms. Feeding the interferometer with a single photon source and studying its output radiation while varying the amplification of the TWPAs might provide more insight on the quantum-classical transition as the unitary amplifiers smoothly turn into unitarity-breaking detectors with increasing gain. In this talk we will share our results on the expected output of the interferometer and, correspondingly, how to observe a collapse of the wavefunction within the experiment. Joseph Cotter In search of multi-path interference using large m I will present results from recent experiments where we search for multi-path interference using a far-field molecule interferometer by comparing the diffraction patterns arising from single, double and triple slits. Using a beam of phthalocyanine molecules, with a mass of 514amu, we have directly bounded the Sorkin-parameter using massive particles. Jamie Vovrosh Controlling the Motion of a Nanoparticle Trapped i Optomechanics in the macroscopic regime has great potential as a platform for testing fundamental principles of quantum physics, in addition to creating a new range of ultra-sensitive sensors. In this work we demonstrate a simple and robust geometry for optical trapping in vacuum of a single nanoparticle based on a parabolic mirror and the optical gradient force. In this trap we demonstrate rapid parametric feedback cooling of all three motional degrees of freedom from room temperature to a few mK. A single laser at 1550nm, and a single photodiode, are used for trapping, position detection, and cooling for all three dimensions. Particles with diameters from 26nm to 160nm are trapped without feedback to 10^?5mbar and with feedback engaged the pressure is reduced to 10^?6mbar. Modifications to the harmonic motion in the presence of feedback is studied, and an experimental mechanical quality factor >4├Ý10^7 is demonstrated. Thomas Durt Non-Linear Quantum Mechanics and de Broglie's Double Solution Program Recently, non-linear modifications of Schroedinger equation, like e.g. the Schroedinger-Newton equation have been studied in relation with the measurement problem. In particular, the presence of a non-linearity would explain why the wave packet associated to a quantum particle does not spread with time. In other words, particles behave as solitons, which is strongly reminiscent of de Broglie's Double Solution Program, elaborated by Louis de Broglie in the twenties, of which the de Broglie-Bohm pilot wave dynamics is a simplified version. It is also reminiscent of Poincare's pressure, invoked by Poincare in 1905 in order to solve the wave-particle duality in the context of classical field theory. We shall describe recent attempts to derive the de Broglie-Bohm pilot wave dynamics from non-linear wave dynamics, and to apply them to quantum corpuscles at one side, and to so-called bouncing oil droplets at the other side. Nalini Gurav Zeno and Anti-Zeno effects in Quantum Mechanics "Schrödinger equation gives us the time evolution operator (unitary) which tells us the dynamics of any quantum mechanical system. We can categorize this evolution with time in three different regimes. For the intermediate time interval, quantum system decays exponentially, whereas for very short or very long time intervals, decay is not exponential. Among them a short time regime is the most interesting because, if the time interval (of the observation) is too short, it never decays! One can completely halt the system from evolving.This is called as Quantum Zeno effect. The ""collapse of wave function"" playing very peculiar role here. Also, it has recently been pointed out that by exploiting the short-time features of the quantal evolution, one can also accelerate the decay. This phenomenon is known as ""Inverse or Anti Zeno effect (IZE)."" However, these effects cannot work macroscopically, because continuous or infinite measurements are physically unattainable. But we can still use it under finite number of measurements." Gregor Weihs Multipath Interference Experiments Probe the Foundations of Quantum Physics Born's rule, the superposition principly, higher-order interference, and hypercomplex representations of quantum mechanics have one thing in common: They can be tested using multipath interference experiments. Going beyond the traditional double slit or interferometer we use sensitive free-space and waveguide multipath interferometers to investigate the foundations of quantum mechanics and generalized probabilistic theories. Ashutosh Singh Manipulation of entanglement sudden death in an all-optical experimental set-up The unavoidable and irreversible interaction between an entangled quantum system and its environment causes decoherence of the individual qubits as well as degradation of the entanglement between them. En-tanglement sudden death (ESD) is the phenomenon wherein disentanglement happens in finite time even when individual qubits decohere only asymptotically in time due to noise. Prolonging the entanglement is essential for the practical realization of entanglement-based quantum information and computation proto-cols. For this purpose, the local NOT operation in the computational basis on one or both qubits has beenproposed. In this talk, I will briefly review the ESD followed by an all-optical implementation of the NOT operations such that it can hasten, delay, or completely avert ESD, all depending on when it is applied during the process of decoherence for the polarization entangled qubits as the system. The simulation results of such manipulations [1] of ESD will be presented with the experimental progress on the same. 1. Ashutosh Singh, Siva Pradyumna, A.R.P. Rau and Urbasi Sinha, “Manipulation of entanglement sudden death in an all-optical experimental set-up”, Submitted, 2016." Sourav Dutta Coupled atom-cavity system: a quantum sensor A technique for non-destructive detection of trapped ions using a strongly coupled atom-cavity system will be discussed. Generalization of the technique to detect generic two-particle interactions will be suggested. Som Kanjilal Probing Aspects of Quantum Non-Locality using Weak Interaction and Post-Selection "A novel application of weak measurement involving pointer state post-selection is proposed in order to study non-locality and its relationship with entanglement. Our scheme starts with Werner-like states( non-maximally entangled state mixed with white noise )which satisfy Bell-CHSH inequality and introduce, in one of the parties, weak interaction involving a coupling between the particle momentum and position. The position acts like a pointer coordinate. The pointer position is then post-selected, which would mean selecting or filtering the particles corresponding to a particular value of the weakly interacting particleΓÇÖs position. The resulting two-party state is in general a mixed entangled state, and its entanglement is more or less than the unfiltered state, depending on what position of the pointer is post-selected. What is particularly interesting is that in the same spirit, we can post-select states that violate the Bell-CHSH inequalities, even though the unfiltered state may not. Hence, this constitutes a demonstration of what has been called ΓÇýhidden non-localityΓÇÖ. Furthermore, we probe, using the filtered states, the relationship between entanglement and non-locality. This is done by comparing the concurrence versus the Bell-CHSH operator as functions of the filtered states corresponding to different values of the post-selected pointer coordinates. It is found that among this set of filtered states, the one that corresponds to maximum concurrence (hence, maximum entanglement) is not the one that displays maximum violation of the Bell-CHSH inequalities ( hence, maximum non-locality) . This demonstrates that maximum entanglement does not necessarily imply maximum non-locality." Fatemeh Ahmadi On a New Formulation of Microphenomena and Relativ "We develop a new formulation of microphenomena based on the principles of reality and causality. This theory provides us with a new formulation of quantum phenomena based on a unified concept of information, matter and energy. We suppose that in a definite microphysicsl context, each particle is enfolded by a probability field whose existence is contingent on the existence of the particle, but it can locally affect the physical status of the particle in a context-dependent manner. The dynamics of the whole particle-field system (PF system) obeys deterministic equations in a manner such that when the particle is subjected to a conservative force, the form of which is determined by the dynamics of the particle. Here, by using Newtonian-like equation for a one-particle system, we show how quantum dynamics will be reconciled with classical rules and find the trajectory of a PF system in terms of the particleΓÇÖs location and time. At last we will talk about the relativistic generalization of the theory. We are going to formulate the equation of motion for PF system in a form which is Lorentz-invariant. This means the description should not allow one to differentiate between frames of reference which are moving relative to each other with a constant uniform velocity." Miles Blencowe An Investigation of the Influence of Gravity on Macroscopic Mechanical Quantum Superpositions We describe our work in progress to address theoretically the influence of gravity on spatial and energy quantum superposition states of macroscopic mass systems. One relevant problem concerns whether it is possible to consistently describe a ‘quantum Cavendish’ type thought experiment where a macroscopic mass in a spatial superposition state is a source for a gravitational field. Another problem concerns whether gravity as an environment causes unavoidable decoherence of such superposition states. We argue that a quantitative analysis can be brought to bear on these two related problems by applying perturbative quantum general relativity as an effective field theory, and by also invoking analogue mechanical, quantum non-inertial reference frame systems as a guide. Dipankar Home Quantum mechanical violation of macrorealism for large spin and for large mass using the harmonic oscillator coherent state This talk seeks to provide an overview of the core ideas and key results of two different types of recent studies concerning the quantum mechanical (QM) violation of macrorealism (MR): (a) For multilevel spin systems, using two different necessary conditions of MR, namely, the Leggett-Garg inequality (LGI) and Wigner’s form of the Leggett – Garg inequality (WLGI), the extent to which the QM violation of MR can be demonstrated in the asymptotic limit of spin, even for arbitrary coarse-grained or unsharp (noisy) measurements, is investigated. It is shown that classicality in the sense of satisfying MR does not emerge in the asymptotic limit of spin, whatever be the unsharpness or coarse-grainedness of measurements. (b) The QM violation of LGI or WLGI in the context of a linear harmonic oscillator is invoked to reveal non-classicality of the state which is considered the most “classical-like” of all quantum states, namely the Schrodinger coherent state. In the macrolimit, the extent to which such nonclassicality persists for large values of mass and classical amplitudes of oscillation is quantitatively investigated, and the relevant results will be presented, hinting a possible experimental setup using nano-objects. Daniel Bedingham Collapse models and spacetime symmetries A relativistic and time-symmetric picture of dynamical collapse of the wave function is presented. The part of the model which exhibits these symmetries is the set of collapse outcomes. These play the role of matter distributed in space and time. It is argued that the dynamically collapsing quantum state, which is both foliation dependent and follows a time asymmetric dynamics, is not fundamental: it represents a state of information about the past matter distribution for the purpose of estimating the future matter distribution. It is also argued from the point of view of collapse models that both special and general relativistic considerations point towards a discrete spacetime structure and that gravity may not need to be quantised to give a theory that is consistent with quantum matter. Apoorva Patel Understanding the Born rule in weak measurements "Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the framework to model quantum measurement as a continuous nonlinear stochastic process. It combines attraction towards the measured operator eigenstates with white noise, and for a specific ratio of the two reproduces the Born rule. We emphasise some striking features this result, which would be important ingredients for understanding the origin of the Born rule in quantum measurements." Suman Chand Quantum Otto heat engine and refrigerator We demonstrate how a quantum Otto engine (QOE) can be implemented how to implement a quantum Otto engine in trapped ion set-up. The existing proposals on implementing quantum heat engine consider `switching off' the interaction between the working fluid and the bath during the cycle. In a quantum system, it is quite challenging. In our work we show that one can implement the quantum Otto engine in a realistic trapped ion set up, without switching off the bath. The electronic state of the ion is chosen as the working fluid while its vibrational degree of freedom works as a cold bath. The adiabatic stages of the Otto cycle involve change in the local magnetic field, while a projective measurement of the electronic state of the ion leads to heat release to the cold bath. Further we are considering two trapped ions in the system. This gives an entanglement effect on the whole study. The the effect of entanglement in work-efficiency of the engine give many interesting results. Shreya Banerjee Quantum discord-tool for comparing collapse models "The quantum to classical transition maybe caused by decoherence or by dynamical collapse of the wave-function. We propose quantum discord as a tool, 1) for comparing and contrasting the role of a collapse model (Continuous Spontaneous Localization) and various sources of decoherence (environmental and fundamental), 2) for detecting collapse model and fundamental decoherence for an experimentally demonstrated macroscopic entanglement (where the effect of environmental decoherence is negligible). We discuss the experimental times which will lead to the detection of either Continuous Spontaneous Localization or fundamental decoherence. We further put bounds on the collapse parameters from this experiment for quantum discord and compare them with those obtained by a similar study of quantum entanglement. [ arXiv:1604.05834]" Ankur Mandal Some importance of "time delay" in quantum theory In general scattering processes involves a time delay to take place. Current state-of-the-art experimental techniques are capable of measuring the time delay in Photoionization, which can be described by the time reversed scattering process. This tiny time delay (of the order of attosecond) is very important, since the quantum description of the interaction of light and matter is to be understood clearly to derive the experimental results with better accuracy. In this presentation, a review of the recent works in this field will be given. Adrian Kent Quantum Reality via Late Time Photodetection I investigate postulates for realist versions of relativistic quantum theory and quantum field theory in Minkowski space and other background space-times. According to these postulates, quantum theory is supplemented by local variables that depend on possible outcomes of hypothetical measurements on the late time electromagnetic field in spacelike separated regions. We illustrate the implications in simple examples using photon wave mechanics, and discuss possible extensions to quantum field theory. Joseph Samuel Exceptional Points and Quantum Information "My talk will address the behaviour of quantum systems near an exceptional point. These are points in the space of Hamiltonians which show remarkable geometric and topological properties. Some of these properties have already been experimental realised in analog classical wave systems. My talk will concern the use of the geometry and topolgy near exceptional points of quantum systems to achieve a degree of control in manipulating quantum information." Rafael D Sorkin The quantum measure (and how to measure it) Sumati Surya Covariant Observables in Causal Set Quantum Gravity The standard formulation and interpretation of quantum theory which depends crucially on the existence of external observers is inadequate to describe closed systems and in particular quantum cosmology. How then are we to understand quantum amplitudes and probabilities associated with the very early universe? The quantum measure formulation is an attempt to address this question, drawing on the close analogy between quantum theory and classical stochastic processes. At its most basic, it requires that events of zero quantum measure are ``precluded'' or do not occur. In this talk I will examine how these ideas can be used to find precluded covariant observables in a simple model of causal set quantum cosmology. Lajos Diosi Gravity-related alterations of non-relativistic quantum theory Possible gravity-related limitations of standard quantum mechanics and a fundational role of gravity in quantum-classical transition have been under consideration for about three decades. Various concepts have been developped, leading to variants of non-linear and stochastic modifications of the standard Schrodinger equation by terms proportional to Newtons's G. Their status will be considered, also in the light of some particular tests, and their perspectives will be outlined." Kinjalk Lochan Quantum Correlations in curved spacetime "Quantum correlations play very decisive role in characterizing a system classical or quantum. The role of correlations has been studied in various systems in standard laboratory settings. We will discuss applications of such quantum correlations in the curved spacetime, particularly in black hole settings, which gives rise to some (classically) counter intuitive phenomenon, such as Unruh radiation for inertial observers without any (classical) source. Reference : arXiv:1603.01964" Souradeep Sasmal A proposed steering criterion using Generalised Uncertainty Relation "Reid steering criterion (Phys. Rev. A 40, 913 (1989)) based on Heisenberg uncertainty relation fails to detect steerability for the states having higher than second order correlation. Here, we have derived a new steering criterion using generalized uncertainty relation. Our steering criterion overcomes the limitation of Reid steering criterion and our derived steering inequality is tighter than Reid steering criterion. The proposed steering criterion is able to detect steerability of LG beam, NOPA state and photon annihilated NOPA states. Furthermore, the steerability of the two mode Werner state in continuous variable systems are investigated for all range of the mixedness parameter." Debarshi Das Probing quantum nonlocality of bipartite qutrits by generalising Wigner's argument "Bell-type local realist inequalities are developed for demonstrating quantum nonlocality of bipartite entangled qutrit states by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment is based upon assuming existence of the overall joint probability distributions for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition, and yielding the measurable marginal probabilities. The salient features of the paper are as follows: a) We first show that such generalised Wigner inequalities (GWI) are violated by quantum mechanics (QM) for both the bipartite qutrit isotropic and singlet states, thereby revealing their nonlocality. b) The efficacy of GWI is then probed by comparing its QM violation with that obtained for the other two types of local realist inequalities that have been used for probing nonlocality of entangled bipartite qutrits. This is done in two differentways, (i) by employing unsharp measurements, and (ii) by incorporating white noise in the isotropic and singlet qutrit states. c) It is shown that in both these cases, contingent upon using z-component spin-1 observables, GWI outperforms the other two types of local realist inequalities in terms of the respective QM violations, by being (i) more robust against the unsharpness of measurement, and (ii) more tolerant to the white noise incorporated in the states considered." Shiladitya Mal Sharing of Nonlocality of a single member of an En We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1/2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other [1]. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing. Anirudh Reddy Entropy and Geometry of Quantum States We derive a metric on the space of density matrices using the relative entropy as the starting point. This metric enables us to distinguish nearby quantum states. We derive an explicit form in the context of a qubit. We notice that there is an advantage in using the quantum relative entropy over its classical counterpart in the context of measurements on quantum states. T. Padmanabhan GR And QG: The Next Hundred Years It appears that the field equations of GR - which describe the dynamical evolution of the spacetime - have the same conceptual status as the equations of fluid mechanics or elasticity, suggesting a paradigm shift in our ubderstanding of gravity. In particular, it may be incorrect to think of Cosmology as a part of GR described by a specific solution to gravitational field equations. I will describe several aspects of these results and how it could lead to a solution of the cosmological constant problem. Daniel Sudarsky Dynamical Reduction in General Relativistic Contexts Spontaneous collapse theories provide one of the most promising approaches to dealing with the measurement problem in Quantum Theory (QT) . Recent advances have provided versions of the theory that are compatible with special relativity. However for a theory to be truly viable, it should also be made compatible with General Relativity (GR). I will describe some of the issues that arise when attempting to follow that path, and discuss some ideas about how these might be addressed. I will then argue that , in fact, it is in various situations where GR and QT come together, that collapse theories exhibit their potential in the most spectacular manner, offering plausible resolutions to issues that have remained unresolved for a long time. Ward Struyve Must space-time be singular? According to Einstein's theory of general relativity space-time singularities such as a big bang typically occur. It has been believed that a quantum theory for gravity might avoid such singularities. The answer will of course depend on which approach to quantum gravity one considers, such as e.g. loop quantum gravity or the Wheeler-DeWitt approach. It will also depend on which version of quantum theory one adopts. I will consider the Bohmian version of quantum mechanics, for which the question in the title is well-posed. For mini-superspace models I will show that there is no singularity for loop quantum gravity, while there may be a singularity in the Wheeler-DeWitt approach Parampreet Singh Consistent quantum histories and the probability for singularity resolution "In this talk, we will apply the methods of generalized quantum mechanics to answer the questions about the fate of singularities in different quantum cosmological models. In particular, we will discuss the way consistent histories formalism can be applied in a rigorous way to extract consistent probabilities for singularities or bounce to occur. We will show that for Wheeler-DeWitt quantum cosmology, an arbitrary superposition of physical states result in probability of singularity to be unity. In loop quantum cosmology, the probability for singularity to occur turns out to be zero. We will also discuss a covariant generalization of these results." Sumanta Chakraborty Information Retrieval from Black Holes "It is generally believed that, when matter collapses to form a black hole, the complete information about the initial state of the matter cannot be retrieved by future asymptotic observers, through local measurements. This is contrary to the expectation from a unitary evolution in quantum theory and leads to (a version of) the black hole information paradox. Classically nothing else, apart from mass, charge and angular momentum is expected to be revealed to such asymptotic observers after the formation of a black hole. Semi-classically, black holes evaporate after their formation through the Hawking radiation. The dominant part of the radiation is expected to be thermal and hence one cannot know anything about the initial data from the resultant radiation. However, there can be sources of distortions which make the radiation non-thermal. Although the distortions are not strong enough to make the evolution unitary, these distortions carry some part of information regarding the in-state. In this work, we show how one can decipher the information about the in-state of the field from these distortions. We show that the distortions of a particular kind --- which we call {\it non-vacuum distortions} --- can be used to \emph{fully} reconstruct the initial data. The asymptotic observer can do this operationally by measuring certain well-defined observables of the quantum field at late times. We demonstrate that a general class of in-states encode all their information content in the correlation of late time out-going modes. Further, using a $1+1$ dimensional CGHS model to accommodate back-reaction self-consistently, we show that observers can also infer and track the information content about the initial data, during the course of evaporation, unambiguously. Implications of such information extraction are discussed." Antoine Tilloy An alternative to the Schrodinger Newton approach The Schrodinger-Newton (SN) equation is often used to model the gravitational interaction between quantum particles. It is well known that its non-linearity introduces fundamental problems such as faster than light signalling and Born rule breakdown. Starting from continuous dynamical reduction models like CSL, we show that it is possible to couple quantum matter and a classical gravitational field in a different way, formally inspired from quantum feedback. The resulting theory is fully explicit, linear at the master equation level, and suffers from no obvious inconsistency. Predictions include the usual Newtonian pair potential, additional gravitational decoherence and, importantly, no one particle self-interaction in contrast with what the SN approach predicts. We will argue that due to its increased consistency and comparable simplicity, our theory could be chosen as a reasonable alternative to the SN approach. Sayantani Bera A New Stochastic Schrodinger Newton equationn We propose a modified stochastic Schrodinger-Newton equation which takes into account the effect of extrinsic spacetime fluctuations. We use this equation to demonstrate gravitationally induced decoherence of two gaussian wave-packets, and obtain a decoherence criterion similar to those obtained in the earlier literature in the context of effects of gravity on the Schrodinger equation. Suratna Das Cosmic inflation and the measurement problem Cosmic inflationary paradigm beautifully explains the origin of the large structures, like galaxies and cluster of galaxies, we observe today in our universe. But the fluctuations during inflation, which seed these large scale structures, were quantum by origin, whereas the structures which were developed due to gravitational instability seeded by these quantum perturbations are all classical. Thus the persisting question of how and when these primordial quantum perturbations become classical still plagues the basic concept behind inflationary paradigm. In this talk we will seek for a plausible answer to this lingering issue by implementing collapse models of quantum mechanics in early universe. Jaffino Stargen Quantum-⁠to-⁠classical transition and imprints of wavefunction collapse in bouncing universes Authors: D. J. Stargen, V. Sreenath and L. Sriramkumar Abstract: The perturbations in the early universe are supposed to have originated due to quantum fluctuations, which turn classical as the universe evolves. The quantum-to-classical transition of the primordial perturbations have been studied to a good extent in the context of inflationary cosmology. A reasonably popular alternative to the inflationary paradigm are the bouncing scenarios, wherein the universe undergoes a phase of contraction until the scale factor reaches a minimum value, before it begins to expand. Such bouncing scenarios can provide well motivated initial conditions at early times during the contracting phase, as inflation does. In this talk, divided into two parts, we consider two issues related to the quantum-to-classical transition of tensor perturbations in bouncing universes. In the first part, we describe the evolution of the quantum state (specifically, the extent of the squeezing) of the tensor modes with the aid of the Wigner function. In the second part, we consider the evolution of the tensor perturbations from the perspective of the quantum measurement problem. In particular, we discuss the effects of wave function collapse, using a phenomenological model known as continuous spontaneous localization, on the tensor power spectra. Madhavan Varadarajan A note on entanglement entropy, coherent states and gravity "We review two recent attempts to explain aspects of gravitational physics through changes in entanglement entropy of quantum fields. The first, due to Bianchi, builds on Sorkin's seminal ideas in the 80's and attempts to connect the change in black hole entropy with that of entanglement entropy without recourse to an explicit UV cutoff. The second, due to Jacobson, seeks to derive the equations of gravitational dynamics itself (which relate matter stress energy to spacetime curvature), through a certain maximal entanglement entropy hypothesis. Common to both attempts is the emergence, purely from quantum entanglement, of (changes in) stress energy. On the other hand, we note that the entanglement entropy of a free quantum field in a coherent state is {\em independent} of its stress energy content. We explore the tension between this fact and Bianchi's and Jacobson's ideas." André Großardt Quantum mechanics for non-inertial observers I discuss the difficulties arising for the definition of centre-of-mass coordinates in a relativistic system. I will further consider how the centre-of-mass motion should be described from the point of view of accelerated observers. I will discuss consequences for alleged decoherence effects of the centre-of-mass of a complex quantum system in the presence of post-Newtonian gravitational forces. Jerome Martin Cosmic Inflation and Quantum Mechanics According to cosmic inflation, the inhomogeneities in our universe are of quantum mechanical origin. This scenario was recently spectacularly confirmed by the data obtained by the European Space Agency (ESA) Planck satellite. In fact, cosmic inflation represents the unique situation in Physics where quantum mechanics and general relativity are needed to establish the predictions of the theory and where, at the same time, we have high accuracy data at our disposal to test the resulting framework. So inflation is not only a phenomenologically very appealing theory but it is also an ideal playground to discuss deep questions in a cosmological context. In this talk, I review and discuss those quantum-mechanical aspects of inflation. In particular, I explain why inflationary quantum perturbations represent a system which is very similar to systems found in quantum optics. But I also point out the limitation of this approach and investigate whether the large squeezing of the perturbations can allow us to observe a genuine observational signature in the sky of the quantum origin of the cosmological fluctuations. Aephraim Steinberg How to count one photon and get a(n average) result of 1000… (in binary) I will present our recent experimental work using electromagnetically induced transparency in laser-cooled atoms to measure the nonlinear phase shift created by a single post-selected photon, and its enhancement through "weak-value amplification." Put simply, due to the striking effects of "post-selective" quantum measurements, a (very uncertain) measurement of photon number can yield an average value much larger than one, even when it is carried out on a single photon. I will say a few words about possible practical applications of this "weak value amplification" scheme, and their limitations. Time permitting, I will also describe other future and past work related to quantum metrology and ultracold atoms – in particular, we have implemented a quantum-information-inspired protocol to beat “Rayleigh’s curse” for resolving closely-separated spots in classical imaging; and we have preliminary evidence of a narrow Fabry-Perot resonance for atoms. T. S. Mahesh Exploring Quantum Physics using Spin Ensembles Nuclear spin ensembles controlled via nuclear magnetic resonance (NMR) techniques have long been used to study various aspects of quantum physics. The nuclear spins with weak magnetic moments are reclusive enough to retain quantum coherences for long durations – seconds to minutes – sufficient for implementing intricate unitary operators. Highly sophisticated modern spectrometers also allow precise control over spin dynamics via digitally modulated radio waves. Accordingly, NMR has been regarded as a convenient testbed for emulating quantum phenomena. After a brief introduction, I will describe some of our recent experiments, particularly – (i) simulating ‘quantum pigeon hole effect’ and (ii) discriminating between von Neuman and Lüders measuring devices. Suvrat Raju The Information Paradox and State-Dependence Over the past few years, our understanding of the information paradox has improved greatly. In flat space, it appears to be now clear that the paradox can be satisfactorily resolved through black-hole complementarity. This is the idea that local operators in the interior of the black hole can be represented as very complicated polynomials of local operators in the exterior, and it can be realized explicitly in some simple examples. However, I will describe how a consideration of the information paradox in anti-de Sitter space leads to some new questions, including whether local operators in quantum gravity may be "state dependent". Herbert Spohn Landau-lifshitz Equations of Radiative Damping In a $\bar{h} = 0$ world radiative friction is very small but the effective equations for the motion of charges are still unsettled. Based on a model for a charge coupled to the Maxwell field I will explain why the equations written down around 1950 by Landau and Lifshitz are the appropriate effective equations.
65a5fdadc422f1fe	(Representation of a helium-4 atom with, appearing pink in the center, the atomic nucleus and, in gradient of gray all around, the electronic cloud. The helium-4 nucleus, enlarged on the right, is formed of two protons and two neutrons.) An atom (ancient Greek ἄτομος [atomos], “non-breaking”) is the smallest part of a single body that can chemically combine with another. Atoms are the elementary constituents of all solid, liquid or gaseous substances. The physical and chemical properties of these substances are determined by the atoms that constitute them as well as by the three-dimensional arrangement of these atoms. Contrary to what their etymology suggests, atoms are not indivisible but are themselves subatomic particles. Atoms comprise a nucleus, which concentrates more than 99.9% of their mass, around which electrons are distributed, which form a cloud 10,000 to 100,000 times larger than the nucleus itself, so that the volume of an atom, roughly spherical, is almost entirely empty. The nucleus is formed of protons, carrying a positive electric charge, and neutrons, electrically neutral; hydrogen is an exception, because the nucleus of its isotope 1H, called protium, contains no neutron. Protons and neutrons, also called nucleons, are held together in the nucleus by the nuclear bond, which is a manifestation of strong interaction. Electrons occupy atomic orbitals interacting with the nucleus via the electromagnetic force. The electronic cloud is stratified into quantized energy levels around the nucleus, levels that define layers and electronic sub-layers; the nucleons are also distributed along nuclear layers, although a fairly convenient approximate model popularizes the nuclear structure from the liquid drop model. Several atoms can establish chemical bonds between them thanks to their electrons. In general, the chemical properties of atoms are determined by their electronic configuration, which results from the number of protons in their nucleus. This number, called an atomic number, defines a chemical element. 118 chemical elements have been recognized by the International Union of Pure and Applied Chemistry (IUPAC) since November 18, 2016. The atoms of different elements have different sizes, so generally different masses, although the atoms of a given chemical element may have different masses depending on the isotopes considered. The heavier atoms, or whose nucleus has too much imbalance between the two types of nucleons, tend to become more unstable, and are then radioactive; lead 208 is the heaviest stable isotope. The atomist theory, which supports the idea of a material composed of indivisible “grains” (against the idea of an indefinitely separable matter), has been known since Antiquity, and was defended by Leucippe and his disciple Democritus, philosophers of ancient Greece, as well as in India, more previously, by one of the six schools of Hindu philosophy, the Vaisheshika, founded by Kanada. It was disputed until the end of the 19th century and has not been questioned since then. Direct observation of atoms became possible only in the middle of the 20th century with transmission electron microscopy and the invention of the tunneling microscope. It is thus on the properties of atoms that all the sciences of modern materials are based, while the elucidation of the nature and structure of atoms has contributed decisively to the development of modern physics, especially quantum mechanics. . Orders of magnitude The estimated diameter of a “free” atom (excluding covalent or crystalline bond) is between 62 μm (6.2×10-11 m) for helium and 596 μm (5.96×10-10 m) for cesium, while that of an atomic nucleus is between 2.4 fm (2.4×10-15 m) for the isotope 1H and 14.8 fm (1.48×10-14 m) approximately for the nuclide 238U: the nucleus of a hydrogen atom is therefore about 40,000 times smaller than the hydrogen atom itself. The core, however, concentrates most of the mass of the atom: the nucleus of lithium 7, for example, is about 4,300 times more massive than the three electrons that surround it, the atom of 7Li itself having a mass of the order of 1.172×10-26 kg. To put it in perspective, the mass of atoms is between 1.674×10-27 kg for protium and 3.953×10-25 kg for uranium 238, sticking to the isotopes that have a significant abundance in the natural environment (There are heavier nuclei but also more unstable than the 238U nuclide). This mass is generally expressed in of atomic mass units (“amu” or “u”), defined as the twelfth part of the mass of an unbound 12C atom and its ground state, ie 1 amu = 1.66054×10-27 kg. An alternative unit also widely used in particle physics is the electron-volt divided by the square of the speed of light (eV/c2), which is homogeneous to a mass under the famous equation E = mc2 of relativity restricted, and which is 1 eV/c2 = 1.783×10-36 kg; in this unit, the mass of the nucleus 238U is equal to 221.7 GeV/c2. Given their size and their mass singularly reduced, the atoms are always in very large number as soon as one handles a quantity of macroscopic matter. We thus define the mole as the quantity of matter constituted by as many elementary units (atoms, molecules, electrons, etc.) as there are atoms in 12 g of carbon 12, not less than 6,022×1023 elementary units, what is called the Avogadro number. Subatomic particles Although its etymology means “indivisible” in ancient Greek, an atom is actually made up of smaller elementary particles, and can therefore be divided; but it is indeed the smallest indivisible unit of a chemical element as such: by breaking, for example, a helium atom, we will obtain electrons, protons and neutrons, but we will no longer have a simple body having the properties of helium. • The electron e is a very small particle (9.1094×10-31 kg, 511,00 keV/c2) and has a negative electric charge of -1,602×10-19 C. • The p+ proton is 1836 times more massive than the electron (1.6726×10-27 kg, or 938.27 MeV/c2) and has a positive electric charge of the same absolute value as the electron (1.602×10-19 C). • The neutron n0 is 1838.5 times more massive than the electron (1.6749×10-27 kg, ie (939.57 MeV/c2), and electrically neutral. The standard model of particle physics describes nucleons as baryons composed of elementary particles called quarks: • the proton consists of two up quarks and one down quark: p+ = uud; • the neutron consists of one up quark and two down quarks: n0 = udd. Electrons, on the other hand, are leptons, which together with quarks constitute the group of fermions. The big difference between quarks and leptons is that only the first ones know all the elementary interactions, including the strong nuclear interaction, whose mediators are gauge bosons called gluons; the leptons know only the weak interaction (via the Z0 and W+ bosons) and the electromagnetic interaction (via the photons). All these particles also know a priori gravitational interaction, but the latter has not yet been integrated into the standard model of particle physics; its intensity on the atomic scale is, in any case, insignificant compared to the intensity of the other three interactions. Electronic cloud O groapă de potențial(Schematic representation of a potential well. The energy V(x) required to occupy each x-coordinate confines to the interval [x1, x2] any particle provided with the energy E on the axis.) The essential physical and chemical properties of atoms are due to their electronic cloud. It is the understanding of the nature and structure of this electronic cloud that has paved the way for understanding the structure of the atom itself and, ultimately, has led to the development of particle physics. The atomic nucleus being positively charged, it forms a potential well for the electrons, which are negatively charged. This potential well consists of energy levels defined by quantum numbers whose combination determines atomic orbitals conferring on the corresponding wave functions dimensions and characteristic shapes. (Atomic orbitals – Periodic table – How atoms are constructed from electron orbitals and link to the periodic table) Introduction to the Schrödinger model The electron manifests, like any quantum object, a wave-particle duality, by virtue of which it behaves sometimes like a geometrically delimited particle occupying a determined position, sometimes like a wave likely to present, for example, phenomena of interferences. These two aspects of the electron coexist in the atom, although the Schrödinger model is exclusively undulatory: • an electron is never located at a precise location on a trajectory defined around the nucleus, but distributed within an atomic orbital with a probability of presence equal to the square of the norm of its wave function, which is correlated to its quantum state, as well as to an electron phase: it is the undulatory aspect; • this distribution is not static, but dynamic, in that the electron is provided, within its stationary atomic orbital, with a quantity of motion and an orbital angular momentum: it is the corpuscular aspect. Atomic orbitals (Illustration of atomic orbitals.) Therefore, an electron can not “fall on the nucleus” as an object falls to the ground, as this would mean that the spatial extension of its wave function would be reduced to a point, which is not the case of no eigenfunction of the Schrödinger equation: the latter requires, on the contrary, that an electron, in the vicinity of the nucleus, be “diluted” in a volume (an orbital) to the geometry determined by the quantum numbers which satisfy this equation. We can therefore consider that an electron in an atom has already fallen on the nucleus, insofar as it is confined in its vicinity by the electrostatic potential well. Moreover, the wave function of an electron is not zero inside the nucleus, although its probability of being there is small, because the nucleus is of very small size compared to that of the atomic orbitals . The possible wave functions for the electrons of an atom being centered on the nucleus, we can thus say that the electron has in fact fallen into the nucleus, although it is there only very rarely: from the quantum view point, several particles can indeed occupy the same space by virtue of their wave nature. An imaginary – but approximate – way of seeing things is to imagine, by analogy, that the wave function of the electron would be as “diffracted” by the atomic nucleus, which would give it different forms, depending on its quantum state , by which the probability of the presence of the electron reaches its maximum in certain zones more or less distant from the nucleus – typically, several tens of thousands of times the nuclear radius. Atomic nucleus Protons and neutrons form an atomic nucleus of femtometric size. The nuclear radius of an atom whose mass number is A is about 1.23√A fm, while the atom itself has a radius of the order of a hundred picometers (about 35,000 to 40,000 times bigger). The protons being positively charged, they repel within the nucleus, but the intensity of this electrostatic repulsion is much lower than that of the attraction between nucleons induced by the strong nuclear interaction at distances less than 2.5 fm. The geometry of atomic nuclei is generally spherical, although some sufficiently massive stable nuclei also adopt spheroidal shapes drawn into a rugby balloon or, on the contrary, flattened. Some unstable nuclei, known as halo nuclei, are characterized by one or more nucleons with very distended wave functions, which give the nucleus fuzzy outlines and a much increased apparent volume; these nuclei have a nuclear cohesion at the extreme limit of the field of action of the strong interaction. In the liquid drop model, the protons tend to repel each other and, consequently, to concentrate outwards from the nuclei (at the “poles” or at the “equator” in the case of spheroids) , while neutrons tend to accumulate in the center of the nucleus. Dozens of models have been proposed to explain experimental data on the nature and structure of atomic nuclei, but none to date is sufficient to account for all observations. The nuclear volume, estimated experimentally by electron beam diffraction techniques, corresponds roughly to the stacking of hard spheres representing the nucleons, with a constant nuclear density, which is very well conceptualized with the model of the liquid drop. Nevertheless, some quantum properties of the nuclear structure seem better described by the layered model, developed by German physicists Maria Goeppert-Mayer and Hans Daniel Jensen, who won the Nobel Prize for Physics in 1963 for this breakthrough. Their model considers the nucleons as fermions subject to the Pauli exclusion principle and distributed on quantified energy levels – the “nuclear layers” – similar to the electrons at the atomic scale. In the nucleus, protons and neutrons constitute two distinct fermion populations with respect to the Pauli exclusion principle. The analogy with the electrons, however, has its limits because, if the electrons interact with each other and with the nucleus via the electromagnetic interaction, the nucleons interact with each other essentially via the strong nuclear interaction and the weak interaction. The energy levels within the nucleus thus have a different distribution than the energy levels of the electrons of an atom. In addition, the spin-orbit coupling phenomena are much more sensitive for the nucleons than for the electrons, which redistributes the nuclear sub-shells according to the spin (indicated in index in the table below): Sub-shell 1s 1/2 2 states 1st shell : magic number = 2 Sub-shell 1p 3/2 4 states Sub-shell 1p 1/2 2 states 2nd shell : magic number = 8 Sub-shell 1d 5/2 6 states Sub-shell 2s 1/2 2 states Sub-shell 1d 3/2 4 states 3rd shell : magic number = 20 Sub-shell 1f 7/2 8 states 4th shell : magic number = 28 Sub-shell 2p 3/2 4 states Sub-shell 1f 5/2 6 states Sub-shell 2p 1/2 2 states Sub-shell 1g 9/2 10 states 5th shell : magic number = 50 Sub-shell 1g 7/2 8 states Sub-shell 2d 5/2 6 states Sub-shell 2d 3/2 4 states Sub-shell 3s 1/2 2 states Sub-shell 1h 11/2 12 states 6th shell : magic number = 82 Sub-shell 1h 9/2 10 states Sub-shell 2f 7/2 8 states Sub-shell 2f 5/2 6 states Sub-shell 3p 3/2 4 states Sub-shell 3p 1/2 2 states Sub-shell 1i 13/2 14 states 7th shell : magic number = 126 Sub-shell 2g 9/2 10 states Sub-shell 3d 5/2 6 states Sub-shell 1i 11/2 12 states Sub-shell 2g 7/2 8 states Sub-shell 4s 1/2 2 states Sub-shell 3d 3/2 4 states Sub-shell 1j 15/2 16 states 8th shell : magic number = 184 The saturation of a nuclear shell confers to the atomic nucleus a stability superior to that calculated by the Weizsäcker formula, resulting from the model of the liquid drop – which is not unlike the chemical inertia of the rare gases, characterized by the saturation of their peripheral electronic sub-shell. The number of nucleons of a given population corresponding to the saturation of a nuclear shell is called the “magic number”; the nucleus of the lead 208, which is the heaviest of the stable isotopes, thus consists of 82 protons and 126 neutrons: 82 and 126 are two magic numbers, which explains the stability of this nuclide compared to those which do not differ from it than with one or two nucleons. Leave a Reply
94a6e657ef4256c3	Aspects of Wave Interaction in Nonlinear Media Doktorsavhandling, 2005 Selected aspects of various types of nonlinear wave interaction are investigated. These aspects are studied in terms of models described by Nonlinear Schrödinger equations, mainly analytically and numerically, but also experimentally. The emphasis is on fundamental phenomena and mathematical methods, aiming to take full advantage of the formal analogies between different physical fields. The thesis contains ten appended papers. Paper I investigates four-wave mixing of femtosecond pulses propagating close to the zero-dispersion wavelength in optical fibers. Papers IIIII investigate how a non-monotonic chirp can split a pulse in an optical fiber. Paper IV determines the ground states of a Bose-Einstein condensate described by the Gross-Pitaevskii equation by means of a variational approximation with a super-Gaussian ansatz. Paper V describes a coupled-mode theory for Bose-Einstein condensates in a harmonic double-well trap. In Papers VI, VII and IX, the Wigner transform method is used for modelling nonlinear propagation of incoherent light, and the method is applied to modulational instability of a single plane wave (Paper VI) and of two interacting plane waves (Papers VII and IX). Paper VIII applies the Wigner transform method to studying the stability of a multistream quantum plasma. Paper X treats the modulational instability of partially coherent waves by obtaining the full solution of the perturbations as an initial value problem. nonlinear Schrödinger equation incoherent solitons partial incoherence nonlinear wave interaction Gross-Pitaevskii equation modulational instability collective dynamics Wigner transform nonlinear guided waves Bose-Einstein condensation Björn Hall Chalmers, Institutionen för radio- och rymdvetenskap Elektroteknik och elektronik Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 2320 Mer information
65033222d94cf39b	Weekly Papers on Quantum Foundations (5) Dawid, Richard (2017) The Significance of Non-Empirical Confirmation in Fundamental Physics. [Preprint] Gomori, Marton and Szabo, Laszlo E. (2017) On the Persistence of the Electromagnetic Field. [Preprint] Authors: Tung-Ho ShiehKun-Yuan WuHsiu-Fen KaoKuan-Ming Hung Frequent Measurements on an unstable particle located at observable initial state freeze the particle on this state, known as quantum Zeno effect [1-14]. Measurements on an observable subspace further open the prelude of quantum Zeno dynamics [15-17]. These phenomena affect the results of quantum measurement that has been widely used in quantum information and quantum computation [18-21]. However, this common argument is insufficient when the initial state is coupled with a continuum reservoir. In such an irreversible system, intrinsic decay property destroys the frozen behavior. Although it has been proven that the decay rate of the initial state will be affected by measurements [7], the knowledge of detailed mechanisms for this measurement-dependent decay is still limited. In this work, we found based on three-subsystem perturbation theory that the indirect correlation between a continuum reservoir and a measurement apparatus through the interaction with a quantum system causes a Rabi-like inter-modulation among these subsystems. The indirect correlation effect from this combination drives the system into quantum Zeno and reactivation regions, which are dependent upon the Rabi strength, pulse duty, and its repetition rate. We also found that the dynamical changes of decay rate are quite different in continuous and pulsed measurements, and, thus, significantly affect the dynamics of the system. Author(s): Gentaro Watanabe, B. Prasanna Venkatesh, Peter Talkner, and Adolfo del Campo Theoretical calculations show that the performance of a quantum heat engine over several cycles can’t be judged by analyzing just a single cycle. [Phys. Rev. Lett. 118, 050601] Published Thu Feb 02, 2017 Authors: John F. DonoghueMikhail M. IvanovAndrey Shkerin These notes are an introduction to General Relativity as a Quantum Effective Field Theory, following the material given in a short course on the subject at EPFL. The intent is to develop General Relativity starting from a quantum field theoretic viewpoint, and to introduce some of the techniques needed to understand the subject. Authors: Björn SchrinskiBenjamin A. SticklerKlaus Hornberger We show how the ro-translational motion of anisotropic particles is affected by the model of Continuous Spontaneous Localization (CSL), the most prominent hypothetical modification of the Schr\”odinger equation restoring realism on the macroscale. We derive the master equation describing collapse-induced spatio-orientational decoherence, and demonstrate how it leads to linear- and angular-momentum diffusion. Since the associated heating rates scale differently with the CSL parameters, the latter can be determined individually by measuring the random motion of a single levitated nanorotor. Authors: Nicolas GisinQuanxin MeiArmin TavakoliMarc Olivier RenouNicolas Brunner The nature of quantum correlations in networks featuring independent sources of entanglement remains poorly understood. Here, focusing on the simplest network of entanglement swapping, we start a systematic characterization of the set of quantum states leading to violation of the so-called “bilocality” inequality. First, we show that all possible pairs of entangled pure states can violate the inequality. Next, we derive a general criterion for violation for arbitrary pairs of mixed two-qubit states. Notably, this reveals a strong connection between the CHSH Bell inequality and the bilocality inequality, namely that any entangled state violating CHSH also violates the bilocality inequality. We conclude with a list of open questions. Authors: Justyna ŁodygaWaldemar KłobusRavishankar RamanathanAndrzej GrudkaMichał Horodecki,Ryszard Horodecki One of the formulations of Heisenberg uncertainty principle, concerning so-called measurement uncertainty, states that the measurement of one observable modifies the statistics of the other. Here, we derive such a measurement uncertainty principle from two comprehensible assumptions: impossibility of instantaneous messaging at a distance (no-signaling), and violation of Bell inequalities (non-locality). The uncertainty is established for a pair of observables of one of two spatially separated systems that exhibit non-local correlations. To this end, we introduce a gentle form of measurement which acquires partial information about one of the observables. We then bound disturbance of the remaining observables by the amount of information gained from the gentle measurement, minus a correction depending on the degree of non-locality. The obtained quantitative expression resembles the quantum mechanical formulations, yet it is derived without the quantum formalism and complements the known qualitative effect of disturbance implied by non-locality and no-signaling. Physicists harness starlight to support the case for entanglement. Nature News doi: 10.1038/nature.2017.21401 Diverse measurements indicate that entropy grows as the universe evolves, we analyze from a quantum point of view plausible scenarios that allow such increase. Authors: Tejinder P. Singh Collapse models possibly suggest the need for a better understanding of the structure of space-time. We argue that physical space, and space-time, are emergent features of the Universe, which arise as a result of dynamical collapse of the wave function. The starting point for this argument is the observation that classical time is external to quantum theory, and there ought to exist an equivalent reformulation which does not refer to classical time. We propose such a reformulation, based on a non-commutative special relativity. In the spirit of Trace Dynamics, the reformulation is arrived at, as a statistical thermodynamics of an underlying classical dynamics in which matter and non-commuting space-time degrees of freedom are matrices obeying arbitrary commutation relations. Inevitable statistical fluctuations around equilibrium can explain the emergence of classical matter fields and classical space-time, in the limit in which the universe is dominated by macroscopic objects. The underlying non-commutative structure of space-time also helps understand better the peculiar nature of quantum non-locality, where the effect of wave-function collapse in entangled systems is felt across space-like separations. Authors: Flavio Del Santo I present the reconstruction of the involvement of Karl Popper in the community of physicists concerned with foundations of quantum mechanics, in the 1980s. At that time Popper gave active contribution to the research in physics, of which the most significant is a new version of the EPR thought experiment, alleged to test different interpretations of quantum mechanics. The genesis of such an experiment is reconstructed in detail, and an unpublished letter by Popper is reproduced in the present paper to show that he formulated his thought experiment already two years before its first publication in 1982. The debate stimulated by the proposed experiment as well as Popper’s role in the physics community throughout 1980s is here analysed in detail by means of personal correspondence and publications. The Structure of the World on 2017-2-01 12:00am GMT Author: Steven French ISBN: 9780198776666 Binding: Paperback Publication Date: 01 February 2017 Price: $45.00 We investigate the violation factor of the Bell-Mermin inequality. Until now, we use an assumption that the results of measurement are ±1. In this case, the maximum violation factor is 2(n−1)/2. The quantum predictions by n-partite Greenberger-Horne-Zeilinger (GHZ) state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory based on the truth values is proposed (Nagata and Nakamura, Int. J. Theor. Phys. 55:3616, 2016). The values of measurement outcome are either +1 or 0. Here we use the new measurement theory. We consider multipartite GHZ state. It turns out that the Bell-Mermin inequality is violated by the amount of 2(n−1)/2. The measurement theory based on the truth values provides the maximum violation of the Bell-Mermin inequality. Author(s): Jing Zhang, Tiancai Zhang, and Jie Li Wave-function collapse models are considered to be the modified theories of standard quantum mechanics at the macroscopic level. By introducing nonlinear stochastic terms in the Schrödinger equation, these models (different from standard quantum mechanics) predict that it is fundamentally impossible… [Phys. Rev. A 95, 012141] Published Mon Jan 30, 2017 International Journal of Modern Physics A Particles and Fields; Gravitation; CosmologyNext Article > Volume 31, Issue 35, 20 December 2016 Tian Yu CaoInt. J. Mod. Phys. A31, 1630061 (2016) [20 pages] DOI: http://dx.doi.org/10.1142/S0217751X16300611 The Englert–Brout–Higgs mechanism — An unfinished project Tian Yu Cao1 1Department of Philosophy, Boston University, 745 Commonwealth Avenue, Boston, MA 02215, USA The conceptual foundation of the Englert–Brout–Higgs (EBH) mechanism (understood as a set of a scalar field’s couplings to a gauge system and a fermion system) is clarified (as being provided by broken symmetry solution of the scalar field and broken symmetry solutions of the gauge and fermion systems induced by the scalar field’s couplings to these systems, which are manifested in massive scalar and vector bosons as a result of reorganizing the physical degrees of freedom in the scalar and gauge sectors, whose original organization renders possible the broken symmetry solution to the scalar sector and symmetrical solutions to the gauge sector); its ontological status, as a physically real mechanism or merely an instrumental device, is examined, and a new ontologically primary entity, the symbiont of scalar–vector moments is suggested to replace the old ontology of scalar field and vector (gauge) field as the physical underpinning for a realistic understanding of the EBH mechanism; with a conclusion that two puzzles, thetransmutation of the Goldstone modes’ dynamic identity and the fixity in reorganizing the physical degrees of freedom within the symbiont, have to be properly addressed before a consistent realist understanding of the mechanism can be developed. 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c25d6a60e643bb00	Archive for the ‘History’ Category Yesterday I introduced Paul Dirac, number 10 in “The Guardian’s” list of the 10 best physicists. I mentioned that his main contributions to physics were (i) predicting antimatter, which he did in 1928, and (ii) producing an equation (now called the Dirac equation) which describes the behaviour of a sub-atomic particle such as an electron travelling at close to the speed of light (a so-called relativistic theory). This equation was also published in 1928. The Dirac Equation In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation. The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light. The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light. This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time. The Schrödinger eqation The time dependent Schrödinger equation which describes the motion of an electron The time dependent Schrödinger equation which describes the motion of an electron The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have i is the imaginary number, remember i = \sqrt{-1}. The next term \hbar is just Planck’s constant divided by two times pi, i.e. \hbar = h/2\pi. The next term \partial/\partial t \text{ } \psi(\vec{r},t) is the partial derivative with respect to time of the wave function \psi(\vec{r},t). Now, moving to the right hand side of the equality, we have m which is the mass of the particle, V is its potential energy, \nabla^{2} is the Laplacian. The Laplacian, \nabla^{2} \psi(\vec{r},t) is simply the divergence of the gradient of the wave function, \nabla \cdot \nabla \psi(\vec{r},t). In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below. Read Full Post » Today (January 30th) marks the 50th anniversary of the last time The Beatles played live together, in the infamous “rooftop” concert in 1969. Although they would go on to make one more studio album, Abbey Road in the summer of 1969; due to contractual and legal wranglings the rooftop concert, which was meant to be the conclusion to the movie they were shooting, would not come out until 1970 in the movie Let it Be. It is also true to say that some of the songs on Abbey Road were performed “live” in the studio with very little overdubbing (as opposed to separate instrument parts being recorded separately as was done on e.g. Sgt. Pepper). But, the rooftop concert was the last time the greatest band in history were seen playing together, and has gone down in infamy. It has been copied by many, including the Irish band U2 who did a similar thing to record the video for their single “Where the Streets Have no Name” in 1987 in Los Angeles. The Beatles were trying to think of a way to finish the movie that they had been shooting throughout January of 1969. They had discussed doing a live performance in all kinds of places; including on a boat, in the Roundhouse in London, and even in an amphitheatre in Greece. Finally, a few days before January 30th 1969, the idea of playing on the roof of their central-London offices was discussed. Whilst Paul and Ringo were in favour of this idea, and John was neutral, George was against it. The decision to go ahead with playing on the roof was not made until the actual day. They took their equipment up onto the roof of their London offices at 3, Saville Row, and just start playing. No announcement was made, only The Beatles and their inner circle knew about the impromptu concert. The concert consisted of the following songs : 1. “Get Back” (take one) 2. “Get Back” (take two) 3. “Don’t Let Me Down” (take one) 4. “I’ve Got a Feeling” (take one) 5. “One After 909” 6. “Dig a Pony” 7. “I’ve Got a Feeling” (take two) 8. “Don’t Let Me Down” (take two) 9. “Get Back” (take three) People in the streets below initially had no idea what the music (“noise”) coming from the top of the building was, but of course younger people knew the building was the Beatles’ offices. However, they would not have recognised any of the songs, as these were not to come out for many more months. After the third song “Don’t Let Me Down”, the Police were called and came to shut the concert down. The band managed nine songs (five different songs, with three takes of “Get Back”, two takes of “Don’t Let Me Down”, and two takes of “I’ve Got a Feeling”) before the Police stopped them. Ringo Starr later said that he wanted to be dragged away from his drums by the Police, but no such dramatic ending happened. At the end of the set John said You can read more about the rooftop concert here. Here is a YouTube video of “Get Back” (which may get taken down at any moment) and here is a video on the Daily Motion website of the whole rooftop concert (again, it may get taken down at any moment). Enjoy watching the greatest band ever perform live for the very last time! Read Full Post » I have seen their eyes, the terrible, empty eyes Of women in a glimmerless dawn, and the hands Of men who have wrestled through long years with the dark Underpinning of the mountains, strong hands that fight In dumb faith that what was once flesh born of their flesh Not that of the devil’s making… A victim in reversion, held beneath A vast, invisible paw… Not a lion to toss A proud, volcano-mane of destruction, crouched Like a rat, it waited… And they walk like victims of a second Flood In a world no longer home, where the void of sky Between tall mountains looms as a cenotaph For a generation of laughter… I have seen them Between nine and ten of the clock, death raised his flute And the children followed… Read Full Post » Today I thought I would share this great anti-war song – “Fortunate Son” by Creedence Clearwater Revival. It was released in September 1969, and is specifically about the lucky men who were born into families which, somehow, meant that they were not called up for the draft to fight in the Vietnam war. These were the senators’ sons, the millionaires’ sons, the fortunate sons. Sons like George W. Bush, who miraculously found himself in the National Guard, far away from any danger, rather than in Vietnam fighting. I wonder why? Oh, maybe because his father, George H. Bush, had the political clout and importance to make sure his precious son didn’t go and fight in the jungles of Vietnam, unlike the poor white and black men who were drafted there. As the draft went on, it became more and more apparent how many fortunate sons were avoiding going to war, thanks to their family’s influence in bending the rules. And how many poor blacks and whites had no choice, they were forced to go and would be jailed should they refuse. The Vietnam war was wrong on so many levels, but the inequity of the draft was certainly one of its wrongs. “Fortunate Son” was released in September 1969, and talks of the privileged few who, somehow, avoided the Vietnam war draft. “Fortunate Son” is rated at 99 in Rolling Stone Magazine’s list of the 500 greatest songs of all time. It really is a great song, I am surprised that I haven’t blogged about it before. Some folks are born, made to wave the flag Ooo, they’re red, white and blue And when the band plays “Hail to the Chief” Ooo, they point the cannon at you, Lord Some folks are born, silver spoon in hand Lord, don’t they help themselves, y’all But when the taxman comes to the door Lord, the house looks like a rummage sale, yeah Yeah, yeah Some folks inherit star spangled eyes Ooh, they send you down to war, Lord Here is a video of the song. Enjoy! Read Full Post » The speech opens with these lines…. Followed immediately by these words… Read Full Post » Fifty years ago yesterday (17 May 1966), one of the seminal moments in 20th Century popular culture took place in the Manchester Free Trade Hall. Bob Dylan, who had burst onto the folk scene a few years before, was playing to a packed crowd towards the end of his gruelling 1966 World tour. The first half of his set was vintage Dylan, just the man (poet) and his guitar. The crowd were enraptured. But, it all turned sour in the second half, when Dylan was joined by his band, The Hawks, and proceeded to do an ‘electric’ set. The crowd became restless. Many left; others booed, stamped their feet or started chanting. When he came back on to do his encore, things came to a head. “Judas!” a man shouted. “I don’t believe you.” Dylan replied. Then he started getting ready for the encore song. A few seconds later Dylan added “You’re a liar!” Then, he turned to his band and said “Play it fucking loud”, and they ripped into an angry version of Like a Rolling Stone. This is the moment as captured on film, it forms the closing scene of Martin Scorsese’s fascinating documentary No Direction Home. There is also a very interesting in-depth audio documentary about this whole seminal incident, Ghosts of Electricity, made by Andy Kershaw for BBC Radio 1 and broadcast in 1999. It is available here on Andy Kershaw’s website. Andy Kershaw’s fascinating documentary about the Bob Dylan “Judas” incident, which was originally broadcast in 1999 on BBC Radio 1. The whole concert was recorded and circulated as a bootleg for many years. For some reason, it became known as the Royal Albert Hall Concert, even though it had happened at the Manchester Free Trade Hall; possibly because the 1966 World tour ended at the Royal Albert Hall on the 26 and 27 May. Dylan sanctioned an official release of the concert in 1998. The cover for Bob Dylan’s “Royal Albert Hall Concert” CD, which includes the “Judas” heckle. In fact, the concert was recorded at the Manchester Free Trade Hall on 17 May 1966. Read Full Post » One of the physicists in our book Ten Physicists Who Transformed Our Understanding of Reality (follow this link for more information on the book) is, not surprisingly, Isaac Newton. In fact, he is number 1 in the list. One could argue that he practically invented the subject of physics. We decided to call him the ‘father of physics’, with Galileo (whose life preceded Newton’s) being given the title of ‘grandfather’. Newton was, clearly, a man of genius. But he was also a nasty, vindictive bastard (not to mince my words!). He didn’t really have any close friends in his life; there were plenty of people who admired him and respected him, and of course he had colleagues. But, apart from a niece whom he seemed to dote on in later life, and two men with whom he probably had love affairs, he was not a man who sought company. He was probably autistic, but lived at a time before such conditions were diagnosed or talked about. Isaac Newton (1643-1727), the ‘father of physics’. He relished in feuding with other scientists One sort of interaction that he did seem to enjoy with other people though was feuds. In fact, he seemed to thrive on feuding with other scientists. He loved to argue with others, which is not uncommon amongst academics. He had strong opinions which he liked to defend; this is normal. But, Newton took these disputes to an extreme; if he fell out with someone he would do everything he could to destroy that person. Although I am sure that he had many ‘minor’ arguments, he had three main feuds with fellow scientists. These three men were • Robert Hooke – curator of experiments at the Royal Society • Gottfried (von) Leibniz – the German mathematician • John Flamsteed – the first Astronomer Royal In each case, he did his level best to destroy the other man. Each of these feuds is discussed in more detail in our book, but in this blogpost I will give a brief summary of his feud with Leibniz. The feud came about because Newton refused to believe that Leibniz had independently come up with the mathematical idea of calculus. It was a recurring theme throughout Newton’s life that he sincerely believed that he was special. He had deep religious views (some would say extreme religious views). As part of these views, he believed that he had been specially chosen by God to understand things that others would never be able to understand. Thus, when he heard that Leibniz had developed a mathematics similar to his own ‘theory of fluxions’ (as Newton called it), he naturally assumed that the German had stolen it from him. There then ensued a 30-year dispute between the two men, with Newton very much the aggressor. Gottfried (von) Leibniz (1646-1716), German mathematician and co-inventor of calculus It escalated from a dispute to a feud, and culminated in the Royal Society commissioning an ‘official investigation’ to establish propriety for the invention of calculus. When the report came out in 1713 it came out in Newton’s favour. But, by this time Newton was not only President of the Royal Society, but he had secretly authored the entire report. It was anything but impartial. Leibniz died the following year, a broken man from Newton’s relentless attacks. One should, of course, be able to to admire a person for their work but not admire them in the least for the person that they were. Newton, in my mind, falls very firmly into this category. His contribution to physics is unparalleled, but I don’t think he was the kind of person one would want to know or even come across if one could help it! What is your favourite story about Newton? Read Full Post » As I mentioned in this blog here, a few months ago I contributed some articles to a book called 30-Second Einstein, which will be published by Ivy Press in the not too distant future. One of the articles I wrote for the book was on Indian mathematical physicist Satyendra Bose. It is after Bose that ‘bosons’ are named (as in ‘the Higgs boson’), and also terms like ‘Bose-Einstein statistics’ and ‘Bose-Einstein condensate’. So, who was Satyendra Bose, and why is his name attached to these things? Satyendra Bose was an Indian mathematical physicist after whom the 'boson' and Bose-Einstein statistics are named Satyendra Bose was an Indian mathematical physicist after whom the ‘boson’ and Bose-Einstein statistics are named Satyendra Bose was born in Calcutta, India, in 1894. He studied applied mathematics at Presidency College, Calcutta, obtaining a BSc in 1913 and an MSc in 1915. On both occasions, he graduated top of his class. In 1919, he made the first English translation of Einstein’s general theory of relativity, and by 1921 he had moved to Dhaka (in present-day Bangladesh) to become Reader (one step below full professor) in in the department of Physics. It was whilst in Dhaka, in 1924, that he came up with the theory of how to count indistinguishable particles, such as photons (light particles). He showed that such particles follow statistics which are different from particles which can be distinguished. All his attempts to get his paper published failed, so in an act of some desperation he sent it to Einstein. The great man recognised the importance of Bose’s work immediately, translated it into German and got it published in Zeitschrift für Physik, one of the premier physics journals of the day. Because of Einstein’s part in getting the theory published, we now know of this way of counting indistinguishable particles as Bose-Einstein statistics. We also name particles which obey this kind of statistics bosons; examples are the photon, the W and Z-particles (which mediate the weak nuclear force), and the most famous boson, the Higgs boson (responsible for mediating the property of mass via the Higgs field). With the imminent partition of India when it was gaining independence from Britain, Bose returned to his native Calcutta where he spent the rest of his career. He died in 1974 at the age of 80. You can read more about Satyendra Bose, Bose-Einstein statistics and Bose-Einstein condensates in 30-second Einstein, out soon from Ivy Press. Read Full Post » In part 3 of this blog series I explained how Max Planck found a mathematical formula to fit the observed Blackbody spectrum, but that when he presented it to the German Physics Society on the 19th of October 1900 he had no physical explanation for his formula. Remember, the formula he found was E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} -1) } \; d\lambda if we express it in terms of wavelength intervals. If we express it in terms of frequency intervals it is E_{\nu} \; d \nu = A^{\prime} \nu^{3} \frac{ 1 }{ (e^{ a^{\prime} \nu / T } - 1) } \; d\nu Planck would spend six weeks trying to find a physical explanation for this equation. He struggled with the problem, and in the process was forced to abandon many aspects of 19th Century physics in both the fields of thermodynamics and electromagnetism which he had long cherished. I will recount his derivation – it is not the only one and maybe in coming blog posts I can show how his formula can be derived from other arguments, but this is the method Planck himself used. Radiation in a cavity As we saw in the derivation of the Rayleigh-Jeans law (see part 3 here, and links in that to parts 1 and 2), blackbody radiation can be modelled as an idealised cavity which radiates through a small hole. Importantly, the system is given enough time for the radiation and the material from which the cavity is made to come into thermal equilibrium with each other. This means that the walls of the cavity are giving energy to the radiation at the same rate that the radiation is giving energy to the walls. Using classical physics, as we did in the derivation of the Rayleigh-Jeans law, we saw that the energy density (the energy per unit volume) is \frac{du}{d\nu} = \left( \frac{ 8 \pi kT }{ c^{3} } \right) \nu^{2} After trying to derive his equation based on standard thermodynamic arguments, which failed, Planck developed a model which, he found, was able to produce his equation. How did he do this? Harmonic Oscillators First, he knew from classical electromagnetic theory that an oscillating electron radiates (as it is accelerating), and he reasoned that when the cavity was in thermal equilibrium with the radiation in the cavity, the electrons in the walls of the cavity would oscillate and it was they that produced the radiation. After much trial and error, he decided upon a model where the electrons were attached to massless springs. He could model the radiation of the electrons by modelling them as a whole series of harmonic oscillators, but with different spring stiffnesses to produce the different frequencies observed in the spectrum. As we have seen (I derived it here), in classical physics the energy of a harmonic oscillator depends on both its amplitude of oscillation squared (E \propto A^{2}); and it also depends on its frequency of oscillation squared (E \propto \nu^{2}). The act of heating the cavity to a particular temperature is what, in Planck’s model, set the electrons oscillating; but whether a particular frequency oscillator was set in motion or not would depend on the temperature. If it were oscillating, it would emit radiation into the cavity and absorb it from the cavity. He knew from the shape of the blackbody curve (and, by now, his equation which fitted it), that the energy density E d\nu at any particular frequency started off at zero for high frequencies (UV), then rose to a peak, and then dropped off again at low frequencies (in the infrared). So, Planck imagined that the number of oscillators with a particular resonant frequency would determine how much energy came out in that frequency interval. He imagined that there were more oscillators with a frequency which corresponded to the maximum in the blackbody curve, and fewer oscillators at higher and lower frequencies. He then had to figure out how the total energy being radiated by the blackbody would be shared amongst all these oscillators, with different numbers oscillating at different frequencies. He found that he could not derive his formula using the physics that he had long accepted as correct. If he assumed that the energy of each oscillator went as the square of the amplitude, as it does in classical physics, his formula was not reproduced. Instead, he could derive his formula for the blackbody radiation spectrum only if the oscillators absorbed and emitted packets of energy which were proportional to their frequency of oscillation, not to the square of the frequency as classical physics argued. In addition, he found that the energy could only come in certain sized chunks, so for an oscillator at frequency \nu, \; E = nh\nu, where n is an integer, and h is now known as Planck’s constant. What does this mean? Well, in classical physics, an oscillator can have any energy, which for a particular oscillator vibrating at a particular frequency can be altered by changing the amplitude. Suppose we have an oscillator vibrating with an amplitude of 1 (in abitrary units), then because the energy goes as the square of the amplitude its energy is E=1^{2} =1. If we increase the amplitude to 2, the energy will now be E=2^{2} = 4. But, if we wanted an energy of 2, we would need an amplitude of \sqrt{2} = 1.414, and if we wanted an energy of 3 we would need an amplitude of \sqrt{3} = 1.73. In classical physics, there is nothing to stop us having an amplitude of 1.74, which would give us an energy of 3.0276 (not 3), or an amplitude of 1.72 whichg would give us an energy of 2.9584 (not 3). But, what Planck found is that this was not allowed for his oscillators, they did not seem to obey the classical laws of physics. The energy could only be integers of h\nu, so E=0h\nu, 1h\nu, 2h\nu, 3h\nu, 4h\nu etc. Then, as I said above, he further assumed that the total energy at a particular frequency was given by the energy of each oscillator at that frequency multiplied by the number of oscillators at that frequency. The frequency of a particular oscillator was, he imagined, determined by its stiffness (Hooke’s constant). The energy of a particular oscillator at a particular frequency could be varied by the amplitude of its oscillations. Let us assume, just to illustrate the idea, that the value of h is 2. If the total energy in the blackbody at a particular frequency of, say, 10 (in arbitrary units) were 800 (also in arbitrary units), this would mean that the energy of each chunk (E=h \nu) was E = 2 \times 10 = 20. So, the number of chunks at that frequency would then be 800/20 = 40. 40 oscillators, each with an energy of 20, would be oscillating to give us our total energy of 800 at that frequency. Because of this quantised energy, we can write that E_{n} = nh \nu, where n=0,1,2,3, \cdots. The number of oscillators at each frequency The next thing Planck needed to do was derive an expression for the number of oscillators at each frequency. Again, after much trial and error he found that he had to borrow an idea first proposed by Austrian physicist Ludwig Boltzmann to describe the most likely distribution of energies of atoms or molecules in a gas in thermal equilibrium. Boltzmann found that the number of atoms or molecules with a particular energy E was given by N_{E} \propto e^{-E/kT} where E is the energy of that state, T is the temperature of the gas and k is now known as Boltzmann’s constant. The equation is known as the Boltzmann distribution, and Planck used it to give the number of oscillators at each frequency. So, for example, if N_{0} is the number of oscillators with zero energy (in the so-called ground-state), then the numbers in the 1st, 2nd, 3rd etc. levels (N_{1}, N_{2}, N_{3},\cdots) are given by N_{1} = N_{0} e^{ -E_{1}/kT }, \; N_{2} = N_{0} e^{ -E_{2}/kT }, \; N_{3} = N_{0} e^{ -E_{3}/kT }, \cdots But, as E_{n} = nh \nu, we can write N_{1} = N_{0} e^{ -h \nu /kT }, \; N_{2} = N_{0} e^{ -2h \nu /kT }, \; N_{3} = N_{0} e^{ -3h \nu /kT }, \cdots Planck modelled blackbody radiation as a series of harmonic oscillators with equally spaced energy levels To make it easier to write, we are going to substitute x = e^{ -h \nu / kT }, so we have N_{1} = N_{0}x, \; N_{2} = N_{0} x^{2}, \; N_{3} = N_{0} x^{3}, \cdots The total number of oscillators N_{tot} is given by N_{tot} = N_{0} + N_{1} + N_{2} + N_{3} + \cdots = N_{0} ( 1 + x + x^{2} + x^{3} + \cdots) Remember, this is the number of oscillators at each frequency, so the energy at each frequency is given by the number at each frequency multiplied by the energy of each oscillator at that frequency. So E_{1}=N_{1} h \nu , \; E_{2} = N_{2} 2h \nu , \; E_{3} = N_{3} 3h \nu, \cdots which we can now write as E_{1} = h \nu N_{0}x, \; E_{2} = 2h \nu N_{0}x^{2}, \; E_{3} = 3h \nu N_{0}x^{3}, \cdots The total energy E_{tot} is given by E_{tot} = E_{0} + E_{1} + E_{2} + E_{3} + \cdots = N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots) The average energy \langle E \rangle is given by \langle E \rangle = \frac{ E_{tot} }{ N_{tot} } = \frac{ N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots) }{ N_{0} ( 1 + x + x^{2} + x^{3} + \cdots ) } The two series inside the brackets can be summed. The sum of the series in the numerator, which we will call S_{1} is given by S_{1} = \frac{ x - (n+1)x^{n+1} + nx^{n+2} }{ (1-x)^{2} } (for the proof of this, see for example here) The series in the denominator, which we will call S_{2}, is just a geometric progression. The sum of such a series is simply S_{2} = \frac{ 1 - x^{n} }{ (1-x) } Both series are in x, but remember x = e^{-h \nu / kT}. Also, both series are from a frequency of \nu = 0 \text{ to } \infty, and e^{-h \nu /kT} < 1, which means the sums converge and can be simplified. S_{1} \rightarrow \frac{x}{ (1-x)^{2} } \text{ and } S_{2} \rightarrow \frac{ 1 }{(1-x)} which means that \langle E \rangle = (h \nu S_{1})/S_{2} is given by \langle E \rangle = \frac{ h \nu x }{ (1-x)^{2} } \times \frac{ (1-x) }{1} = \frac{h \nu x}{ (1-x) } and so we can write that the average energy is \boxed{ \langle E \rangle = \frac{h \nu}{( 1/x - 1) } = \frac{h \nu}{ (e^{h \nu/kT} - 1) } } The radiance per frequency interval In our derivation of the Rayleigh-Jeans law (in this blog here), we showed that, using classical physics, the energy density du per frequency interval was given by du = \frac{ 8 \pi }{ c^{3} } kT \nu^{2} \, d \nu where kT was the energy of each mode of the electromagnetic radiation. We need to replace the kT in this equation with the average energy for the harmonic oscillators that we have just derived above. So, we re-write the energy density as du = \frac{ 8 \pi }{ c^{3} } \frac{ h \nu }{ (e^{h\nu/kT} - 1) } \nu^{2} \; d\nu = \frac{ 8 \pi h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu du is the energy density per frequency interval (usually measured in Joules per metre cubed per Hertz), and by replacing kT with the average energy that we derived above the radiation curve does not go as \nu^{2} as in the Rayleigh-Jeans law, but rather reaches a maximum and turns over, avoiding the ultraviolet catastrophe. It is more common to express the Planck radiation law in terms of the radiance per unit frequency, or the radiance per unit wavelength, which are written B_{\nu} and B_{\lambda} respectively. Radiance is the power per unit solid angle per unit area. So, as a first step to go from energy density to radiance we will divide by 4 \pi, the total solid angle. This gives We want the power per unit area, not the energy per unit volume. To do this we first note that power is energy per unit time, and second that to go from unit volume to unit area we need to multiply by length. But, for EM radiation, length is just ct. So, we need to divide by t and multiply by ct, giving us that the radiance per frequency interval is \boxed{ B_{\nu} = \frac{ 2h \nu^{3} }{ c^{2} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu } which is the way the Planck radiation law per frequency interval is usually written. Radiance per unit wavelength interval If you would prefer the radiance per wavelength interval, we note that \nu = c/\lambda and so d\nu = -c/\lambda^{2} \; d\lambda. Ignoring the minus sign (which is just telling us that as the frequency increases the wavelength decreases), and substituting for \nu and d\nu in terms of \lambda and d\lambda, we can write B_{\lambda} = \frac{ 2h }{ c^{2} } \frac{ c^{3} }{ \lambda^{3} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \frac{ c }{ \lambda^{2} } \; d\lambda Tidying up, this gives \boxed{ B_{\lambda} = \frac{ 2hc^{2} }{ \lambda^{5} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \; d\lambda } which is the way the Planck radiation law per wavelength interval is usually written. To summarise, in order to reproduce the formula which he had empirically derived and presented in October 1900, Planck found that he he could only do so if he assumed that the radiation was produced by oscillating electrons, which he modelled as oscillating on a massless spring (so-called “harmonic oscillators”). The total energy at any given frequency would be given by the energy of a single oscillator at that frequency multiplied by the number of oscillators oscillating at that frequency. However, he had to assume that 1. The energy of each oscillator was not related to either the square of the amplitude of oscillation or the square of the frequency of oscillation (as it would be in classical physics), but rather to the square of the amplitude and the frequency, E \propto \nu. 2. The energy of each oscillator could only be a multiple of some fundamental “chunk” of radiation, h \nu, so E_{n} = nh\nu where n=0,1,2,3,4 etc. 3. The number of oscillators with each energy E_{n} was given by the Boltzmann distribution, so N_{n} = N_{0} e^{-nh\nu/kT} where N_{0} is the number of oscillators in the lowest energy state. In a way, we can imagine that the oscillators at higher frequencies (to the high-frequency side of the peak of the blackbody) are “frozen out”. The quantum of energy for a particular oscillator, given by E_{n}=nh\nu, is just too large to exist at the higher frequencies. This avoids the ultraviolet catastrophe which had stumped physicists up until this point. By combining these assumptions, Planck was able in November 1900 to reproduce the exact equation which he had derived empirically in October 1900. In doing so he provided, for the first time, a physical explanation for the observed blackbody curve. • Part 1 of this blogseries is here. • Part 2 is here. • Part 3 is here. Read Full Post » There has been quite a bit of mention in the media this last week or so that it is 100 years since Albert Einstein published his ground-breaking theory of gravity – the general theory of relativity. Yet, there seems to be some confusion as to when this theory was first published, in some places you will see 1915, in others 1916. So, I thought I would try and clear up this confusion by explaining why both dates appear. Albert Einstein in Berlin circa 1915 when his General Theory of Relativity was first published Albert Einstein in Berlin circa 1915/16 when his General Theory of Relativity was first published From equivalence to the field equations Everyone knew that Einstein was working on a new theory of gravity. As I blogged about here, he had his insight into the equivalence between acceleration and gravity in 1907, and ever since then he had been developing his ideas to create a new theory of gravity. He had come up with his principle of equivalence when he was asked in the autumn of 1907 to write a review article of his special theory of relativity (his 1905 theory) for Jahrbuch der Radioaktivitätthe (the Yearbook of Electronics and Radioactivity). That paper appeared in 1908 as Relativitätsprinzip und die aus demselben gezogenen Folgerungen (On the Relativity Principle and the Conclusions Drawn from It) (Jahrbuch der Radioaktivität, 4, 411–462). In 1908 he got his first academic appointment, and did not return to thinking about a generalisation of special relativity until 1911. In 1911 he published a paper Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravitation on the Propagation of Light) (Annalen der Physik (ser. 4), 35, 898–908), in which he calculated for the first time the deflection of light produced by massive bodies. But, he also realised that, to properly develop his ideas of a new theory of gravity, he would need to learn some mathematics which was new to him. In 1912, he moved to Zurich to work at the ETH, his alma mater. He asked his friend Marcel Grossmann to help him learn this new mathematics, saying “You’ve got to help me or I’ll go crazy.” Grossmann gave Einstein a book on non-Euclidean geometry. Euclidean geometry, the geometry of flat surfaces, is the geometry we learn in school. The geometry of curved surfaces, so-called Riemann geometry, had first been developed in the 1820s by German mathematician Carl Friedrich Gauss. By the 1850s another German mathematician, Bernhard Riemann developed this geometry of curved surfaces even further, and this was the Riemann geometry textbook which Grossmann gave to Einstein in 1912. Mastering this new mathematics proved very difficult for Einstein, but he knew that he needed to master it to be able to develop the equations for general relativity. These equations were not ready until late 1915. Everyone knew Einstein was working on them, and in fact he was offered and accepted a job in Berlin in 1914 as Berlin wanted him on their staff when the new theory was published. The equations of general relativity were first presented on the 25th of November 1915, to the Prussian Academy of Sciences. The lecture Feldgleichungen der Gravitation (The Field Equations of Gravitation) was the fourth and last lecture that Einstein gave to the Prussian Academy on his new theory (Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847), the previous three lectures, given on the 4th, 11th and 18th of November, had been leading up to this. But, in fact, Einstein did not have the field equations ready until the last few days before the fourth lecture! The peer-reviewed paper of the theory (which also contains the field equations) did not appear until 1916 in volume 49 of Annalen der PhysikGrundlage der allgemeinen Relativitätstheorie (The Foundation of the General Theory of Relativity) Annalen der Physik (ser. 4), 49, 769–822. The paper was submitted by Einstein on the 20th of March 1916. The beginning of Einstein's first paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916 and The beginning of Einstein’s first peer-reviewed paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916 In a future blog, I will discuss Einstein’s field equations, but hopefully I have cleared up the confusion as to why some people refer to 1915 as the year of publication of the General Theory of Relativity, and some people choose 1916. Both are correct, which allows us to celebrate the centenary twice! You can read more about Einstein’s development of the general theory of relativity in our book 10 Physicists Who Transformed Our Understanding of Reality. Order your copy here Read Full Post » Older Posts »
030653572ba0a574	Difference between revisions of "RAZ Glossary" From Lesswrongwiki Jump to: navigation, search (moving entries to the talk page that are some combination of 'pretty obvious' and/or 'pretty unimportant') Line 418: Line 418: * '''utilon'''. Yudkowsky’s name for a unit of utility, i.e., something that satisfies a goal. The term is deliberately vague, to permit discussion of desired and desirable things without relying on imperfect proxies such as monetary value and self-reported happiness. Revision as of 10:32, 29 July 2020 This is a list of brief explanations and definitions for terms that Eliezer Yudkowsky uses in the book Rationality: From AI to Zombies, an edited version of the Sequences. The glossary is a community effort, and you're welcome to improve on the entries here, or add new ones. See the Talk page for some ideas for unwritten entries. • a priori. Before considering the evidence. Similarly, "a posteriori" means "after considering the evidence"; compare prior and posterior probabilities. In philosophy, "a priori" often refers to the stronger idea of something knowable in the absence of any experiential evidence (outside of the evidence needed to understand the claim). • affect heuristic. People's general tendency to reason based on things' felt goodness or badness. • affective death spiral. Yudkowsky's term for a halo effect that perpetuates and exacerbates itself over time. • AGI. See “artificial general intelligence.” • AI-Box Experiment. A demonstration by Yudkowsky that people tend to overestimate how hard it is to manipulate people, and therefore underestimate the risk of building an Unfriendly AI that can only interact with its environment by verbally communicating with its programmers. One participant role-plays an AI, while another role-plays a human whose job it is interact with the AI without voluntarily releasing the AI from its “box”. Yudkowsky and a few other people who have role-played the AI have succeeded in getting the human supervisor to agree to release them, which suggests that a superhuman intelligence would have an even easier time escaping. • akrasia. • alien god. One of Yudkowsky's pet names for natural selection. • ambiguity aversion. Preferring small certain gains over much larger uncertain gains. • amplitude. A quantity in a configuration space, represented by a complex number. Many sources misleadingly refer to quantum amplitudes as "probability amplitudes", even though they aren't probabilities. Amplitudes are physical, not abstract or formal. The complex number’s modulus squared (i.e., its absolute value multiplied by itself) yields the Born probabilities, but the reason for this is unknown. • amplitude distribution. See “wavefunction.” • anchoring. The cognitive bias of relying excessively on initial information after receiving relevant new information. • anthropics. Problems related to reasoning well about how many observers like you there are. • artificial general intelligence. Artificial intelligence that is "general-purpose" in the same sense that human reasoning is general-purpose. It's hard to crisply state what this kind of reasoning consists in—if we knew how to fully formalize it, we would already know how to build artificial general intelligence. However, we can gesture at (e.g.) humans' ability to excel in many different scientific fields, even though we did not evolve in an ancestral environment containing particle accelerators. • Aumann's Agreement Theorem. • availability heuristic. The tendency to base judgments on how easily relevant examples come to mind. • average utilitarianism. • Backward chaining. • Base rate. • Bayes's Theorem. The equation stating how to update a hypothesis H in light of new evidence E. In its simplest form, Bayes's Theorem says that a hypothesis' probability given the evidence, written P(H\|E), equals the likelihood of the evidence given that hypothesis, multiplied by your prior probability P(H) that the hypothesis was true, divided by the prior probability P(E) that you would see that evidence regardless. I.e.: Also known as Bayes's Rule. See "odds ratio" for a simpler way to calculate a Bayesian update. • Bayesian. (a) Optimally reasoned; reasoned in accordance with the laws of probability. (b) An optimal reasoner, or a reasoner that approximates optimal inference unusually well. (c) Someone who treats beliefs as probabilistic and treats probability theory as a relevant ideal for evaluating reasoners. (d) Related to probabilistic belief. (e) Related to Bayesian statistical methods. • Bayesian updating. Revising your beliefs in a way that's fully consistent with the information available to you. Perfect Bayesian updating is wildly intractable in realistic environments, so real-world agents have to rely on imperfect heuristics to get by. As an optimality condition, however, Bayesian updating helps make sense of the idea that some ways of changing one's mind work better than others for learning about the world. • beisutsukai. Japanese for "Bayes user." A fictional order of high-level rationalists, also known as the Bayesian Conspiracy. • Bell's Theorem. • Berkeleian idealism. The belief, espoused by George Berkeley, that things only exist in various minds (including the mind of God). • bias. (a) A cognitive bias. In Rationality: From AI to Zombies, this will be the default meaning. (b) A statistical bias. (c) An inductive bias. (d) Colloquially: prejudice or unfairness. • bit. (a) A binary digit, taking the value 0 or 1. (b) The logarithm (base 1/2) of a probability—the maximum information that can be communicated using a binary digit, averaged over the digit's states. Rationality: From AI to Zombies usually uses "bit" in the latter sense. • black box. Any process whose inner workings are mysterious or poorly understood. • Black Swan. • blind god. One of Yudkowsky's pet names for natural selection. • Blue and Green. Rival sports teams and political factions in ancient Rome. • Born rule. • calibration. Assigning probabilities to beliefs in a way that matches how often those beliefs turn out to be right. E.g., if your assignment of "70% confidence" to claims is well-calibrated, then you will get such claims right about 70% of the time. • causal decision theory. The theory that the right way to make decisions is by picking the action with the best causal consequences. • causal graph. A directed acyclic graph in which an arrow going from node A to node B is interpreted as "changes in A can directly cause changes in B." • cognitive bias. A systematic error stemming from the way human reasoning works. This can be contrasted with errors due to ordinary ignorance, misinformation, brain damage, etc. • collapse. • comparative advantage. An ability to produce something at a lower cost than some other actor could. This is not the same as having an absolute advantage over someone: you may be a better cook than someone across-the-board, but that person will still have a comparative advantage over you at cooking some dishes. This is because your cooking skills make your time more valuable; the worse cook may have a comparative advantage at baking bread, for example, since it doesn’t cost them much to spend a lot of time on baking, whereas you could be spending that time creating a large number of high-quality dishes. Baking bread is more costly for the good cook than for the bad cook because the good cook is paying a larger opportunity cost, i.e., is giving up more valuable opportunities to be doing other things. • complex. (a) Colloquially, something with many parts arranged in a relatively specific way. (b) In information theory, something that's relatively hard to formally specify and that thereby gets a larger penalty under Occam's razor; measures of this kind of complexity include Kolmogorov complexity. (c) Complex-valued, i.e., represented by the sum of a real number and an imaginary number. • conditional independence. • conditional probability. The probability that a statement is true on the assumption that some other statement is true. E.g., the conditional probability P(A\|B) means "the probability of A given that B." • configuration space. • confirmation bias. The cognitive bias of giving more weight to evidence that agrees with one's current beliefs. • conjunction. A sentence that asserts multiple things. "It's raining and I'm eating a sandwich" is a conjunction; its conjuncts are "It's raining" and "I'm eating a sandwich." • conjunction fallacy. The fallacy of treating a conjunction as though it were more likely than its conjuncts. • consequentialism. (a) The ethical theory that the moral rightness of actions depends only on what outcomes result. Consequentialism is normally contrasted with ideas like deontology, which says that morality is about following certain rules (e.g., "don't lie") regardless of the consequences. (b) Yudkowsky's term for any reasoning process that selects actions based on their consequences. • Copenhagen Interpretation. • correspondence bias. Drawing conclusions about someone's unique disposition from behavior that can be entirely explained by the situation in which it occurs. When we see someone else kick a vending machine, we think they are "an angry person," but when we kick the vending machine, it's because the bus was late, the train was early, and the machine ate our money. • Cox's Theorem. • cryonics. The low-temperature preservation of brains. Cryonics proponents argue that cryonics should see more routine use for people whose respiration and blood circulation have recently stopped (i.e., people who qualify as clinically deceased), on the grounds that future medical technology may be able to revive such people. • de novo. Entirely new; produced from scratch. • decibel. • decision theory. (a) The mathematical study of correct decision-making in general, abstracted from an agent's particular beliefs, goals, or capabilities. (b) A well-defined general-purpose procedure for arriving at decisions, e.g., causal decision theory. • decoherence. • deontology. The theory that moral conduct is about choosing actions that satisfy specific rules like "don't lie" or "don't steal." • directed acyclic graph. A graph that is directed (its edges have a direction associated with them) and acyclic (there's no way to follow a sequence of edges in a given direction to loop around from a node back to itself). • dukkha. • Dutch book. • edge. See “graph.” • élan vital. "Vital force." A term coined in 1907 by the philosopher Henri Bergson to refer to a mysterious force that was held to be responsible for life's "aliveness" and goal-oriented behavior. • entanglement. (a) Causal correlation between two things. (b) In quantum physics, the mutual dependence of two particles' states upon one another. Entanglement in sense (b) occurs when a quantum amplitude distribution cannot be factorized. • entropy. (a) In thermodynamics, the number of different ways a physical state may be produced (its Boltzmann entropy). E.g., a slightly shuffled deck has lower entropy than a fully shuffled one, because there are many more configurations a fully shuffled deck is likely to end up in. (b) In information theory, the expected value of the information contained in a message (its Shannon entropy). That is, a random variable’s Shannon entropy is how many bits of information one would be missing (on average) if one did not know the variable’s value. Boltzmann entropy and Shannon entropy have turned out to be equivalent; that is, a system’s thermodynamic disorder corresponds to the number of bits needed to fully characterize it. • epistemic. Concerning knowledge. • epistemology. (a) A world-view or approach to forming beliefs. (b) The study of knowledge. • eudaimonia. • Eurisko. • eutopia. Yudkowsky’s term for a utopia that’s actually nice to live in, as opposed to one that’s unpleasant or unfeasible. • Everett branch. A "world" in the many-worlds interpretation of quantum mechanics. • existential risk. Something that threatens to permanently and drastically reduce the value of the future, such as stable global totalitarianism or human extinction. • expected utility. The expected value of a utility function given some action. Roughly: how much an agent’s goals will tend to be satisfied by some action, given uncertainty about the action's outcome. A sure $1 will usually lead to more utility than a 10% chance of $1 million. Yet in all cases, the 10% shot at $1 million has more expected utility, assuming you assign more than ten times as much utility to winning $1 million. Expected utility is an idealized mathematical framework for making sense of the idea "good bets don't have to be sure bets." • expected value. The sum of all possible values of a variable, each multiplied by its probability of being the true value. • FAI. See “friendly AI.” • falsificationism. • Fermi paradox. The puzzle of reconciling "on priors, we should expect there to be many large interstellar civilizations visible in the night sky" and "we see no clear signs of such civilizations." Some reasons many people find it puzzling that there are no visible alien civilizations include: "the elements required for life on Earth seem commonplace"; "life had billions of years to develop elsewhere before we evolved"; "high intelligence seems relatively easy to evolve (e.g., many of the same cognitive abilities evolved independently in humans, octopuses, crows)"; and "although some goals favor hiddenness, many different possible goals favor large-scale extraction of resources, and we only require there to exist one old species of the latter type." • fitness. See “inclusive fitness.” • foozality. See "rationality." • frequentism. (a) The view that the Bayesian approach to probability—i.e., treating probabilities as belief states—is unduly subjective. Frequentists instead propose treating probabilities as frequencies of events. (b) Frequentist statistical methods. • Friendly AI. Artificial general intelligence systems that are safe and useful. "Friendly" is a deliberately informal descriptor, intended to signpost that "Friendliness" still has very little technical content and needs to be further developed. Although this remains true in many respects as of this writing (2018), Friendly AI research has become much more formally developed since Yudkowsky coined the term "Friendly AI" in 2001, and the research area is now more often called "AI alignment research." • Fun Theory. • graph. In graph theory, a mathematical object consisting of simple atomic objects ("vertices," or "nodes") connected by lines (or "edges"). When edges have an associated direction, they are also called "arrows." • gray goo. • Gricean implication. • group selection. Natural selection at the level of groups, as opposed to individuals. Historically, group selection used to be viewed as a more central and common part of evolution—evolution was thought to frequently favor self-sacrifice "for the good of the species." • halo effect. The tendency to assume that something good in one respect must be good in other respects. • halting oracle. An abstract agent that is stipulated to be able to reliably answer questions that no algorithm can reliably answer. Though it is provably impossible for finite rule-following systems (e.g., Turing machines) to answer certain questions (e.g., the halting problem), it can still be mathematically useful to consider the logical implications of scenarios in which we could access answers to those questions. • happy death spiral. See “affective death spiral.” • hedonic. Concerning pleasure. • heuristic. An imperfect method for achieving some goal. A useful approximation. Cognitive heuristics are innate, humanly universal brain heuristics. • hindsight bias. The tendency to exaggerate how well one could have predicted things that one currently believes. • humility. Not being arrogant or overconfident. Yudkowsky defines humility as "taking specific actions in anticipation of your own errors." He contrasts this with "modesty," which he views as a social posture for winning others' approval or esteem, rather than as a form of epistemic humility. • inclusive fitness. The degree to which a gene causes more copies of itself to exist in the next generation. Inclusive fitness is the property propagated by natural selection. Unlike individual fitness, which is a specific organism’s tendency to promote more copies of its genes, inclusive fitness is held by the genes themselves. Inclusive fitness can sometimes be increased at the expense of the individual organism’s overall fitness. • inductive bias. The set of assumptions a learner uses to derive predictions from a data set. The learner is "biased" in the sense that it's more likely to update in some directions than in others, but unlike with other conceptions of "bias", the idea of "inductive bias" doesn't imply any sort of error. • instrumental. Concerning usefulness or effectiveness. • instrumental value. A goal that is only pursued in order to further some other goal. • intelligence explosion. A scenario in which AI systems rapidly improve in cognitive ability because they see fast, consistent, sustained returns on investing work into such improvement. This could happen via AI systems using their intelligence to rewrite their own code, improve their hardware, or acquire more hardware, then leveraging their improved capabilities to find more ways to improve. • intentionality. The ability of things to represent, or refer to, other things. Not to be confused with "intent." • isomorphism. A two-way mapping between objects in a category. Informally, two things are often called "isomorphic" if they're identical in every relevant respect. • Iterated Prisoner’s Dilemma. A series of Prisoner’s Dilemmas between the same two players. Because players can punish each other for defecting on previous rounds, they will usually more reason to cooperate than in the one-shot Prisoner’s Dilemma. • joint probability distribution. A probability distribution that assigns probabilities to combinations of claims. E.g., if the claims in question are "Is it cold?" and "Is it raining?", a joint probability distribution could assign probabilities to "it's cold and rainy," "it's cold and not rainy," "it's not cold but is rainy," and "it's neither cold nor rainy." • just-world fallacy. The cognitive bias of systematically overestimating how much reward people get for good deeds, and how much punishment they get for bad deeds. • koan. In Zen Buddhism, a short story or riddle aimed at helping the hearer break through various preconceptions. • Kolmogorov complexity. A formalization of the idea of complexity. Given a programming language, a computable string's Kolmogorov complexity is the length of the shortest computer program in that language that outputs the string. • likelihood. In Bayesian probability theory, how much probability a hypothesis assigns to a piece of evidence. Suppose we observe the evidence E = "Mr. Boddy was knifed," and our hypotheses are HP = "Professor Plum killed Boddy" and HW = "Mrs. White killed Boddy." If we think there's a 25% chance that Plum would use a knife in the worlds where he chose to kill Boddy, then we can say HP assigns a likelihood of 25% to E. Suppose that there's only a 5% chance Mrs. White would use a knife if she killed Boddy. Then we can say that the likelihood ratio between HP and HW is 25/5 = 5. This means that the evidence supports "Plum did it" five times as strongly as it supports "White did it," which tells us how to update upon observing E. (See "odds ratio" for a simple example.) • magisterium. Stephen Gould’s term for a domain where some community or field has authority. Gould claimed that science and religion were separate and non-overlapping magisteria. On his view, religion has authority to answer questions of "ultimate meaning and moral value" (but not empirical fact) and science has authority to answer questions of empirical fact (but not meaning or value). • many-worlds interpretation. The idea that the basic posits in quantum physics (complex-valued amplitudes) are objectively real and consistently evolve according to the Schrödinger equation. Opposed to anti-realist and collapse interpretations. Many-worlds holds that the classical world we seem to inhabit at any given time is a small component of an ever-branching amplitude. The "worlds" of the many-worlds interpretation are not discrete or fundamental to the theory. Speaking of "many worlds" is, rather, a way of gesturing at the idea that the ordinary objects of our experience are part of a much larger whole that contains enormously many similar objects. • map and territory. A metaphor for the relationship between beliefs (or other mental states) and the real-world things they purport to refer to. • materialism. The belief that all mental phenomena can in principle be reduced to physical phenomena. • maximum-entropy probability distribution. A probability distribution which assigns equal probability to every event. • Maxwell’s Demon. A hypothetical agent that knows the location and speed of individual molecules in a gas. James Maxwell used this demon in a thought experiment to show that such knowledge could decrease a physical system’s entropy, “in contradiction to the second law of thermodynamics.” The demon’s ability to identify faster molecules allows it to gather them together and extract useful work from them. Leó Szilárd later pointed out that if the demon itself were considered part of the thermodynamic system, then the entropy of the whole would not decrease. The decrease in entropy of the gas would require an increase in the demon’s entropy. Szilárd used this insight to simplify Maxwell’s scenario into a hypothetical engine that extracts work from a single gas particle. Using one bit of information about the particle (e.g., whether it’s in the top half of a box or the bottom half), a Szilárd engine can generate log2(kT) joules of energy, where T is the system’s temperature and k is Boltzmann’s constant. • meta level. A domain that is more abstract or derivative than the object level. • metaethics. A theory about what it means for ethical statements to be correct, or the study of such theories. Whereas applied ethics speaks to questions like "Is murder wrong?" and "How can we reduce the number of murders?", metaethics speaks to questions like "What does it mean for something to be wrong?" and "How can we generally distinguish right from wrong?" • Mind Projection Fallacy. • Minimum Message Length Principle. A formalization of Occam’s Razor that judges the probability of a hypothesis based on how long it would take to communicate the hypothesis plus the available data. Simpler hypotheses are favored, as are hypotheses that can be used to concisely encode the data. • modesty. Yudkowsky's term for the social impulse to appear deferential or self-effacing, and resultant behaviors. Yudkowsky contrasts this with the epistemic virtue of humility. • monotonicity. Roughly, the property of never reversing direction. A monotonic function is any function between ordered sets that either preserves the order, or completely flips it. A non-monotonic function, then, is one that at least once takes an a<b input and outputs a>b, and at least once takes a c>d input and outputs c<d. A monotonic logic is one that will always continue to assert something as true if it ever asserted it as true. If "2+2=4" is proved, then in a monotonic logic no subsequent operation can make it impossible to derive that theorem again in the future. In contrast, non-monotonic logics can "forget" past conclusions and lose the ability to derive them. • Moore’s Law. A 1965 observation and prediction by Intel co-founder Gordon Moore: roughly every two years (originally every one year), engineers are able to double the number of transistors that can be fit on an integrated circuit. This projection held true into the 2010s. Other versions of this "law" consider other progress metrics for computing hardware. • motivated cognition. Reasoning that is driven by some goal or emotion that's at odds with accuracy. Examples include non-evidence-based inclinations to reject a claim (motivated skepticism), to believe a claim (motivated credulity), to continue evaluating an issue (motivated continuation), or to stop evaluating an issue (motivated stopping). • Murphy’s law. The saying “Anything that can go wrong will go wrong.” • mutual information. For two variables, the amount that knowing about one variable tells you about the other's value. If two variables have zero mutual information, then they are independent; knowing the value of one does nothing to reduce uncertainty about the other. • nanotechnology. (a) Fine-grained control of matter on the scale of individual atoms, as in Eric Drexler's writing. This is the default meaning in Rationality: From AI to Zombies. (b) Manipulation of matter on a scale of nanometers. • Nash equilibrium. A situation in which no individual would benefit by changing their own strategy, assuming the other players retain their strategies. Agents often converge on Nash equilibria in the real world, even when they would be much better off if multiple agents simultaneously switched strategies. For example, mutual defection is the only Nash equilibrium in the standard one-shot Prisoner’s Dilemma (i.e., it is the only option such that neither player could benefit by changing strategies while the other player’s strategy is held constant), even though it is not Pareto-optimal (i.e., each player would be better off if the group behaved differently). • negentropy. Negative entropy. A useful concept because it allows one to think of thermodynamic regularity as a limited resource one can possess and make use of, rather than as a mere absence of entropy. • Newcomb’s Problem. A central problem in decision theory. Imagine an agent that understands psychology well enough to predict your decisions in advance, and decides to either fill two boxes with money, or fill one box, based on their prediction. They put $1,000 in a transparent box no matter what, and they then put $1 million in an opaque box if (and only if) they predicted that you’d only take the opaque box. The predictor tells you about this, and then leaves. Which do you pick? If you take both boxes ("two-boxing"), you get only the $1000, because the predictor foresaw your choice and didn’t fill the opaque box. On the other hand, if you only take the opaque box, you come away with $1 million. So it seems like you should take only the opaque box. However, causal decision theorists object to this strategy on the grounds that you can’t causally control what the predictor did in the past; the predictor has already made their decision by the time you make yours, and regardless of whether or not they placed the $1 million in the opaque box, you’ll be throwing away a free $1000 if you choose not to take it. For the same reason, causal decision theory prescribes defecting in one-shot Prisoner’s Dilemmas, even if you’re playing against a perfect atom-by-atom copy of yourself. • nonmonotonic logic. See “monotonic logic.” • normalization. Adjusting values to meet some common standard or constraint, often by adding or multiplying a set of values by a constant. E.g., adjusting the probabilities of hypotheses to sum to 1 again after eliminating some hypotheses. If the only three possibilities are A, B, and C, each with probability 1/3, then evidence that ruled out C (and didn’t affect the relative probability of A and B) would leave us with A at 1/3 and B at 1/3. These values must be adjusted (normalized) to make the space of hypotheses sum to 1, so A and B change to probability 1/2 each. • normative. Good, or serving as a standard for desirable behavior. • NP-complete. The hardest class of decision problems within the class NP, where NP consists of the problems that an ideal computer (specifically, a deterministic Turing machine) could efficiently verify correct answers to. The difficulty of NP-complete problems is such that if an algorithm were discovered to efficiently solve even one NP-complete problem, that algorithm would allow one to efficiently solve every NP problem. Many computer scientists hypothesize that this is impossible, a conjecture called “P ≠ NP.” • null-op. A null operation; an action that does nothing in particular. • object level. The level of concrete things, as contrasted with the "meta" level. The object level tends to be a base case or starting point, while the meta level is comparatively abstract, recursive, or indirect in relevance. • Occam’s Razor. The principle that, all else being equal, a simpler claim is more probable than a relatively complicated one. Formalizations of Occam’s Razor include Solomonoff induction and the Minimum Message Length Principle. • odds ratio. A way of representing how likely two events are relative to each other. E.g., if I have no information about which day of the week it is, the odds are 1:6 that it’s Sunday. This is the same as saying that "it's Sunday" has a prior probability of 1/7. If x:y is the odds ratio, the probability of x is x / (x + y). Likewise, to convert a probability p into an odds ratio, I can just write p : (1 - p). For a percent probability p%, this becomes p : (100 - p). E.g., if my probability of winning a race is 40%, my odds are 40:60, which can also be written 2:3. Odds ratios are useful because they're usually the easiest way to calculate a Bayesian update. If I notice the mall is closing early, and that’s twice as likely to happen on a Sunday as it is on a non-Sunday (a likelihood ratio of 2:1), I can simply multiply the left and right sides of my prior it’s Sunday (1:6) by the evidence’s likelihood ratio (2:1) to arrive at a correct posterior probability of 2:6, or 1:3. • Omega. A hypothetical arbitrarily powerful agent used in various thought experiments. • one-boxing. Taking only the opaque box in Newcomb's Problem. • ontology. An account of the things that exist, especially one that focuses on their most basic and general similarities. Things are "ontologically distinct" if they are of two fundamentally different kinds. • opportunity cost. The value lost from choosing not to acquire something valuable. If I choose not to make an investment that would have earned me $10, I don’t literally lose $10 -- if I had $100 at the outset, I’ll still have $100 at the end, not $90. Still, I pay an opportunity cost of $10 for missing a chance to gain something I want. I lose $10 relative to the $110 I could have had. Opportunity costs can result from making a bad decision, but they also occur when you make a good decision that involves sacrificing the benefits of inferior options for the different benefits of a superior option. Many forms of human irrationality involve assigning too little importance to opportunity costs. • optimization process. Yudkowsky’s term for a process that performs searches through a large search space, and manages to hit very specific targets that would be astronomically unlikely to occur by chance. E.g., the existence of trees is much easier to understand if we posit a search process, evolution, that iteratively comes up with better and better solutions to cognitively difficult problems. A well-designed dam, similarly, is easier to understand if we posit an optimization process searching for designs or policies that meet some criterion. Evolution, humans, and beavers all share this property, and can therefore be usefully thought of as optimization processes. In contrast, the processes that produce mountains and stars are easiest to describe in other terms. • orthogonality. The independence of two (or more) variables. If two variables are orthogonal, then knowing the value of one doesn't help you learn the value of the other. • P ≠ NP. A widely believed conjecture in computational complexity theory. NP is the class of mathematically specifiable questions with input parameters (e.g., “can a number list A be partitioned into two number lists B and C whose numbers sum to the same value?”) such that one could always in principle efficiently confirm that a correct solution to some instance of the problem (e.g., “the list {3,2,7,3,5} splits up into the lists {3,2,5} and {7,3}, and the latter two lists sum to the same number”) is in fact correct. More precisely, NP is the class of decision problems that a deterministic Turing machine could verify answers to in a polynomial amount of computing time. P is the class of decision problems that one could always in principle efficiently solve -- e.g., given {3,2,7,3,5} or any other list, quickly come up with a correct answer (like “{3,2,5} and {7,3}”) should one exist. Since all P problems are also NP problems, for P to not equal NP would mean that some NP problems are not P problems; i.e., some problems cannot be efficiently solved even though solutions to them, if discovered, could be efficiently verified. • Pareto optimum. A situation in which no one can be made better off without making at least one person worse off. • phase space. A mathematical representation of physical systems in which each axis of the space is a degree of freedom (a property of the system that must be specified independently) and each point is a possible state. • phlogiston. A substance hypothesized in the 17th entity to explain phenomena such as fire and rust. Combustible objects were thought by late alchemists and early chemists to contain phlogiston, which evaporated during combustion. • physicalism. See “materialism.” • Planck units. Natural units, such as the Planck length and the Planck time, representing the smallest physically significant quantized phenomena. • positive bias. Bias toward noticing what a theory predicts you’ll see, instead of noticing what a theory predicts you won’t see. • possible world. A way the world could have been. One can say “there is a possible world in which Hitler won World War II” in place of “Hitler could have won World War II,” making it easier to contrast the features of multiple hypothetical or counterfactual scenarios. Not to be confused with the worlds of the many-worlds interpretation of quantum physics or Max Tegmark's Mathematical Universe Hypothesis, which are claimed (by their proponents) to be actual. • posterior probability. An agent's beliefs after acquiring evidence. Contrasted with its prior beliefs, or priors. • prior probability. An agent’s beliefs prior to acquiring some evidence. • Prisoner’s Dilemma. A game in which each player can choose to either "cooperate" with or "defect" against the other. The best outcome for each player is to defect while the other cooperates; and the worst outcome is to cooperate while the other defects. Each player views mutual cooperation as the second-best option, and mutual defection as the second-worst. Traditionally, game theorists have argued that defection is always the correct move in one-shot dilemmas; it improves your reward if the other player independently cooperates, and it lessens your loss if the other player independently defects. Yudkowsky is one of a minority of decision theorists who argue that rational cooperation is possible in the one-shot Prisoner's Dilemma, provided the two players' decision-making is known to be sufficiently similar. "My opponent and I are both following the same decision procedure, so if I cooperate, my opponent will cooperate too; and if I defect, my opponent will defect. The former seems preferable, so this decision procedure hereby outputs `cooperate." • probability amplitude. See “amplitude.” • probability distribution. A function which assigns a probability (i.e., a number representing how likely something is to be true) to every possibility under consideration. Discrete and continuous probability distributions are generally encoded by, respectively, probability mass functions and probability density functions. Thinking of probability as a "mass" that must be divided up between possibilities can be a useful way to keep in view that reducing the probability of one hypothesis always requires increasing the probability of others, and vice versa. Probability, like (classical) mass, is conserved. • probability theory. The branch of mathematics concerned with defining statistical truths and quantifying uncertainty. • problem of induction. In philosophy, the question of how we can justifiably assert that the future will resemble the past without relying on evidence that presupposes that very fact. • quark. An elementary particle of matter. • quine. A program that outputs its own source code. • rationalist. (a) Related to rationality. (b) A person who tries to apply rationality concepts to their real-world decisions. • rationality. Making systematically good decisions (instrumental rationality) and achieving systematically accurate beliefs (epistemic rationality). • reductio ad absurdum. Refuting a claim by showing that it entails a claim that is more obviously false. • reduction. An explanation of a phenomenon in terms of its origin or parts, especially one that allows you to redescribe the phenomenon without appeal to your previous conception of it. • reductionism. (a) The practice of scientifically reducing complex phenomena to simpler underpinnings. (b) The belief that such reductions are generally possible. • representativeness heuristic. A cognitive heuristic where one judges the probability of an event based on how well it matches some mental prototype. • Ricardo’s Law of Comparative Advantage. See “comparative advantage.” • satori. In Zen Buddhism, a non-verbal, pre-conceptual apprehension of the ultimate nature of reality. • Schrödinger equation. A fairly simple partial differential equation that defines how quantum wavefunctions evolve over time. This equation is deterministic; it is not known why the Born rule, which converts the wavefunction into an experimental prediction, is probabilistic, though there have been many attempts to make headway on that question. • scope insensitivity. A cognitive bias where people tend to disregard the size of certain phenomena. • screening off. Making something evidentially irrelevant. A piece of evidence A screens off a piece of evidence B from a hypothesis C if, once you know about A, learning about B doesn’t affect the probability of C. • search tree. A graph with a root node that branches into child nodes, which can then either terminate or branch once more. The tree data structure is used to locate values; in chess, for example, each node can represent a move, which branches into the other player’s possible responses, and searching the tree is intended to locate winning sequences of moves. • self-anchoring. Anchoring to oneself. Treating one’s own qualities as the default, and only weakly updating toward viewing others as different when given evidence of differences. • Shannon entropy. See “entropy.” • Shannon mutual information. See “mutual information.” • Simulation Hypothesis. The hypothesis that the world as we know it is a computer program designed by some powerful intelligence. • Singularity. One of several scenarios in which artificial intelligence systems surpass human intelligence in a large and dramatic way. • skyhook. An attempted explanation of a complex phenomenon in terms of a deeply mysterious or miraculous phenomenon -- often one of even greater complexity. • Solomonoff induction. An attempted definition of optimal (albeit computationally unfeasible) inference. Bayesian updating plus a simplicity prior that assigns less probability to percept-generating programs the longer they are. • stack trace. A retrospective step-by-step report on a program's behavior, intended to reveal the source of an error. • statistical bias. A systematic discrepancy between the expected value of some measure, and the true value of the thing you're measuring. • superintelligence. Something vastly smarter than present-day humans. This can be a predicted future technology, like smarter-than-human AI; or it can be a purely hypothetical agent, such as Omega or Laplace's Demon. • System 1. The processes behind the brain’s fast, automatic, emotional, and intuitive judgments. • System 2. The processes behind the brain’s slow, deliberative, reflective, and intellectual judgments. • Szilárd engine. See “Maxwell’s Demon.” • Taboo. A game by Hasbro where you try to get teammates to guess what word you have in mind while avoiding conventional ways of communicating it. Yudkowsky uses this as an analogy for the rationalist skill of linking words to the concrete evidence you use to decide when to apply them. Ideally, one should be know what one is saying well enough to paraphrase the message in several different ways, and to replace abstract generalizations with concrete observations. • Tegmark world. A universe contained in a vast multiverse of mathematical objects. The idea comes from Max Tegmark's Mathematical Universe Hypothesis, which holds that our own universe is a mathematical object contained in an ensemble in which all possible computable structures exist. • terminal value. A goal that is pursued for its own sake, and not just to further some other goal. • Tit for Tat. A strategy in which one cooperates on the first round of an Iterated Prisoner’s Dilemma, then on each subsequent rounds mirrors what the opponent did the previous round. • Traditional Rationality. Yudkowsky’s term for the scientific norms and conventions espoused by thinkers like Richard Feynman, Carl Sagan, and Charles Peirce. Yudkowsky contrasts this with the ideas of rationality in contemporary mathematics and cognitive science. • transhuman. (a) Entities that are human-like, but much more capable than ordinary biological humans. (b) Related to radical human enhancement. Transhumanism is the view that humans should use technology to radically improve their lives—e.g., curing disease or ending aging. • truth-value. A proposition’s truth or falsity. • Turing-computability. The ability to be executed, at least in principle, by a simple process following a finite set of rules. "In principle" here means that a Turing machine could perform the computation, though we may lack the time or computing power to build a real-world machine that does the same. Turing-computable functions cannot be computed by all Turing machines, but they can be computed by some. In particular, they can be computed by all universal Turing machines. • Turing machine. An abstract machine that follows rules for manipulating symbols on an arbitrarily long tape. • two-boxing. Taking both boxes in Newcomb's Problem. • Unfriendly AI. A hypothetical smarter-than-human artificial intelligence that causes a global catastrophe by pursuing a goal without regard for humanity’s well-being. Yudkowsky predicts that superintelligent AI will be “Unfriendly” by default, unless a special effort goes into researching how to give AI stable, known, humane goals. Unfriendliness doesn’t imply malice, anger, or other human characteristics; a completely impersonal optimization process can be “Unfriendly” even if its only goal is to make paperclips. This is because even a goal as innocent as ‘maximize the expected number of paperclips’ could motivate an AI to treat humans as competitors for physical resources, or as threats to the AI’s aspirations. • uniform probability distribution. A distribution in which all events have equal probability; a maximum-entropy probability distribution. • universal Turing machine. A Turing machine that can compute all Turing-computable functions. If something can be done by any Turing machine, then it can be done by every universal Turing machine. A system that can in principle do anything a Turing machine could is called “Turing-complete." • updating. Revising one’s beliefs. See also "Bayesian updating." • utilitarianism. An ethical theory asserting that one should act in whichever causes the most benefit to people, minus how much harm results. Standard utilitarianism argues that acts can be justified even if they are morally counter-intuitive and harmful, provided that the benefit outweighs the harm. • utility function. A function that ranks outcomes by "utility," i.e., by how well they satisfy some set of goals or constraints. Humans are limited and imperfect reasoners, and don't consistently optimize any endorsed utility function; but the idea of optimizing a utility function helps us give formal content to "what it means to pursue a goal well," just as Bayesian updating helps formalize "what it means to learn well." • wavefunction. A complex-valued function used in quantum mechanics to explain and predict the wave-like behavior of physical systems at small scales. Realists about the wavefunction treat it as a good characterization of the way the world really is, more fundamental than earlier (e.g., atomic) models. Anti-realists disagree, although they grant that the wavefunction is a useful tool by virtue of its mathematical relationship to observed properties of particles (the Born rule). • wu wei. “Non-action.” The concept, in Daoism, of effortlessly achieving one’s goals by ceasing to strive and struggle to reach them. • XML. Extensible Markup Language, a system for annotating texts with tags that can be read both by a human and by a machine. • ZF. The Zermelo–Fraenkel axioms, an attempt to ground standard mathematics in set theory. ZFC (the Zermelo–Fraenkel axioms supplemented with the Axiom of Choice) is the most popular axiomatic set theory. • zombie. In philosophy, a perfect atom-by-atom replica of a human that lacks a human’s subjective awareness. Zombies behave exactly like humans, but they lack consciousness. Some philosophers argue that the idea of zombies is coherent -- that zombies, although not real, are at least logically possible. They conclude from this that facts about first-person consciousness are logically independent of physical facts, that our world breaks down into both physical and nonphysical components. Most philosophers reject the idea that zombies are logically possible, though the topic continues to be actively debated.
b409282b26f9c35b	Sign up Here's how it works: 1. Anybody can ask a question 2. Anybody can answer The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis of objects. I wonder, is my intuition true? What is the physical meaning of Laplace-Beltrami eigenfunctions? For now, I only know that the eigenfunctions of the Laplace-Beltrami operator are real and orthogonal, thus they could be used as the basis of functions on the manifold where the functions are defined. share\|cite\|improve this question Cross-posted to – Qmechanic Dec 24 '12 at 9:25 Eigenfunctions of the Laplacian can be used to construct solutions to differential equations involving the Laplacian, most notably the wave equation, the heat equation, and the Schrödinger equation. The wave equation gives the collection to oscillation; Laplacian eigenfunctions describe standing wave solutions and their eigenvalues describe the period of the oscillation. See also hearing the shape of a drum. share\|cite\|improve this answer Your Answer
f1ed9f3b93a549b7	Take the 2-minute tour × When finding the discrete energy states of a operator I have been taught to use the time-independent Schrodinger equation which restates the definition of eigenvalues and eigenvectors. What I don’t understand is why the eigenvalues are the energy states, is there firstly a mathematical reason and secondly a physical reason? Does this arise from Hamiltonian or Lagrangian mechanics which I am not familiar with? share\|improve this question Sorry I mean eigenvalues of the operator not wave funciton – Josh Apr 6 '11 at 17:00 Keep in mind that we are only dealing with Hermetian operators, because their eigenvalues are real, and hence correspond to positive definite probabilities. – Matt Calhoun Apr 8 '11 at 16:44 7 Answers 7 up vote 5 down vote accepted As has been remarked by others and explained clearly, and mathematically, the eigenvalues are important because a) they allow you to solve the time-dependent equation, i.e., solve for the evolution of the system and b) a state which belongs to the eigenvalue $E$, i.e., as we say, a state which is an eigenstate with eigenvalue $E$, has an expectation value of the energy operator which is easy to see has to be $E$ itself. But those explanations are advanced and rely on the maths. And they do not explain why $E$ should be considered 'an energy level'. At some risk, I will try to answer your question more physically. What is the physical reason why the energy states of a system, e.g., an atom, are the eigenvalues of the operator $H$ that appears in the time-independent Schroedinger equation? Well, first, note that it's absolutely the same $H$ that appears in the time-dependent Schrodinger equation, $$H\cdot \psi = -i{\partial \psi \over \partial t}$$ which controls the rate of change of $\psi$. The answer doesn't come from the classical Hamiltonian or Lagrangian mechanics, but from the then-new quantum properties of Nature. A non-classical feature of QM is that some states are stationary, which means they do not change in time. E.g., the electron in a Bohr orbit is actually not moving, not orbiting at all, and this solves the classical paradoxes about the atom (why the rotating charge doesn't radiate its energy away and fall into the centre). The first key point is that an eigenstate is a stationary state: what is the explanation for this? well, Schroedinger's time dependent equation clearly says that, up to a constant of proportionality, the time-rate of change of any state $\psi$ is found by applying the operator $H$ (the Hamiltonian: we do not yet know it is also the energy operator) to it: the new vector or function $H\cdot\psi$ is the change in $\psi$ per unit time. Obviously if this is zero, $\psi$ does not change (this was the only classical possibility). But also if $H\cdot\psi$ is even a non-zero multiple of $\psi$, call it $E\psi$, then $\psi$ plus this rate of change is still a multiple of $\psi$, so as time goes on, $\psi$ changes in a trivial fashion: just to another multiple of itself. In QM, a multiple of the wave function represents the same quantum state, so we see the quantum state does not change. Now the next key point is that a state with a definite energy value must be stationary. Why? In QM, it is not automatic that a system has a definite value of a physical quantity, but if it does, that means its measurement always leads to the same answer, so there is no uncertainty. So if there is no uncertainty in the energy, by Heisenberg's uncertainty principle there must be infinite uncertainty in something else, whatever is 'conjugate' to energy. And that is time. You cannot tell the time using this system, which implies it is not changing. So it is stationary. (remember, we are not assuming that $H$ is also the energy operator and we are not assuming the formula for expectations). Thus being an eigenstate of $H$ implies $\psi$ is stationary. And having a definite energy value implies it is stationary. Being physicists, we now conclude that being an eigenstate implies it has a definite energy value, which answers your question, and these are the 'energy levels' of a system such as an atom: a system, even an atom, might not possess a definite energy, but if it doesn't, it won't be stationary, and being microscopic, the time-scale in which it will evolve will be so rapid we are unlikely to be able to observe its energy, or even care (since it won't be relevant to molecules or chemistry). So, 'most' atoms for which we can actually measure their energy must be stationary: this is 'why' the definite values of energy which a stationary state can possess are called the 'energy levels' of the system, and historically were discovered first, before Schroedingers equation. From a human perspective, most atoms that we care about spend most of their time that matters to us in an approximately stationary state. In case you are wondering why time is the conjugate to energy, whereas Heisenberg's original analysis of his uncertainty principle showed that position was conjugate to momentum, we rely on relativity: time is just another coordinate of space-time, and so is analogous to position. And in relativistic mechanics, momentum in a spatial direction is analogous to energy (or mass, same thing). In the standard relativistic equation $$p^2-m^2=E^2,$$ we see that momentum ($p$) and mass $m$ are symmetric (except for the negative sign) with each other. So since momentum is conjugate to position, $m$ or energy must be conjugate to time. For this reason, Bohr was able to extend Heisenberg's analysis, of the uncertainty relations between measurements of position and measurements of momentum, to show the same relations between energy and time. share\|improve this answer Both eigenvalues and eigenstates belong to some operator. In your case, this is the Hamiltonian operator $\hat H$. It's fundamental because of many reasons. First is that it is indeed an operator that represents energy in a sense that possible energy levels are encoded in its spectrum (i.e. a set of eigenvalues). The second important reason is that it is the operator that can be found in Schrodinger equation $i \hbar \partial_t \left \| \psi(t) \right > = \hat H \left \| \psi(t) \right >$. This equation can then be solved by writing $\left \| \psi(t) \right >$ as superposition of eigenstates of $\hat H$: $\left \| \psi(t) \right > = \sum_n c_n(t) \left \| \psi_n \right >$. If we can find these states, we are done as $c_n(t) = \exp({-iE_n t \over \hbar}) c_n(t=0)$ solves the equation (and it also shows the importance of these eigenstates because they are preserved by time evolution). So this means the problem of time evolution in quantum mechanics can be reduced to the problem of finding the eigenvalues and eigenstates of $\hat H$, the equation for that being $\hat H \left \| \psi_n \right> = E_n \left\| \psi_n \right>$. Note: the above assumes that $\hat H$ is time-independent. If it's not (as is the case in basically all practical applications, using perturbation theory) then we use different techniques, e.g. of path integration, or various scattering formulas. share\|improve this answer You seem to be confusing two things, namely the eigenstates of an operator and Schrödinger's equation. A priori, these two have nothing to do with each other. In Quantum Mechanics, measurable quantities are represented by (hermitian) operators on a Hilbert space. For instance there is an operator $P$ corresponding to the momentum. In general, when measuring the momentum of a state $\|\psi \rangle$, the result will not be deterministic. However, the average over several measurements will be equal to the expection value $$ \langle \psi \| P \| \psi \rangle $$ However, when $\|\psi\rangle$ is an eigenvector of the operator, $P\|\psi\rangle = \lambda \|\psi\rangle$, then the measurement will always be the same value $\lambda$. In particular, there is an operator corresponding to the total energy, the Hamiltionian $H$. The form of this operator can be obtained from classical physics if you replace momentum and location by their corresponding operators. For instance, the Hamiltonian of an electron in an electric potential $V$ is $$ H = \frac1{2m} P^2 + eV(X) .$$ Thus, the expectation value for the energy of a state $\|\psi\rangle$ is $\langle \psi\|H\|\psi\rangle$. Now, the Hamiltionian is a very interesting operator because it features prominently in the equation of motion, the Schrödinger equation. $$ i\hbar \partial_t \|\psi(t)\rangle = H \|\psi(t)\rangle .$$ What does this have to do with the eigenvalues of the Hamiltionian? A priori nothing, but the point is that knowing the eigenvectors and -values of $H$ allow you to solve this equation. Namely, if you have an eigenvator $\|\psi_n\rangle$, then you have $$ i\hbar \partial_t \|\psi_n(t)\rangle = H \|\psi_n(t)\rangle = E_n\|\psi(t)\rangle$$ which can be solved to $$ \|\psi_n(t)\rangle = e^{-\frac{i}{\hbar} E_nt} \|\psi(0)\rangle $$ To summarize, the eigenvalues of an operator tells you something about what happens when you perform measurements, but in addition, the eigenvalues of the energy operator help you solve the equations of motion. share\|improve this answer The reason why it is the eigenvalues of the Hamiltonian and not some other operator that will give you the energy states is that in classical Mechanics, the Hamiltonian function is just the energy of your system, expressed as a function of position $x$ and momentum $p$. As a simple example, the Hamiltonian for a harmonic oscillator is $$H(x,p) = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ Note that this really is just the sum of kinetic and potential energy, so we could write $$H(x,p) = E$$. To get to quantum mechanics, one now performs what is called canonical quantization. There is no mathematically rigorous reason why this will give you a correct quantum mechanics. Since quantum isn't classical, we cannot really expect to find a seamless and watertight derivation of the former from the latter. To my knowledge, this approach has, however, always given correct results. So, in canonical quantization, what one does is to replace the variables of the Hamiltonian, i.e., $x$ and $p$, with their operator versions, $\hat x$ and $\hat p$. Now we cannot simply write $H(x,p) = E$ anymore, since the energy is a scalar, but the Hamiltonian $H$ is now an operator. Operators are functions that take a wavefunction and modify it in some way and give you a new wavefunction. Now, another postulate of quantum mechanics is that you get the expectation value of an operator $\hat A$ in a given state $\Psi$ by calculation the integral $$ \int dx \Psi^(x) \hat A \Psi(x) $$ Hence, we get the expectation value of the energy by calculating $$ \int dx \Psi^(x) H \Psi(x)$$ Obviously, if $H\Psi(x) = E\Psi(x)$, then the expectation value yields $E$, and it is not hard to show that for such an eigenstate, the variance of $E$ will be $0$, i.e. every measurement of the energy in state $\Psi$ will yield the same value $E$. share\|improve this answer A transformation from one set to another can be regarded as a matrix if we define a particular basis. Likewise, an operator can be thought of as a matrix. What is the matrix equation that relates eigenvalues and eigenvectors? You are solving an eigenvalue problem when you are solving the time independent Schrodinger equation. share\|improve this answer The basic experimental fact the inventors of QM had to deal with was the uncertainty principle. The mathematics behind this principle has two major parts, one involving linear algebra and another involving Fourier analysis. In other words, the operator algebra of QM is necessary in order to have a theory which obeys the uncertainty principle, and if you want to know why this is true, you have to study the mathematics. share\|improve this answer I think you should specify you answer. – Self-Made Man Nov 14 '13 at 14:29 The physics of this is the DeBroglie relation for particles, which relates the energy to the frequency of some wave. The energy of a photon is the frequency of the emitted electromagnetic wave. When a quantum mechanical atom is weakly interacting with the photon field, and goes from a state with frequency f to a state with frequency f', it can only emit photons with frequency f-f'. The reason is that the transition process is only resonant with waves of frequency equal to the beat frequency \Delta f= f-f'. The atomic relative phases during the transition process recurs with time $1\over \Delta f$, and for any outgoing wave whose frequency does not match this, the process will be cancelling at long times, and no wave will be emitted. This means that atomic transitions from f to f' are accompanied by a loss of energy of $h\Delta f$, so that one must identify the frequency with the energy in general quantum systems. The Schrodinger waves of definite frequency are the solutions of the time independent problem, since when $$i{d\over dt} \psi = H \psi $$ and $H\psi = E\psi$, that is, if $\psi$ is an eigenvector of H, then $\psi(t) = e^{-iEt} \psi(0)$, so the time dependence of the wave has a definite frequency. I am giving a physical argument here, because the notion that energy is frequency is engrained into the foundation of quantum mechanics, and it is hard to argue that it is true using a formalism built upon this as a foundation. share\|improve this answer Your Answer
f180a1f556e9765b	From Here to Eternity Imagine a universe with no past or future, where time is an illusion and everyone is immortal. Welcome to that world, says physicist Julian Barbour By Tim Folger\|Friday, December 01, 2000 Time seems to stand still in South Newington, a secluded village ringed by rolling green hills about 20 miles north of Oxford, England. The 1,000-year-old baptismal font in the town's church, the thatch-roofed houses, and the tidy gardens along narrow lanes all appear unchanged by the passage of centuries. Standing on the roof of the church's bell tower on a warm, late-summer day, Julian Barbour, a theoretical physicist with some extraordinary notions about the nature of time, points to his home, known as College Farm, which borders the ancient church. "It looks almost exactly as it did when it was built 340 years ago," says Barbour. "The barn is also from the 17th century. Virtually all the houses you see around are from about 1640 to 1720. The long, low house is the one I grew up in. That's my parents' house. It dates from about 1710 to 1720." The entire scene is so placid one can't help but imagine that Barbour's childhood home, as well as the village and the surrounding landscape, will remain unchanged for the next 340 years. Such utter quiescence suits Barbour, who is convinced the static harmony of South Newington extends past the horizon to the universe at large. In his view, this moment and all it holds— Barbour himself, his American visitor, Earth, and everything beyond to the most distant galaxies— will never change. There is no past and no future. Indeed, time and motion are nothing more than illusions. In Barbour's universe, every moment of every individual's life— birth, death, and everything in between— exists forever. "Each instant we live," Barbour says, "is, in essence, eternal." That means each and every one of us is immortal. Like the perpetually unmoving lovers in Keats's "Ode on a Grecian Urn," we are "for ever panting, and for ever young." We are also for ever aged and decrepit, on our deathbeds, in the dentist's chair, at Thanksgivings with our in-laws, and reading these words. Barbour fully realizes how outrageous the notion of a world without time sounds. "I still have trouble accepting it," he says. But then, common sense has never been a reliable guide to understanding the universe— physicists have been confounding our perceptions since Copernicus first suggested that the sun does not revolve around Earth. After all, we don't feel the slightest movement as the spinning Earth hurtles through the void at some 67,000 miles per hour. Our sense of the passage of time, Barbour argues, is just as wrongheaded as the credo of the Flat Earth Society. Barbour has been preoccupied with studying the basic properties of time for four decades. It's an issue he believes most theoretical physicists have ignored."Given what a fascinating thing time is, it's surprising how few physicists have made a serious attempt to study time and say exactly what it is," he says. "It's an unusual gap." At the outset Barbour didn't think he would have any fresh insights he could bring to the topic. "I don't regard myself as being at all talented. I struggle to do equations," he says, laughing. "But I just got very interested in the subject and found that very few people have really thought seriously about it." Perhaps Barbour himself wouldn't have been able to devote nearly 40 years of his, well, time to the problem if it hadn't been for his unique background. Unlike most of his colleagues, he doesn't work at a university or a government lab— he is one of the world's few freelance theoretical physicists. Nevertheless, his credentials are solid, and prominent physicists take him— and his unconventional ideas— quite seriously. "He has some wild ideas, but he definitely knows what he's talking about when it comes to these fundamental issues," says Carlo Rovelli, who works at the Center for Theoretical Physics in Luminy, France. Lee Smolin, a theoretical physicist at Pennsylvania State University, agrees: "Barbour is one of the few people I know who went out on their own and succeeded in doing several things that were important and would not have been easy to do in a conventional career." After receiving his doctorate in physics from the University of Cologne in 1968, Barbour, who is now 63, decided he didn't want to follow a traditional academic career, with the inevitable pressure to publish or perish. So he supported his wife and four children by translating Russian scientific articles and worked on physics on the side, publishing scholarly papers every few years. Outside academia, he was free to explore his interest in time without worrying about tenure or funding for what might seem an arcane pursuit. Until recently, Barbour's provocative work was little known beyond a rarefied circle of physicists. That changed earlier this year with the publication of his latest book, The End of Time, in which he presents his case for a universe where time, despite all appearances to the contrary, plays no role. Barbour is not alone in recognizing that the pictures of time in general relativity and quantum mechanics are fundamentally incompatible. Theoretical physicists around the world, spurred by Nobel dreams, sweat over the problem. But Barbour has taken perhaps the most unorthodox approach by proposing that the way to solve the conundrum is to leave time out of the equations that describe the universe entirely. He has been obsessed with this solution for more than 10 years, since he learned of a vexing mathematical tour de force by a young American physicist named Bryce DeWitt. DeWitt, with the help of the eminent American physicist John Wheeler, developed an equation in 1967 that apparently melded quantum mechanics with general relativity. He did this by taking the principles from quantum mechanics that describe the interactions of atoms and molecules and applying them to the entire universe, a mind-bending feat not unlike trying to make a jockey's suit fit Michael Jordan. Specifically, DeWitt hijacked the Schrödinger equation, named for the great Austrian physicist who created it. In its original form, the equation reveals how the arrangement of electrons determines the geometrical shapes of atoms and molecules. As modified by DeWitt, the equation describes different possible shapes for the entire universe and the position of everything in it. The key difference between Schrödinger's quantum and DeWitt's cosmic version of the equation— besides the scale of the things involved— is that atoms, over time, can interact with other atoms and change their energies. But the universe has nothing to interact with except itself and has only a fixed total energy. Because the energy of the universe doesn't change with time, the easiest of the many ways to solve what has become known as the Wheeler-DeWitt equation is to eliminate time. Most physicists balk at that solution, believing it couldn't possibly describe the real universe. But a number of respected theorists, Barbour and Stephen Hawking among them, take DeWitt's work seriously. Barbour sees it as the best path to a real theory of everything, even with its staggering implication that we live in a universe without time, motion, or change of any kind. Strolling in the meadows of oxford's Christ Church College with Julian Barbour, time and motion seem undeniable. Towering cumulus clouds float overhead, ferried by a gentle breeze. Children run and shout in the same field where Alice Liddell, the girl who inspired Lewis Carroll's Alice's Adventures in Wonderland, often played. How can there be no time, no movement? Barbour settles his tall, lean frame into the grass, readying himself for a long explanation to yet another skeptic. He begins with what seems a most straightforward proposition: Time is nothing but a measure of the changing positions of objects. A pendulum swings, the hands on a clock advance. Objects— and their positions— he argues, are therefore more fundamental than time. The universe at any given instant simply consists of many different objects in many different positions. That sounds reasonable, as it should, coming from a thoughtful gentleman like Barbour. But the next part of his argument— the crux of his view— is much harder to swallow: Every possible configuration of the universe, past, present, and future, exists separately and eternally. We don't live in a single universe that passes through time. Instead, we— or many slightly different versions of ourselves— simultaneously inhabit a multitude of static, everlasting tableaux that include everything in the universe at any given moment. Barbour calls each of these possible still-life configurations a "Now." Every Now is a complete, self-contained, timeless, unchanging universe. We mistakenly perceive the Nows as fleeting, when in fact each one persists forever. Because the word universe seems too small to encompass all possible Nows, Barbour coined a new word for it: Platonia. The name honors the ancient Greek philosopher who argued that reality is composed of eternal and changeless forms, even though the physical world we perceive through our senses appears to be in constant flux. Before allowing himself to be interrupted by the stream of questions he knows will come, Barbour continues to press his point. He likens his view of reality to a strip of movie film. Each frame captures one possible Now, which may include blades of grass, clouds in a blue sky, Julian Barbour, a baffled Discover writer, and distant galaxies. But nothing moves or changes in any one frame. And the frames— the past and future— don't disappear after they pass in front of the lens. "This corresponds to the way you remember highlights of your life," Barbour says. "You remember very vividly certain scenes as snapshots. I remember once, very tragically, I had to go to a man who had shot himself. And I still have no difficulty in recalling the scene of opening the door just to where he was at the foot of the stairs and seeing him there with the gun and the blood. It's still imprinted as a photograph on my mind. Many other memories I have take that form. People have strong visual memories. If it's not just a snapshot, it might be a few stills of a movie you recall. Think of perhaps your most vivid memories. You don't think of them as just lasting a second. You see them as snapshots in your mind's eye, don't you? They don't fade— they don't seem to have any duration. They're just there, like the pages of a book. You wouldn't ask how many seconds a page lasts. It doesn't last a millisecond, or a second; it just is." Barbour calmly awaits the inevitable sputtering objections. Don't we then somehow shift from one "frame" to another? No. There is no movement from one static arrangement of the universe to the next. Some configurations of the universe simply contain little patches of consciousness— people— with memories of what they call a past that are built into the Now. The illusion of motion occurs because many slightly different versions of us— none of which move at all— simultaneously inhabit universes with slightly different arrangements of matter. Each version of us sees a different frame— a unique, motionless, eternal Now. "My position is that we are never the same in any two instants," Barbour says. "Obviously, as macroscopic human beings, we don't change much from second to second. And there's no question that we're the same people. I mean only an extreme madman would deny that," he says reassuringly. "To that extent, it's true that we do move from one Now to another. But in what sense can you say we're moving? The way I see it, not exactly the same information content, but nearly the same information content, is present in many different Nows." Nothing really moves, he says. "The information content or the consciousness that makes us aware of being ourselves, of having a certain identity, is just present in many different Nows. There are two things that distinguish my position from what people might just intuitively think. First of all, the Nows are not on one timeline. They're just there. And second, there is nothing corresponding to motion. I'm taking a very radical position on that. I'm saying the Nows are really like snapshots. The impression of motion only arises because the snapshots have got an extraordinarily special structure." We are part of that special structure. For all the apparent complexity of his scheme, Barbour believes that it provides the simplest way to merge quantum mechanics and relativity into a single theory of the universe. Like all physicists, he strongly believes that mathematically elegant explanations tend to be true, even if they conflict with common sense. "I think the approach I'm proposing does deserve to be taken seriously," he says. "It would be extremely rash and stupid to say it's definitely right, but there's an inner logic to these ideas. They're very natural. If we want to put quantum mechanics and general relativity together, what is the simplest way that could be done? I believe it is the way I've proposed. And I believe it is essentially the way that Bryce DeWitt discovered in 1967 when he found his infamous equation." Barbour stands and brushes some grass from his pants. He has to meet his wife, Verena, for dinner and looks at his watch, grinning as he does so. "This is what comes of saying there is no time— I have to pull my own leg sometimes," he says. Walking to a fashionable new restaurant on Oxford's old High Street, Barbour talks about how his ideas have changed his perceptions of the world. "I think it's completely wrong to say that the world was created in the Big Bang and that it was the unique creation event." Barbour hastens to add that there exists an eternal Now that contains the Big Bang, but he sees it as just one of an infinite array of Nows existing alongside this instant on High Street. "Immortality is all around us," he says. "Our task is to recognize it." How does the physics community react to such ideas? Physicists who know Barbour's work agree that it shouldn't be dismissed out of hand. At a physics conference in Spain, Barbour conducted an informal poll. He asked how many of the physicists believed that time would not be a part of a final, complete description of the universe. A majority were inclined to agree. Don Page, a cosmologist at the University of Alberta in Edmonton who frequently collaborates with Stephen Hawking, raised his hand that day. "I think Julian's work clears up a lot of misconceptions," says Page. "Physicists might not need time as much as we might have thought before. He is really questioning the basic nature of time, its nonexistence. You can't make technical advances if you're stuck in a conceptual muddle." Strangely enough, Page feels that Barbour might actually be too conservative. When physicists finally iron out a new theory of the universe, Page suspects that time won't be the only casualty. "I think space will go too," he says cryptically. Like Page, Carlo Rovelli applauds Barbour for forcing physicists to think about things they may have taken for granted. "It's time to go back to the big questions," he says. "We need a new way to think about the world. There are major philosophical challenges, and Julian is a part of that."Barbour, meanwhile, is still developing his theory. With Niall Ó Murchadha, an Irish physicist, he is attempting to formulate a modification of general relativity in which not only time but also distance plays no role. In particular, his theory would predict that the universe, being static, is not expanding. The main evidence that physicists have for the expansion— the pervasive stretching of the spectra of light from distant galaxies known as the cosmic redshift— would instead be explained by the gravitational effects of neutron stars and black holes. "If you want the wildly optimistic scenario," he says, "in which the Irishman and I develop this theory, make this prediction, and it turns out to agree with observations, then we would really be in the big time." The parish church next to Barbour's home contains some of the rarest murals in England. One painting, completed in about 1340, shows the murder of Thomas à Becket, the 12th-century archbishop whose beliefs clashed with those of King Henry II. The mural captures the instant when a knight's sword cleaves Becket's skull. Blood spurts from the gash. If Barbour's theory is correct, then the moment of Becket's martyrdom still exists as an eternal Now in some configuration of the universe, as do our own deaths. But in Barbour's cosmos, the hour of our death is not an end; it is but one of the numberless components of an inconceivably vast, frozen structure. All the experiences we've ever had and ever will have lie forever fixed, set like crystalline facets in some infinite, immortal jewel. Our friends, our parents, our children, are always there. In many ways it's a beautiful and comforting vision. But the question still nags: Could it possibly be true?Only time will tell. Is There Life After Death? Julian Barbour is convinced we are all immortal. Unfortunately, in a timeless universe immortality does not come with the same kind of perks that it does on Mount Olympus. In Barbour's vision, we are not like Greek gods who remain forever young. We still have to buy life insurance, and we will certainly seem to age and die. And instead of life after death, there is life alongside death. "We're always locked within one Now," Barbour says. We do not pass through time. Instead, each new instant is an entirely different universe. In all of these universes, nothing ever moves or ages, since time is not present in any of them. One universe might contain you as a baby staring at your mother's face. In that universe you will never move from that one, still scene. In yet another universe, you'll be forever just one breath away from death. All of those universes, and infinitely many more, exist permanently, side by side, in a cosmos of unimaginable size and variety. So there is not one immortal you, but many: the toddler, the cool dude, the codger. The tragedy— or perhaps it's a blessing— is that no one version recognizes its own immortality. Would you really want to be 14 for eternity, waiting for your civics class to end? As odd as this vision of a timeless world might seem, Barbour believes there is something stranger still to ponder: the very fact of our existence. "Creation and the fact that anything is— this for me is the complete mystery," he says. "The fact that we are here is totally mysterious." — T.F. Barbour's home page: a brief biography and an online interview with Barbour, see Comment on this article Collapse bottom bar Log in to your account Email address: Remember me Forgot your password? No problem. Click here to have it emailed to you. Not registered yet?
68ede8782ab78446	Psychology Wiki Schrödinger equation 34,190pages on this wiki Psychology Wiki does not yet have a page about Schrödinger equation, even though this subject is highly linked to it (This is due to the initial use of content from Wikipedia). If not, you may wish to see Wikipedia's article on Schrödinger equation. In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only atomic and subatomic systems, atoms and electrons, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger, who constructed it in 1926. The Schrödinger equation Edit The Schrödinger equation takes several different forms, depending on the physical situation. This section presents the equation for the general case and for the simple case encountered in many textbooks. For a general quantum system: For a single particle in three dimensions: • is the particle's position in three-dimensional space, • is the wavefunction, which is the amplitude for the particle to have a given position r at any given time t. • is the mass of the particle. • is the time independent external with respect to the particle potential energy of the particle at each position r (see Self-action in a system of elementary particles). • is the Laplace operator. Historical background and development Edit Einstein interpreted Planck's quanta as photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, a mysterious wave-particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in relativity, it followed that the momentum of a photon is proportional to its wavenumber. DeBroglie hypothesized that this is true for all particles, for electrons as well as photons, that the energy and momentum of an electron are the frequency and wavenumber of a wave. Assuming that the waves travel roughly along classical paths, he showed that they form standing waves only for certain discrete frequencies, discrete energy levels which reproduced the old quantum condition. Following up on these ideas, Schrödinger decided to find a proper wave equation for the electron. He was guided by Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system--- the trajectories of light rays become sharp tracks which obey an analog of the principle of least action. Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves. This is what Schrödinger did, and a modern version of his reasoning is reproduced in the next section. The equation he found is (in natural units): : Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V, created by the positively charged proton. This computation reproduced the energy levels of the Bohr model. But this was not enough, since Sommerfeld had already seemingly correctly reproduced relativistic corrections. Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential: He found the standing-waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin with a lover. While there, Schrödinger decided that the earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. He put together his wave equation and the spectral analysis of hydrogen in a paper in 1926. The paper was enthusiastically endorsed by Einstein, who saw the matter-waves as the visualizable antidote to what he considered to be the overly formal matrix mechanics. The Schrödinger equation tells you the behaviour of , but does not say what is. Schrödinger tried unsuccessfully, in his fourth paper, to interpret it as a charge density. In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted as a probability amplitude. Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities; like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory, Schrödinger was never reconciled to the Copenhagen interpretation. Derivation Edit (1) The total energy E of a particle is This is the classical expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. The momentum of the particle is p, or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. Note that the energy E and momentum p appear in the following two relations: (2) Einstein's light quanta hypothesis of 1905, which asserts that the energy E of a photon is proportional to the frequency f of the corresponding electromagnetic wave: where the frequency f of the quanta of radiation (photons) are related by Planck's constant h,:::and is the angular frequency of the wave. (3) The de Broglie hypothesis of 1924, which states that any particle can be associated with a wave, represented mathematically by a wavefunction Ψ, and that the momentum p of the particle is related to the wavelength λ of the associated wave by: where is the wavelength and is the wavenumber of the wave. Expressing p and k as vectors, we have Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: and to realize that since and similarly since we find: so that, again for a plane wave, he obtained: And by inserting these expressions for the energy and momentum into the classical formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V: The particle is described by a wave; the frequency is the energy E of the particle, while the momentum p is the wavenumber k. Because of special relativity, these are not two separate assumptions. The total energy is the same function of momentum and position as in classical mechanics: where the first term T(p) is the kinetic energy and the second term V(x) is the potential energy. Schrödinger required that a Wave packet at position x with wavenumber k will move along the trajectory determined by Newton's laws in the limit that the wavelength is small. Consider first the case without a potential, V=0. So that a plane wave with the right energy/frequency relationship obeys the free Schrödinger equation: and by adding together plane waves, you can make an arbitrary wave. When there is no potential, a wavepacket should travel in a straight line at the classical velocity. The velocity v of a wavepacket is: which is the momentum over the mass as it should be. This is one of Hamilton's equations from mechanics: after identifying the energy and momentum of a wavepacket as the frequency and wavenumber. To include a potential energy, consider that as a particle moves the energy is conserved, so that for a wavepacket with approximate wavenumber k at approximate position x the quantity must be constant. The frequency doesn't change as a wave moves, but the wavenumber does. So where there is a potential energy, it must add in the same way: This is the time dependent Schrödinger equation. It is the equation for the energy in classical mechanics, turned into a differential equation by substituting: Schrödinger studied the standing wave solutions, since these were the energy levels. Standing waves have a complicated dependence on space, but vary in time in a simple way: substituting, the time-dependent equation becomes the standing wave equation: Which is the original time-independent Schrödinger equation. In a potential gradient, the k-vector of a short-wavelength wave must vary from point to point, to keep the total energy constant. Sheets perpendicular to the k-vector are the wavefronts, and they gradually change direction, because the wavelength is not everywhere the same. A wavepacket follows the shifting wavefronts with the classical velocity, with the acceleration equal to the force divided by the mass. An easy modern way to verify that Newton's second law holds for wavepackets is to take the Fourier transform of the time dependent Schrödinger equation. For an arbitrary polynomial potential this is called the Schrödinger equation in the momentum representation: The group-velocity relation for the fourier transformed wave-packet gives the second of Hamilton's equations. Versions Edit There are several equations which go by Schrödinger's name: Where is a linear operator acting on the wavefunction . takes as input one and produces another in a linear way, a function-space version of a matrix multiplying a vector. For the specific case of a single particle in one dimension moving under the influence of a potential V. and the operator H can be read off: it is a combination of the operator which takes the second derivative, and the operator which pointwise multiplies by V(x). When acting on it reproduces the right hand side. For a particle in three dimensions, the only difference is more derivatives: and for N particles, the difference is that the wavefunction is in 3N-dimensional configuration space, the space of all possible particle positions. This last equation is in a very high dimension, so that the solutions are not easy to visualize. This is the equation for the standing waves, the eigenvalue equation for H. In abstract form, for a general quantum system, it is written: For a particle in one dimension, But there is a further restriction--- the solution must not grow at infinity, so that it has a finite L^2-norm: For example, when there is no potential, the equation reads: which has oscillatory solutions for E>0 (the C's are arbitrary constants): and exponential solutions for E<0 For a constant potential V the solution is oscillatory for E>V and exponential for E<V, corresponding to energies which are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, to quantum tunneling. If the potential V grows at infinity, the motion is classically confined to a finite region, which means that in quantum mechanics every solution becomes an exponential far enough away. The condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies. A solution of the time independent equation is called an energy eigenstate with energy E: To find the time dependence of the state, consider starting the time-dependent equation with an initial condition . The time derivative at t=0 is everywhere proportional to the value: So that at first the whole function just gets rescaled, and thus maintains the property that its time derivative is proportional to itself, for being linear. So for all times, So that the solution of the time-dependent equation with this initial condition is: This is a restatement of the fact that solutions of the time-independent equation are the standing wave solutions of the time dependent equation. They only get multiplied by a phase as time goes by, and otherwise are unchanged. Since is time-independent they are called stationary states. The nonlinear Schrödinger equation is the partial differential equation for the complex field ψ. This equation arises from the Hamiltonian with the Poisson brackets It must be noted that this is a classical field equation. Unlike its linear counterpart, it never describes the time evolution of a quantum state. Properties Edit The Schrödinger equation describes the time evolution of a quantum state, and must determine the future value from the present value. A classical field equation can be second order in time derivatives, the classical state can include the time derivative of the field. But a quantum state is a full description of a system, so that the Schrödinger equation is always first order in time. The Schrödinger equation is linear in the wavefunction: if and are solutions to the time dependent equation, then so is , where a and b are any complex numbers. In quantum mechanics, the time evolution of a quantum state is always linear, for fundamental reasons. Although there are nonlinear versions of the Schrödinger equation, these are not equations which describe the evolution of a quantum state, but classical field equations like Maxwell's equations or the Klein-Gordon equation. The Schrödinger equation itself can be thought of as the equation of motion for a classical field not for a wavefunction, and taking this point of view, it describes a coherent wave of nonrelativistic matter, a wave of a Bose condensate or a superfluid with a large indefinite number of particles and a definite phase and amplitude. The time-independent equation is also linear, but in this case linearity has a slightly different meaning. If two wavefunctions and are solutions to the time-independent equation with the same energy E, then any linear combination of the two is a solution with energy E. Two different solutions with the same energy are called degenerate. In an arbitrary potential, there is one obvious degeneracy: if a wavefunction solves the time-independent equation, so does . By taking linear combinations, the real and imaginary part of are each solutions. So that restricting attention to real valued wavefunctions does not affect the time-independent eigenvalue problem. In the time-dependent equation, complex conjugate waves move in opposite directions. Given a solution to the time dependent equation , the replacement: produces another solution, and is the extension of the complex conjugation symmetry to the time-dependent case. The symmetry of complex conjugation is called time-reversal. The Schrödinger equation is Unitary, which means that the total norm of the wavefunction, the sum of the squares of the value at all points: has zero time derivative. The derivative of is according to the complex conjugate equations where the operator is defined as the continuous analog of the Hermitian conjugate, For a discrete basis, this just means that the matrix elements of the linear operator H obey: The derivative of the inner product is: and is proportional to the imaginary part of H. If H has no imaginary part, if it is self-adjoint, then the probability is conserved. This is true not just for the Schrödinger equation as written, but for the Schrödinger equation with nonlocal hopping: so long as: the particular choice: reproduces the local hopping in the ordinary Schrödinger equation. On a discrete lattice approximation to a continuous space, H(x,y) has a simple form: whenever x and y are nearest neighbors. On the diagonal where n is the number of nearest neighbors. If the potential is bounded from below, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the variational principle, as follows. (See also below.) For any linear operator bounded from below, the eigenvector with the smallest eigenvalue is the vector that minimizes the quantity over all which are normalized: For the Schrödinger Hamiltonian bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of (we used an integration by parts). The right hand side is never smaller than the smallest value of ; in particular, the ground state energy is positive when is everywhere positive. For potentials that are bounded below and are not infinite in such a way that will divide space into regions which are inaccessible by quantum tunneling, there is a ground state which minimizes the integral above. The lowest energy wavefunction is real and nondegenerate and has the same sign everywhere. To prove this, let the ground state wavefunction be . The real and imaginary parts are separately ground states, so it is no loss of generality to assume the is real. Suppose now, for contradiction, that changes sign. Define to be the absolute value of . η = \| ψ \| The potential and kinetic energy integral for is equal to psi, except that has a kink wherever changes sign. The integrated-by-parts expression for the kinetic energy is the sum of the squared magnitude of the gradient, and it is always possible to round out the kink in such a way that the gradient gets smaller at every point, so that the kinetic energy is reduced. This also proves that the ground state is nondegenerate. If there were two ground states and not proportional to each other and both everywhere nonnegative then a linear combination of the two is still a ground state, but it can be made to have a sign change. For one-dimensional potentials, every eigenstate is nondegenerate, because the number of sign changes is equal to the level number. Already in two dimensions, it is easy to get a degeneracy--- for example, if a particle is moving in a separable potential: V(x,y) = U(x) + W(y), then the energy levels are the sum of the energies of the one-dimensional problem. It is easy to see that by adjusting the overall scale of U and W that the levels can be made to collide. For standard examples, the three-dimensional harmonic oscillator and the central potential, the degeneracies are a consequence of symmetry. The probability density of a particle is . The probability flux is defined as: The probability flux satisfies the continuity equation: where is the probability density and measured in units of (probability)/(volume) = r. This equation is the mathematical equivalent of the probability conservation law. For a plane wave: So that not only is the probability of finding the particle the same everywhere, but the probability flux is as expected from an object moving at the classical velocity . The reason that the Schrödinger equation admits a probability flux is because all the hopping is local and forward in time. There are many linear operators which act on the wavefunction, each one defines a Heisenberg matrix when the energy eigenstates are discrete. For a single particle, the operator which takes the derivative of the wavefunction in a certain direction: Is called the momentum operator. Multiplying operators is just like multiplying matrices, the product of A and B acting on is A acting on the output of B acting on . An eigenstate of p obeys the equation: for a number k, and for a normalizable wavefunction this restricts k to be real, and the momentum eigenstate is a wave with frequency k. The position operator x multiplies each value of the wavefunction at the position x by x: So that in order to be an eigenstate of x, a wavefunction must be entirely concentrated at one point: In terms of p, the Hamiltonian is: It is easy to verify that p acting on x acting on psi: while x acting on p acting on psi reproduces only the first term: so that the difference of the two is not zero: or in terms of operators: Since the time derivative of a state is: while the complex conjugate is The time derivative of a matrix element obeys the Heisenberg equation of motion. This establishes the equivalence of the Schrödinger and Heisenberg formalisms, ignoring the mathematical fine points of the limiting procedure for continuous space. The Schrödinger equation satisfies the correspondence principle. In the limit of small wavelength wavepackets, it reproduces Newton's laws. This is easy to see from the equivalence to matrix mechanics. All operators in Heisenberg's formalism obey the quantum analog of Hamilton's equations: So that in particular, the equations of motion for the X and P operators are: in the Schrödinger picture, the interpretation of this equation is that it gives the time rate of change of the matrix element between two states when the states change with time. Taking the expectation value in any state shows that Newton's laws hold not only on average, but exactly, for the quantities: The Schrödinger equation does not take into account relativistic effects; as a wave equation, it is invariant under a Galilean transformation, but not under a Lorentz transformation. But in order to include relativity, the physical picture must be altered in a radical way. The Klein–Gordon equation uses the relativistic mass-energy relation: to produce the differential equation: which is relativistically invariant, but second order in , and so cannot be an equation for the quantum state. This equation also has the property that there are solutions with both positive and negative frequency, a plane wave solution obeys: which has two solutions, one with positive frequency the other with negative frequency. This is a disaster for quantum mechanics, because it means that the energy is unbounded below. A more sophisticated attempt to solve this problem uses a first order wave equation, the Dirac equation, but again there are negative energy solutions. In order to solve this problem, it is essential to go to a multiparticle picture, and to consider the wave equations as equations of motion for a quantum field, not for a wavefunction. The reason is that relativity is incompatible with a single particle picture. A relativistic particle cannot be localized to a small region without the particle number becoming indefinite. When a particle is localized in a box of length L, the momentum is uncertain by an amount roughly proportional to h/L by the uncertainty principle. This leads to an energy uncertainty of hc/L, when \|p\| is large enough so that the mass of the particle can be neglected. This uncertainty in energy is equal to the mass-energy of the particle when and this is called the Compton wavelength. Below this length, it is impossible to localize a particle and be sure that it stays a single particle, since the energy uncertainty is large enough to produce more particles from the vacuum by the same mechanism that localizes the original particle. But there is another approach to relativistic quantum mechanics which does allow you to follow single particle paths, and it was discovered within the path-integral formulation. If the integration paths in the path integral include paths which move both backwards and forwards in time as a function of their own proper time, it is possible to construct a purely positive frequency wavefunction for a relativistic particle. This construction is appealing, because the equation of motion for the wavefunction is exactly the relativistic wave equation, but with a nonlocal constraint that separates the positive and negative frequency solutions. The positive frequency solutions travel forward in time, the negative frequency solutions travel backwards in time. In this way, they both analytically continue to a statistical field correlation function, which is also represented by a sum over paths. But in real space, they are the probability amplitudes for a particle to travel between two points, and can be used to generate the interaction of particles in a point-splitting and joining framework. The relativistic particle point of view is due to Richard Feynman. Feynman's method also constructs the theory of quantized fields, but from a particle point of view. In this theory, the equations of motion for the field can be interpreted as the equations of motion for a wavefunction only with caution--- the wavefunction is only defined globally, and in some way related to the particle's proper time. The notion of a localized particle is also delicate--- a localized particle in the relativistic particle path integral corresponds to the state produced when a local field operator acts on the vacuum, and exactly which state is produced depends on the choice of field variables. Some general techniques are: In some special cases, special methods can be used: When the potential is zero, the Schrödinger equation is linear with constant coefficients: The solution for any initial condition can be found by Fourier transforms. Because the coefficients are constant, an initial plane wave stays a plane wave. Only the coefficient changes: So that A is also oscillating in time: and the solution is: Where , a restatement of DeBroglie's relations. To find the general solution, write the initial condition as a sum of plane waves by taking its Fourier transform: The equation is linear, so each plane waves evolves independently: Which is the general solution. When complemented by an effective method for taking Fourier transforms, it becomes an efficient algorithm for finding the wavefunction at any future time--- Fourier transform the initial conditions, multiply by a phase, and transform back. An easy and instructive example is the Gaussian wavepacket: where a is a positive real number, the square of the width of the wavepacket. The total normalization of this wavefunction is: The Fourier transform is a Gaussian again in terms of the wavenumber k: With the physics convention which puts the factors of in Fourier transforms in the k-measure. Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is: The inverse Fourier transform is still a Gaussian, but the parameter a has become complex, and there is an overall normalization factor. The branch of the square root is determined by continuity in time--- it is the value which is nearest to the positive square root of a. It is convenient to rescale time to absorb m, replacing t/m by t. The integral of over all space is invariant, because it is the inner product of with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy state, with wavefunction , the inner product: only changes in time in a simple way: its phase rotates with a frequency determined by the energy of . When has zero energy, like the infinite wavelength wave, it doesn't change at all. The sum of the absolute square of is also invariant, which is a statement of the conservation of probability. Explicitly in one dimension: Which gives the norm: which has preserved its value, as it must. The width of the Gaussian is the interesting quantity, and it can be read off from the form of : The width eventually grows linearly in time, as . This is wave-packet spreading--- no matter how narrow the initial wavefunction, a Schrödinger wave eventually fills all of space. The linear growth is a reflection of the momentum uncertainty--- the wavepacket is confined to a narrow width and so has a momentum which is uncertain by the reciprocal amount , a spread in velocity of , and therefore in the future position by , where the factor of m has been restored by undoing the earlier rescaling of time. Galilean boosts are transformations which look at the system from the point of view of an observer moving with a steady velocity -v. A boost must change the physical properties of a wavepacket in the same way as in classical mechanics: So that the phase factor of a free Schrödinger plane wave: is only different in the boosted coordinates by a phase which depends on x and t, but not on p. An arbitrary superposition of plane wave solutions with different values of p is the same superposition of boosted plane waves, up to an overall x,t dependent phase factor. So any solution to the free Schrödinger equation, , can be boosted into other solutions: produces a boosted wave: Boosting the spreading Gaussian wavepacket: produces the moving Gaussian: Which spreads in the same way. The narrow-width limit of the Gaussian wavepacket solution is the propagator K. For other differential equations, this is sometimes called the Green's function, but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K. When a is the infinitesimal quantity , the Gaussian initial condition, rescaled so that its integral is one: becomes a delta function, so that its time evolution: gives the propagator. Note that a very narrow initial wavepacket instantly becomes infinitely wide, with a phase which is more rapidly oscillatory at large values of x. This might seem strange--- the solution goes from being concentrated at one point to being everywhere at all later times, but it is a reflection of the momentum uncertainty of a localized particle. Also note that the norm of the wavefunction is infinite, but this is also correct since the square of a delta function is divergent in the same way. The factor of is an infinitesimal quantity which is there to make sure that integrals over K are well defined. In the limit that becomes zero, K becomes purely oscillatory and integrals of K are not absolutely convergent. In the remainder of this section, it will be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit is to only to be taken after the final state is calculated. The propagator is the amplitude for reaching point x at time t, when starting at the origin, x=0. By translation invariance, the amplitude for reaching a point x when starting at point y is the same function, only translated: In the limit when t is small, the propagator converges to a delta function: but only in the sense of distributions. The integral of this quantity multiplied by an arbitrary differentiable test function gives the value of the test function at zero. To see this, note that the integral over all space of K is equal to 1 at all times: since this integral is the inner-product of K with the uniform wavefunction. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit is taken after everything else. So the propagation kernel is the future time evolution of a delta function, and it is continuous in a sense, it converges to the initial delta function at small times. If the initial wavefunction is an infinitely narrow spike at position : it becomes the oscillatory wave: Since every function can be written as a sum of narrow spikes: the time evolution of every function is determined by the propagation kernel: And this is an alternate way to express the general solution. The interpretation of this expression is that the amplitude for a particle to be found at point x at time t is the amplitude that it started at times the amplitude that it went from to x, summed over all the possible starting points. In other words, it is a convolution of the kernel K with the initial condition. Since the amplitude to travel from x to y after a time can be considered in two steps, the propagator obeys the identity: Which can be interpreted as follows: the amplitude to travel from x to z in time t+t' is the sum of the amplitude to travel from x to y in time t multiplied by the amplitude to travel from y to z in time t', summed over all possible intermediate states y. This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a path integral. The spreading of wavepackets in quantum mechanics is directly related to the spreading of probability densities in diffusion. For a particle which is random walking, the probability density function at any point satisfies the diffusion equation: where the factor of 2, which can be removed by a rescaling either time or space, is only for convenience. A solution of this equation is the spreading gaussian: and since the integral of , is constant, while the width is becoming narrow at small times, this function approaches a delta function at t=0: again, only in the sense of distributions, so that for any smooth test function f. The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity: Which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H: which is the infinitesimal diffusion operator. A matrix has two indices, which in continuous space makes it a function of x and x'. In this case, because of translation invariance, the matrix element K only depend on the difference of the position, and a convenient abuse of notation is to refer to the operator, the matrix elements, and the function of the difference by the same name: Translation invariance means that continuous matrix multiplication: is really convolution: The exponential can be defined over a range of t's which include complex values, so long as integrals over the propagation kernel stay convergent. As long as the real part of z is positive, for large values of x K is exponentially decreasing and integrals over K are absolutely convergent. The limit of this expression for z coming close to the pure imaginary axis is the Schrödinger propagator: and this gives a more conceptual explanation for the time evolution of Gaussians. From the fundamental identity of exponentiation, or path integration: holds for all complex z values where the integrals are absolutely convergent so that the operators are well defined. So that quantum evolution starting from a Gaussian, which is the diffusion kernel K: gives the time evolved state: This explains the diffusive form of the Gaussian solutions: The variational principle asserts that for any Hermitian matrix A, the eigenvector corresponding to the lowest eigenvalue minimizes the quantity: on the unit sphere . This follows by the method of Lagrange multipliers, at the minimum the gradient of the function is parallel to the gradient of the constraint: which is the eigenvalue condition so that the extreme values of a quadratic form A are the eigenvalues of A, and the value of the function at the extreme values is just the corresponding eigenvalue: When the hermitian matrix is the Hamiltonian, the minimum value is the lowest energy level. In the space of all wavefunctions, the unit sphere is the space of all normalized wavefunctions , the ground state minimizes or, after an integration by parts, All the stationary points come in complex conjugate pairs since the integrand is real. Since the stationary points are eigenvalues, any linear combination is a stationary point, and the real and imaginary part are both stationary points. For a particle in a positive definite potential, the ground state wavefunction is real and positive, and has a dual interpretation as the probability density for a diffusion process. The analogy between diffusion and nonrelativistic quantum motion, originally discovered and exploited by Schrödinger, has led to many exact solutions. A positive definite wavefunction: is a solution to the time-independent Schrödinger equation with m=1 and potential: with zero total energy. W, the logarithm of the ground state wavefunction. The second derivative term is higher order in , and ignoring it gives the semi-classical approximation. The form of the ground state wavefunction is motivated by the observation that the ground state wavefunction is the Boltzmann probability for a different problem, the probability for finding a particle diffusing in space with the free-energy at different points given by W. If the diffusion obeys detailed balance and the diffusion constant is everywhere the same, the Fokker Planck equation for this diffusion is the Schrödinger equation when the time parameter is allowed to be imaginary. This analytic continuation gives the eigenstates a dual interpretation--- either as the energy levels of a quantum system, or the relaxation times for a stochastic equation. W should grow at infinity, so that the wavefunction has a finite integral. The simplest analytic form is: with an arbitrary constant , which gives the potential: This potential describes a Harmonic oscillator, with the ground state wavefunction: The total energy is zero, but the potential is shifted by a constant. The ground state energy of the usual unshifted Harmonic oscillator potential: is then the additive constant: which is the zero point energy of the oscillator. Another simple but useful form is where W is proportional to the radial coordinate. This is the ground state for two different potentials, depending on the dimension. In one dimension, the corresponding potential is singular at the origin, where it has some nonzero density: and, up to some rescaling of variables, this is the lowest energy state for a delta function potential, with the bound state energy added on. with the ground state energy: and the ground state wavefunction: In higher dimensions, the same form gives the potential: which can be identified as the attractive Coulomb law, up to an additive constant which is the ground state energy. This is the superpotential that describes the lowest energy level of the Hydrogen atom, once the mass is restored by dimensional analysis: where is the Bohr radius, with energy The ansatz modifies the Coulomb potential to include a quadratic term proportional to , which is useful for nonzero angular momentum. In the mathematical formulation of quantum mechanics, a physical system is fully described by a vector in a complex Hilbert space, the collection of all possible normalizable wavefunctions. The wavefunction is just an alternate name for the vector of complex amplitudes, and only in the case of a single particle in the position representation is it a wave in the usual sense, a wave in space time. For more complex systems, it is a wave in an enormous space of all possible worlds. Two nonzero vectors which are multiples of each other, two wavefunctions which are the same up to rescaling, represent the same physical state. The wavefunction vector can be written in several ways: 1. as an abstract ket vector: ::2. As a list of complex numbers, the components relative to a discrete list of normalizable basis vectors : ::3. As a continuous superposition of non-normalizable basis vectors, like position states : The divide between the continuous basis and the discrete basis can be bridged by limiting arguments. The two can be formally unified by thinking of each as a measure on the real number line. In the most abstract notation, the Schrödinger equation is written: which only says that the wavefunction evolves linearly in time, and names the linear operator which gives the time derivative the Hamiltonian H. In terms of the discrete list of coefficients: which just reaffirms that time evolution is linear, since the Hamiltonian acts by matrix multiplication. In a continuous representation, the Hamiltonian is a linear operator, which acts by the continuous version of matrix multiplication: Taking the complex conjugate: In order for the time-evolution to be unitary, to preserve the inner products, the time derivative of the inner product must be zero: for an arbitrary state , which requires that H is Hermitian. In a discrete representation this means that . When H is continuous, it should be self-adjoint, which adds some technical requirement that H does not mix up normalizable states with states which violate boundary conditions or which are grossly unnormalizable. The formal solution of the equation is the matrix exponential (natural units): For every time-independent Hamiltonian operator, , there exists a set of quantum states, , known as energy eigenstates, and corresponding real numbers satisfying the eigenvalue equation. This is the time-independent Schrödinger equation. For the case of a single particle, the Hamiltonian is the following linear operator (natural units): which is a Self-adjoint operator when V is not too singular and does not grow too fast. Self-adjoint operators have the property that their eigenvalues are real in any basis, and their eigenvectors form a complete set, either discrete or continuous. Expressed in a basis of Eigenvectors of H, the Schrödinger equation becomes trivial: Which means that each energy eigenstate is only multiplied by a complex phase: Which is what matrix exponentiation means--- the time evolution acts to rotate the eigenfunctions of H. When H is expressed as a matrix for wavefunctions in a discrete energy basis: so that: The physical properties of the C's are extracted by acting by operators, matrices. By redefining the basis so that it rotates with time, the matrices become time dependent, which is the Heisenberg picture. Galilean symmetry requires that H(p) is quadratic in p in both the classical and quantum Hamiltonian formalism. In order for Galilean boosts to produce a p-independent phase factor, px - Ht must have a very special form--- translations in p need to be compensated by a shift in H. This is only true when H is quadratic. The infinitesimal generator of Boosts in both the classical and quantum case is: where the sum is over the different particles, and B,x,p are vectors. The poisson bracket/commutator of with x and p generate infinitesimal boosts, with v the infinitesimal boost velocity vector: Iterating these relations is simple, since they add a constant amount at each step. By iterating, the dv's incrementally sum up to the finite quantity V: In other words, B/M is the current guess for the position that the center of mass had at time zero. Since B is explicitly time dependent, H does not commute with B, rather: this gives the transformation law for H under infinitesimal boosts: the interpretation of this formula is that the change in H under an infinitesimal boost is entirely given by the change of the center of mass kinetic energy, which is the dot product of the total momentum with the infinitesimal boost velocity. The two quantities (H,P) form a representation of the Galilean group with central charge M, where only H and P are classical functions on phase-space or quantum mechanical operators, while M is a parameter. The transformation law for infinitesimal v: can be iterated as before--- P goes from P to P+MV in infinitesimal increments of v, while H changes at each step by an amount proportional to P, which changes linearly. The final value of H is then changed by the value of P halfway between the starting value and the ending value: The factors proportional to the central charge M are the extra wavefunction phases. Boosts give too much information in the single-particle case, since Galilean symmetry completely determines the motion of a single particle. Given a multi-particle time dependent solution: See also Edit Notes Edit 1. ^ 2. ^ Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), Main paper 3. ^ Schrödinger: Life and Thought by Walter John Moore, Cambridge University Press 1992 ISBN 0-521-43767-9, page 219 (hardback version) 5. ^ Schrödinger: Life and Thought by Walter John Moore, Cambridge University Press 1992 ISBN 0-521-43767-9, page 479 (hardback version) makes it clear that even in his last year of life, in a letter to Max Born, he never accepted the Copenhagen Interpretation. cf pg 220 References Edit • Paul Adrien Maurice Dirac (1958). The Principles of Quantum Mechanics (4th ed.). Oxford University Press. • David J. Griffiths (2004). Introduction to Quantum Mechanics (2nd ed.). Benjamin Cummings. • Richard Liboff (2002). Introductory Quantum Mechanics (4th ed.). Addison Wesley. • David Halliday (2007). Fundamentals of Physics (8th ed.). Wiley. • Serway, Moses, and Moyer (2004). Modern Physics (3rd ed.). Brooks Cole. • Walter John Moore (1992). Schrödinger: Life and Thought. Cambridge University Press. • Schrödinger, Erwin (December 1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Phys. Rev. 28 (6) 28: 1049–1070. doi:10.1103/PhysRev.28.1049. External links Edit Around Wikia's network Random Wiki
04c7f5b7afdf3f39	Name: Robert Parrish Age: 23 Born: Miami Nationality: U.S. Current position: Graduate student, Georgia Institute of Technology Education: B.S., mechanical engineering, Georgia Institute of Technology What is your field of research? I apply quantum mechanics to simulate the motions of electrons in molecules, using computers. Accurate simulations of this type provide in silico chemical predictions about whether a molecule might make a good drug candidate, reaction catalyst, etc. I have always been fascinated that the beautiful complexities of phenomena ranging from weather patterns to the evolution of the universe each emerge from a simple governing equation which can be written in a page or less. In my undergraduate work in engineering, I learned that the hard bit is solving those equations, and discovered that I was a natural at finding new approximations to speed up those solutions. Of all the equations I studied, the electronic Schrödinger equation of quantum chemistry was easily the most difficult, and therefore the most fun to work on. Where do you see yourself in 10 years? Do you have specific research goals, or a particular problem or puzzle that you really want to solve? In 10 years, I hope to be a professor. Thinking about a really tough problem 24/7, working flexible hours (albeit 100 of them per week!), and having amazing friends as research collaborators is the kind of lifestyle I would cultivate even if I was not paid to do it. As far as research goals, I am extremely interested in compression algorithms to treat the correlated motions of electrons. If an efficient scheme could be devised, we could run fully quantum-mechanical simulations of chemical systems as large as proteins. This would move a lot of chemical discovery away from the lab bench and onto the computer, in the same way that computational fluid dynamics has revolutionized the design of aircraft. Who are your scientific heroes? Horst Störmer of Columbia University, and formerly Bell Labs, who discovered the fractional quantum Hall effect. I saw him speak about nanotechnology when I was in high school, and was struck by how much he obviously enjoyed going to work every day, and how that enthusiasm naturally led to an amazing discovery. Also, my father, Jack Parrish, who is a flight meteorologist studying hurricanes with NOAA. As a very practical scientist, he can fly into a hurricane to eyeball it, and gather just as much information as a supercomputer simulation. This reminds me that all the pretty mathematics I work on should eventually boil down to something useful. The funny thing about my field is that we already know how to exactly solve for the motions of the electrons for any system, but we would need an exponential amount of computer time to do it. We often joke about writing an article titled “Exact solutions of the electronic Schrödinger equation with a time machine,” where we send a workstation back in time about 150 million years, and then go pick the results of the simulation up yesterday. What activities outside of physics do you most enjoy? I really enjoy travel and am looking forward to seeing some of Germany and Austria after the Lindau conference. Also, being a Floridian, I am experiencing severe beach withdrawal here in Atlanta, and I look forward to finding someplace to reacquire my windsurfing skills during my postdoc. What do you hope to gain from this year’s Lindau meeting? Popular culture often seems to think that science is done at 3:00 A.M. by a solo grad student in a white coat slaving over a lab bench. While I have certainly had my fair share of evenings spent in front of green-on-black windows of C++ source code, all of my best ideas have come from having a chat over a beer with a friend. Lindau is a great opportunity to make friends like this, who might eventually become colleagues. In particular, I hope to have the opportunity to talk with many young scientists and laureates who are working in areas orthogonal to my own. After finding success in chemistry following an undergraduate in engineering, I am a strong believer that ideas can often cross from one field to another, and Lindau is the perfect place for that to happen. I have spent a considerable amount of time over the last two years writing a code for a method called density functional theory, for which Walter Kohn won the Nobel Prize in 1998. His development of the method has caused a renaissance in electronic structure theory over the past 25 years, but we are becoming increasingly aware of spectacular failures for some chemical systems. I am very interested to hear his take on how we might fix these errors firsthand. « Previous 17. Ulrika Forsberg 30 Under 30: Lindau Nobel Laureate Meeting Next » 19. Matteo Lucchini
40296b3bfa4e4014	Study your flashcards anywhere! Download the official Cram app for free > • Shuffle Toggle On Toggle Off • Alphabetize Toggle On Toggle Off • Front First Toggle On Toggle Off • Both Sides Toggle On Toggle Off • Read Toggle On Toggle Off How to study your flashcards. H key: Show hint (3rd side).h key A key: Read text to speech.a key Play button Play button Click to flip 108 Cards in this Set • Front • Back Atoms and molecules are governed by same or different laws? Atoms and molecules are not governed by the same physical laws as larger objects Max Planck discovered? Energy is continous, and Atoms and molecules emit energy only in certain discrete quantities or quanta. What is a wave? vibrating disturbance by which energy is transmitted How do water waves travel? Wave repeats itself at regular intervals. Waves can be characterized by? length, height, and number of waves per second What is wavelength? distance between identical points on successive waves What is frequency? number of waves that pass through a particular point in 1 second What is amplitude? What is speed? Depends on the type of wave and nature of the medium. Speed is a product of wavelength and frequency u = yV What is wavelenght and frequency measured in? m, cm, nm. 1 Hz = 1 cycle/second Visbile light is made up of? Electromagnetic waves What are Electromagnetic waves? Waves with an electrical field component and a magnetic field component The two components of electromagnetic waves differ in? Same speed, but travel in perpendicular planes What is electromagnetic radiation? Emission and transmission of energy in the form of electromagnetic waves What speed do electromagnetic waves travel at? 3.00 * 10^8 m/s Long waves Are emitted from large antennas (radio, cell phones). Radio waves lowest frequency Short waves Have higher energy radiation. Gamma rays have the shortest wavelenght and highest frequency What is quantum? smallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation What is the wavelength (in m) of an electromagnetic wave whose frequency is 3.64 * 10^7 Hz? What is the frequency (in Hz) of a wave whose speed is 713 m/s and wavelength is 1.14 m? What is Planck's constant? H= 6.63 * 10^-34 What is the photoelectric effect? Electrons are ejected from the surface of certain metals exposed to light of at least a certain minimum frequency called the Threshold frequency Number of electrons depends on? intensity of light but energy of electrons does not Einstein theorized light is made up of? A stream of particles called photons. What do you need to break electrons free from a metal? Requires light of sufficiently high frequency What is KE and BE? E = hn = KE + BE KE = kinetic energy of electron BE = binding energy of the electron in the metal The higher the frequency the greater the KE KE is dependent on? Frequency of the light. Ejected Light behaves as? Both as a particle and wave depending on the property being measured. All matter actually exhibits this dual nature. The energy of a photon is 5.87 * 10-20 J. What is the wavelength in nm? h = 6.63 * 10-34 Js A photon has a wavelength of 624 nm. Calculate the energy of the photon in J. What is an emission spectra? Either continous or line spectra of radiation emitted by substances What is a line spectra? The light emmission only at specific wavelenghts. Produced by atoms Emission spectra of the sun or heated solids Why is Bohr's model no accurate? Because it does not explain the spectral lines What is Rydberg's constant? Rydberg constant (RH) = 2.18 10^-18 J Energy of electron Where are free electrons? infinitely far from the nucleus Why is there a negative sign in equation of Rydberg's constant? Negative sign – is to assign a lower energy of electron in an atom than the energy of the free electron (arbitrarily assigned a value of zero) What happens as the electron gets closer to the nucleus? Becomes more stable and E becomes more negative, which corresponds to the most stable state. What is the ground state? The lowest energy state of a system or most stable What is the excited state? Higher energy than the ground state When is radiant energy emitted? When electrons drop from a higher energy orbital to a lower energy orbital The quantity of energy produced is dependent only on what? Initial and final states If an electron starts at ni and drops to a lower energy state of nf the change in energy is given by Equation= What happen in the states when energy is given off. Ni>Nf change in energy is negative Each line on the emission spectrum corresponds to? To a transition in the H atom When a large number of H atoms are examined all the lines of the spectrum are visible Electrons bound to a nucleus behave like? Electrons bound to a nucleus behave like a standing wave Waves that can be generated by plucking a string Some points on a standing wave are Do not move at all The amplitude at this point is zero Nodes are located at the end of the string and maybe in the middle De Broglie says that if an electron does not behave like a wave then? De Broglie said that if an electron does behave like as wave then the wave must perfectly fit the circumference of the orbit The circumference of the orbit is related to the wavelength by the equation When can a particle be a wave and a wave be a particle? A particle in motion can be treated as a wave A wave can also exhibit properties of a particle Protons can be accelerated to speeds near the speed of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at 2.90 * 108 m/s. (mp = 1.673 * 10-27 kg) A baseball has a mass of about 255 g. Calculate the wavelength of the baseball if it is thrown 100. mph. What is Heisenberg uncertainity principle? It is impossible to know simultaneously both the momentum p and the position of a particle Applying the Heisenberg uncertainty principle to the H atom? We see that the electron can not orbit the nucleus in a circular orbital. If this were the case then we could know both the position and momentum of the electron at the same time. Erwin schrodinger formulated what? Formulated an equation to describe the behavior and energies of submicroscopic objects This equation is very complicated and requires Calculus to solve The equation incorporated? The Schrödinger equation incorporates particle behavior of electrons in the form of mass and wave behavior in the form of wave function (y) Why is the wave funciton significant? Is that the square of the wave function (y2) is proportional to the probability of where the electron is located Where is an electron most likely to be? The most likely place an electron will be is where y2 is greatest What tells Schrodinger equation tell us? The Schrödinger equation gives possible energy states and identifies the wave function of the electrons. These are characterized by quantum numbers. Quantum mechanics gives probability of an electron in a particular region of the atom (electron density) In quantum mechanics the orbits are called? Atomic orbitals What is an atomic orbital? The wave function of an electron in an atom. This is to differentiate from the orbits in Bohr’s model. Each atomic orbital has a characteristic energy and therefore a characteristic distribution of electron density. An assumptions must be made The difference between hydrogen and atoms with more than one electron is not that large. What are quantum numbers? Describe the distribution of electrons in hydrogen and other atoms. What are three quantum numbers that describe the distributioni of electrons? Principal quantum number Angular momentum quantum number Magnetic quantum number What is the fourth quantum number that describes the behavior of a specific electron? Spin quantum number Principal quantum numbers Represented by n Can have values of integers Relates the average distance from the electron to the nucleus in a particular orbital. Larger the n what happens? The greater distance of an electron in the orbital from the nucleus and therefor the larger the orbital. Angular Momentum Quantum Number Represent by l Tells the shape of the orbital Dependent on n For any n, l = any integer from 0 to (n-1) For n = 1, l = 0 l is normally designated by the letter that symbolizes the different atomic orbitals Magnetic quantum number Represented by ml Depends on l For any value l, there are (2l +1) values for ml If l = o then ml = 0 If l = 1 then there are three possible values of ml, (-1, 0, 1) The value of ml indicates the number of orbitals in the subshell with value l Spin quantum numbers Represented by ms It was noticed that the application of a magnetic field could split the lines in an emission spectra The only way this could be explain is if electron behave as tiny magnets If the electrons are thought of as spinning on their own then the magnetic field can be explained The spinning charge generates a magnetic field Spin quantum numbers are always? –½ or +½ Atomic orbital related to angular momentum What is a shell? A collection of orbitals with the same value of n What are subshells? Orbitals with the same n and l value are called subshells For example n = 2 Two subshells: l = 0 and l = 1 Called the 2s and 2p subshells What is the shape of the orbitals? Shapes are not well defined Wave function defining orbital extends from the nucleus to infinity It is convenient to think of the orbitals as having a shape It is especially helpful when talking about chemical bonds What shape and size are s orbitals? All s orbitals are spherical The s orbitals do change in size however The increase in size is the reason for the increase in the principal quantum number Electron density Falls off rapidly as the electron get farther from the nucleus. p orbitals start with the principal quantum number? What happen when n=2, and l=1? three possible orbital so there are 3 2p orbitals What are the shape, and size of orbitals? They are oriented along the axes of a 3-d plot The three orbitals differ only in the orientation They are identical in shape, size and energy p orbital are thought of as two lobes on either side of the nucleus d orbitals n must equal at least 3 and l must equal 2 for the d orbitals to exist How many d orbitals are there? There are 5 d orbitals The orbitals differ in orientation as well as one having a different shape. All 3d orbitals have the same energy d orbitals which have a larger n value have similarly shaped orbitals that are larger Why are f orbitals important? are important for accounting for the behavior element with atomic number > 57 In this class we are not concerned with orbitals having l > 3 Give the quantum numbers associated with the following orbitals: Energy of orbitals increase as? n increases Does electron density change for 2s and 2p orbitals? Although the electron density is different for 2s and 2p orbitals the energy remains the same Which is orbital is the most stable? The total energy depends on what? The total energy depends on the sum of orbital energies as well as the repulsive forces It turns ends up that the total energy is lower when the 4s orbital fills before the 3d Order of atomic orbitals filling 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s 7p 2s quantum numbers has no affect on energy, size, shape, or orientation of the orbital but determines how electrons are arranged in an orbital What is electron configuration? How the electrons of an atom are distributed among the orbitals This is how the electrons are distributed in a ground state atom The number of electrons in an atom is equal to? Atomic number Ground State H Pauli exclusion principle? If they have the same n, l, and ml, then they must have different ms. These two electrons would be in the same orbital but have opposite spins. Thus each orbital can contain only two electrons Contain net unpaired spins and are attracted by a magnet Do not contain unpaired spins and are slightly repelled by a magnet. A He atom with opposite spins in the orbital What happens if the spins in an orbital do match up? The magnetic fields reinforce each other. This would make an atom paramagnetic. Odd and even numbered atoms have? Odd numbered atoms always have one or more unpaired electrons Even numbered atoms may or may not have unpaired electrons Which orbital is filled first? The 1s orbital is filled before electrons are start to fill the 2s or 2p orbital 2s and 2p orbitals Both the 2s and 2p orbitals have electrons that spend more time away from the nucleus than electrons in the 1s orbital The electrons of the 2s and 2p orbitals are shielded from the attractive forces of the nucleus by the 1s electrons This reduces the electrostatic interactions between the nucleus and the 2s and 2p electrons Experimentally the 2s orbital gives us a lower energy than the 2p Although the 2s electron spend more time on average farther from the nucleus than a 2p electron; the denisty near the nucleus is greater for a 2s electron So the 2s orbital is more penetrating The 2s orbital is less shielded For the same values of n, the penetrating power decreases as l increases How is the stability of the electron is determined? The stability of the electron is determined by the strength of attraction to the nucleus Shielding effect 2s orbitals are less shielded than 2p orbitals so it follows that 2s orbitals have lower energy. Less energy is required to remove a 2p electron than a 2s electron Hund's Rule The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins Rules for assigning electrons to orbitals Each shell (or principal quantum number) n contains n subshells Each subshell consists of quantum number l contain (2l + 1) orbitals No more than 2 electrons can be placed in an orbital The maximum number of electrons in principal level n is equal to 2n2 How many electrons can be present in the prinicpal level n=4? What are the principal quantum numbers for the last electron in boron (B)? Aufbau principle As protons are added one by one to the nucleus to build up elements, electrons are similarly added to the atomic orbitals Noble gas core a method of showing electron configurations where the noble gas most nearly preceding the element being considered is used.The noble gas is followed by the electron configuration of the most highly filled subshells Transition metals have either Have either incompletely filled d orbitals or give rise to cations that have incompletely filled d subshells Two irregularities in the fourth preiod Chromium – [Ar]4s13d5 Copper – [Ar]4s13d10 The reason for this is that there is actually more stability in a half-filled or filled d orbital Lanthanides and Actinide Lanthanides – have incompletely filled 4f orbitals or readily give rise to cations with incompletely filled 4f subshells Actinide series – last row of elements, most are not found in nature but have been synthesized Write the ground-state electron configuration for Sr. Write the ground-state electron configuration for Ga.
b6b6ac287b203b83	Nuclear Astrophysics with Radioactive Beams C. A. Bertulani Carlos_B A. Gade Department of Physics, Texas A & M University, Commerce, Texas 75429, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824, USA The quest to comprehend how nuclear processes influence astrophysical phenomena is driving experimental and theoretical research programs worldwide. One of the main goals in nuclear astrophysics is to understand how energy is generated in stars, how elements are synthesized in stellar events and what the nature of neutron stars is. New experimental capabilities, the availability of radioactive beams and increased computational power paired with new astronomical observations have advanced the present knowledge. This review summarizes the progress in the field of nuclear astrophysics with a focus on the role of indirect methods and reactions involving beams of rare isotopes. nuclear astrophysics, rare isotopes, reactions. 24.10.-i, 26., 29.38.-c 1 Introduction Nuclear reactions in stars and stellar explosions are responsible for the ongoing synthesis of the elements [2, 3, 4, 5, 6, 7, 8, 9, 10]. Nuclear physics plays an important role as it determines the signatures of isotopic and elemental abundances found in the spectra of stars, novae, supernovae, and X-ray burst as well as in characteristic -ray radiation from nuclear decays, or in the composition of meteorites and presolar grains (see [11] for a review on the nuclear structure input to nuclear astrophysics). The rapid neutron capture process (r process) is responsible for the existence of about half of the stable nuclei heavier than iron; yet a site that can produce the observed elements self-consistently has not been identified [12, 13]. Capture cross sections for most of the nuclei involved are hard if not impossible to measure in the laboratory and indirect experimental approaches have to be employed to gather the relevant nuclear structure information. Nuclear masses and -decay half-lives are among the few direct observables that are input for calculations that model nucleosynthesis in the r process. X-ray bursts provide a unique window into the physics of neutron stars. They are the most frequent thermonuclear explosions known. The brightness, frequency and opportunity to be observed with different telescopes makes them unique laboratories for explosive nuclear burning at extreme temperatures and densities [14, 15, 16]. The reaction sequence during an X-ray burst proceeds through nuclei at or close to the proton drip line mainly by and reactions and decays (p and rp process) [17]. Most rp-process reaction rates are still based exclusively on theory. Energy in explosive hydrogen burning events such as X-ray bursts is initially generated in the CNO cycle and as the temperature increases, -capture on unstable oxygen and neon ( and ) leads to a break out and an ensuing chain of proton captures that can go as far as tin. Supernovae play a crucial role in the understanding of the universe as they are the major source of nucleosynthesis and possibly of cosmic rays. Core-collapse supernovae [18] are one of the proposed sites of the r process. Thermonuclear supernovae (type Ia) are powered by explosive carbon and oxygen burning of a white dwarf that has reached the Chandrasekhar mass limit. For both types of supernovae, the driving processes are not well understood and weak interaction rates play a key role [19]. Temperatures and densities are so high that electron captures on unstable nuclei become crucial. Most aspects in the study of nuclear physics demand beams of energetic particles to induce nuclear reactions on the nuclei of target atoms. It was from this need that accelerators were developed. Over the years, many ways of accelerating charged particles to ever increasing energies have been devised. Today we have ion beams of all elements from protons to uranium available at energies well beyond those needed for the study of atomic nuclei. The quantities used in nucleosynthesis calculations are reaction rates. A thermonuclear reaction rate is a function of the density of the interacting nuclei, their relative velocity and the reaction cross section. Extrapolation procedures are often needed to derive cross sections in the energy or temperature region of astrophysical relevance. While non-resonant cross sections can be rather well extrapolated to the low-energy region, the presence of continuum, or sub-threshold resonances, can complicate these extrapolations. We will mention some of the important examples. In the Sun, the reaction BeB plays a major role for the production of high energy neutrinos from the -decay of B. These neutrinos come directly from the center of the Sun and are ideal probes of the Sun’s structure. John Bahcall frequently said that this was the most important reaction in nuclear astrophysics [20]. Our understanding of this reaction has improved considerably with the advent of rare-isotope beam facilities. The reaction CO is extremely relevant for the fate of massive stars. It determines if the remnant of a supernova explosion becomes a black-hole or a neutron star [21]. These two reactions are only two examples of a large number of reactions, which are not yet known with the accuracy needed in astrophysics. In this review, we summarize recent developments and achievements in nuclear astrophysics with a focus on theoretical approaches and experimental techniques that are applicable to or utilize rare-isotope beams, respectively. Section 2 will cover reactions within stars, Section 3 is devoted to nuclear reaction models, Section 4 reviews the effect of environment electrons, Section 5 outlines approaches with indirect methods and Section 6 summarizes recent nuclear astrophysics experiments with rare-isotope beams. Finally, in Section 7 we present our outlook for the present and future of this field. 2 Reactions within stars 2.1 Thermonuclear cross sections and reaction rates The nuclear cross section for a reaction between a nuclear target and a nuclear projectile is defined by where the target number density is given by , the projectile number density is given by , and is the relative velocity between target and projectile nuclei. The number of reactions per unit volume and time can be expressed as , or, more generally, by The evaluation of this integral depends on the type of particles and their distributions. For nuclei and in an astrophysical plasma, obeying a Maxwell-Boltzmann distribution (MB), Eq. (2) simplifies to , where the reaction rate is the average of over the temperature distribution in (3). More specifically, Here denotes the reduced mass of the target-projectile system. 2.1.1 Photons When in Eq. (2) particle is a photon, the relative velocity is always and there is no need to integrate quantities over . Thus, one obtains where results from an integration of the photodisintegration cross section over a Planck distribution for photons of temperature which leads to There is, however, no direct need to evaluate photodisintegration cross sections, because, due to detailed balance, they can be expressed by the capture cross sections for the inverse reaction [22] where is the mass unit, is the reaction -value, is the temperature, is the inverse reaction rate, are partition functions, and are the mass numbers of the participating nuclei in a thermal bath of temperature . 2.1.2 Electron, positron and neutrino capture The electron is about 2000 times less massive than a nucleon. Thus, the velocity of the nucleus is negligible in the center of mass system in comparison to the electron velocity (), and there is no need to integrate quantities over . The electron capture cross section has to be integrated either over a Boltzmann or a Fermi distribution of electrons, depending on the astrophysical scenario. The electron capture rates are a function of and the electron number density, [23]. In a completely ionized plasma, , i.e., the electron abundance is equal to the total proton abundance in nuclei. Here denotes the abundance of nucleus defined by , where is the number density of nuclei per unit volume and is Avogadro’s number. Therefore, This treatment can be generalized for the capture of positrons, which are in a thermal equilibrium with photons, electrons, and nuclei. At high densities ( g/cm) the neutrino scattering cross sections on nuclei and electrons are large enough to thermalize the neutrino distribution. Inverse electron (neutrino) capture can also occur and the neutrino capture rate can be expressed similarly to Eqs. (7) or (9), integrating over the neutrino distribution. 2.1.3 Beta-decay For normal decays, like or decays with half-life , we obtain an equation similar to Eqs. (7) or (9) with a decay constant and 2.1.4 Charged particles Nuclear cross sections for charged particles are suppressed at low energies due to the Coulomb barrier. The product of the penetration factor and the Maxwell-Boltzmann (MB) distribution at a given temperature yields an energy window in which most of the reactions occur, known as the Gamow window. Experimentally, it is more convenient to work with the astrophysical factor with being the Sommerfeld parameter, describing the s-wave barrier penetration, , and energy and the relative velocity of the ions. In this case, the steep increase of the cross section is transformed into a rather flat, energy dependent function. One can easily see the two contributions of the velocity distribution and the penetrability in the integral where and is the reduced mass in units of (unit atomic mass). Experimentally it is very difficult to perform direct measurements of fusion reactions involving charged particles at very small energies. The experimental data at higher energies can be guided by a theoretical model for the cross section, which can then be extrapolated down to the Gamow energy. However, the extrapolation can be inadequate due to the presence of resonances and subthreshold resonances, for example. Schematic representation of the energy dependence of a fusion reaction involving charged particles (Courtesy of C. Spitaleri). Figure 1: Schematic representation of the energy dependence of a fusion reaction involving charged particles (Courtesy of C. Spitaleri). A simple result can be obtained by assuming a constant -factor, i.e., . In this case, the first derivative of the integrand in Eq. (12) yields the location of the Gamow peak, and the effective width of the energy window, i.e. carrying the dependence on the charges , , the reduced mass of the involved nuclei in units of , and the temperature given in 10 K. Figure 1 outlines one of the main challenges in astrophysical reactions with charged particles. The experimental data can be guided by a theoretical model for the cross section, which can then be extrapolated to the Gamow energy. The solid curve is a theoretical prediction, which supposedly describes the data at high energies. Its extrapolation to lower energies yields the desired value of the -factor, or cross section, at the Gamow energy . The extrapolation can be complicated by the presence of unknown resonances. 2.1.5 Neutron-induced reactions For neutron-induced reactions, the effective energy window for s-wave neutrons () is given by the location and width of the peak of the MB distribution function. For , the penetrability of the centrifugal barrier shifts the effective energy to higher values. For neutrons with energies less than the height of the centrifugal barrier one gets [24] Usually, is not much different (in magnitude) from the neutron separation energy. 2.2 Reaction networks The time evolution of the number densities, , of each of the species in an astrophysical plasma (at constant density) is obtained by solving equations of the type where the can be positive or negative numbers that specify how many particles of species are created or destroyed in a reaction . The reactions fall in three categories: 1. decays, photodisintegrations, electron and positron captures and neutrino-induced reactions, , 2. two-particle reactions, , and 3. three-particle reactions, , like the triple- process . The ’s are given by: where the products in the denominators run over the different species destroyed in the reaction and avoid double counting when identical particles react with each other. Example of reaction networks (pp-chains and CNO-cycles). A particular nucleus on the Segrè chart can take different paths along the reaction network, as shown in the inset at the lower right side. (Courtesy of S. Typel) Figure 2: Example of reaction networks (pp-chains and CNO-cycles). A particular nucleus on the Segrè chart can take different paths along the reaction network, as shown in the inset at the lower right side. (Courtesy of S. Typel) In terms of the nuclear abundances, such that for a nucleus with atomic weight , represents the mass fraction of this nucleus, and the reaction network equations can be rewritten as The energy generation per unit volume in a time interval is expressed in terms of the mass excess of the participating nuclei The solution of the above group of equations allows to deduce the path for the r process until the heavier elements are reached. The relative abundances of elements are also obtained theoretically by means of these equations by using stellar models for the initial conditions, as the neutron density and the temperature. Nuclear physics has to contribute with -decay half-lives, electron and positron capture rates, photo-nuclear and neutrino cross sections. Simple examples of reaction networks are shown in Figure 2 for typical pp-chains and CNO-cycles. On the right we show a particular nucleus on the Segrè chart from where different paths can start along the reaction network. 3 Nuclear reaction models Explosive nuclear burning in astrophysical environments produces short-lived, exotic nuclei, which again can be targets for subsequent reactions. In addition, it involves a very large number of stable nuclei, which are still not fully explored by experiments. Thus, it is necessary to be able to predict reaction cross sections and thermonuclear rates with the aid of theoretical models. Especially during the hydrostatic burning stages of stars, charged-particle induced reactions proceed at such low energies that a direct cross-section measurement is often not possible with existing experimental techniques. Hence extrapolations down to the stellar energies of the cross sections measured at the lowest possible energies in the laboratory are usually applied. To be trustworthy, such extrapolations should have as strong of a theoretical foundation as possible. Theory is even more mandatory when excited nuclei are involved in the entrance channel, or when unstable, very neutron-rich or neutron-deficient nuclides (many of them being even impossible to produce with present-day experimental techniques) have to be considered. Such situations are often encountered in the modelling of explosive astrophysical scenarios. 3.1 Potential and DWBA models Potential models assume that the physically important degrees of freedom are the relative motion between structureless nuclei in the entrance and exit channels. The only microscopic information is introduced in terms of spectroscopic factors and parameters of the optical potential. The weakness of the models is that the nucleus-nucleus potentials adopted for calculating the initial and final wavefunctions from the Schrödinger equation cannot be unambiguously defined. Single-particle wavefunctions are calculated using nuclear potentials of the form where and are the central and spin-orbit interactions, respectively, and is the Coulomb potential of a uniform distribution of charges. The potentials and , are usually given in terms of a Woods-Saxon (WS) parameterization. The parameters of the potentials – their depth, range and diffuseness, are chosen to reproduce the ground state energy (or the energy of an excited state). For knockout reactions, they are also adjusted to reproduce the orbital radius of the nucleon. Most often, the same parameters do not reproduce the proper continuum wavefunctions, do not yield location and widths of resonances, etc. These can be obtained by readjusting the strengths of the potentials, effectively increasing the number of parameters at hand. The WS parameterization is well suited to describe any reaction of interest, except perhaps for those cases in which one of the partners is a neutron-rich halo nucleus. Then, the extended radial dependence leads to unusual forms for the potentials. Also, for capture reactions in which the light partner is either a deuteron, triton, -particle or a heavier nucleus, folding models are more appropriate. The central part of the potential is obtained by a folding of an effective interaction with the ground state densities, and , of the nuclei and : with . is a normalization factor which is close to unity. Folding models are based on an effective nucleon-nucleon interaction, , and nuclear densities, , which are either obtained experimentally (not really, because only charge densities can be accurately determined from electron scattering), or calculated from some microscopic model (typically Hartree-Fock or relativistic mean field models). The effective interactions as well as the nuclear densities are subject of intensive theoretical studies. Potential models have been applied to all kinds of calculations for nuclear astrophysics. For simplicity, let us consider radiative capture reactions involving a target nucleus and a nucleon. The wavefunctions for the nucleon () + nucleus () system are calculated by solving the radial Schrödinger equation The nucleon , the nucleus , and the –system have intrinsic spins labeled by , and , respectively. The orbital angular momentum for the relative motion of is described by . Angular momenta are usually coupled as and , where is called the channel spin. In Eq. (18), for one uses and in Eq. (20) denotes the set of quantum numbers, for the bound state, and for the continuum states. The bound-state wavefunctions are normalized to unity, , whereas the continuum wavefunctions have boundary conditions at large distances given by where , with and being the nuclear and the Coulomb phase shifts, respectively. In Eq. (21), , where and are the regular and irregular Coulomb wavefunctions. For neutrons, the Coulomb functions reduce to the usual spherical Bessel functions, and . With these definitions, the continuum wavefunctions are normalized as Potential model calculation Figure 3: Potential model calculation [25] for the reaction O(p,F. The dotted line and the dashed line are for the capture to the ground state and to the first excited state respectively. The experimental data are from Refs. [26, 27, 28]. The dotted-dashed lines are the result of shell model calculations published in Ref. [29]. For and (electric (magnetic) -pole) transitions, the cross sections are obtained from where is the binding energy and is the multipole matrix element. For electric multipole transitions, where is the effective charge, which takes into account the displacement of the center-of-mass, . In comparison with electric dipole transitions, the cross sections for magnetic dipole transitions are reduced by a factor of , where is the relative velocity of the system. At very low energies, , transitions will be much smaller than the electric transitions. Only in the case of sharp resonances, the transitions play a significant role. The total radiative capture cross section is obtained by adding all multipolarities and final spins of the bound state (), where are spectroscopic factors. As an example, Figure 3 shows a potential model calculation [25] for the -factor of the reaction. The rate of this reaction influences sensitively the O/O isotopic ratio predicted by models of massive () AGB stars, where proton captures occur at the base of the convective envelope (hot bottom burning). A fine-tuning of the reaction rate may account for the measured anomalous O/O abundance ratio in small grains which are formed by the condensation of the material ejected from the surface of AGB stars via strong stellar winds [30]. The agreement of the potential model calculation with the experimental data seen in Figure 3 is very good and comparable to more elaborated calculations [29]. 3.2 Microscopic models In microscopic models, nucleons are grouped into clusters and the completely antisymmetrized relative wavefunctions between the various clusters are determined by solving the Schrödinger equation for a many-body Hamiltonian with an effective nucleon-nucleon interaction. Typical cluster models are based on the Resonating Group Method (RGM) or the Generator Coordinate Method (GCM). They are based on a set of coupled integro-differential equations of the form In these equations is the Hamiltonian for the system of two nuclei (A and B) with the energy , is the wavefunction of nucleus A (and B), and is a function to be found by numerical solution of Eq. (25), which describes the relative motion of A and B in channel . Full antisymmetrization between nucleons of A and B are implicit. Microscopic calculations for the reaction Figure 4: Microscopic calculations for the reaction Be. The dashed line is the no-core shell-model calculation of Ref. [34] and the dotted line is from the resonant group method calculation of Ref. [35]. Experimental data are from Refs. [36, 37, 38, 33, 39, 40, 41, 42]. Modern nuclear shell-model calculations, such as the Monte-Carlo shell model, or the no-core shell model, are able to provide the wavefunctions for light nuclei. But so far they cannot describe scattering wavefunctions with a full account of anti-symmetrization. Moreover, the road to an effective interaction which can simultaneously describe bound and continuum states has not been an easy one. Thus, methods based on Eq. (25) seem to be the best way to obtain scattering wavefunctions needed for astrophysical purposes. Old interactions, such as Volkov interactions, are still used for practical purposes. It is also worth mentioning that this approach has provided the best description of bound, resonant, and scattering states of nuclear systems [31]. As an example of applications of this method, we again give a radiative capture reaction. The creation and destruction of Be in astrophysical environments is essential for the description of several stellar and cosmological processes and is not well understood. B also plays an essential role in understanding our Sun. High energy neutrinos produced by B decay in the Sun oscillate into other active species on their way to earth [32]. Precise predictions of the production rate of B solar neutrinos are important for testing solar models, and for limiting the allowed neutrino mixing parameters. The most uncertain reaction leading to B formation in the Sun is the radiative capture reaction [33]. Additionally, the Coulomb dissociation method, discussed later in this review, has given some new insights about the electromagnetic matrix elements for this reaction. Figure 4 shows a comparison of microscopic calculations for the reaction Be with experimental data. The dashed-dotted line is the no-core shell-model calculation of Ref. [34] and the dotted line is for the resonant group method calculation of Ref. [35]. Experimental data are from Refs. [36, 37, 38, 33, 39, 40, 41, 42]. It is evident that both, theory and experiment, need improvements for this important reaction. 3.2.1 Asymptotic normalization coefficients Although the potential model works well for many nuclear reactions of interest in astrophysics, it is often necessary to pursue a more microscopic approach to reproduce experimental data. Instead of the single-particle wavefunctions one often makes use of overlap integrals, , and a many-body wavefunction for the relative motion, . Both and might be very complicated to calculate, depending on how elaborate the microscopic model is. The variable is the relative coordinate between the nucleon and the nucleus , with all the intrinsic coordinates of the nucleons in being integrated out. The radiative capture cross sections are obtained from the calculation of . Comparison of various radial overlap integrals Figure 5: Comparison of various radial overlap integrals for with the normalized Whittaker function (dashed curve). Most of the contribution to the rms radius comes from the region outside the core, with radius . The imprints of many-body effects will eventually disappear at large distances between the nucleon and the nucleus. One thus expects that the overlap function asymptotically matches the solution of the Schrödinger equation (20), with for protons and for neutrons. That is, when , where the binding energy of the system is related to by means of , is the Whittaker function and is the modified Bessel function. In Eq. (26), is the asymptotic normalization coefficient (ANC). In Figure 5 we show the comparison of the ANC for F () as a function of the distance , with the Whittaker function, Eq. (26). As can be seen, most of the contribution to the rms radius comes from the region outside the core. In the calculation of above, one often meets the situation in which only the asymptotic part of and contributes significantly to the integral over . In these situations, is also well described by a simple two-body scattering wave (e.g. Coulomb waves). Therefore, the radial integration in can be done accurately and the only remaining information from the many-body physics at short distances is contained in the asymptotic normalization coefficient , i.e. . We thus run into an effective theory for radiative capture cross sections, in which the constants carry all information about the short-distance physics, where the many-body aspects are relevant. It is worthwhile to mention that these arguments are reasonable for proton capture at very low energies because of the Coulomb barrier. The asymptotic normalization coefficients, , can also be obtained from the analysis of peripheral transfer and breakup reactions. As the overlap integral, Eq. (26), asymptotically becomes a Whittaker function, so does the single-particle bound-state wavefunction , calculated with Eq. (20). If we label the single-particle ANC by , then the relation between the ANC obtained from experiment, or a microscopic model, with the single-particle ANC given by (this becomes clear from Eq. (24)). The values of and obtained with the simple potential model are useful telltales of the complex short-range many-body physics of radiative capture reactions. One can also invert this argumentation and obtain spectroscopic factors if the are deduced from a many-body model, or from experiment, and the are calculated from a single-particle potential model [43]. Microscopic calculations of ANCs rely on obtaining the projection, or overlap, of the many-body wave functions of nuclei and . The overlap integral must have correct asymptotic behavior with respect to the variable which is the distance between the nucleon and the c.m. of the nucleus . The most common methods are: (a) the resonating group method (RGM), as described above, (b) the Fadeev method for three-body systems, (c) a combination of microscopic cluster method and -matrix approaches, to be discussed later, (d) Green’s function Monte-Carlo method, (e) no-core shell model, or (f) hyperspherical functions method. As an example, early applications of the ANC method have obtained eV b for the Be(p,)B reaction using ANCs calculated with oscillator wave functions and M3Y(E) effective potential as a model for and [44]. The M3Y interaction is an effective interaction constructed as in Eq. (19), with given in terms of sums of (3) Yukawa functions. 3.2.2 Threshold behavior and the r-process The threshold behavior of radiative capture cross sections is fundamental in nuclear astrophysics because of the small projectile energies in the thermonuclear region. For example, for neutron capture near the threshold, the cross section can be written as [45] where is the logarithmic derivative for the wave at a channel radius. Since is only weakly dependent on the projectile energy, one obtains for low energies the well-known -behavior. With increasing neutron energy, higher partial waves with contribute more significantly to the radiative capture cross section. Thus the product becomes a slowly varying function of the neutron velocity and one can expand this quantity in terms of or around zero energy, The quantity is the astrophysical -factor for neutron–induced reactions and the dotted quantities represent derivatives with respect to , i.e., The astrophysical -factor for neutron-induced reactions is different from that for charged-particle induced reactions. In the astrophysical -factor for charged–particle induced reactions also the penetration factor through the Coulomb barrier has to be considered (Eq. (11)). Inserting this into Eq. (5), one obtains for the reaction rate of neutron-induced reactions In most astrophysical neutron-induced reactions, neutron s-waves will dominate, resulting in a cross section showing a 1/–behavior (i.e., ). In this case, the reaction rate will become independent of temperature, . Therefore it will suffice to measure the cross section at one temperature in order to calculate the rates for a wider range of temperatures. The rate can then be computed very easily by using with . The mean lifetime of a nucleus against neutron capture, i.e., the mean time between subsequent neutron captures, is inversely proportional to the available number of neutrons and the reaction rate , . If this time is shorter than the -decay half-life of the nucleus, it will be likely to capture a neutron before decaying (r process). In this manner, more and more neutrons can be captured to build up nuclei along an isotopic chain until the -decay half-life of an isotope finally becomes shorter than . With the very high neutron densities encountered in several astrophysical scenarios, isotopes very far off stability can be synthesized. 3.2.3 Halo nuclei For low values of the binding energy , e.g. for halo-nuclei, the simple -law does not apply anymore. A significant deviation can be observed if the neutron energy is of the order of the -value. For radiative capture to weakly-bound final states, the bound-state wave function in Eq. (23) decreases very slowly in the nuclear exterior, so that the contributions come predominantly from far outside the nuclear region, i.e., from the nuclear halo. For this asymptotic region, the scattering and bound wave functions in Eq. (23) can be approximated by their asymptotic expressions neglecting the nuclear potential, , and , where and are the spherical Bessel, and the Hankel function of the first kind, respectively. The separation energy in the exit channel is related to the parameter by . Performing the calculations of the radial integrals in Eq. (23), one readily obtains the energy dependence of the radiative capture cross section for halo nuclei [46]. For example, for a transition it becomes while a transition has the energy dependence If , the conventional energy dependence is recovered. From the above equations one obtains that the reaction rate is not constant (for s-wave capture) or proportional to (for p-wave capture) in the case of small -values. In the case of charged particles, is expected to be a slowly varying function in energy for non-resonant nuclear reactions. In this case, can be expanded in a McLaurin series, as was done to obtain Eq. (27). Using the expansion in Eq. (12) and approximating the product of the exponentials and by a Gaussian centered at the energy , Eq. (12) can be evaluated as [47] The quantity defines the effective mean energy for thermonuclear fusion and is given by Eq. (13). The quantity is given by , and is given by Eq. (13). 3.2.4 Resonances For the case of resonances, where is the resonance energy, we can approximate by a Breit-Wigner resonance formula, where , , and are the spins of the resonance and the nuclei and , respectively, and the total width is the sum of the particle decay partial width and the -ray partial width . The particle partial width, or entrance channel width, , can be expressed in terms of the single-particle spectroscopic factor and the single-particle width of the resonance state, . The single-particle width can be calculated from the scattering phase shifts of a scattering potential with the potential parameters being determined by matching the resonance energy. The partial widths are calculated from the reduced electromagnetic transition probabilities which carry the nuclear structure information of the resonance states and the final bound states. The reduced transition rates are usually computed within the framework of the nuclear shell model. Most of the typical transitions are or transitions. For these, the relations are For the case of narrow resonances, with width , the Maxwellian exponent can be taken out of the integral, and one finds where the resonance strength is defined by For broad resonances, Eq. (12) is usually calculated numerically. An interference term has to be added. The total capture cross section is then given by [48] In this equation is the resonance phase shift. Only the contributions with the same angular momentum of the incoming wave interfere in Eq. (38). 3.3 -matrix theory Reaction rates dominated by the contributions from a few resonant or bound states are often extrapolated to energies of astrophysical interest in terms of - or -matrix fits. The appeal of these methods rests on the fact that analytical expressions can be derived from underlying formal reaction theories that allow for a rather simple parametrization of the data. However, the relation between the parameters of the -matrix model and the experimental data (resonance energies and widths) is only quite indirect. The -matrix formalism solves this problem, but suffers from other drawbacks [49]. 3.4 Elastic and inelastic scattering reactions In the -matrix formalism, the eigenstates of the nuclear Hamiltonian in the interior region of a nucleus are denoted by , with energy , and are required to satisfy the boundary condition at the channel radius , where the constant is a real number. The true nuclear wavefunction for the compound system is not stationary, but since the form a complete set, it is possible to expand in terms of , i.e. The differential equations for and are (for s-wave neutrons) Figure 6: Total factor data (filled-in circles) [53] for C()O compared with E1 (open triangles) and E2 (open squares) contributions [54, 55]. The solid line represents the sum of the single amplitudes of an -matrix fit [27] (the dotted and dashed lines are the and amplitudes, respectively). In addition, the matrix fit of [56] to their data (dotted-dashed line) is shown. (Adapted from Ref. [57]). Multiplying Eq. (39) by and Eq. (40) by , subtracting and integrating, we have where the prime indicates the differentiation with respect to . This result, together with the definition of , gives where the function relates the value of at the surface to its derivative at the surface: Rearranging Eq. (41) we have which is just the logarithmic derivative which can be used to determine the -matrix element in terms of the function. This gives Finally, we assume that is near to a particular say , neglect all terms in Eq. (42), and define so that the -matrix element becomes and the scattering cross section is We see that the procedure of imposing the boundary conditions at the channel radius leads to isolated s-wave resonances of Breit-Wigner form. If the constant is non-zero, the position of the maximum in the cross section is shifted. The level shift does not appear in the simple form of the Breit-Wigner formula because is defined as the resonance energy. In general, a nucleus can decay through many channels and when the formalism is extended to take this into account, the -function becomes a matrix. In this -matrix theory the constant is real and and can be chosen to be real so that the eigenvalue problem is Hermitian [50]. The -matrix theory can be easily generalized to account for higher partial waves and spin-channels. If we define the reduced width by , which is a property of a particular state and not dependent of the scattering energy of the scattering system, we can write where is the channel label. , and are treated as parameters in fitting the experimental data. If we write the wavefunction for any channel as , where and are incoming and outgoing waves, Eq. (41) means Thus, as in Eq. (43), the -matrix is related to the -matrix and from the above relation we obtain that, The total cross sections for states with angular momenta and spins given by , and is where are spin geometric factors. In the statistical model, it can be argued that because the -matrix elements vary rapidly with energy, the statistical assumption implies that there is a random phase relation between the different components of the -matrix. The process of energy averaging then eliminates the cross terms and gives where the symmetry properties of the
b0c462017af2f6d2	Towards an Ontology for Unified Knowledge: The Hypothesis of Logical Quanta. Print Friendly, PDF & Email The weirdness of quantum mechanics From the very beginning of quantum mechanics it was seen that there was something absurd about it. A hundred years later, we are still speaking about quantum paradoxes.1 There is a difference albeit; we now know that these paradoxes govern the way things are at the most fundamental level. The quantum paradox can be described with one phrase; Things in quantum world behave in a strongly different way than in our everyday world.2 There is a loophole in our understanding of the (real) nature of the physical world. This is not the only one. There are also loopholes in our understanding of the nature of life and, even more, the nature of consciousness. There is also more weirdness about the special abilities of the inner world of human beings. Confronting this situation, we usually avoid the problem by dividing it in no compatible sections and dismissing all evidence not fitting to our bias3 3 This vein of thinking helped a lot in solving many scientific problems of everyday world, 4 but there are strong indications that it ends at a deadlock. Working on the opposite direction, there is a hard temptation to follow a confusing way of thinking, like this; quantum mechanics is weird, spiritual life is also unusual, so these are similar in this respect, or it is possible to interpret the second through the first. 5 This paper presents a synthesis of quantum mechanics and the ontology based on the notion of logos, as this has been developed in ancient and Medieval Greek philosophy and theology. We acknowledge the danger just mentioned and we try to overcome it, by clarifying as possible the way we work. On the other hand, philosophy of logos has been developed in a theological context, and theology, nowadays, is strongly ideological. An ontological proposal for unifying the knowledge, the spiritual and the scientific one, has to be accepted both by believers and non believers. This restriction demands a special interpretation of theology. In fact, there is a conceptual tool useful for both cases. This is the distinction that should be made, between empirical data connected with physical or spiritual facts, their explanation and, finally, the ontology that one can construct by using them, in other words, the metaphysics that one could attach to these data. Empirical data, explanation and ontology. In physics we usually attach a set of empirical data to a theory that explains them. Such a theory is a conceptual construction explaining the causes of these data and predicts their evolution in time and /or space. The data are correlated with entities and, usually, when we know the data and the theory we think that we know the entities and the ontological state of these entities. In our everyday life, the theory that predicts the evolution of the entities and the theory, i.e. the ontology that describe their ontological state, coincide. In Philosophy of Science, it is well known that the adoption of the right theory that describes the evolution of an entity, or a phenomenon, is very complicated. However, the distinction between the theory and the description of the ontological state is less obvious, and in our everyday life it is expelled. The interconnection of the facts and the theory that explains them is well studied by the philosophy of science and we know well that a data set can be explained by more than one theory, and we can find related examples in many scientific fields. 6 In certain scientific fields, we have different theories, which describe the same ontological states of an entity. A scientific theory is expressed by a mathematical formalism.7 In our everyday world physics, i.e. the classical physics, the entities that formalism describes are well defined. There is a rigid connection between data, formalism and entity. This is not the case in quantum physics, as we will clarify afterwards. The same distinction is very useful, when we work at interpreting theology. Theology starts by determining the ontological status of the entities, and then develops a theological theory about them and connects them with empirical data. Traditional theologies work like classical physics. The interconnections between the three stages are very rigid. I have in mind traditional monotheistic theologies. In our globalized world, this attitude of monotheistic mainstream theologies has been proved insufficient. The problem is that the same or very similar data, like religious and mystical experiences or miracles, are explained by different theologies in various ways, all of them claiming the same credibility with reference to the same ontological state of a fundamental conceptual entity, named God. This situation is probably hard for traditional theologies, but allows us a very fertile approach to any theological system. We can accept the truthfulness even the objectiveness, of spiritual or mystical empirical data and can distinguish them from any subjective theological system and the almost arbitrary ontology that this system produces. It is possible to accept certain parts of a theological system and introduce them to our interpretation of physical phenomena. The result could be a synthesis of an old tradition with contemporary philosophical or scientific research. Through this procedure, there could be a great gain. A unification of our understanding of spiritual and physical world. Interpreting Quantum Mechanics It is quite important to clarify the conceptual framework of the interpretative problem of quantum mechanics. It is about the behavior of a quantum entity, being in a very special condition, in a superposition state. 8 Such a quantum entity is a microscopic particle that we study per se, when it is not correlated with macroscopic environment. This happens, generally talking, when such an entity exists between two successive measurements. 9 Quantum weirdness appears, when such an entity interacts with macroscopic environment at the end of the second measurement. “Quantum mechanics is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had. Mathematically, the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to quantum mechanics. Minimally interpreted, the theory describes a set of facts about the way the microscopic world impinges on the macroscopic one, how it affects our measuring instruments, described in everyday language or the language of classical mechanics. Disagreement centers on the question of what a microscopic world, which affects our apparatuses in the prescribed manner, is, or even could be, like intrinsically; or how those apparatuses could themselves be built out of microscopic parts of the sort the theory describes.” 10 In common English, a quantum entity appears to be either a particle, or a strange kind of wave. It appears with a different “personality”, which is supposed to be depended on the structure of the measurement apparatus we use. 11 It responds instantly to any change we make to apparatus, sometimes even before we make our decision, as if it knows what we (will) have in our mind. 12 Somehow, it changes its condition and it is transformed into to a regular particle. This transformation obeys to strictly defined rules that are statistical. When such a transformation occurs, we, by no means, know what exactly will happen. A quantum entity appears to communicate instantly with the whole universe. 13 After all, there is the famous Uncertainty Principle of Heisenberg: “According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum.” 14 It is obvious that no entity of our world can have such a behavior. Using the distinction we have made, when we study a quantum phenomenon, we have a well defined set of empirical data, a set of explanations for them, but we have no ontology, that could be well accepted by physics and philosophers, describing what a quantum entity is. We have a behavior that is well observed, well explained and calculated by mathematical formalism of quantum mechanics, but we cannot adapt the nature of a quantum entity with any entity of our everyday word. There are various interpretations of quantum mechanics aiming to reconcile our observations of quantum world and our observation of our everyday world. 15 From our point of view, these interpretations follow two main ways. The first is to avoid, somehow, the ontological problem and focus on the explanatory part of quantum theory. These are based on what we call Copenhagen’s interpretation. There are various alternatives but, in fact, it is still impossible to avoid ontology. They introduce a number of principles aiming to explain the experimental data. The most famous among them is the Complementarily Principle, proposed by Niels Bohr, 16 or the Projection Postulate proposed by von Neumann. 17 Modal interpretations refute the rigid ‘eigenstate-eigenvalue link’ 18 and so on. In any case, those principles express an ontology which is radically different from our every day understanding of our world. The second way is more radical and develops new ontology describing the whole physical world. This path follows Bohmian mechanics, 19 Many Worlds 20 interpretations, or Collapse theories. 21 They explicitly introduce new ontology, either at the quantum level, or at a cosmic level. Physicists do not like the concept of Metaphysics. Any quantum interpretation is strictly and necessarily metaphysical, but this, however, is not how physicists like to think. They question the problem through mathematics; develop their ontology by giving ontological meaning at certain parts of quantum mechanical formalism. They achieve the development, more or less, a self consistent explanation, but none of which could be preferable, because their ontology is not integrated with the rest experience of the human civilization. Confronting this problem, we propose an alternative approach. Our point of departure is not the formalism, but an already developed ontology. This is an ontology still based on the observation of the physical world, but uses different methods than contemporary science. This method is not completely analytical, but it is based on a combination of intuitive, conceptual and analytical approach to the problems. It is the way a basketball player computes and makes a shooting, the way ancient Greeks built Acropolis. Ancient and Medieval Greeks developed the ontology of Logos to communicate their understanding of how physical world works. This usage of the notion of logos usually passes unnoticed, as it is overridden by the intense use of divine Logos in Christian Theology. But in the background of theological conflicts, ontology of logos of a natural being has been developed to a complete system that was able to describe both spiritual and the physical world of sences. The ontology of logos It is usual to say that father of the concept of logos (in Greek λόγος, which is translated in English as word, but we will prefer to use type logos and his plural logoi), is Greek philosopher Heraclitus from Ephesus (535475 BC). It’s hard to believe that a single person could conceive such a revolutionary, in those times, thought, as the following: “This world-order [cosmos], the same of all, no neither god nor man did create, but it ever was and is and will be: ever living fire, kindling in measures and being quenched in measures.” 22 Heraclitus and others pre-Socratic philosophers expelled divine action from the world and formulated for the universe a natural way of being and evolving. Heraclitus was a step forward from his ancestors. Among others, he first made a basic distinction between the “stuff” universe is made and the principle that controls the way this stuff evolves and the beings become to existence. This stuff was fire and the principle was Logos. 23 Everything is becoming according to Logos and, if we speak in contemporary terms, Logos includes all information that controls life and evolution of all beings. We can call this information “active information” because it is strongly connected with beings and constitutes them, it makes them exist. As far as Heraclitus is concerned, fire and logos are not divine, they are somehow material. 24 Soon after the stuff of the universe was separated from the formatting principle, the latter became divine, immaterial and even constituted a completely separated world, the Plato world of ideas. Ideas were not only separated from beings, but they had a more analytical structure. There was not an abstract idea that controls the world, but there were many ideas, and each one, controls all the similar beings. Plato’s system was more complicated than Heraclitus, but yet not enough. The emphasis was put on the separation and superiority of the divine world of ideas, from the physical word, the separation of the principle that controls the universe, from itself. 25 Stoics rejoined the controlling principle with the physical world, reusing the concept of logos for their ontology. Logos is inside beings and it is divine even though it was material. Beings and God are completely united, this was typical pantheism. “In accord with this ontology, the Stoics, like the Epicureans, make God material. But while the Epicureans think the gods are too busy being blessed and happy to be bothered with the governance of the universe, the Stoic God is imminent throughout the whole of creation and directs its development down to the smallest detail. God is identical with one of the two ungenerated and indestructible first principles (archai) of the universe. One principle is matter which they regard as utterly unqualified and inert. It is that which is acted upon. God is identified with an eternal reason (logos, Diog. Laert. 44B ) or intelligent designing fire (Aetius, 46A) which structures matter in accordance with Its plan.” 26 The major contribution to the evolution of the concept of logos was made by Philo of Alexandria, 20 BC – 50 AD. He and his contemporary Judith theologians, tried to harmonize Judith theology with Greek philosophy. He combined the concept of Judith God with the concepts of logos and ideas. He joined logos with ideas and distinguished Logos, as the principle of all beings, from idea-logos the ontological principle of every separate being. Logos was connected with God, and became the ultimate power of God, the Son of God. Ideas were renamed to logoi and were the ontological background of every being. Logoi were pictures of beings, established at the mind of God, and Logos created beings according to these logoi.27 “For the world has been created, and has by all means derived its existence from some extraneous cause. But the word (logos) itself of the Creator is the seal by which each of existing things is invested with form. In accordance with which fact perfect species also does from the very beginning follow things when created, as being an impression and image of the perfect word.” 28 Logos is expressed through logoi, and logoi are unified in Logos. From then on, the ontology of logos follows this scheme: Logos is the ontological background of logoi, which are the ontological background of beings. The dissociation of Logos to logoi was developed by Christian theology. Logos became the second person of Holy Trinity and monopolized their interest. Albeit, they used the concept of logos quite often, when they tried to describe God’s connection with beings. That was a major problem for ancient Christian theology, which confronted the problem of evil, as a result of the tight connection of Creator with creation. Origen (185–254 AD) did not used logoi to solve the problem of evil, he preferred the concept of souls, 30 but he confirmed definitely that, for every being, there is his logos and he associated logoi of being, with epistemology. He taught that human mind can “see” the logos of being through “φυσική θεωρία”, which can be translated as natural contemplation. Heraclitus first associated logos, with a certain state of human mind, but it was Origen and his pupil Evagrios Pontikos, who developed in details the interaction of state of human mind and the “vision” of logoi of being. The theory of logoi of Maximus the Confessor Maximus the Confessor (580-662 A.D.), was the Christian theologian who used the most the concept of logos in his work. We owe him the detailed and subtle record of the use of logos of natural being. He didn’t make any radical contribution to it, but he pushed to the end the various properties of logoi of being that were previously introduced, as he intended to develop his theological framework. He uses the concept of logos of natural being for two major goals. The first was to correct theology of Origen 31 and the second, to express the ascetical and mystical experience of religious life.32 The problem of Origen is correlated with the problem of Evil. Origen taught that Logos-Creator created a spiritual world that consisted of souls. This world was (almost) perfect, but somehow, the souls got bored and tried to rebel against the Creator who punished them to be imprisoned to bodies and matter and so had been produced all beings we see. Logos has been embodied to Jesus Christ to give manhood a second chance and, finally, at the end of time, all beings will recover their spiritual nature. There were many problems, in this scheme of Origen. The most important was that there was confusion between Creator and creation, because in this scheme God and souls are co-eternal. The radical distinction between God and World is the strongest characteristic of Judith-Christian theology. Another characteristic is that God is perfect and everything He does (must be) perfect. The world we observe is not perfect, so there is a problem. Origen tried to solve this problem with the teaching of the fall of souls but confused Creator with creation. To avoid these problems, Maximus uses the ancient distinction between the stuff that the beings are made of and the principle or pattern that shapes this stuff. So he used the concept of logoi that govern the way that beings are made and evolved. 33 If logoi constitute a world outside God, there must be a time when they didn’t existed, so we must assume that there was a change at the state of God. The time before the creation of logoi, God was not a Creator and after that time, He became a Creator. That was unacceptable for Maximus and his contemporary people’s vision of God. So he declared that logoi are God’s wills which are co-eternal in God’s mind.34 At a point that is timeless, God created the beginning of time and logoi started to be expressed as beings. With this scheme, God is always a Creator and the material creation is not co-eternal. But the problem of Evil remains. Logoi and beings are very strongly correlated, and logoi are very strongly joined with Logos. Logoi and beings are interacting and continuously evolving and the whole creation is moving to a certain point, which is Logos.35 So Logos is simultaneously, the beginning and the end of the motion and evolution of all beings. Logos, as the end of evolution, offers a kind of restoration of everything, and Maximus believed that the problem of Evil is solved. 36 However, it is not, because there still is a lot of suffering that cannot be explained. Maximus offered an explanation; all that we suffer is given by God to make for us necessary the spiritual world. 37 A medieval person could accept that, but such an image of how God acts, is hardly acceptable by a contemporary man. Maximus supported his theological scheme by taking advantage of the ontology of logos as it was developed by previous philosophers and theologians. By doing so, he gave us many details about it. He declared that logos of every being is the ontological background of all of his physical properties. 38 He described the hierarchical levels that exist in every logos, a scheme that we call tree-structure of logoi of beings. More specifically a logos which is the result of synthesis of other partial logoi, it is the ontological background of the synthesis of the partial logoi, he controls them as they evolve to constitute him. 39 This property of logoi was very important for him, because he believed that the power that pushes the evolution and motion of beings is not at the beginning of history, but at the end. For Maximus it is God-Logos who attracts the beings to Him and makes them move. Maximus understood logoi as God’s wills that are inside His mind, but he also believed that human mind is capable to “view” them through natural contemplation. 40 Ascetical life refines human mind and it passes from natural contemplation to mystical contemplation, 41 which assures that logoi have a real existence. Maximus established his “logical” realism to the ascetical experience. This is quite important for us, because it allows us to use the distinction between facts, explanation and ontology that we’ve mentioned previously. We can accept the empirical core of Maximus Theory and interpret differently the explanation and the ontology. Such an interpretation of Maximus teachings, leads us to summarize that logos is a hidden pattern that controls the beings and reality, logos in his original meaning that has been introduced by Heraclitus is information that is active, that is expressed as a being. Logoi are not concepts, but they are real, information has self existence. Logoi (information) have inner structure, they are organized at hierarchical levels and these levels make the tree of logoi, the ontological tree of our universe, which has a construction from bottom to top. The top of it, it is down, it is his foundation. The top of this tree supports the whole tree, it is a reversed tree. For Maximus, the top of the tree which supports it is God-Logos, the basis, the beginning and the end of everything. 42 This property has important physical consequences. Logos of a being, which is constituted by other beings, controls the logos of these beings and makes them to constitute it. The cause of a fact can be in the future. In theology we call it eschatology. This can be understood only if we interpret Maximus doctrine that logoi are sited at God’s Mind. Orthodox theology determines logoi as “aktistoi” because they co-exist with God. That means that they are not simply eternal. Eternal is something that remains the same as the time passes. Logoi do not remain the same, they evolve, but they are outside time and space. About logoi there is no meaning for before and after. A composite logos controls the logoi which consist him. It is the cause of their evolution, but when it is expressed at space-time, the (composite) entity that it controls, appears in time, after its components. Causality is independent from the arrow of time. Every being is attached with its logos. It is more accurate to say that a being, is a composite being, it is logos-information expressed as a (material) being in space-time. Logos interacts with other logoi but this life of logoi, is taking place outside time and space. Logoi have an inner structure, which is inverted from a point of view inside space-time. Life of logoi gives to beings special properties that are revealed to human mind under special conditions. A human mind that is properly exercised, can feel all these. Throughout human civilization, there are evidences of deep feeling of an inner side of all beings. This experience is interpreted in Medieval Greek philosophy with ontology based on logos. This ontology was strongly correlated with Christian theology but ontology of logos, pre-exists Christianity. It is a common denominator of the whole Ancient and Medieval Greek Philosophy. If it is necessary to introduce metaphysics in Physics, ontology of logos is a appropriate candidate. The Hypothesis of logical quanta To visualize ontology of logos, we used the scheme of an inverted ontological tree. As we are going up we can find logoi of fundamental elements of our world. We can find logoi of elementary particles. So we can speak about logos of quantum particle. Such a particle is an entity that it is not correlated with macroscopic environment. The Hypothesis of Logical Quanta (HLQ) says that a quantum particle is a logos disconnected from the ontological tree, it is a pure logos not connected with an entity that exists at space-time, it is pure information which has not yet been expressed at space-time. Such a pure logos is a potential entity. HLQ answers the basic question of any interpretation of quantum mechanics; what a quantum particle is, and the answer is that it is a logos, that a quantum particle is pure, yet unexpressed, information. Quantum entities as logoi, have the properties of logoi. They “exist” in a special space; we can call it “logical space” with no spatial or time coordinates. Even so, they evolve and interact with other logoi, both with pure logoi, other quantum entities, and logoi connected with beings, macroscopic entities. The projection of pure logos to space-time is expressed by Schrödinger equation. 43 Schrödinger equation does not describe the evolution of a “real” entity, but the projection to “real” world of the timeless evolution of a logical entity. It is important to emphasize that logical space and space-time are rigidly connected and that ontological cause, lies in logical space. Ontological background of every physical entity is his logos. Every entity has its own logos, and as every entity is constituted by other entities, every logos is a synthesis of other logoi. We can say that beings float at a sea of logoi, they are the visible top of an “iceberg”. With the conceptual equipments that HLQ gives us, we can interpret various quantum mechanical issues. First, we can explain the collapse of the wave function. It is equivalent with the question what and why happens, when a quantum entity ends being in superposition and it changes into a classical entity. HLQ explains that it happens, when a quantum entity-logos is connected with the ontological tree. A composite logos controls the logoi that compose it. When a “free” logos is connected with the ontological tree, it is no longer free and is under the action of composite logos. This action causes the collapse of wave function. Because of this action, a pure logos is expressed to an entity, and is correlated with the composite macroscopic entity, the measurement apparatus. This statement entails that we have a phase transition that happens as a quantum entity is correlated with a macroscopic entity. The wave-particle duality is well understood, if we consider that the logoi of every quantum entity, more accurately of every (elementary) particle, however massive it could be, are all together within the same dimensional space, a space without spatial and time coordinates, and constitute a “logical fluid” with defined wave properties. In two slit experiment, there may be always one entity at a time, but the logoi of all particles are all together, so that it behaves like a wave. Our interpretation contradicts complementary principle, in our case an electron is neither wave, nor particle, it is logos that interacts with logoi of apparatus and appears to be either a particle, or a wave even both as wave and particle. Non locality and delayed decision, or non catastrophic measurements are easily understood by the non spatial or time coordinates of logoi. At every instant, a quantum entity through its logos communicates with every single part of the experimental apparatus and it corresponds instantly with anything that happens in it. From the point of view of an observer that stands in space-time, it looks as if the quantum entity knows what will happen, or observer’s action changes the past. Entangled particles are particles that have the same logos, or better, their logoi are tightly connected. As far as we can consider HLQ, we cannot explain the values of probabilities that we take by the Schrödinger’s equation solution. But no other interpretation does it. We can only comment that, if the wave function ψ were a real function and quantum entities were localized at phase space, it is hard to think how the diversity and complexity of our world could arise from quantum world. Schrödinger’s equation probabilities and Uncertainty Principle loosen the connection between quantum entity and information which is included in its logos. All elementary particles are undistinguishable and their logoi include the same information. It is necessary, for our world to exist, that this information should be expressed in various ways. HLQ arises from a metaphysical background, but it is not more metaphysical than other interpretations. One could notice that, every particular implementation of HLQ exists in similar form in other interpretations. Bohm’s dynamic and quantum potential have common properties with logoi. But there is a very important difference. Logoi is a characteristic of every being and not only of quantum entities. Logoi is not a set of hidden variables, but includes every variable. Modal interpretations give primary role to the apparatus, even if they offer no ontology for their claims. Other interpretations suggest actions which reverse the time arrow 44 and so on. There are strong indications supporting HLQ from other scientific fields. Many physicists suggest that information is crucial for the structure of Universe. 45 There is also Holographic Principle that potentially gives a mathematical meaning to “logical dimensions.” 46 The creative role that Ilya Prigogine gives to the arrow of time and the concept of emergence, 47 that is very popular nowadays, has a lot in common with the action of logos. Most supporting to our Hypothesis is the work of Roland Omnès, who concludes his analysis with the necessity of distinction between reality and logos, the formatting principle, but he says: “The notion of logos is obviously insufficiently developed and is rather questionable. We shall see however that it offers a possible way out of several problems.” 48 I think that Omnès is not familiar with the complete ontology of logos as it was developed by Medieval Greek philosophy. Human mind and Unification of Knowledge The greatest merit with HLQ that was developed by ancient philosophers and finally, declared by Maximus, is the aspect that human mind is created or evolved with the ability to “see” logoi of being. Reality has many levels of organization and many points of view. HLQ suggests that all levels and all perspectives of Reality are based on information. This information is not the kind that contemporary science of information studies. As Antony Zeilinger proved, there is information that cannot be expressed by bits.49 This is a strong indication that there is information of a different kind than the usual we know in our everyday life. Information, we are talking about, has inner structure and is self existent. These points drive us to our next step of understanding. Information of a composite logos, is more than the sum of partial information that is included at the logoi that compose it. This is a result of quantum mechanical formalism, but it can be extended to logoi of macroscopic entities.50 A composite logos has new functionality and new relations to other logoi of beings. As we move downwards the ontological tree, from one level to the other, an active information excess is always produced. The more complicate is a being, the more information excess it includes. Talking about the human brain, which is the most complicated structure in the known universe, we can consider the information excess it possesses. This excess could be the cause for whatever we call free will. Considering the above it seems very tenable to suppose that a mind is the result of the “logical structure” of brain, the logoi of entities of the physical structure of brain. Memories could be stored and processed in it. It is not the biochemical structure of brain that stores and processes information and produces the mind, but the structure of logoi beneath it. Connections and interactions between logoi of neurons are more stable than connections of neurons. Procession of information could be made by logoi of whole parts of brain. This model is flexible enough to explain the way mind arises from brain. HLQ shows us new ways of research in this field. They are ways that are established on physical structure of brain, but are not restricted by it. If this model is valid, it follows that the mind has access to “logical space”. Whatever we call spiritual life or activity, is taking place in it. This scheme, if developed, can give us answers about the nature of mathematics, intuition, art and every phenomenon we characterize as spiritual. We can develop a unified approach of various aspects of human civilization based on a certain interpretation of quantum mechanics. That doesn’t mean that spiritual phenomena have a quantum mechanical structure or explanation, as it is often said. HLQ gives a special role to information. This role opens new ways of understanding the way brain works. These ways need to be explored with scientific method to find out what is really going on. Science and religion HLQ offers us an opportunity to understand scientifically religion, without denying his experiential reality. It allows us to distinguish experiential reality of God, from ontological reality of Him. Traditional theologies interpret God in terms of Creation. God is a concept that explains the existence of the world and deep feelings and facts of communication with Him. Every civilization develops an explanatory model about God, based on its knowledge about how it isthe world and the man. This model is thought to be an ontological reality and evolved to a doctrine believed by the particular civilization. Nowadays, reality of world and human nature, have been proved very complicated and contradictory. All these models about God transfer these contradictions to God’s nature. It is the well known problem of evil. Religions cannot overcome it with a rational way and they are driven to logical deadlock. This deadlock drives contemporary man to reject religion edifice, leaving a serious psychological emptiness. HLQ help us to construct a model about God that is logically consistent and includes religion experiences accepting them as real. As we have noticed in previously, causality lies in logical space and is outside time. Causality follows the arrow of time only phenomenological and has nothing to do with it. The necessity of Creator is due only to human perception. The question of who created the world is pointless. Medieval theologians thought that logoi are inside God’s mind. We can unite God with His mind, the logical space. It is not a complete answer to the question, what or who God is, but it is flexible enough and gives us the possibility to understand religious phenomena, like prayer and mystical experience. Human mind has the ability to access logical space, in other words, God’s mind. This ability, as all abilities of human beings, can be cultivated and developed and can produce strong feelings to the person that practices it. These feelings produce the mystical vein of every religion. The act of accessing logical space is understood, as a special kind of communication and it is described, as prayer. Every religion and civilization expresses all these empirical data with its own theological and philosophical concepts. It is not hard to understand that a person with a special gift can develop his ability to communicate or interact through logical space with other persons, or with previous or future facts. These and many others, quite unusual facts, can be explained with the aid of HLQ, without denying our naturalistic view of the world. HLQ is a proposal that it is strictly defined at the field of quantum mechanics. It is indisputable that it is a metaphysical one, but there is no self consistent way to avoid metaphysics, if one aims to face the question of what a quantum entity is. By accepting HLQ, we grasp a powerful tool to explain emergence of life and mind. We give information the status of matter and energy, but we need a formalism to describe its inner structure. If we achieve this, we could construct a proper model about the connection of mind with brain. At this time, HLQ is a way that needs to be explored towards the various directions that are in front of us. Afshar, S. S., “Sharp complementary wave and particle behaviours in the same welcher weg experiment”, IRIMS preprint, May 2003. Albert, David, Quantum mechanics and Experience, Harvard, 1994 Atmanspacher, Harald and Primas, Hans (2002), “Epistemic and Ontic Quantum Realities”. PhilSci Archive, ID cod 938 Barrett, Jeffrey, “Everett’s Relative-State Formulation of Quantum Mechanics”, Standford Ecyclopedia of Philosophy, Barrow, J., P. Davies, Jr. Harper, Science Ultimate Reality, Quantum Theory, Cosmology and Complexity, Cambridge University Press, 2004 Bayer von Hans Christian, Information the New Language of Science, Phoenix Orion Books, 2004 Bohr (1935), “Quantum Mechanics and physical reality”, Quantum Theory and Measurment, Princeton Series in Physics, Princeton, New Jersey 1983, p. 144 Borgen, Peder, Philo of Alexandria, Brill Academic Publishers, Brill 1996 Cartwright, Nancy, How the Laws of Physics Lie, Claredon Press, 19863 Clauser, John F., “De Broglie wave interference of small rocks and live viruses”, Experimental metaphysics, Quantum Mechanical Studies for Abner Shimony, Kluwer Academic Publishers Vol 193, p. 1 Cramer, John G., “An Overview of the Transactional Interpretation of Quantum Mechanics”, International Journal of Theoretical Physics 27, 227 (1988) Cramer, John G., “Velocity Reversal and the Arrow of Time”, published in Foundations of Physics 18, 1205 (1988) Cushing, James T. Philosophical concepts in Physics, The historical relation between philosophy and scientific theories, Cambridge University Press, 1998 Durr, D. S. Goldstein and N. Zanghi, “Bohmian mechanics and the wave Function”, Experimental metaphysics, Quantum Mechanical Studies for Abner Shimony, Kluwer Academic Publishers, p. 32 Egan,Harvey D., AnAnthology of Christian Mysticism, Pueblo Book,Liturgical Press 1991 Einstain, Podolsky, Rosen (1935), “Can quantum-mechanical description of physical reality be considered complete?” , Quantum Theory and Measurment, Princeton Series in Physics, Princeton, New Jersey 1983, p. 138 Everett, H., 1957a, On the Foundations of Quantum Mechanics, thesis submitted to Princeton University, March 1, 1957 Feynman, Richard, QED, Princeton University Press, 1985, translated in Greek, Trochalia 1988 Ghirardi, Giancarlo “Collapse Theories”, Standford Ecyclopedia of Philosophy, Ghose, Partha, Testing Quantum Mechanics on New Ground, Cambrige University Press, 1999 Hawking, Stephen, Roger Penrose, The nature of Space and Time, Princeton University Press, 1996 Heisenberg, Werner, Encounters with Einstein, and other essays on people, places and particles, Seabury Press, 1983 Heisenberg, Werner, Physic and Philosophy, Penguin Classic3, Great Britain 2000 Hey, Tony and Patrick Walters, The Quantum Universe, Cambridge University Press, 1987 Jammer, M., The Philosophy of Quantum Mechanics, New York, Wiley 1974 Kraut, Richard, The Cambridge Companion to Plato, Cambridge University Press Landaou, Robert and Menas Kafatos, The Non-local Universe, Oxford University Press, 1999 Laughlin, Robert B., A Different Universe, Reinventing Physics from the Bottom Down, Basic Books, New York, 2005 Long, A.A., The Cambridge Companion to Early Greek Philosophy, Cambridge University Press, 1999 Louth, Andrew, Maximus the Confessor, Routledge 1996 Messiah, Albert, Quantum Mechanics, Dover Publications, Inc. Mineola, New York, 1999 Migne, Patrologia Graeca volume PG 90 and 91 Minkel, J. R., “The hollow universe”, New Scientist, issue 2340, 27 April 2002, page 22 Minkel, J.R., “The top-down Universe”, New Scientist, issue 2355, 10 August 2002, page 28 Moore, Edward, Origen of Alexandria and St. Maximus the Confessor, Boca Raton, Florida 2005 Mouraviev, Serge, “The Hidden Patterns of the Logos”, The Philosophy of Logos, Volume 1 Athens 1996, p. 148 Murchu, Diarmuid o’, Quantum Theology, Spiritual Implications of the New Physics, The Crossroad Publishing Company, New York, 2004 Omnès, Roland, Interpretation of Quantum Mechanics, Princeton University Press, 1994, Omnès, Roland, Quantum Philosophy: Understanding and Interpreting Contemporary Science, Princeton University Press, 1999 Penrose, Roger, Abner Shimony, Nancy Cartwright, Stephen Hawking, The Large, the Small and the Human Mind, Cambridge University Press, 1997 Penrose, Roger, The Road to Reality, Jonathan Cape London, 2004 Penrose, Roger, The Shadows of the Mind, Oxford University Press, 1994 Pierris, A. L., “Logos as ontological principle of reality”, The Philosophy of Logos, International Center for Greek Philosophy and Culture, Athens 1996, Volume II Powers, Jonathan, Philosophy and New Physics, Routledge 1991 Redhead, Michel, From Physics to Metaphysics, Cambridge University Press, 1995 Runia, D.T., Philo and the Church Fathers: A Collection of Papers, Brill Academic Publishers, Brill 1995 Runia, David T., “Philo, Alexandrian and Jew,” Idem, Exegesis and Philosophy: Studies on Philo of Alexandria (Variorum, Aldershot, 1990) Sandywell, Barry, Presocratic Reflexivity: Logological Investigations, Routledge (February, 1996 Selleri, Franco, Die Debatte um die Quantentheorie (Facetten der Physik), F. Vieweg (1983), Translated in Greek, Gutenberg, 1986 Sherwood, Polycarp, St. Maximus the Confessor, The ascetic life, the four centuries on charity, Newman Press, N.Y. Stonier, Tom, Information and the Internal Structure of the Universe, An Exploration into Information Physics, Springer-Verlag, 1990 Talbot, O Michael, The Holographic Universe, Harper Collins Publishers, London 1996 Tanona, Scott, “Idealization and Formalism in Bohr’s Approach to Quantum Mechanics”, Philosophy of Science, Vol. 71, number 5 p. 683 Thunberg, Lars, Microcosm and Mediator, The Theological Anthropology of Maximus the Confessor, second edition, Open Court, Chicago and La Salle, Illinois, 1995 Valtasar, von Hans Urs, Cosmic Liturgy, The Universe According to Maximus the Confessor, Communio Ignatius, translated from German, Brian E. Daley, S.J. 2003 Wheeler, John Archibald and Wojiech Zurek, Quantum Theory and Measurment, Princeton Series in Physics, Princeton, New Jersey 1983 1 Roger Penrose 1994, p. 305, ( g.e.) 2 Rolland Omnès 1999, p. 161 3 Alison George, Lone voices special: Take nobody’s word for it, An interview with the Nobel winner Brian Josephson, issue 2581 of New Scientist magazine, 09 December 2006, page 56-57 4 Physics and the Real World, Ellis, George F R. This paper was prepared for “Science and Religion: Global Perspectives”, June 4-8, 2005, in Philadelphia, PA, USA, 5 There are many such examples, like Diarmuid o’ Murchu, 2004 6 In quantum mechanics we talk about and distinguish formalism and interpretation. James T. Cushing, 1998 p. 439 (g.e.) 7 Roland Omnès 1999, p. 124. 8David Albert, 1992, p. 30 9 Werner Heisenberg, 2000, p. 14 10 Jenann Ismael, “Quantum Mechanics”, Online Stanford Encyclopedia of Philosophy, 11 This effect is well shown in the two-split experiment, Penrose, 2004, p. 504 and a very good description at Tony Hey and Patric Walters 1987 p.22 (g.e). 12 It is the known Wheeler’s delayed-choice experiment, Robert Nadeau and Menas Kafatos 1999, 50 and Penrose 2004, p. 512. 13 Detailed description Penrose 1994, p.309 (g.e.) 14 Jan Hilgevoord and Jos Uffink, The Uncertainty Principle, First published Mon Oct 8, 2001; substantive revision Mon Jul 3, 2006, SEP, 15 Roland Omnès 1999, p. 149. 16 Jonathan Powers, 1995, p 177 (g.e.) 17 David Albert, 1994, p. 80. 18 Michael Dickson, 1998, p. 88 19 David Albert, 1994, p.135, Franco Selleri, 1983 p. 59 (g.e.) 20 Lev Vaidman, “Many World Interpretation”, Stanford Encyclopedia of Philosophy, First published Sun 24 Mar, 2002 21 Giancarlo Ghirardi, “Collapse Theories”, Stanford Encyclopedia of Philosophy, First published Thu 7 Mar, 2002, 22 Greek text and English translation can be found at, B30 23Edward Hussey, Heraclitus, A. A. Long 1999, p. 154 (g. e.) The most characteristic text is B1; “Though this Word (logos) is true evermore, yet men are as unable to understand it when they hear it for the first time as before they have heard it at all. For, though all things come to pass in accordance with this Word (logos), men seem as if they had no experience of them, when they make trial of words and deeds such as I set forth, dividing each thing according to its kind and showing how it is what it is. But other men know not what they are doing when awake, even as they forget what they do in sleep”. 24 Edward Hussey, Heraclitus, A. A. Long 1999. p. 164 (g.e.) 25 Richard Kraut, “Plato”, Stanford Encyclopedia of Philosophy, first published Sat 20 Mar, 2004, . It deserve to mention that now days it is controversial if Plato endorses this understanding of his writings. Mark Balaguer, Platonism in Metaphysics”, first published Wed 12 May, 2004, Stanford Encyclopedia of Philosophy, 26 Dirk Baltzly, “Stoicism”, Stanford Encyclopedia of Philosophy, first published Mon Apr 15, 1996; substantive revision Mon Dec 13, 2004 27 David T. Runia, “Philo, Alexandrian and Jew,” Idem, Exegesis and Philosophy: Studies on Philo of Alexandria (Variorum, Aldershot, 1990), 28 ON FLIGHT AND FINDING 12.1, translated by Charles Duke Yonge London, H. G. Bohn, 1854-1890. 30 Edward Moore, “Origen of Alexandria” (185 – 254 A.D.), The Internet Encyclopedia of Philosophy, 31 Hans Urs von Valtasar, 2003, p. 127. 32 Polycarp Sherwood, p. 81. 33 This scheme is known as “double creation”, Lars Thunberg, 1995, p. 151. 34 Lars Thunberg, 1995, p. 64. 35 Hans Urs von Valtasar, 2003, p. 135. 36 Lars Thunberg, 1995, p 145. 37 PG 91, 1104A 38 PG 91 1229 39 PG 90, 447 40 PG 91, 1228 41 PG 90, 1133 A 42 It is a reverse of Origen’s myth; Hans Urs von Valtasar, 2003, p. 133 and p. 154. 43 Roger Penrose 2004, p. 498 44 John G. Cramer, 1988 45 Tom Stonier, 1990 46 By Amanda Gefter, “The elephant and the event horizon”, From issue 2575 of New Scientist magazine, 26 October 2006, page 36-39, 47 Robert Nadeau and Menas Kafatos, 1999, p. 113. 48 Roland Omnès, 1994, p. 527. 49 Anton Zeilinger, University of Vienna, Why The Quantum? “It” from “Bit”? A participatory universe? Three far-reaching challenges from John Archibald Wheeler and their relation to experiment, Science Ultimate Reality, Quantum Theory, Cosmology and Complexity, p. 211. 50 Michael Redhead, 1995, p. 51. (g.e.)
f8691738f4e5e656	Properties of Water Get Properties of Water essential facts below. View Videos or join the Properties of Water discussion. Add Properties of Water to your PopFlock.com topic list for future reference or share this resource on social media. Properties of Water The water molecule has this basic geometric structure Ball-and-stick model of a water molecule Space filling model of a water molecule A drop of water falling towards water in a glass IUPAC name water, oxidane Other names Hydrogen hydroxide (HH or HOH), hydrogen oxide, dihydrogen monoxide (DHMO) (systematic name[1]), dihydrogen oxide, hydric acid, hydrohydroxic acid, hydroxic acid, hydrol,[2] ?-oxido dihydrogen, ?1-hydroxyl hydrogen(0) 3D model (JSmol) RTECS number • ZC0110000 Molar mass 18.01528(33) g/mol Appearance White crystalline solid, almost colorless liquid with a hint of blue, colorless gas[3] Odor None Density Liquid:[4] 0.9998396 g/mL at 0 °C 0.9970474 g/mL at 25 °C 0.961893 g/mL at 95 °C 0.9167 g/ml at 0 °C Melting point 0.00 °C (32.00 °F; 273.15 K) [a] Boiling point 99.98 °C (211.96 °F; 373.13 K) [6][a] Solubility Poorly soluble in haloalkanes, aliphatic and aromatic hydrocarbons, ethers.[7] Improved solubility in carboxylates, alcohols, ketones, amines. Miscible with methanol, ethanol, propanol, isopropanol, acetone, glycerol, 1,4-dioxane, tetrahydrofuran, sulfolane, acetaldehyde, dimethylformamide, dimethoxyethane, dimethyl sulfoxide, acetonitrile. Partially miscible with Diethyl ether, Methyl Ethyl Ketone, Dichloromethane, Ethyl Acetate, Bromine. Vapor pressure 3.1690 kilopascals or 0.031276 atm at 25 °C[8] Acidity (pKa) 13.995[9][10][b] Basicity (pKb) 13.995 Conjugate acid Hydronium Conjugate base Hydroxide Thermal conductivity 0.6065 W/(m·K)[13] 1.3330 (20 °C)[14] Viscosity 0.890 mPa·s (0.890 cP)[15] 1.8546 D[16] 75.385 ± 0.05 J/(mol·K)[17] 69.95 ± 0.03 J/(mol·K)[17] -285.83 ± 0.04 kJ/mol[7][17] -237.24 kJ/mol[7] Main hazards Drowning Avalanche (as snow) Water intoxication (see also Dihydrogen monoxide parody) Safety data sheet SDS NFPA 704 (fire diamond) Flammability code 0: Will not burn. E.g. waterHealth code 0: Exposure under fire conditions would offer no hazard beyond that of ordinary combustible material. E.g. sodium chlorideReactivity code 0: Normally stable, even under fire exposure conditions, and is not reactive with water. E.g. liquid nitrogenSpecial hazards (white): no codeNFPA 704 four-colored diamond Flash point Non-flammable Related compounds Other cations Hydrogen sulfide Hydrogen selenide Hydrogen telluride Hydrogen polonide Hydrogen peroxide Related solvents Supplementary data page Refractive index (n), Dielectric constant (?r), etc. Phase behaviour ?Y verify (what is ?Y?N ?) Infobox references Water is a polar inorganic compound that is at room temperature a tasteless and odorless liquid, which is nearly colorless apart from an inherent hint of blue. It is by far the most studied chemical compound[18] and is described as the "universal solvent"[19] and the "solvent of life."[20] It is the most abundant substance on Earth[21] and the only common substance to exist as a solid, liquid, and gas on Earth's surface.[22] It is also the third most abundant molecule in the universe (behind molecular hydrogen and carbon monoxide).[21] Water molecules form hydrogen bonds with each other and are strongly polar. This polarity allows it to dissociate ions in salts and bond to other polar substances such as alcohols and acids, thus dissolving them. Its hydrogen bonding causes its many unique properties, such as having a solid form less dense than its liquid form,[c] a relatively high boiling point of 100 °C for its molar mass, and a high heat capacity. Water is amphoteric, meaning that it can exhibit properties of an acid or a base, depending on the pH of the solution that it is in; it readily produces both and ions.[c] Related to its amphoteric character, it undergoes self-ionization. The product of the activities, or approximately, the concentrations of and is a constant, so their respective concentrations are inversely proportional to each other.[23] Physical properties Water is the chemical substance with chemical formula ; one molecule of water has two hydrogen atoms covalently bonded to a single oxygen atom.[24] Water is a tasteless, odorless liquid at ambient temperature and pressure. Liquid water has weak absorption bands at wavelengths of around 750 nm which cause it to appear to have a blue colour.[3] This can easily be observed in a water-filled bath or wash-basin whose lining is white. Large ice crystals, as in glaciers, also appear blue. Under standard conditions, water is primarily a liquid, unlike other analogous hydrides of the oxygen family, which are generally gaseous. This unique property of water is due to hydrogen bonding. The molecules of water are constantly moving in relation to each other, and the hydrogen bonds are continually breaking and reforming at timescales faster than 200 femtoseconds (2×10-13 seconds).[25] However, these bonds are strong enough to create many of the peculiar properties of water, some of which make it integral to life. Water, ice, and vapour Within the Earth's atmosphere and surface, the liquid phase is the most common and is the form that is generally denoted by the word "water". The solid phase of water is known as ice and commonly takes the structure of hard, amalgamated crystals, such as ice cubes, or loosely accumulated granular crystals, like snow. Aside from common hexagonal crystalline ice, other crystalline and amorphous phases of ice are known. The gaseous phase of water is known as water vapor (or steam). Visible steam and clouds are formed from minute droplets of water suspended in the air. Water also forms a supercritical fluid. The critical temperature is 647 K and the critical pressure is 22.064 MPa. In nature this only rarely occurs in extremely hostile conditions. A likely example of naturally occurring supercritical water is in the hottest parts of deep water hydrothermal vents, in which water is heated to the critical temperature by volcanic plumes and the critical pressure is caused by the weight of the ocean at the extreme depths where the vents are located. This pressure is reached at a depth of about 2200 meters: much less than the mean depth of the ocean (3800 meters).[26] Heat capacity and heats of vaporization and fusion Heat of vaporization of water from melting to critical temperature Water has a very high specific heat capacity of 4.1814 J/(g·K) at 25 °C - the second highest among all the heteroatomic species (after ammonia), as well as a high heat of vaporization (40.65 kJ/mol or 2257 kJ/kg at the normal boiling point), both of which are a result of the extensive hydrogen bonding between its molecules. These two unusual properties allow water to moderate Earth's climate by buffering large fluctuations in temperature. Most of the additional energy stored in the climate system since 1970 has accumulated in the oceans.[27] The specific enthalpy of fusion (more commonly known as latent heat) of water is 333.55 kJ/kg at 0 °C: the same amount of energy is required to melt ice as to warm ice from -160 °C up to its melting point or to heat the same amount of water by about 80 °C. Of common substances, only that of ammonia is higher. This property confers resistance to melting on the ice of glaciers and drift ice. Before and since the advent of mechanical refrigeration, ice was and still is in common use for retarding food spoilage. The specific heat capacity of ice at -10 °C is 2.03 J/(g·K)[28] and the heat capacity of steam at 100 °C is 2.08 J/(g·K).[29] Density of water and ice Density of ice and water as a function of temperature The density of water is about 1 gram per cubic centimetre (62 lb/cu ft): this relationship was originally used to define the gram.[30] The density varies with temperature, but not linearly: as the temperature increases, the density rises to a peak at 3.98 °C (39.16 °F) and then decreases;[31] this is unusual.[d] Regular, hexagonal ice is also less dense than liquid water--upon freezing, the density of water decreases by about 9%.[34][e] These effects are due to the reduction of thermal motion with cooling, which allows water molecules to form more hydrogen bonds that prevent the molecules from coming close to each other.[31] While below 4 °C the breakage of hydrogen bonds due to heating allows water molecules to pack closer despite the increase in the thermal motion (which tends to expand a liquid), above 4 °C water expands as the temperature increases.[31] Water near the boiling point is about 4% less dense than water at 4 °C (39 °F).[34][f] Under increasing pressure, ice undergoes a number of transitions to other polymorphs with higher density than liquid water, such as ice II, ice III, high-density amorphous ice (HDA), and very-high-density amorphous ice (VHDA).[35][36] Temperature distribution in a lake in summer and winter The unusual density curve and lower density of ice than of water is vital to life--if water were most dense at the freezing point, then in winter the very cold water at the surface of lakes and other water bodies would sink, lakes could freeze from the bottom up, and all life in them would be killed.[34] Furthermore, given that water is a good thermal insulator (due to its heat capacity), some frozen lakes might not completely thaw in summer.[34] The layer of ice that floats on top insulates the water below.[37] Water at about 4 °C (39 °F) also sinks to the bottom, thus keeping the temperature of the water at the bottom constant (see diagram).[34] Density of saltwater and ice WOA surface density The density of salt water depends on the dissolved salt content as well as the temperature. Ice still floats in the oceans, otherwise they would freeze from the bottom up. However, the salt content of oceans lowers the freezing point by about 1.9 °C[38] (see here for explanation) and lowers the temperature of the density maximum of water to the former freezing point at 0 °C. This is why, in ocean water, the downward convection of colder water is not blocked by an expansion of water as it becomes colder near the freezing point. The oceans' cold water near the freezing point continues to sink. So creatures that live at the bottom of cold oceans like the Arctic Ocean generally live in water 4 °C colder than at the bottom of frozen-over fresh water lakes and rivers. As the surface of salt water begins to freeze (at -1.9 °C[38] for normal salinity seawater, 3.5%) the ice that forms is essentially salt-free, with about the same density as freshwater ice. This ice floats on the surface, and the salt that is "frozen out" adds to the salinity and density of the sea water just below it, in a process known as brine rejection. This denser salt water sinks by convection and the replacing seawater is subject to the same process. This produces essentially freshwater ice at -1.9 °C[38] on the surface. The increased density of the sea water beneath the forming ice causes it to sink towards the bottom. On a large scale, the process of brine rejection and sinking cold salty water results in ocean currents forming to transport such water away from the Poles, leading to a global system of currents called the thermohaline circulation. Miscibility and condensation Red line shows saturation Water is miscible with many liquids, including ethanol in all proportions. Water and most oils are immiscible usually forming layers according to increasing density from the top. This can be predicted by comparing the polarity. Water being a relatively polar compound will tend to be miscible with liquids of high polarity such as ethanol and acetone, whereas compounds with low polarity will tend to be immiscible and poorly soluble such as with hydrocarbons. As a gas, water vapor is completely miscible with air. On the other hand, the maximum water vapor pressure that is thermodynamically stable with the liquid (or solid) at a given temperature is relatively low compared with total atmospheric pressure. For example, if the vapor's partial pressure is 2% of atmospheric pressure and the air is cooled from 25 °C, starting at about 22 °C water will start to condense, defining the dew point, and creating fog or dew. The reverse process accounts for the fog burning off in the morning. If the humidity is increased at room temperature, for example, by running a hot shower or a bath, and the temperature stays about the same, the vapor soon reaches the pressure for phase change, and then condenses out as minute water droplets, commonly referred to as steam. A saturated gas or one with 100% relative humidity is when the vapor pressure of water in the air is at equilibrium with vapor pressure due to (liquid) water; water (or ice, if cool enough) will fail to lose mass through evaporation when exposed to saturated air. Because the amount of water vapor in air is small, relative humidity, the ratio of the partial pressure due to the water vapor to the saturated partial vapor pressure, is much more useful. Vapor pressure above 100% relative humidity is called super-saturated and can occur if air is rapidly cooled, for example, by rising suddenly in an updraft.[g] Vapor pressure Vapor pressure diagrams of water The compressibility of water is a function of pressure and temperature. At 0 °C, at the limit of zero pressure, the compressibility is . At the zero-pressure limit, the compressibility reaches a minimum of around 45 °C before increasing again with increasing temperature. As the pressure is increased, the compressibility decreases, being at 0 °C and 100 megapascals (1,000 bar).[39] The bulk modulus of water is about 2.2 GPa.[40] The low compressibility of non-gases, and of water in particular, leads to their often being assumed as incompressible. The low compressibility of water means that even in the deep oceans at 4 km depth, where pressures are 40 MPa, there is only a 1.8% decrease in volume.[40] Triple point The Solid/Liquid/Vapour triple point of liquid water, ice Ih and water vapor in the lower left portion of a water phase diagram. The temperature and pressure at which ordinary solid, liquid, and gaseous water coexist in equilibrium is a triple point of water. Since 1954, this point had been used to define the base unit of temperature, the kelvin[41][42] but, starting in 2019, the kelvin is now defined using the Boltzmann constant, rather than the triple point of water.[43] Due to the existence of many polymorphs (forms) of ice, water has other triple points, which have either three polymorphs of ice or two polymorphs of ice and liquid in equilibrium.[42]Gustav Heinrich Johann Apollon Tammann in Göttingen produced data on several other triple points in the early 20th century. Kamb and others documented further triple points in the 1960s.[44][45][46] The various triple points of water Phases in stable equilibrium Pressure Temperature liquid water, ice Ih, and water vapor 611.657 Pa[47] 273.16 K (0.01 °C) liquid water, ice Ih, and ice III 209.9 MPa 251 K (-22 °C) liquid water, ice III, and ice V 350.1 MPa -17.0 °C liquid water, ice V, and ice VI 632.4 MPa 0.16 °C ice Ih, Ice II, and ice III 213 MPa -35 °C ice II, ice III, and ice V 344 MPa -24 °C ice II, ice V, and ice VI 626 MPa -70 °C Melting point The melting point of ice is 0 °C (32 °F; 273 K) at standard pressure; however, pure liquid water can be supercooled well below that temperature without freezing if the liquid is not mechanically disturbed. It can remain in a fluid state down to its homogeneous nucleation point of about 231 K (-42 °C; -44 °F).[48] The melting point of ordinary hexagonal ice falls slightly under moderately high pressures, by 0.0073 °C (0.0131 °F)/atm[h] or about 0.5 °C (0.90 °F)/70 atm[i][49] as the stabilization energy of hydrogen bonding is exceeded by intermolecular repulsion, but as ice transforms into its polymorphs (see crystalline states of ice) above 209.9 MPa (2,072 atm), the melting point increases markedly with pressure, i.e., reaching 355 K (82 °C) at 2.216 GPa (21,870 atm) (triple point of Ice VII[50]). Electrical properties Electrical conductivity Pure water containing no exogenous ions is an excellent insulator, but not even "deionized" water is completely free of ions. Water undergoes auto-ionization in the liquid state, when two water molecules form one hydroxide anion and one hydronium cation . Because water is such a good solvent, it almost always has some solute dissolved in it, often a salt. If water has even a tiny amount of such an impurity, then the ions can carry charges back and forth, allowing the water to conduct electricity far more readily. It is known that the theoretical maximum electrical resistivity for water is approximately 18.2 M?·cm (182 k?·m) at 25 °C.[51] This figure agrees well with what is typically seen on reverse osmosis, ultra-filtered and deionized ultra-pure water systems used, for instance, in semiconductor manufacturing plants. A salt or acid contaminant level exceeding even 100 parts per trillion (ppt) in otherwise ultra-pure water begins to noticeably lower its resistivity by up to several k?·m.[] In pure water, sensitive equipment can detect a very slight electrical conductivity of 0.05501 ± 0.0001 ?S/cm at 25.00 °C.[51] Water can also be electrolyzed into oxygen and hydrogen gases but in the absence of dissolved ions this is a very slow process, as very little current is conducted. In ice, the primary charge carriers are protons (see proton conductor).[52] Ice was previously thought to have a small but measurable conductivity of 1×10-10 S/cm, but this conductivity is now thought to be almost entirely from surface defects, and without those, ice is an insulator with an immeasurably small conductivity.[31] Polarity and hydrogen bonding A diagram showing the partial charges on the atoms in a water molecule An important feature of water is its polar nature. The structure has a bent molecular geometry for the two hydrogens from the oxygen vertex. The oxygen atom also has two lone pairs of electrons. One effect usually ascribed to the lone pairs is that the H-O-H gas phase bend angle is 104.48°,[53] which is smaller than the typical tetrahedral angle of 109.47°. The lone pairs are closer to the oxygen atom than the electrons sigma bonded to the hydrogens, so they require more space. The increased repulsion of the lone pairs forces the O-H bonds closer to each other.[54] Another consequence of its structure is that water is a polar molecule. Due to the difference in electronegativity, a bond dipole moment points from each H to the O, making the oxygen partially negative and each hydrogen partially positive. A large molecular dipole, points from a region between the two hydrogen atoms to the oxygen atom. The charge differences cause water molecules to aggregate (the relatively positive areas being attracted to the relatively negative areas). This attraction, hydrogen bonding, explains many of the properties of water, such as its solvent properties.[55] Although hydrogen bonding is a relatively weak attraction compared to the covalent bonds within the water molecule itself, it is responsible for a number of water's physical properties. These properties include its relatively high melting and boiling point temperatures: more energy is required to break the hydrogen bonds between water molecules. In contrast, hydrogen sulfide , has much weaker hydrogen bonding due to sulfur's lower electronegativity. is a gas at room temperature, in spite of hydrogen sulfide having nearly twice the molar mass of water. The extra bonding between water molecules also gives liquid water a large specific heat capacity. This high heat capacity makes water a good heat storage medium (coolant) and heat shield. Cohesion and adhesion Dew drops adhering to a spider web Water molecules stay close to each other (cohesion), due to the collective action of hydrogen bonds between water molecules. These hydrogen bonds are constantly breaking, with new bonds being formed with different water molecules; but at any given time in a sample of liquid water, a large portion of the molecules are held together by such bonds.[56] Water also has high adhesion properties because of its polar nature. On extremely clean/smooth glass the water may form a thin film because the molecular forces between glass and water molecules (adhesive forces) are stronger than the cohesive forces. In biological cells and organelles, water is in contact with membrane and protein surfaces that are hydrophilic; that is, surfaces that have a strong attraction to water. Irving Langmuir observed a strong repulsive force between hydrophilic surfaces. To dehydrate hydrophilic surfaces--to remove the strongly held layers of water of hydration--requires doing substantial work against these forces, called hydration forces. These forces are very large but decrease rapidly over a nanometer or less.[57] They are important in biology, particularly when cells are dehydrated by exposure to dry atmospheres or to extracellular freezing.[58] Rain water flux from a canopy. Among the forces that govern drop formation: Surface tension, Cohesion (chemistry), Van der Waals force, Plateau-Rayleigh instability. Surface tension This paper clip is under the water level, which has risen gently and smoothly. Surface tension prevents the clip from submerging and the water from overflowing the glass edges. Temperature dependence of the surface tension of pure water Water has an unusually high surface tension of 71.99 mN/m at 25 °C[59] which is caused by the strength of the hydrogen bonding between water molecules.[60] This allows insects to walk on water.[60] Capillary action Because water has strong cohesive and adhesive forces, it exhibits capillary action.[61] Strong cohesion from hydrogen bonding and adhesion allows trees to transport water more than 100 m upward.[60] Water as a solvent Presence of colloidal calcium carbonate from high concentrations of dissolved lime turns the water of Havasu Falls turquoise. Water is an excellent solvent due to its high dielectric constant.[62] Substances that mix well and dissolve in water are known as hydrophilic ("water-loving") substances, while those that do not mix well with water are known as hydrophobic ("water-fearing") substances.[63] The ability of a substance to dissolve in water is determined by whether or not the substance can match or better the strong attractive forces that water molecules generate between other water molecules. If a substance has properties that do not allow it to overcome these strong intermolecular forces, the molecules are precipitated out from the water. Contrary to the common misconception, water and hydrophobic substances do not "repel", and the hydration of a hydrophobic surface is energetically, but not entropically, favorable. When an ionic or polar compound enters water, it is surrounded by water molecules (hydration). The relatively small size of water molecules (~ 3 angstroms) allows many water molecules to surround one molecule of solute. The partially negative dipole ends of the water are attracted to positively charged components of the solute, and vice versa for the positive dipole ends. In general, ionic and polar substances such as acids, alcohols, and salts are relatively soluble in water, and non-polar substances such as fats and oils are not. Non-polar molecules stay together in water because it is energetically more favorable for the water molecules to hydrogen bond to each other than to engage in van der Waals interactions with non-polar molecules. An example of an ionic solute is table salt; the sodium chloride, NaCl, separates into cations and anions, each being surrounded by water molecules. The ions are then easily transported away from their crystalline lattice into solution. An example of a nonionic solute is table sugar. The water dipoles make hydrogen bonds with the polar regions of the sugar molecule (OH groups) and allow it to be carried away into solution. Quantum tunneling The quantum tunneling dynamics in water was reported as early as 1992. At that time it was known that there are motions which destroy and regenerate the weak hydrogen bond by internal rotations of the substituent water monomers.[64] On 18 March 2016, it was reported that the hydrogen bond can be broken by quantum tunneling in the water hexamer. Unlike previously reported tunneling motions in water, this involved the concerted breaking of two hydrogen bonds.[65] Later in the same year, the discovery of the quantum tunneling of water molecules was reported.[66] Electromagnetic absorption Water is relatively transparent to visible light, near ultraviolet light, and far-red light, but it absorbs most ultraviolet light, infrared light, and microwaves. Most photoreceptors and photosynthetic pigments utilize the portion of the light spectrum that is transmitted well through water. Microwave ovens take advantage of water's opacity to microwave radiation to heat the water inside of foods. Water's light blue colour is caused by weak absorption in the red part of the visible spectrum.[3][67] Model of hydrogen bonds (1) between molecules of water A single water molecule can participate in a maximum of four hydrogen bonds because it can accept two bonds using the lone pairs on oxygen and donate two hydrogen atoms. Other molecules like hydrogen fluoride, ammonia and methanol can also form hydrogen bonds. However, they do not show anomalous thermodynamic, kinetic or structural properties like those observed in water because none of them can form four hydrogen bonds: either they cannot donate or accept hydrogen atoms, or there are steric effects in bulky residues. In water, intermolecular tetrahedral structures form due to the four hydrogen bonds, thereby forming an open structure and a three-dimensional bonding network, resulting in the anomalous decrease in density when cooled below 4 °C. This repeated, constantly reorganizing unit defines a three-dimensional network extending throughout the liquid. This view is based upon neutron scattering studies and computer simulations, and it makes sense in the light of the unambiguously tetrahedral arrangement of water molecules in ice structures. However, there is an alternative theory for the structure of water. In 2004, a controversial paper from Stockholm University suggested that water molecules in liquid form typically bind not to four but to only two others; thus forming chains and rings. The term "string theory of water" (which is not to be confused with the string theory of physics) was coined. These observations were based upon X-ray absorption spectroscopy that probed the local environment of individual oxygen atoms.[68] Molecular structure The repulsive effects of the two lone pairs on the oxygen atom cause water to have a bent, not linear, molecular structure,[69] allowing it to be polar. The hydrogen-oxygen-hydrogen angle is 104.45°, which is less than the 109.47° for ideal sp3 hybridization. The valence bond theory explanation is that the oxygen atom's lone pairs are physically larger and therefore take up more space than the oxygen atom's bonds to the hydrogen atoms.[70] The molecular orbital theory explanation (Bent's rule) is that lowering the energy of the oxygen atom's nonbonding hybrid orbitals (by assigning them more s character and less p character) and correspondingly raising the energy of the oxygen atom's hybrid orbitals bonded to the hydrogen atoms (by assigning them more p character and less s character) has the net effect of lowering the energy of the occupied molecular orbitals because the energy of the oxygen atom's nonbonding hybrid orbitals contributes completely to the energy of the oxygen atom's lone pairs while the energy of the oxygen atom's other two hybrid orbitals contributes only partially to the energy of the bonding orbitals (the remainder of the contribution coming from the hydrogen atoms' 1s orbitals). Chemical properties In liquid water there is some self-dissociation giving hydronium ions and hydroxide ions. 2 ? + The equilibrium constant for this reaction, known as the ionic product of water,Kw, has a value of about 10-14 at 25 °C. At neutral pH, the concentration of the hydroxide ion equal to that of the (solvated) hydrogen ion , with a value close to 10-7 mol L-1 at 25 °C.[71] See data page for values at other temperatures. Action of water on rock over long periods of time typically leads to weathering and water erosion, physical processes that convert solid rocks and minerals into soil and sediment, but under some conditions chemical reactions with water occur as well, resulting in metasomatism or mineral hydration, a type of chemical alteration of a rock which produces clay minerals. It also occurs when Portland cement hardens. Water ice can form clathrate compounds, known as clathrate hydrates, with a variety of small molecules that can be embedded in its spacious crystal lattice. The most notable of these is methane clathrate, 4 , naturally found in large quantities on the ocean floor. Acidity in nature Rain is generally mildly acidic, with a pH between 5.2 and 5.8 if not having any acid stronger than carbon dioxide.[72] If high amounts of nitrogen and sulfur oxides are present in the air, they too will dissolve into the cloud and rain drops, producing acid rain. Several isotopes of both hydrogen and oxygen exist, giving rise to several known isotopologues of water. Vienna Standard Mean Ocean Water is the current international standard for water isotopes. Naturally occurring water is almost completely composed of the neutron-less hydrogen isotope protium. Only 155 ppm include deuterium ( or D), a hydrogen isotope with one neutron, and fewer than 20 parts per quintillion include tritium ( or T), which has two neutrons. Oxygen also has three stable isotopes, with present in 99.76%, in 0.04%, and in 0.2% of water molecules.[73] Deuterium oxide, , is also known as heavy water because of its higher density. It is used in nuclear reactors as a neutron moderator. Tritium is radioactive, decaying with a half-life of 4500 days; exists in nature only in minute quantities, being produced primarily via cosmic ray-induced nuclear reactions in the atmosphere. Water with one protium and one deuterium atom occurs naturally in ordinary water in low concentrations (~0.03%) and in far lower amounts (0.000003%) and any such molecules are temporary as the atoms recombine. The most notable physical differences between and , other than the simple difference in specific mass, involve properties that are affected by hydrogen bonding, such as freezing and boiling, and other kinetic effects. This is because the nucleus of deuterium is twice as heavy as protium, and this causes noticeable differences in bonding energies. The difference in boiling points allows the isotopologues to be separated. The self-diffusion coefficient of at 25 °C is 23% higher than the value of .[74] Because water molecules exchange hydrogen atoms with one another, hydrogen deuterium oxide (DOH) is much more common in low-purity heavy water than pure dideuterium monoxide . Consumption of pure isolated may affect biochemical processes - ingestion of large amounts impairs kidney and central nervous system function. Small quantities can be consumed without any ill-effects; humans are generally unaware of taste differences,[75] but sometimes report a burning sensation[76] or sweet flavor.[77] Very large amounts of heavy water must be consumed for any toxicity to become apparent. Rats, however, are able to avoid heavy water by smell, and it is toxic to many animals.[78] Light water refers to deuterium-depleted water (DDW), water in which the deuterium content has been reduced below the standard level. Water is the most abundant substance on Earth and also the third most abundant molecule in the universe, after and .[21] 0.23 ppm of the earth's mass is water and 97.39% of the global water volume of 1.38×109 km3 is found in the oceans.[79] Acid-base reactions Water is amphoteric: it has the ability to act as either an acid or a base in chemical reactions.[80] According to the Brønsted-Lowry definition, an acid is a proton donor and a base is a proton acceptor.[81] When reacting with a stronger acid, water acts as a base; when reacting with a stronger base, it acts as an acid.[81] For instance, water receives an ion from HCl when hydrochloric acid is formed: + ? + In the reaction with ammonia, , water donates a ion, and is thus acting as an acid: + ? + Because the oxygen atom in water has two lone pairs, water often acts as a Lewis base, or electron pair donor, in reactions with Lewis acids, although it can also react with Lewis bases, forming hydrogen bonds between the electron pair donors and the hydrogen atoms of water. HSAB theory describes water as both a weak hard acid and a weak hard base, meaning that it reacts preferentially with other hard species: + -> + -> + -> When a salt of a weak acid or of a weak base is dissolved in water, water can partially hydrolyze the salt, producing the corresponding base or acid, which gives aqueous solutions of soap and baking soda their basic pH: + ? NaOH + Ligand chemistry Water's Lewis base character makes it a common ligand in transition metal complexes, examples of which include metal aquo complexes such as to perrhenic acid, which contains two water molecules coordinated to a rhenium center. In solid hydrates, water can be either a ligand or simply lodged in the framework, or both. Thus, consists of [Fe2(H2O)6]2+ centers and one "lattice water". Water is typically a monodentate ligand, i.e., it forms only one bond with the central atom.[82] Some hydrogen-bonding contacts in FeSO4.7H2O. This metal aquo complex crystallizes with one molecule of "lattice" water, which interacts with the sulfate and with the [Fe(H2O)6]2+ centers. Organic chemistry As a hard base, water reacts readily with organic carbocations; for example in a hydration reaction, a hydroxyl group and an acidic proton are added to the two carbon atoms bonded together in the carbon-carbon double bond, resulting in an alcohol. When addition of water to an organic molecule cleaves the molecule in two, hydrolysis is said to occur. Notable examples of hydrolysis are the saponification of fats and the digestion of proteins and polysaccharides. Water can also be a leaving group in SN2 substitution and E2 elimination reactions; the latter is then known as a dehydration reaction. Water in redox reactions Water contains hydrogen in the oxidation state +1 and oxygen in the oxidation state -2.[83] It oxidizes chemicals such as hydrides, alkali metals, and some alkaline earth metals.[84][85] One example of an alkali metal reacting with water is:[86] 2 Na + 2 -> + 2 + 2 Some other reactive metals, such as aluminum and beryllium, are oxidized by water as well, but their oxides adhere to the metal and form a passive protective layer.[87] Note that the rusting of iron is a reaction between iron and oxygen[88] that is dissolved in water, not between iron and water. Water can be oxidized to emit oxygen gas, but very few oxidants react with water even if their reduction potential is greater than the potential of . Almost all such reactions require a catalyst.[89] An example of the oxidation of water is: 4 + 2 -> 4 AgF + 4 HF + Water can be split into its constituent elements, hydrogen and oxygen, by passing an electric current through it.[90] This process is called electrolysis. The cathode half reaction is: 2 + 2 The anode half reaction is: 2 -> + 4 + 4 The gases produced bubble to the surface, where they can be collected or ignited with a flame above the water if this was the intention. The required potential for the electrolysis of pure water is 1.23 V at 25 °C.[90] The operating potential is actually 1.48 V or higher in practical electrolysis. Henry Cavendish showed that water was composed of oxygen and hydrogen in 1781.[91] The first decomposition of water into hydrogen and oxygen, by electrolysis, was done in 1800 by English chemist William Nicholson and Anthony Carlisle.[91][92] In 1805, Joseph Louis Gay-Lussac and Alexander von Humboldt showed that water is composed of two parts hydrogen and one part oxygen.[93] Gilbert Newton Lewis isolated the first sample of pure heavy water in 1933.[94] The properties of water have historically been used to define various temperature scales. Notably, the Kelvin, Celsius, Rankine, and Fahrenheit scales were, or currently are, defined by the freezing and boiling points of water. The less common scales of Delisle, Newton, Réaumur and Rømer were defined similarly. The triple point of water is a more commonly used standard point today. The accepted IUPAC name of water is oxidane or simply water,[95] or its equivalent in different languages, although there are other systematic names which can be used to describe the molecule. Oxidane is only intended to be used as the name of the mononuclear parent hydride used for naming derivatives of water by substituent nomenclature.[96] These derivatives commonly have other recommended names. For example, the name hydroxyl is recommended over oxidanyl for the -OH group. The name oxane is explicitly mentioned by the IUPAC as being unsuitable for this purpose, since it is already the name of a cyclic ether also known as tetrahydropyran.[97][98] The simplest systematic name of water is hydrogen oxide. This is analogous to related compounds such as hydrogen peroxide, hydrogen sulfide, and deuterium oxide (heavy water). Using chemical nomenclature for type I ionic binary compounds, water would take the name hydrogen monoxide,[99] but this is not among the names published by the International Union of Pure and Applied Chemistry (IUPAC).[95] Another name is dihydrogen monoxide, which is a rarely used name of water, and mostly used in the dihydrogen monoxide parody. Other systematic names for water include hydroxic acid, hydroxylic acid, and hydrogen hydroxide, using acid and base names.[j] None of these exotic names are used widely. The polarized form of the water molecule, , is also called hydron hydroxide by IUPAC nomenclature.[100] Water substance is a term used for hydrogen oxide (H2O) when one does not wish to specify whether one is speaking of liquid water, steam, some form of ice, or a component in a mixture or mineral. See also 1. ^ a b Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C, and boils at 373.1339 K (99.9839 °C). Other isotopic compositions melt or boil at slightly different temperatures. 2. ^ A commonly quoted value of 15.7 used mainly in organic chemistry for the pKa of water is incorrect.[11][12] 3. ^ a b H+ represents and more complex ions that form. 4. ^ Negative thermal expansion is also observed in molten silica.[32] Also, fairly pure silicon has a negative coefficient of thermal expansion for temperatures between about 18 and 120 kelvins.[33] 5. ^ Other substances that expand on freezing are silicon (melting point of 1,687 K (1,414 °C; 2,577 °F)), gallium (melting point of 303 K (30 °C; 86 °F), germanium (melting point of 1,211 K (938 °C; 1,720 °F)), antimony (melting point of 904 K (631 °C; 1,168 °F)), and bismuth (melting point of 545 K (272 °C; 521 °F)) 6. ^ (1-0.95865/1.00000) × 100% = 4.135% 7. ^ Adiabatic cooling resulting from the ideal gas law. 8. ^ The source gives it as 0.0072°C/atm. However the author defines an atmosphere as 1,000,000 dynes/cm2 (a bar). Using the standard definition of atmosphere, 1,013,250 dynes/cm2, it works out to 0.0073°C/atm. 9. ^ Using the fact that 0.5/0.0073 = 68.5. 10. ^ Both acid and base names exist for water because it is amphoteric (able to react both as an acid or an alkali). 1. ^ "naming molecular compounds". www.iun.edu. Retrieved 2018. Sometimes these compounds have generic or common names (e.g., H2O is "water") and they also have systematic names (e.g., H2O, dihydrogen monoxide). 2. ^ "Definition of Hydrol". Merriam-Webster. Retrieved 2019. 3. ^ a b c Braun, Charles L.; Smirnov, Sergei N. (1993-08-01). "Why is water blue?" (PDF). Journal of Chemical Education. 70 (8): 612. Bibcode:1993JChEd..70..612B. doi:10.1021/ed070p612. ISSN 0021-9584. 4. ^ Riddick 1970, Table of Physical Properties, Water 0b. pg 67-8. 5. ^ Lide 2003, Properties of Ice and Supercooled Water in Section 6. 6. ^ Water in Linstrom, Peter J.; Mallard, William G. (eds.); NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology, Gaithersburg (MD), http://webbook.nist.gov (retrieved 2016-5-27) 7. ^ a b c Anatolievich, Kiper Ruslan. "Properties of substance: water". 8. ^ Lide 2003, Vapor Pressure of Water From 0 to 370° C in Sec. 6. 9. ^ Lide 2003, Chapter 8: Dissociation Constants of Inorganic Acids and Bases. 10. ^ Weingärtner et al. 2016, p. 13. 11. ^ "What is the pKa of Water". University of California, Davis. 2015-08-09. 12. ^ Silverstein, Todd P.; Heller, Stephen T. (17 April 2017). "pKa Values in the Undergraduate Curriculum: What Is the Real pKa of Water?". Journal of Chemical Education. 94 (6): 690-695. Bibcode:2017JChEd..94..690S. doi:10.1021/acs.jchemed.6b00623. 13. ^ Ramires, Maria L. V.; Castro, Carlos A. Nieto de; Nagasaka, Yuchi; Nagashima, Akira; Assael, Marc J.; Wakeham, William A. (1995-05-01). "Standard Reference Data for the Thermal Conductivity of Water". Journal of Physical and Chemical Reference Data. 24 (3): 1377-1381. Bibcode:1995JPCRD..24.1377R. doi:10.1063/1.555963. ISSN 0047-2689. 14. ^ Lide 2003, 8--Concentrative Properties of Aqueous Solutions: Density, Refractive Index, Freezing Point Depression, and Viscosity. 15. ^ Lide 2003, 6.186. 16. ^ Lide 2003, 9--Dipole Moments. 17. ^ a b c Water in Linstrom, Peter J.; Mallard, William G. (eds.); NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology, Gaithersburg (MD), http://webbook.nist.gov (retrieved 2014-06-01) 18. ^ Greenwood & Earnshaw 1997, p. 620. 19. ^ "Water, the Universal Solvent". USGS. 20. ^ Reece et al. 2013, p. 48. 21. ^ a b c Weingärtner et al. 2016, p. 2. 22. ^ Reece et al. 2013, p. 44. 23. ^ "Autoprotolysis constant". IUPAC Compendium of Chemical Terminology. IUPAC. 2009. doi:10.1351/goldbook.A00532. ISBN 978-0-9678550-9-7. 24. ^ Campbell, Williamson & Heyden 2006. 25. ^ Smith, Jared D.; Christopher D. Cappa; Kevin R. Wilson; Ronald C. Cohen; Phillip L. Geissler; Richard J. Saykally (2005). "Unified description of temperature-dependent hydrogen bond rearrangements in liquid water" (PDF). Proc. Natl. Acad. Sci. USA. 102 (40): 14171-14174. Bibcode:2005PNAS..10214171S. doi:10.1073/pnas.0506899102. PMC 1242322. PMID 16179387. 26. ^ Deguchi, Shigeru; Tsujii, Kaoru (2007-06-19). "Supercritical water: a fascinating medium for soft matter". Soft Matter. 3 (7): 797. Bibcode:2007SMat....3..797D. doi:10.1039/b611584e. ISSN 1744-6848. 27. ^ Rhein, M.; Rintoul, S.R. (2013). "3: Observations: Ocean" (PDF). IPCC WGI AR5 (Report). p. 257. Ocean warming dominates the global energy change inventory. Warming of the ocean accounts for about 93% of the increase in the Earth's energy inventory between 1971 and 2010 (high confidence), with warming of the upper (0 to 700 m) ocean accounting for about 64% of the total. Melting ice (including Arctic sea ice, ice sheets and glaciers) and warming of the continents and atmosphere account for the remainder of the change in energy. 28. ^ Lide 2003, Chapter 6: Properties of Ice and Supercooled Water. 29. ^ Lide 2003, 6. Properties of Water and Steam as a Function of Temperature and Pressure. 30. ^ "Decree on weights and measures". April 7, 1795. Gramme, le poids absolu d'un volume d'eau pure égal au cube de la centième partie du mètre, et à la température de la glace fondante. 31. ^ a b c d Greenwood & Earnshaw 1997, p. 625. 32. ^ Shell, Scott M.; Debenedetti, Pablo G.; Panagiotopoulos, Athanassios Z. (2002). "Molecular structural order and anomalies in liquid silica" (PDF). Phys. Rev. E. 66 (1): 011202. arXiv:cond-mat/0203383. Bibcode:2002PhRvE..66a1202S. doi:10.1103/PhysRevE.66.011202. PMID 12241346. S2CID 6109212. Archived from the original (PDF) on 2016-06-04. Retrieved . 33. ^ Bullis, W. Murray (1990). "Chapter 6". In O'Mara, William C.; Herring, Robert B.; Hunt, Lee P. (eds.). Handbook of semiconductor silicon technology. Park Ridge, New Jersey: Noyes Publications. p. 431. ISBN 0-8155-1237-6. Retrieved . 34. ^ a b c d e Perlman, Howard. "Water Density". The USGS Water Science School. Retrieved . 35. ^ Loerting, Thomas; Salzmann, Christoph; Kohl, Ingrid; Mayer, Erwin; Hallbrucker, Andreas (2001-01-01). "A second distinct structural "state" of high-density amorphous ice at 77 K and 1 bar". Physical Chemistry Chemical Physics. 3 (24): 5355-5357. Bibcode:2001PCCP....3.5355L. doi:10.1039/b108676f. ISSN 1463-9084. 36. ^ Greenwood & Earnshaw 1997, p. 624. 37. ^ Zumdahl & Zumdahl 2013, p. 493. 38. ^ a b c "Can the ocean freeze?". National Ocean Service. National Oceanic and Atmospheric Administration. Retrieved . 39. ^ Fine, R.A.; Millero, F.J. (1973). "Compressibility of water as a function of temperature and pressure". Journal of Chemical Physics. 59 (10): 5529. Bibcode:1973JChPh..59.5529F. doi:10.1063/1.1679903. 40. ^ a b Nave, R. "Bulk Elastic Properties". HyperPhysics. Georgia State University. Retrieved . 41. ^ "Base unit definitions: Kelvin". National Institute of Standards and Technology. Retrieved 2018. 42. ^ a b Weingärtner et al. 2016, p. 5. 43. ^ Proceedings of the 106th meeting (PDF). International Committee for Weights and Measures. Sèvres. 16-20 October 2017. 44. ^ Schlüter, Oliver (2003-07-28). "Impact of High Pressure -- Low Temperature Processes on Cellular Materials Related to Foods" (PDF). Technischen Universität Berlin. Archived from the original (PDF) on 2008-03-09. Cite journal requires \|journal= (help) 45. ^ Tammann, Gustav H.J.A (1925). "The States Of Aggregation". Constable And Company. Cite journal requires \|journal= (help) 46. ^ Lewis & Rice 1922. 47. ^ Murphy, D. M. (2005). "Review of the vapour pressures of ice and supercooled water for atmospheric applications". Quarterly Journal of the Royal Meteorological Society. 131 (608): 1539-1565. Bibcode:2005QJRMS.131.1539M. doi:10.1256/qj.04.94. 48. ^ Debenedetti, P. G.; Stanley, H. E. (2003). "Supercooled and Glassy Water" (PDF). Physics Today. 56 (6): 40-46. Bibcode:2003PhT....56f..40D. doi:10.1063/1.1595053. 49. ^ Sharp 1988, p. 27. 50. ^ "Revised Release on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance" (PDF). IAPWS. September 2011. Retrieved . 51. ^ a b Light, Truman S.; Licht, Stuart; Bevilacqua, Anthony C.; Morash, Kenneth R. (2005-01-01). "The Fundamental Conductivity and Resistivity of Water". Electrochemical and Solid-State Letters. 8 (1): E16-E19. doi:10.1149/1.1836121. ISSN 1099-0062. 52. ^ Crofts, A. (1996). "Lecture 12: Proton Conduction, Stoichiometry". University of Illinois at Urbana-Champaign. Retrieved . 53. ^ Hoy, AR; Bunker, PR (1979). "A precise solution of the rotation bending Schrödinger equation for a triatomic molecule with application to the water molecule". Journal of Molecular Spectroscopy. 74 (1): 1-8. Bibcode:1979JMoSp..74....1H. doi:10.1016/0022-2852(79)90019-5. 54. ^ Zumdahl & Zumdahl 2013, p. 393. 55. ^ Campbell & Farrell 2007, pp. 37-38. 56. ^ Campbell & Reece 2009, p. 47. 57. ^ Chiavazzo, Eliodoro; Fasano, Matteo; Asinari, Pietro; Decuzzi, Paolo (2014). "Scaling behaviour for the water transport in nanoconfined geometries". Nature Communications. 5: 4565. Bibcode:2014NatCo...5.4565C. doi:10.1038/ncomms4565. PMC 3988813. PMID 24699509. 58. ^ "Physical Forces Organizing Biomolecules" (PDF). Biophysical Society. Archived from the original on August 7, 2007.CS1 maint: unfit url (link) 59. ^ Lide 2003, Surface Tension of Common Liquids. 60. ^ a b c Reece et al. 2013, p. 46. 61. ^ Zumdahl & Zumdahl 2013, pp. 458-459. 62. ^ Greenwood & Earnshaw 1997, p. 627. 63. ^ Zumdahl & Zumdahl 2013, p. 518. 64. ^ Pugliano, N. (1992-11-01). "Vibration-Rotation-Tunneling Dynamics in Small Water Clusters". Lawrence Berkeley Lab., CA (United States): 6. doi:10.2172/6642535. OSTI 6642535. Cite journal requires \|journal= (help) 65. ^ Richardson, Jeremy O.; Pérez, Cristóbal; Lobsiger, Simon; Reid, Adam A.; Temelso, Berhane; Shields, George C.; Kisiel, Zbigniew; Wales, David J.; Pate, Brooks H.; Althorpe, Stuart C. (2016-03-18). "Concerted hydrogen-bond breaking by quantum tunneling in the water hexamer prism". Science. 351 (6279): 1310-1313. Bibcode:2016Sci...351.1310R. doi:10.1126/science.aae0012. ISSN 0036-8075. PMID 26989250. 66. ^ Kolesnikov, Alexander I. (2016-04-22). "Quantum Tunneling of Water in Beryl: A New State of the Water Molecule". Physical Review Letters. 116 (16): 167802. Bibcode:2016PhRvL.116p7802K. doi:10.1103/PhysRevLett.116.167802. PMID 27152824. 67. ^ Pope; Fry (1996). "Absorption spectrum (380-700nm) of pure water. II. Integrating cavity measurements". Applied Optics. 36 (33): 8710-23. Bibcode:1997ApOpt..36.8710P. doi:10.1364/ao.36.008710. PMID 18264420. 68. ^ Ball, Philip (2008). "Water--an enduring mystery". Nature. 452 (7185): 291-292. Bibcode:2008Natur.452..291B. doi:10.1038/452291a. PMID 18354466. S2CID 4365814. 69. ^ Gonick, Larry; Criddle, Craig (2005-05-03). "Chapter 3 Togetherness". The cartoon guide to chemistry (1st ed.). HarperResource. p. 59. ISBN 9780060936778. Water, H2O, is similar. It has two electron pairs with nothing attached to them. They, too, must be taken into account. Molecules like NH3 and H2O are called bent. 70. ^ Theodore L. Brown; et al. (2015). "9.2 The Vsepr Model". Chemistry : the central science (13 ed.). p. 351. ISBN 978-0-321-91041-7. Retrieved 2019. Notice that the bond angles decrease as the number of nonbonding electron pairs increases. A bonding pair of electrons is attracted by both nuclei of the bonded atoms, but a nonbonding pair is attracted primarily by only one nucleus. Because a nonbonding pair experiences less nuclear attraction, its electron domain is spread out more in space than is the electron domain for a bonding pair (Figure 9.7). Nonbonding electron pairs therefore take up more space than bonding pairs; in essence, they act as large and fatter balloons in our analogy of Figure 9.5. As a result, electron domains for nonbonding electron pairs exert greater repulsive forces on adjacent electron domains and tend to compress bond angles 71. ^ Boyd 2000, p. 105. 72. ^ Boyd 2000, p. 106. 73. ^ "Guideline on the Use of Fundamental Physical Constants and Basic Constants of Water" (PDF). IAPWS. 2001. 74. ^ Hardy, Edme H.; Zygar, Astrid; Zeidler, Manfred D.; Holz, Manfred; Sacher, Frank D. (2001). "Isotope effect on the translational and rotational motion in liquid water and ammonia". J. Chem. Phys. 114 (7): 3174-3181. Bibcode:2001JChPh.114.3174H. doi:10.1063/1.1340584. 75. ^ Urey, Harold C.; et al. (15 Mar 1935). "Concerning the Taste of Heavy Water". Science. 81 (2098). New York: The Science Press. p. 273. Bibcode:1935Sci....81..273U. doi:10.1126/science.81.2098.273-a. 76. ^ "Experimenter Drinks 'Heavy Water' at $5,000 a Quart". Popular Science Monthly. 126 (4). New York: Popular Science Publishing. Apr 1935. p. 17. Retrieved 2011. 77. ^ Müller, Grover C. (June 1937). "Is 'Heavy Water' the Fountain of Youth?". Popular Science Monthly. 130 (6). New York: Popular Science Publishing. pp. 22-23. Retrieved 2011. 78. ^ Miller, Inglis J., Jr.; Mooser, Gregory (Jul 1979). "Taste Responses to Deuterium Oxide". Physiology & Behavior. 23 (1): 69-74. doi:10.1016/0031-9384(79)90124-0. PMID 515218. S2CID 39474797. 79. ^ Weingärtner et al. 2016, p. 29. 80. ^ Zumdahl & Zumdahl 2013, p. 659. 81. ^ a b Zumdahl & Zumdahl 2013, p. 654. 82. ^ Zumdahl & Zumdahl 2013, p. 984. 83. ^ Zumdahl & Zumdahl 2013, p. 171. 84. ^ "Hydrides". Chemwiki. UC Davis. Retrieved . 85. ^ Zumdahl & Zumdahl 2013, pp. 932, 936. 86. ^ Zumdahl & Zumdahl 2013, p. 338. 87. ^ Zumdahl & Zumdahl 2013, p. 862. 88. ^ Zumdahl & Zumdahl 2013, p. 981. 89. ^ Charlot 2007, p. 275. 90. ^ a b Zumdahl & Zumdahl 2013, p. 866. 91. ^ a b Greenwood & Earnshaw 1997, p. 601. 92. ^ "Enterprise and electrolysis..." Royal Society of Chemistry. August 2003. Retrieved . 93. ^ "Joseph Louis Gay-Lussac, French chemist (1778-1850)". 1902 Encyclopedia. Footnote 122-1. Retrieved . 94. ^ Lewis, G. N.; MacDonald, R. T. (1933). "Concentration of H2 Isotope". The Journal of Chemical Physics. 1 (6): 341. Bibcode:1933JChPh...1..341L. doi:10.1063/1.1749300. 95. ^ a b Leigh, Favre & Metanomski 1998, p. 34. 96. ^ IUPAC 2005, p. 85. 97. ^ Leigh, Favre & Metanomski 1998, p. 99. 98. ^ "Tetrahydropyran". Pubchem. National Institutes of Health. Retrieved . 99. ^ Leigh, Favre & Metanomski 1998, pp. 27-28. 100. ^ "Compound Summary for CID 22247451". Pubchem Compound Database. National Center for Biotechnology Information. Further reading External links Music Scenes
a18fdf74bcd9287d	Stanford Encyclopedia of Philosophy The Uncertainty Principle Quantum mechanics is generally regarded as the physical theory that is our best candidate for a fundamental and universal description of the physical world. The conceptual framework employed by this theory differs drastically from that of classical physics. Indeed, the transition from classical to quantum physics marks a genuine revolution in our understanding of the physical world. One striking aspect of the difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, the prime example being the position and momentum of a particle. According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum. The uncertainty principle played an important role in many discussions on the philosophical implications of quantum mechanics, in particular in discussions on the consistency of the so-called Copenhagen interpretation, the interpretation endorsed by the founding fathers Heisenberg and Bohr. This should not suggest that the uncertainty principle is the only aspect of the conceptual difference between classical and quantum physics: the implications of quantum mechanics for notions as (non)-locality, entanglement and identity play no less havoc with classical intuitions. 1. Introduction The uncertainty principle is certainly one of the most famous and important aspects of quantum mechanics. It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world. Roughly speaking, the uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system. Rather, these quantities can only be determined with some characteristic ‘uncertainties’ that cannot become arbitrarily small simultaneously. But what is the exact meaning of this principle, and indeed, is it really a principle of quantum mechanics? (In his original work, Heisenberg only speaks of uncertainty relations.) And, in particular, what does it mean to say that a quantity is determined only up to some uncertainty? These are the main questions we will explore in the following, focusssing on the views of Heisenberg and Bohr. The notion of ‘uncertainty’ occurs in several different meanings in the physical literature. It may refer to a lack of knowledge of a quantity by an observer, or to the experimental inaccuracy with which a quantity is measured, or to some ambiguity in the definition of a quantity, or to a statistical spread in an ensemble of similary prepared systems. Also, several different names are used for such uncertainties: inaccuracy, spread, imprecision, indefiniteness, indeterminateness, indeterminacy, latitude, etc. As we shall see, even Heisenberg and Bohr did not decide on a single terminology for quantum mechanical uncertainties. Forestalling a discussion about which name is the most appropriate one in quantum mechanics, we use the name ‘uncertainty principle’ imply because it is the most common one in the literature. 2. Heisenberg 2.1 Heisenberg's road to the uncertainty relations Heisenberg introduced his now famous relations in an article of 1927, entitled "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". A (partial) translation of this title is: "On the anschaulich content of quantum theoretical kinematics and mechanics". Here, the term anschaulich is particularly notable. Apparently, it is one of those German words that defy an unambiguous translation into other languages. Heisenberg's title is translated as "On the physical content …" by Wheeler and Zurek (1983). His collected works (Heisenberg, 1984) translate it as "On the perceptible content …", while Cassidy's biography of Heisenberg (Cassidy, 1992), refers to the paper as "On the perceptual content …". Literally, the closest translation of the term anschaulich is ‘visualizable’. But, as in most languages, words that make reference to vision are not always intended literally. Seeing is widely used as a metaphor for understanding, especially for immediate understanding. Hence, anschaulich also means ‘intelligible’ or ‘intuitive’.[1] Why was this issue of the Anschaulichkeit of quantum mechanics such a prominent concern to Heisenberg? This question has already been considered by a number of commentators (Jammer, 1977; Miller 1982; de Regt, 1997; Beller, 1999). For the answer, it turns out, we must go back a little in time. In 1925 Heisenberg had developed the first coherent mathematical formalism for quantum theory (Heisenberg, 1925). His leading idea was that only those quantities that are in principle observable should play a role in the theory, and that all attempts to form a picture of what goes on inside the atom should be avoided. In atomic physics the observational data were obtained from spectroscopy and associated with atomic transitions. Thus, Heisenberg was led to consider the ‘transition quantities’ as the basic ingredients of the theory. Max Born, later that year, realized that the transition quantities obeyed the rules of matrix calculus, a branch of mathematics that was not so well-known then as it is now. In a famous series of papers Heisenberg, Born and Jordan developed this idea into the matrix mechanics version of quantum theory. Formally, matrix mechanics remains close to classical mechanics. The central idea is that all physical quantities must be represented by infinite self-adjoint matrices (later identified with operators on a Hilbert space). It is postulated that the matrices q and p representing the canonical position and momentum variables of a particle satisfy the so-called canonical commutation rule qppq = ihbar (1) where hbar = h/2π, h denotes Planck's constant, and boldface type is used to represent matrices. The new theory scored spectacular empirical success by encompassing nearly all spectroscopic data known at the time, especially after the concept of the electron spin was included in the theoretical framework. It came as a big surprise, therefore, when one year later, Erwin Schrödinger presented an alternative theory, that became known as wave mechanics. Schrödinger assumed that an electron in an atom could be represented as an oscillating charge cloud, evolving continuously in space and time according to a wave equation. The discrete frequencies in the atomic spectra were not due to discontinuous transitions (quantum jumps) as in matrix mechanics, but to a resonance phenomenon. Schrödinger also showed that the two theories were equivalent.[2] Even so, the two approaches differed greatly in interpretation and spirit. Whereas Heisenberg eschewed the use of visualizable pictures, and accepted discontinuous transitions as a primitive notion, Schrödinger claimed as an advantage of his theory that it was anschaulich. In Schrödinger's vocabulary, this meant that the theory represented the observational data by means of continuously evolving causal processes in space and time. He considered this condition of Anschaulichkeit to be an essential requirement on any acceptable physical theory. Schrödinger was not alone in appreciating this aspect of his theory. Many other leading physicists were attracted to wave mechanics for the same reason. For a while, in 1926, before it emerged that wave mechanics had serious problems of its own, Schrödinger's approach seemed to gather more support in the physics community than matrix mechanics. Understandably, Heisenberg was unhappy about this development. In a letter of 8 June 1926 to Pauli he confessed that "The more I think about the physical part of Schrödinger's theory, the more disgusting I find it", and: "What Schrödinger writes about the Anschaulichkeit of his theory, … I consider Mist (Pauli, 1979, p. 328)". Again, this last German term is translated differently by various commentators: as "junk" (Miller, 1982) "rubbish" (Beller 1999) "crap" (Cassidy, 1992), and perhaps more literally, as "bullshit" (de Regt, 1997). Nevertheless, in published writings, Heisenberg voiced a more balanced opinion. In a paper in Die Naturwissenschaften (1926) he summarized the peculiar situation that the simultaneous development of two competing theories had brought about. Although he argued that Schrödinger's interpretation was untenable, he admitted that matrix mechanics did not provide the Anschaulichkeit which made wave mechanics so attractive. He concluded: "to obtain a contradiction-free anschaulich interpretation, we still lack some essential feature in our image of the structure of matter". The purpose of his 1927 paper was to provide exactly this lacking feature. 2.2 Heisenberg's argument Let us now look at the argument that led Heisenberg to his uncertainty relations. He started by redefining the notion of Anschaulichkeit. Whereas Schrödinger associated this term with the provision of a causal space-time picture of the phenomena, Heisenberg, by contrast, declared: We believe we have gained anschaulich understanding of a physical theory, if in all simple cases, we can grasp the experimental consequences qualitatively and see that the theory does not lead to any contradictions. Heisenberg, 1927, p. 172) His goal was, of course, to show that, in this new sense of the word, matrix mechanics could lay the same claim to Anschaulichkeit as wave mechanics. To do this, he adopted an operational assumption: terms like ‘the position of a particle’ have meaning only if one specifies a suitable experiment by which ‘the position of a particle’ can be measured. We will call this assumption the ‘measurement=meaning principle’. In general, there is no lack of such experiments, even in the domain of atomic physics. However, experiments are never completely accurate. We should be prepared to accept, therefore, that in general the meaning of these quantities is also determined only up to some characteristic inaccuracy. As an example, he considered the measurement of the position of an electron by a microscope. The accuracy of such a measurement is limited by the wave length of the light illuminating the electron. Thus, it is possible, in principle, to make such a position measurement as accurate as one wishes, by using light of a very short wave length, e.g., γ-rays. But for γ-rays, the Compton effect cannot be ignored: the interaction of the electron and the illuminating light should then be considered as a collision of at least one photon with the electron. In such a collision, the electron suffers a recoil which disturbs its momentum. Moreover, the shorter the wave length, the larger is this change in momentum. Thus, at the moment when the position of the particle is accurately known, Heisenberg argued, its momentum cannot be accurately known: At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum. This change is the greater the smaller the wavelength of the light employed, i.e., the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum therefore can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely (Heisenberg, 1927, p. 174-5). This is the first formulation of the uncertainty principle. In its present form it is an epistemological principle, since it limits what we can know about the electron. From "elementary formulae of the Compton effect" Heisenberg estimated the ‘imprecisions’ to be of the order δpδqh (2) He continued: "In this circumstance we see the direct anschaulich content of the relation qp − pq = ihbar". He went on to consider other experiments, designed to measure other physical quantities and obtained analogous relations for time and energy: δt δEh (3) and action J and angle w δw δJh (4) which he saw as corresponding to the "well-known" relations tEEt = ihbar or wJ − Jw = ihbar (5) However, these generalisations are not as straightforward as Heisenberg suggested. In particular, the status of the time variable in his several illustrations of relation (3) is not at all clear (Hilgevoord 2005). See also on Section 2.5. Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in the realm of atomic processes. But, as a pure fact of experience ("rein erfahrungsgemäß"), experiments that serve to provide such a definition for one quantity are subject to particular indeterminacies, obeying relations (2)-(4) which prohibit them from providing a simultaneous definition of two canonically conjugate quantities. Note that in this formulation the emphasis has slightly shifted: he now speaks of a limit on the definition of concepts, i.e. not merely on what we can know, but what we can meaningfully say about a particle. Of course, this stronger formulation follows by application of the above measurement=meaning principle: if there are, as Heisenberg claims, no experiments that allow a simultaneous precise measurement of two conjugate quantities, then these quantities are also not simultaneously well-defined. Heisenberg's paper has an interesting "Addition in proof" mentioning critical remarks by Bohr, who saw the paper only after it had been sent to the publisher. Among other things, Bohr pointed out that in the microscope experiment it is not the change of the momentum of the electron that is important, but rather the circumstance that this change cannot be precisely determined in the same experiment. An improved version of the argument, responding to this objection, is given in Heisenberg's Chicago lectures of 1930. Here (Heisenberg, 1930, p. 16), it is assumed that the electron is illuminated by light of wavelength λ and that the scattered light enters a microscope with aperture angle ε. According to the laws of classical optics, the accuracy of the microscope depends on both the wave length and the aperture angle; Abbe's criterium for its ‘resolving power’, i.e. the size of the smallest discernable details, gives δq ∼ λ/sin ε (6) On the other hand, the direction of a scattered photon, when it enters the microscope, is unknown within the angle ε, rendering the momentum change of the electron uncertain by an amount δph sin ε/λ (7) leading again to the result (2). Let us now analyse Heisenberg's argument in more detail. First note that, even in this improved version, Heisenberg's argument is incomplete. According to Heisenberg's ‘measurement=meaning principle’, one must also specify, in the given context, what the meaning is of the phrase ‘momentum of the electron’, in order to make sense of the claim that this momentum is changed by the position measurement. A solution to this problem can again be found in the Chicago lectures (Heisenberg, 1930, p. 15). Here, he assumes that initially the momentum of the electron is precisely known, e.g. it has been measured in a previous experiment with an inaccuracy δpi, which may be arbitrarily small. Then, its position is measured with inaccuracy δq, and after this, its final momentum is measured with an inaccuracy δpf. All three measurements can be performed with arbitrary precision. Thus, the three quantities δpi, δq, and δpf can be made as small as one wishes. If we assume further that the initial momentum has not changed until the position measurement, we can speak of a definite momentum until the time of the position measurement. Moreover we can give operational meaning to the idea that the momentum is changed during the position measurement: the outcome of the second momentum measurement (say pf) will generally differ from the initial value pi. In fact, one can also show that this change is discontinuous, by varying the time between the three measurements. Let us now try to see, adopting this more elaborate set-up, if we can complete Heisenberg's argument. We have now been able to give empirical meaning to the ‘change of momentum’ of the electron, pf − pi. Heisenberg's argument claims that the order of magnitude of this change is at least inversely proportional to the inaccuracy of the position measurement: \| pfpi \| δqh (8) However, can we now draw the conclusion that the momentum is only imprecisely defined? Certainly not. Before the position measurement, its value was pi, after the measurement it is pf. One might, perhaps, claim that the value at the very instant of the position measurement is not yet defined, but we could simply settle this by an assignment by convention, e.g., we might assign the mean value (pi + pf)/2 to the momentum at this instant. But then, the momentum is precisely determined at all instants, and Heisenberg's formulation of the uncertainty principle no longer follows. The above attempt of completing Heisenberg's argument thus overshoots its mark. A solution to this problem can again be found in the Chicago Lectures. Heisenberg admits that position and momentum can be known exactly. He writes: If the velocity of the electron is at first known, and the position then exactly measured, the position of the electron for times previous to the position measurement may be calculated. For these past times, δpδq is smaller than the usual bound. (Heisenberg 1930, p. 15) Indeed, Heisenberg says: "the uncertainty relation does not hold for the past". Apparently, when Heisenberg refers to the uncertainty or imprecision of a quantity, he means that the value of this quantity cannot be given beforehand. In the sequence of measurements we have considered above, the uncertainty in the momentum after the measurement of position has occurred, refers to the idea that the value of the momentum is not fixed just before the final momentum measurement takes place. Once this measurement is performed, and reveals a value pf, the uncertainty relation no longer holds; these values then belong to the past. Clearly, then, Heisenberg is concerned with unpredictability: the point is not that the momentum of a particle changes, due to a position measurement, but rather that it changes by an unpredictable amount. It is, however always possible to measure, and hence define, the size of this change in a subsequent measurement of the final momentum with arbitrary precision. Although Heisenberg admits that we can consistently attribute values of momentum and position to an electron in the past, he sees little merit in such talk. He points out that these values can never be used as initial conditions in a prediction about the future behavior of the electron, or subjected to experimental verification. Whether or not we grant them physical reality is, as he puts it, a matter of personal taste. Heisenberg's own taste is, of course, to deny their physical reality. For example, he writes, "I believe that one can formulate the emergence of the classical ‘path’ of a particle pregnantly as follows: the ‘path’ comes into being only because we observe it" (Heisenberg, 1927, p. 185). Apparently, in his view, a measurement does not only serve to give meaning to a quantity, it creates a particular value for this quantity. This may be called the ‘measurement=creation’ principle. It is an ontological principle, for it states what is physically real. This then leads to the following picture. First we measure the momentum of the electron very accurately. By ‘measurement= meaning’, this entails that the term "the momentum of the particle" is now well-defined. Moreover, by the ‘measurement=creation’ principle, we may say that this momentum is physically real. Next, the position is measured with inaccuracy δq. At this instant, the position of the particle becomes well-defined and, again, one can regard this as a physically real attribute of the particle. However, the momentum has now changed by an amount that is unpredictable by an order of magnitude \| pf − pi \| ∼ hq. The meaning and validity of this claim can be verified by a subsequent momentum measurement. The question is then what status we shall assign to the momentum of the electron just before its final measurement. Is it real? According to Heisenberg it is not. Before the final measurement, the best we can attribute to the electron is some unsharp, or fuzzy momentum. These terms are meant here in an ontological sense, characterizing a real attribute of the electron. 2.3 The interpretation of Heisenberg's relation The relations Heisenberg had proposed were soon considered to be a cornerstone of the Copenhagen interpretation of quantum mechanics. Just a few months later, Kennard (1927) already called them the "essential core" of the new theory. Taken together with Heisenberg's contention that they provided the intuitive content of the theory and their prominent role in later discussions on the Copenhagen interpretation, a dominant view emerged in which the uncertainty relations were regarded as a fundamental principle of the theory. The interpretation of these relations has often been debated. Do Heisenberg's relations express restrictions on the experiments we can perform on quantum systems, and, therefore, restrictions on the information we can gather about such systems; or do they express restrictions on the meaning of the concepts we use to describe quantum systems? Or else, are they restrictions of an ontological nature, i.e., do they assert that a quantum system simply does not possess a definite value for its position and momentum at the same time? The difference between these interpretations is partly reflected in the various names by which the relations are known, e.g. as ‘inaccuracy relations’, or: ‘uncertainty’, ‘indeterminacy’ or ‘unsharpness relations’. The debate between these different views has been addressed by many authors, but it has never been settled completely. Let it suffice here to make only two general observations. First, it is clear that in Heisenberg's own view all the above questions stand or fall together. Indeed, we have seen that he adopted an operational "measurement=meaning" principle according to which the meaningfulness of a physical quantity was equivalent to the existence of an experiment purporting to measure that quantity. Similarly, his "measurement=creation" principle allowed him to attribute physical reality to such quantities. Hence, Heisenberg's discussions moved rather freely and quickly from talk about experimental inaccuracies to epistemological or ontological issues and back again. However, ontological questions seemed to be of somewhat less interest to him. For example, there is a passage (Heisenberg, 1927, p. 197), where he discusses the idea that, behind our observational data, there might still exist a hidden reality in which quantum systems have definite values for position and momentum, unaffected by the uncertainty relations. He emphatically dismisses this conception as an unfruitful and meaningless speculation, because, as he says, the aim of physics is only to describe observable data. Similarly, in the Chicago Lectures (Heisenberg 1930, p. 11), he warns against the fact that the human language permits the utterance of statements which have no empirical content at all, but nevertheless produce a picture in our imagination. He notes, "One should be especially careful in using the words ‘reality’, ‘actually’, etc., since these words very often lead to statements of the type just mentioned." So, Heisenberg also endorsed an interpretation of his relations as rejecting a reality in which particles have simultaneous definite values for position and momentum. The second observation is that although for Heisenberg experimental, informational, epistemological and ontological formulations of his relations were, so to say, just different sides of the same coin, this is not so for those who do not share his operational principles or his view on the task of physics. Alternative points of view, in which e.g. the ontological reading of the uncertainty relations is denied, are therefore still viable. The statement, often found in the literature of the thirties, that Heisenberg had proved the impossibility of associating a definite position and momentum to a particle is certainly wrong. But the precise meaning one can coherently attach to Heisenberg's relations depends rather heavily on the interpretation one favors for quantum mechanics as a whole. And because no agreement has been reached on this latter issue, one cannot expect agreement on the meaning of the uncertainty relations either. 2.4 Uncertainty relations or uncertainty principle? Let us now move to another question about Heisenberg's relations: do they express a principle of quantum theory? Probably the first influential author to call these relations a ‘principle’ was Eddington, who, in his Gifford Lectures of 1928 referred to them as the ‘Principle of Indeterminacy’. In the English literature the name uncertainty principle became most common. It is used both by Condon and Robertson in 1929, and also in the English version of Heisenberg's Chicago Lectures (Heisenberg, 1930), although, remarkably, nowhere in the original German version of the same book (see also Cassidy, 1998). Indeed, Heisenberg never seems to have endorsed the name ‘principle’ for his relations. His favourite terminology was ‘inaccuracy relations’ (Ungenauigkeitsrelationen) or ‘indeterminacy relations’ (Unbestimmtheitsrelationen). We know only one passage, in Heisenberg's own Gifford lectures, delivered in 1955-56 (Heisenberg, 1958, p. 43), where he mentioned that his relations "are usually called relations of uncertainty or principle of indeterminacy". But this can well be read as his yielding to common practice rather than his own preference. But does the relation (2) qualify as a principle of quantum mechanics? Several authors, foremost Karl Popper (1967), have contested this view. Popper argued that the uncertainty relations cannot be granted the status of a principle on the grounds that they are derivable from the theory, whereas one cannot obtain the theory from the uncertainty relations. (The argument being that one can never derive any equation, say, the Schrödinger equation, or the commutation relation (1), from an inequality.) Popper's argument is, of course, correct but we think it misses the point. There are many statements in physical theories which are called principles even though they are in fact derivable from other statements in the theory in question. A more appropriate departing point for this issue is not the question of logical priority but rather Einstein's distinction between ‘constructive theories’ and ‘principle theories’. Einstein proposed this famous classification in (Einstein, 1919). Constructive theories are theories which postulate the existence of simple entities behind the phenomena. They endeavour to reconstruct the phenomena by framing hypotheses about these entities. Principle theories, on the other hand, start from empirical principles, i.e. general statements of empirical regularities, employing no or only a bare minimum of theoretical terms. The purpose is to build up the theory from such principles. That is, one aims to show how these empirical principles provide sufficient conditions for the introduction of further theoretical concepts and structure. The prime example of a theory of principle is thermodynamics. Here the role of the empirical principles is played by the statements of the impossibility of various kinds of perpetual motion machines. These are regarded as expressions of brute empirical fact, providing the appropriate conditions for the introduction of the concepts of energy and entropy and their properties. (There is a lot to be said about the tenability of this view, but that is not the topic of this entry.) Now obviously, once the formal thermodynamic theory is built, one can also derive the impossibility of the various kinds of perpetual motion. (They would violate the laws of energy conservation and entropy increase.) But this derivation should not misguide one into thinking that they were no principles of the theory after all. The point is just that empirical principles are statements that do not rely on the theoretical concepts (in this case entropy and energy) for their meaning. They are interpretable independently of these concepts and, further, their validity on the empirical level still provides the physical content of the theory. A similar example is provided by special relativity, another theory of principle, which Einstein deliberately designed after the ideal of thermodynamics. Here, the empirical principles are the light postulate and the relativity principle. Again, once we have built up the modern theoretical formalism of the theory (the Minkowski space-time) it is straightforward to prove the validity of these principles. But again this does not count as an argument for claiming that they were no principles after all. So the question whether the term ‘principle’ is justified for Heisenberg's relations, should, in our view, be understood as the question whether they are conceived of as empirical principles. One can easily show that this idea was never far from Heisenberg's intentions. We have already seen that Heisenberg presented the relations as the result of a "pure fact of experience". A few months after his 1927 paper, he wrote a popular paper with the title "Ueber die Grundprincipien der Quantenmechanik" ("On the fundamental principles of quantum mechanics") where he made the point even more clearly. Here Heisenberg described his recent break-through in the interpretation of the theory as follows: "It seems to be a general law of nature that we cannot determine position and velocity simultaneously with arbitrary accuracy". Now actually, and in spite of its title, the paper does not identify or discuss any ‘fundamental principle’ of quantum mechanics. So, it must have seemed obvious to his readers that he intended to claim that the uncertainty relation was a fundamental principle, forced upon us as an empirical law of nature, rather than a result derived from the formalism of the theory. This reading of Heisenberg's intentions is corroborated by the fact that, even in his 1927 paper, applications of his relation frequently present the conclusion as a matter of principle. For example, he says "In a stationary state of an atom its phase is in principle indeterminate" (Heisenberg, 1927, p. 177, [emphasis added]). Similarly, in a paper of 1928, he described the content of his relations as: "It has turned out that it is in principle impossible to know, to measure the position and velocity of a piece of matter with arbitrary accuracy. (Heisenberg, 1984, p. 26, [emphasis added])" So, although Heisenberg did not originate the tradition of calling his relations a principle, it is not implausible to attribute the view to him that the uncertainty relations represent an empirical principle that could serve as a foundation of quantum mechanics. In fact, his 1927 paper expressed this desire explicitly: "Surely, one would like to be able to deduce the quantitative laws of quantum mechanics directly from their anschaulich foundations, that is, essentially, relation [(2)]" (ibid, p. 196). This is not to say that Heisenberg was successful in reaching this goal, or that he did not express other opinions on other occasions. Let us conclude this section with three remarks. First, if the uncertainty relation is to serve as an empirical principle, one might well ask what its direct empirical support is. In Heisenberg's analysis, no such support is mentioned. His arguments concerned thought experiments in which the validity of the theory, at least at a rudimentary level, is implicitly taken for granted. Jammer (1974, p. 82) conducted a literature search for high precision experiments that could seriously test the uncertainty relations and concluded they were still scarce in 1974. Real experimental support for the uncertainty relations in experiments in which the inaccuracies are close to the quantum limit have come about only more recently. (See Kaiser, Werner and George 1983, Uffink 1985, Nairz, Andt, and Zeilinger, 2001.) Third, it is remarkable that in his later years Heisenberg put a somewhat different gloss on his relations. In his autobiography Der Teil und das Ganze of 1969 he described how he had found his relations inspired by a remark by Einstein that "it is the theory which decides what one can observe" -- thus giving precedence to theory above experience, rather than the other way around. Some years later he even admitted that his famous discussions of thought experiments were actually trivial since "… if the process of observation itself is subject to the laws of quantum theory, it must be possible to represent its result in the mathematical scheme of this theory" (Heisenberg, 1975, p. 6). 2.5 Mathematical elaboration When Heisenberg introduced his relation, his argument was based only on qualitative examples. He did not provide a general, exact derivation of his relations.[3] Indeed, he did not even give a definition of the uncertainties δq, etc., occurring in these relations. Of course, this was consistent with the announced goal of that paper, i.e. to provide some qualitative understanding of quantum mechanics for simple experiments. The first mathematically exact formulation of the uncertainty relations is due to Kennard. He proved in 1927 the theorem that for all normalized state vectors \|ψ> the following inequality holds: Δψp Δψqhbar/2 (9) Here, Δψp and Δψq are standard deviations of position and momentum in the state vector \|ψ>, i.e., ψp)² = <p²>ψ − (<p>ψ)², (Δψq)² = <q²>ψ − (<q>ψ)². (10) where <·>ψ = <ψ\|·\|ψ> denotes the expectation value in state \|ψ>. The inequality (9) was generalized in 1929 by Robertson who proved that for all observables (self-adjoint operators) A and B ΔψA ΔψB ≥ ½\|<[A,B]> ψ\| (11) where [A, B] := AB − BA denotes the commutator. This relation was in turn strengthened by Schrödinger (1930), who obtained: ψA)² (ΔψB)² ≥ ¼\|<[A,B]> ψ\|² + ¼\|<{A−<A> ψ, B−<B> ψ}>ψ (12) where {A, B} := (AB + BA) denotes the anti-commutator. Since the above inequalities have the virtue of being exact and general, in contrast to Heisenberg's original semi-quantitative formulation, it is tempting to regard them as the exact counterpart of Heisenberg's relations (2)-(4). Indeed, such was Heisenberg's own view. In his Chicago Lectures (Heisenberg 1930, pp. 15-19), he presented Kennard's derivation of relation (9) and claimed that "this proof does not differ at all in mathematical content" from the semi-quantitative argument he had presented earlier, the only difference being that now "the proof is carried through exactly". But it may be useful to point out that both in status and intended role there is a difference between Kennard's inequality and Heisenberg's previous formulation (2). The inequalities discussed in the present section are not statements of empirical fact, but theorems of the quantum mechanical formalism. As such, they presuppose the validity of this formalism, and in particular the commutation relation (1), rather than elucidating its intuitive content or to create ‘room’ or ‘freedom’ for the validity of this relation. At best, one should see the above inequalities as showing that the formalism is consistent with Heisenberg's empirical principle. This situation is similar to that arising in other theories of principle where, as noted in Section 2.4, one often finds that, next to an empirical principle, the formalism also provides a corresponding theorem. And similarly, this situation should not, by itself, cast doubt on the question whether Heisenberg's relation can be regarded as a principle of quantum mechanics. There is a second notable difference between (2) and (9). Heisenberg did not give a general definition for the ‘uncertainties’ δp and δq. The most definite remark he made about them was that they could be taken as "something like the mean error". In the discussions of thought experiments, he and Bohr would always quantify uncertainties on a case-to-case basis by choosing some parameters which happened to be relevant to the experiment at hand. By contrast, the inequalities (9)-(12) employ a single specific expression as a measure for ‘uncertainty’: the standard deviation. At the time, this choice was not unnatural, given that this expression is well-known and widely used in error theory and the description of statistical fluctuations. However, there was very little or no discussion of whether this choice was appropriate for a general formulation of the uncertainty relations. A standard deviation reflects the spread or expected fluctuations in a series of measurements of an observable in a given state. It is not at all easy to connect this idea with the concept of the ‘inaccuracy’ of a measurement, such as the resolving power of a microscope. In fact, even though Heisenberg had taken Kennard's inequality as the precise formulation of the uncertainty relation, he and Bohr never relied on standard deviations in their many discussions of thought experiments, and indeed, it has been shown (Uffink and Hilgevoord, 1985; Hilgevoord and Uffink, 1988) that these discussions cannot be framed in terms of standard deviation. Another problem with the above elaboration is that the ‘well-known’ relations (5) are actually false if energy E and action J are to be positive operators (Jordan 1927). In that case, self-adjoint operators t and w do not exist and inequalities analogous to (9) cannot be derived. Also, these inequalities do not hold for angle and angular momentum (Uffink 1990). These obstacles have led to a quite extensive literature on time-energy and angle-action uncertainty relations (Muga et al. 2002, Hilgevoord 2005). 3. Bohr In spite of the fact that Heisenberg's and Bohr's views on quantum mechanics are often lumped together as (part of) ‘the Copenhagen interpretation’, there is considerable difference between their views on the uncertainty relations. 3.1 From wave-particle duality to complementarity Long before the development of modern quantum mechanics, Bohr had been particularly concerned with the problem of particle-wave duality, i.e. the problem that experimental evidence on the behaviour of both light and matter seemed to demand a wave picture in some cases, and a particle picture in others. Yet these pictures are mutually exclusive. Whereas a particle is always localized, the very definition of the notions of wavelength and frequency requires an extension in space and in time. Moreover, the classical particle picture is incompatible with the characteristic phenomenon of interference. His long struggle with wave-particle duality had prepared him for a radical step when the dispute between matrix and wave mechanics broke out in 1926-27. For the main contestants, Heisenberg and Schrödinger, the issue at stake was which view could claim to provide a single coherent and universal framework for the description of the observational data. The choice was, essentially between a description in terms of continuously evolving waves, or else one of particles undergoing discontinuous quantum jumps. By contrast, Bohr insisted that elements from both views were equally valid and equally needed for an exhaustive description of the data. His way out of the contradiction was to renounce the idea that the pictures refer, in a literal one-to-one correspondence, to physical reality. Instead, the applicability of these pictures was to become dependent on the experimental context. This is the gist of the viewpoint he called ‘complementarity’. Bohr first conceived the general outline of his complementarity argument in early 1927, during a skiing holiday in Norway, at the same time when Heisenberg wrote his uncertainty paper. When he returned to Copenhagen and found Heisenberg's manuscript, they got into an intense discussion. On the one hand, Bohr was quite enthusiastic about Heisenberg's ideas which seemed to fit wonderfully with his own thinking. Indeed, in his subsequent work, Bohr always presented the uncertainty relations as the symbolic expression of his complementarity viewpoint. On the other hand, he criticized Heisenberg severely for his suggestion that these relations were due to discontinuous changes occurring during a measurement process. Rather, Bohr argued, their proper derivation should start from the indispensability of both particle and wave concepts. He pointed out that the uncertainties in the experiment did not exclusively arise from the discontinuities but also from the fact that in the experiment we need to take into account both the particle theory and the wave theory. It is not so much the unknown disturbance which renders the momentum of the electron uncertain but rather the fact that the position and the momentum of the electron cannot be simultaneously defined in this experiment. (See the "Addition in Proof" to Heisenberg's paper.) We shall not go too deeply into the matter of Bohr's interpretation of quantum mechanics since we are mostly interested in Bohr's view on the uncertainty principle. For a more detailed discussion of Bohr's philosophy of quantum physics we refer to Scheibe (1973), Folse (1985), Honner (1987) and Murdoch (1987). It may be useful, however, to sketch some of the main points. Central in Bohr's considerations is the language we use in physics. No matter how abstract and subtle the concepts of modern physics may be, they are essentially an extension of our ordinary language and a means to communicate the results of our experiments. These results, obtained under well-defined experimental circumstances, are what Bohr calls the "phenomena". A phenomenon is "the comprehension of the effects observed under given experimental conditions" (Bohr 1939, p. 24), it is the resultant of a physical object, a measuring apparatus and the interaction between them in a concrete experimental situation. The essential difference between classical and quantum physics is that in quantum physics the interaction between the object and the apparatus cannot be made arbitrarily small; the interaction must at least comprise one quantum. This is expressed by Bohr's quantum postulate: [… the] essence [of the formulation of the quantum theory] may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity or rather individuality, completely foreign to classical theories and symbolized by Planck's quantum of action. (Bohr, 1928, p. 580) A phenomenon, therefore, is an indivisible whole and the result of a measurement cannot be considered as an autonomous manifestation of the object itself independently of the measurement context. The quantum postulate forces upon us a new way of describing physical phenomena: In this situation, we are faced with the necessity of a radical revision of the foundation for the description and explanation of physical phenomena. Here, it must above all be recognized that, however far quantum effects transcend the scope of classical physical analysis, the account of the experimental arrangement and the record of the observations must always be expressed in common language supplemented with the terminology of classical physics. (Bohr, 1948, p. 313) This is what Scheibe (1973) has called the "buffer postulate" because it prevents the quantum from penetrating into the classical description: A phenomenon must always be described in classical terms; Planck's constant does not occur in this description. Together, the two postulates induce the following reasoning. In every phenomenon the interaction between the object and the apparatus comprises at least one quantum. But the description of the phenomenon must use classical notions in which the quantum of action does not occur. Hence, the interaction cannot be analysed in this description. On the other hand, the classical character of the description allows to speak in terms of the object itself. Instead of saying: ‘the interaction between a particle and a photographic plate has resulted in a black spot in a certain place on the plate’, we are allowed to forgo mentioning the apparatus and say: ‘the particle has been found in this place’. The experimental context, rather than changing or disturbing pre-existing properties of the object, defines what can meaningfully be said about the object. Because the interaction between object and apparatus is left out in our description of the phenomenon, we do not get the whole picture. Yet, any attempt to extend our description by performing the measurement of a different observable quantity of the object, or indeed, on the measurement apparatus, produces a new phenomenon and we are again confronted with the same situation. Because of the unanalyzable interaction in both measurements, the two descriptions cannot, generally, be united into a single picture. They are what Bohr calls complementary descriptions: [the quantum of action]...forces us to adopt a new mode of description designated as complementary in the sense that any given application of classical concepts precludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena. (Bohr, 1929, p. 10) The most important example of complementary descriptions is provided by the measurements of the position and momentum of an object. If one wants to measure the position of the object relative to a given spatial frame of reference, the measuring instrument must be rigidly fixed to the bodies which define the frame of reference. But this implies the impossibility of investigating the exchange of momentum between the object and the instrument and we are cut off from obtaining any information about the momentum of the object. If, on the other hand, one wants to measure the momentum of an object the measuring instrument must be able to move relative to the spatial reference frame. Bohr here assumes that a momentum measurement involves the registration of the recoil of some movable part of the instrument and the use of the law of momentum conservation. The looseness of the part of the instrument with which the object interacts entails that the instrument cannot serve to accurately determine the position of the object. Since a measuring instrument cannot be rigidly fixed to the spatial reference frame and, at the same time, be movable relative to it, the experiments which serve to precisely determine the position and the momentum of an object are mutually exclusive. Of course, in itself, this is not at all typical for quantum mechanics. But, because the interaction between object and instrument during the measurement can neither be neglected nor determined the two measurements cannot be combined. This means that in the description of the object one must choose between the assignment of a precise position or of a precise momentum. Similar considerations hold with respect to the measurement of time and energy. Just as the spatial coordinate system must be fixed by means of solid bodies so must the time coordinate be fixed by means of unperturbable, synchronised clocks. But it is precisely this requirement which prevents one from taking into account of the exchange of energy with the instrument if this is to serve its purpose. Conversely, any conclusion about the object based on the conservation of energy prevents following its development in time. The conclusion is that in quantum mechanics we are confronted with a complementarity between two descriptions which are united in the classical mode of description: the space-time description (or coordination) of a process and the description based on the applicability of the dynamical conservation laws. The quantum forces us to give up the classical mode of description (also called the ‘causal’ mode of description by Bohr[4]): it is impossible to form a classical picture of what is going on when radiation interacts with matter as, e.g., in the Compton effect. Any arrangement suited to study the exchange of energy and momentum between the electron and the photon must involve a latitude in the space-time description sufficient for the definition of wave-number and frequency which enter in the relation [E = hν and p = hσ]. Conversely, any attempt of locating the collision between the photon and the electron more accurately would, on account of the unavoidable interaction with the fixed scales and clocks defining the space-time reference frame, exclude all closer account as regards the balance of momentum and energy. (Bohr, 1949, p. 210) A causal description of the process cannot be attained; we have to content ourselves with complementary descriptions. "The viewpoint of complementarity may be regarded", according to Bohr, "as a rational generalization of the very ideal of causality". In addition to complementary descriptions Bohr also talks about complementary phenomena and complementary quantities. Position and momentum, as well as time and energy, are complementary quantities.[5] We have seen that Bohr's approach to quantum theory puts heavy emphasis on the language used to communicate experimental observations, which, in his opinion, must always remain classical. By comparison, he seemed to put little value on arguments starting from the mathematical formalism of quantum theory. This informal approach is typical of all of Bohr's discussions on the meaning of quantum mechanics. One might say that for Bohr the conceptual clarification of the situation has primary importance while the formalism is only a symbolic representation of this situation. This is remarkable since, finally, it is the formalism which needs to be interpreted. This neglect of the formalism is one of the reasons why it is so difficult to get a clear understanding of Bohr's interpretation of quantum mechanics and why it has aroused so much controversy. We close this section by citing from an article of 1948 to show how Bohr conceived the role of the formalism of quantum mechanics: The entire formalism is to be considered as a tool for deriving predictions, of definite or statistical character, as regards information obtainable under experimental conditions described in classical terms and specified by means of parameters entering into the algebraic or differential equations of which the matrices or the wave-functions, respectively, are solutions. These symbols themselves, as is indicated already by the use of imaginary numbers, are not susceptible to pictorial interpretation; and even derived real functions like densities and currents are only to be regarded as expressing the probabilities for the occurrence of individual events observable under well-defined experimental conditions. (Bohr, 1948, p. 314) 3.2 Bohr's view on the uncertainty relations In his Como lecture, published in 1928, Bohr gave his own version of a derivation of the uncertainty relations between position and momentum and between time and energy. He started from the relations E = hν and p = h (13) which connect the notions of energy E and momentum p from the particle picture with those of frequency ν and wavelength λ from the wave picture. He noticed that a wave packet of limited extension in space and time can only be built up by the superposition of a number of elementary waves with a large range of wave numbers and frequencies. Denoting the spatial and temporal extensions of the wave packet by Δx and Δt, and the extensions in the wave number σ := 1/λ and frequency by Δσ and Δν, it follows from Fourier analysis that in the most favorable case Δx Δσ ≈ Δt Δν ≈ 1, and, using (13), one obtains the relations Δt ΔE ≈ Δx Δph (14) Note that Δx, Δσ, etc., are not standard deviations but unspecified measures of the size of a wave packet. (The original text has equality signs instead of approximate equality signs, but, since Bohr does not define the spreads exactly the use of approximate equality signs seems more in line with his intentions. Moreover, Bohr himself used approximate equality signs in later presentations.) These equations determine, according to Bohr: "the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wave field" (Bohr 1928, p. 571). He noted, "This circumstance may be regarded as a simple symbolic expression of the complementary nature of the space-time description and the claims of causality" (ibid).[6] We note a few points about Bohr's view on the uncertainty relations. First of all, Bohr does not refer to discontinuous changes in the relevant quantities during the measurement process. Rather, he emphasizes the possibility of defining these quantities. This view is markedly different from Heisenberg's. A draft version of the Como lecture is even more explicit on the difference between Bohr and Heisenberg: These reciprocal uncertainty relations were given in a recent paper of Heisenberg as the expression of the statistical element which, due to the feature of discontinuity implied in the quantum postulate, characterizes any interpretation of observations by means of classical concepts. It must be remembered, however, that the uncertainty in question is not simply a consequence of a discontinuous change of energy and momentum say during an interaction between radiation and material particles employed in measuring the space-time coordinates of the individuals. According to the above considerations the question is rather that of the impossibility of defining rigourously such a change when the space-time coordination of the individuals is also considered. (Bohr, 1985 p. 93) Indeed, Bohr not only rejected Heisenberg's argument that these relations are due to discontinuous disturbances implied by the act of measuring, but also his view that the measurement process creates a definite result: Nor did he approve of an epistemological formulation or one in terms of experimental inaccuracies: […] a sentence like "we cannot know both the momentum and the position of an atomic object" raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the mutual exclusive conditions for an unambiguous use of space-time concepts, on the one hand, and dynamical conservation laws on the other hand. (Bohr, 1948, p. 315; also Bohr 1949, p. 211) It would in particular not be out of place in this connection to warn against a misunderstanding likely to arise when one tries to express the content of Heisenberg's well-known indeterminacy relation by such a statement as ‘the position and momentum of a particle cannot simultaneously be measured with arbitrary accuracy’. According to such a formulation it would appear as though we had to do with some arbitrary renunciation of the measurement of either the one or the other of two well-defined attributes of the object, which would not preclude the possibility of a future theory taking both attributes into account on the lines of the classical physics. (Bohr 1937, p. 292) Instead, Bohr always stressed that the uncertainty relations are first and foremost an expression of complementarity. This may seem odd since complementarity is a dichotomic relation between two types of description whereas the uncertainty relations allow for intermediate situations between two extremes. They "express" the dichotomy in the sense that if we take the energy and momentum to be perfectly well-defined, symbolically ΔE = Δp = 0, the postion and time variables are completely undefined, Δx = Δt = ∞, and vice versa. But they also allow intermediate situations in which the mentioned uncertainties are all non-zero and finite. This more positive aspect of the uncertainty relation is mentioned in the Como lecture: At the same time, however, the general character of this relation makes it possible to a certain extent to reconcile the conservation laws with the space-time coordination of observations, the idea of a coincidence of well-defined events in space-time points being replaced by that of unsharply defined individuals within space-time regions. (Bohr 1928, p. 571) However, Bohr never followed up on this suggestion that we might be able to strike a compromise between the two mutually exclusive modes of description in terms of unsharply defined quantities. Indeed, an attempt to do so, would take the formalism of quantum theory more seriously than the concepts of classical language, and this step Bohr refused to take. Instead, in his later writings he would be content with stating that the uncertainty relations simply defy an unambiguous interpretation in classical terms: These so-called indeterminacy relations explicitly bear out the limitation of causal analysis, but it is important to recognize that no unambiguous interpretation of such a relation can be given in words suited to describe a situation in which physical attributes are objectified in a classical way. (Bohr, 1948, p.315) It must here be remembered that even in the indeterminacy relation [Δq Δph] we are dealing with an implication of the formalism which defies unambiguous expression in words suited to describe classical pictures. Thus a sentence like "we cannot know both the momentum and the position of an atomic object" raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the conditions for an unambiguous use of space-time concepts, on the one hand, and dynamical conservation laws on the other hand. (Bohr, 1949, p. 211) Finally, on a more formal level, we note that Bohr's derivation does not rely on the commutation relations (1) and (5), but on Fourier analysis. These two approaches are equivalent as far as the relationship between position and momentum is concerned, but this is not so for time and energy since most physical systems do not have a time operator. Indeed, in his discussion with Einstein (Bohr, 1949), Bohr considered time as a simple classical variable. This even holds for his famous discussion of the ‘clock-in-the-box’ thought-experiment where the time, as defined by the clock in the box, is treated from the point of view of classical general relativity. Thus, in an approach based on commutation relations, the position-momentum and time-energy uncertainty relations are not on equal footing, which is contrary to Bohr's approach in terms of Fourier analysis (Hilgevoord 1996 and 1998). 4. The Minimal Interpretation In the previous two sections we have seen how both Heisenberg and Bohr attributed a far-reaching status to the uncertainty relations. They both argued that these relations place fundamental limits on the applicability of the usual classical concepts. Moreover, they both believed that these limitations were inevitable and forced upon us. However, we have also seen that they reached such conclusions by starting from radical and controversial assumptions. This entails, of course, that their radical conclusions remain unconvincing for those who reject these assumptions. Indeed, the operationalist-positivist viewpoint adopted by these authors has long since lost its appeal among philosophers of physics. So the question may be asked what alternative views of the uncertainty relations are still viable. Of course, this problem is intimately connected with that of the interpretation of the wave function, and hence of quantum mechanics as a whole. Since there is no consensus about the latter, one cannot expect consensus about the interpretation of the uncertainty relations either. Here we only describe a point of view, which we call the ‘minimal interpretation’, that seems to be shared by both the adherents of the Copenhagen interpretation and of other views. In quantum mechanics a system is supposed to be described by its quantum state, also called its state vector. Given the state vector, one can derive probability distributions for all the physical quantities pertaining to the system such as its position, momentum, angular momentum, energy, etc. The operational meaning of these probability distributions is that they correspond to the distribution of the values obtained for these quantities in a long series of repetitions of the measurement. More precisely, one imagines a great number of copies of the system under consideration, all prepared in the same way. On each copy the momentum, say, is measured. Generally, the outcomes of these measurements differ and a distribution of outcomes is obtained. The theoretical momentum distribution derived from the quantum state is supposed to coincide with the hypothetical distribution of outcomes obtained in an infinite series of repetitions of the momentum measurement. The same holds, mutatis mutandis, for all the other physical quantities pertaining to the system. Note that no simultaneous measurements of two or more quantities are required in defining the operational meaning of the probability distributions. Uncertainty relations can be considered as statements about the spreads of the probability distributions of the several physical quantities arising from the same state. For example, the uncertainty relation between the position and momentum of a system may be understood as the statement that the position and momentum distributions cannot both be arbitrarily narrow -- in some sense of the word "narrow" -- in any quantum state. Inequality (9) is an example of such a relation in which the standard deviation is employed as a measure of spread. From this characterization of uncertainty relations follows that a more detailed interpretation of the quantum state than the one given in the previous paragraph is not required to study uncertainty relations as such. In particular, a further ontological or linguistic interpretation of the notion of uncertainty, as limits on the applicability of our concepts given by Heisenberg or Bohr, need not be supposed. Indeed, this minimal interpretation leaves open whether it makes sense to attribute precise values of position and momentum to an individual system. Some interpretations of quantum mechanics, e.g. those of Heisenberg and Bohr, deny this; while others, e.g. the interpretation of de Broglie and Bohm insist that each individual system has a definite position and momentum (see the entry on Bohmian mechanics). The only requirement is that, as an empirical fact, it is not possible to prepare pure ensembles in which all systems have the same values for these quantities, or ensembles in which the spreads are smaller than allowed by quantum theory. Although interpretations of quantum mechanics, in which each system has a definite value for its position and momentum are still viable, this is not to say that they are without strange features of their own; they do not imply a return to classical physics. We end with a few remarks on this minimal interpretation. First, it may be noted that the minimal interpretation of the uncertainty relations is little more than filling in the empirical meaning of inequality (9), or an inequality in terms of other measures of width, as obtained from the standard formalism of quantum mechanics. As such, this view shares many of the limitations we have noted above about this inequality. Indeed, it is not straightforward to relate the spread in a statistical distribution of measurement results with the inaccuracy of this measurement, such as, e.g. the resolving power of a microscope. Moreover, the minimal interpretation does not address the question whether one can make simultaneous accurate measurements of position and momentum. As a matter of fact, one can show that the standard formalism of quantum mechanics does not allow such simultaneous measurements. But this is not a consequence of relation (9). If one feels that statements about inaccuracy of measurement, or the possibility of simultaneous measurements, belong to any satisfactory formulation of the uncertainty principle, the minimal interpretation may thus be too minimal. Other Internet Resources Related Entries quantum mechanics \| quantum mechanics: Bohmian mechanics \| quantum mechanics: Copenhagen interpretation of
562161ca9c48439c	Given that the global phases of states cannot be physically discerned, why is it that quantum circuits are phrased in terms of unitaries and not special unitaries? One answer I got was that it is just for convenience but I'm still unsure. A related question is this: are there any differences in the physical implementation of a unitary $U$ (mathematical matrix) and $ V: =e^{i\alpha}U$, say in terms of some elementary gates? Suppose there isn't (which is my understanding). Then the physical implementation of $c\text{-}U$ and $c\text{-}V$ should be the same (just add controls to the elementary gates). But then I get into the contradiction that $c\text{-}U$ and $c\text{-}V$ of these two unitaries may not be equivalent up to phase (as mathematical matrices), so it seems plausible they correspond to different physical implementations. What have I done wrong in my reasoning here, because it suggests now that $U$ and $V$ must be implemented differently even though they are equivalent up to phase? Another related question (in fact the origin of my confusion, I'd be extra grateful for an answer to this one): it seems that one can use a quantum circuit to estimate both the modulus and phase of the complex overlap $\langle\psi\|U\|\psi\rangle$ (see https://arxiv.org/abs/quant-ph/0203016). But doesn't this imply again that $U$ and $e^{i\alpha}U$ are measurably different? • 2 $\begingroup$ It is more philosophically accurate to say projective unitary group $PU$ instead. That is because the operation is to take an arbitrary unitary matrix and lose the phase vs the subset for which that phase is $1$. The maps go $SU \to U \to PU$ so they are on opposite sides of the arrows. $\endgroup$ – AHusain May 12 '18 at 22:04 • $\begingroup$ @AHusain Which are "The maps"? In terms of quotienting out, it will go $U\to SU\to PU$. $\endgroup$ May 13 '18 at 10:48 • 1 $\begingroup$ No. SU is the subset with determinant 1, so it includes with a map into U. PU is the quotienting out. You can take a projective unitary and give a representative in SU with determinant 1, but that is not automatic. $\endgroup$ – AHusain May 14 '18 at 15:31 Even if you only limit yourself to special-unitary operations, states will still accumulate global phase. For example, $Z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$ is special-unitary but $Z \cdot \|0\rangle = i \|0\rangle \neq \|0\rangle$. If states are going to accumulate unobservable global phase anyways, what benefit do we get out of limiting ourselves to special unitary operations? As long you're not doing anything that could make the global phases relevant, they can have the same implementation. But if you're going to do something like, uh- add controls to the elementary gates Yeah, like that. If you do stuff like that, then you can't ignore global phases. Controls turn global phases into relative phases. If you want to completely ignore global phase, you can't have a black box "add a control" operation modifier. • $\begingroup$ Thanks, but doesn't an "add a control" modifier exist for gates in a universal gate set and you could first decompose $U$ and $V$ into these gates in order to add control, e.g. c-$X$ is the CNOT gate. $\endgroup$ – wdaochen May 18 '18 at 20:59 • 1 $\begingroup$ @Daochen Yes you can do that, but it's not an example of adding a control while ignoring the sub-operation's global phase. You will have to explicitly decide on the global phase of the sub-operation when deciding what exactly the overall controlled operation should do and how to decompose it. $\endgroup$ May 18 '18 at 21:58 The fact that quantum gates are unitary, is rooted in the fact that the evolution of (closed) quantum systems is by the Schrödiner equation. For a time interval in which we are trying to realise a particular unitary transformation at a constant rate, we use the time-independent Schrödinger equation: $$ \tfrac{\mathrm d}{\mathrm dt} \lvert \psi(t) \rangle = \tfrac {1}{i\hbar}H \lvert \psi(t) \rangle, $$ where $H$ is the Hamiltonian of the system: a Hermitian matrix, whose eigenvalues describe energy eigenvalues. In particular, the eigenvalues of $H $ are real. The solution to this equation is $$ \lvert \psi(t) \rangle = \exp\bigl(-i H t/\hbar\bigr) \lvert \psi(0) \rangle $$ where $U = \exp(-iHt/\hbar)$ is the matrix which you obtain by taking the eigenvectors of $H$, and replacing their eigenvalues $E$ with $\mathrm{e}^{iEt/\hbar}$. Thus, from a matrix with real eigenvalues, we get a matrix whose eigenvalues are complex numbers with unit norm. What would it take for this evolution to specifically be a special unitary matrix? A special unitary matrix is one whose determinant is precisely $1$; that is, whose eigenvalues all multiply to $1$. This corresponds to the restriction that the eigenvalues of $H$ all sum to zero. Furthermore, because the eigenvalues of $H$ are energy levels, whether the sum of its eigenvalues is equal to zero depends on how you have decided to fix what your zero energy point is — which is in effect a subjective choice of reference frame. (In particular, if you decide to adopt the convention that all of your energy levels are non-negative, this implies that no interesting system will ever have the property of the energy eigenvalues summing to zero.) In short, gates are unitary rather than special unitary, because the determinant of a gate does not correspond to physically meaningful properties — in the explicit sense that the gate arises from the physics, and the conditions which correspond to the determinant of the gate being 1 is a condition of one's own reference frame and not the physical dynamics. When writing gates for, for example, a quantum circuit diagram, you could always write them using the convention of having determinant one (from the special unitary group), but it's just a convention. It makes no physical difference to the circuit that you implement. As said elsewhere, whether what you naturally produce corresponds directly to the special unitary is really a choice of convention, and where you define your 0 energy to be. As for the issue when you start implementing controlled-$U$, there is an interesting comparison to be made. Let's say we define $V=e^{i\alpha}$. How can we implement controlled-$V$ in terms of controlled-$U$? You apply controlled-$U$ and then, on the control qubit, you apply the phase gate $\left(\begin{array}{cc} 1 & 0 \\ 0 & e^{i\alpha} \end{array}\right)$. There are two things to observe here. First, the difference is on the control qubit rather than the target qubit. The target qubit, where you're implementing the $U$, doesn't really care about the difference in phase. It's the control-qubit that's hit by the phase gate. The second is that I didn't write the phase gate as a special unitary. Of course, I could have written it as $\left(\begin{array}{cc} e^{-i\alpha/2} & 0 \\ 0 & e^{i\alpha/2}\end{array}\right)$ but I didn't because the way that I chose to write it was notationally more convenient - less writing for me, and hopefully more immediately obvious to you why it works. Plain and simple answer: In the absence of decoherence, state vectors evolve according to $\|\psi(t)\rangle = e^{-iHt}\|\psi(0)\rangle$ for a Hamiltonian $H$. This is what a "gate" is doing. Hamiltonians have to be Hermitian, so this transformation is unitary. Hamiltonians do not have to have eigenvalues that sum to 0, so the transformation does not have to be special unitary. Your Answer
c3db9e81c4700749	Until recently, scientists didn't think it could be done. They thought the fundamental laws of physics would forbid it. But a persistent group of scientists at the University of Warsaw have now accomplished the impossible: They created a hologram of a solitary particle of light. This accomplishment is ushering in a new era of quantum holography, which will give scientists a new way of looking at quantum phenomena. Quantum holograms Unlike photography, holography recreates the spatial structure of objects, giving us their 3-D shapes. The technique takes advantage of something called classical interference, which is when two waves meet and form a new wave. But classical interference is impossible with photons, since their phases (a property of waves) are constantly fluctuating. So the Warsaw physicists tried to give quantum holograms a taste of their own medicine by using quantum interference, in which photons' wave functions (which have to do with the probability of the particle being in a particular state) interact. "Wave function is a fundamental concept in quantum mechanics and the core of its most important principles, the Schrödinger equation," according to "In the hands of a skilled physicist, the function could be compared to putty in the hands of a sculptor. When expertly shaped, it can be used to 'mould' a model of a quantum particle system." So why photons? While filming the behavior pairs of photons, Radoslaw Chrapkiewicz and Michal Jachura, two of the researchers, noticed something called two-photon interference. In two-photon interference, pairs of distinguishable photons act randomly when entering a beam splitter (which divides a ray of light). But nondistinguishable photons exhibit quantum interference, which affects their behavior. The pairs are always either transmitted or reflected together. "Following this experiment, we were inspired to ask whether two-photon quantum interference could be used similarly to classical interference in holography in order to use known-state photons to gain further information about unknown-state photons. Our analysis led us to a surprising conclusion: It turned out that when two photons exhibit quantum interference, the course of this interference depends on the shape of their wavefronts [an imaginary surface joining all adjacent points with the same phase]," Chrapkiewicz told Understanding quantum mechanics This experiment has huge implications for our understanding of the fundamental laws of quantum mechanics, a field of physics that has been perplexing scientists for more than a century. It allows scientists to gain valuable information about the phase of a photon's wave function. "Our experiment is one of the first allowing us to directly observe one of the fundamental parameters of photon's wave function — its phase — bringing us a step closer to understanding what the wave function really is," Jachura said. The researchers hope to apply this method to create holograms of more complex quantum objects, which might have implications that stretch beyond fundamental science into real world applications. "All of us — I mean physicists — must first get our heads around this new tool," said Konrad Banaszek, a researcher in the experiment. "It's likely that real applications of quantum holography won't appear for a few decades yet, but if there's one thing we can be sure of, it's that they will be surprising." 2021 Will Set an All-Time Record for New Renewable Energy Freedom and Safety (4 days ago) The Metaverse Will Need 1,000x More Computing Power, Says Intel Freedom and Safety (4 days ago) Texas just had its hottest December since 1889 Freedom and Safety (4 days ago) The Fourth Industrial Revolution in the World:
a0fadedceab79b2b	Spectroscopy: Theory and Application Science & Technology Course Code CHEM 2360 Semester Length Max Class Size Method Of Instruction Typically Offered To be determined Course Description This course introduces the principles of quantum mechanics as they apply to atomic and molecular spectroscopy. The following techniques are covered from both a theoretical and practical perspective: infrared spectroscopy, Raman spectroscopy, UV-VIS spectroscopy, NMR spectroscopy, atomic spectroscopy and GC-mass spectrometry. The experimental application of this material is covered in both wet-bench and computational laboratory techniques. Course Content Introduction and Principles of Quantum Mechanics • the electromagnetic spectrum • wavefunctions • the Schrödinger equation • particle in a box • atomic and molecular energy levels • electronic transitions • Boltzmann distribution Symmetry and Spectroscopy • symmetry operators and point groups and their relation to spectroscopy Atomic Structure and Spectroscopy • H atom and multi-electron atoms • atomic orbitals • term symbols • atomic spectra Molecular Rotations • rotational motion • selection rules • Raman spectroscopy Molecular Vibrations • vibrational motion • selection rules • energy levels • infrared spectroscopy Magnetic Resonance Spectroscopy • nuclear spin • splitting of energy levels in an applied magnetic field • molecular structure and chemical shifts • NMR spectroscopy of proton and multi-nuclear species Mass Spectrometry • magnetic sector, quadrupole and ion-trap techniques • high resolution spectral interpretation • fragmentation patterns UV-Visible Spectroscopy • energy levels and transitions in organic and coordination compounds • spectra of pure compounds and mixtures Laboratory Content: Experiments are selected from the following list: 1. Quantitative UV/Vis Spectroscopy of a Mixture 2. Determination of Keto-Enol Equilibrium Constants by NMR 3. Geometric Isomers of a Cr(III) Complex 4. Preparation and Identification of Co(III) Complexes 5. Paramagnetic Susceptibility by NMR 6. Computer Lab I: Use of EXCEL 7. Computer Lab II: Molecular Modeling 8. Computer Lab III: Spectral Simulation and Interpretation 9. Infrared Spectroscopy of Liquid Samples 10. Solid Sample Preparation Methods for Infrared Spectroscopy 11. GC-MS Analysis of a Volatile Mixture 12. Structural Determination by NMR: Esterification Reactions of Vanillin 13. Term Project: Determination of an Unknown by IR, NMR, and GC-MS 14. Atomic Absorption: Quantitative Analysis of a Metal Methods Of Instruction The course is presented using lectures, classroom demonstrations, problem sessions, on-line quizzes, and class discussions. Audio-visual materials, including molecular modeling software, are used where appropriate. The laboratory is used to illustrate the practical aspects of the course material, using both traditional wet labs and computer techniques. Close coordination will be maintained between laboratory and classroom work whenever possible. Means of Assessment Evaluation will be carried out in accordance with Douglas College policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following: Lecture Material 70% • Two or three in-class tests will be given during the semester (30%) • A final exam covering the entire semester’s work will be given during the final examination period (30%) • Any or all of the following: problem assignments, on-line quizzes, class participation (10%) Laboratory 30% • Each experiment and will be evaluated based on results and a written report. A formal report based on evaluation of an unknown will also be submitted. A student who misses three or more laboratory experiments will earn a maximum P grade. A student who achieves less than 50% in either the lecture or laboratory portion of the course will earn a maximum P grade. Learning Outcomes 1. Describe the properties of the electromagnetic spectrum and calculate energy, wavelength, and frequency of light 2. Describe the photoelectric effect and blackbody radiation 3. Describe wave particle duality and the Heisenberg Uncertainty Principle 4. Apply the Schrödinger equation to wavefunctions in order to understand how electronic energy levels are determined 5. Apply the concepts of moment of inertia and particle in a box to determine allowed rotational excitations in a molecule 6. Explain spin and the allowed transitions of electron and nuclear spin in an applied magnetic field 7. Sketch the shapes of atomic orbitals from a set of quantum numbers 8. Explain the Aufbau Principle, Pauli Exclusion Principle and the use of term symbols for multi-electron atoms 9. Describe bond formation in a diatomic molecule through the use of the Born-Oppenheimer approximation 10. Apply the simple harmonic oscillator model to understand vibrations in diatomic molecules and simple polyatomics 11. Apply selection rules to molecules to determine allowed vibrational modes 12. Sketch simple vibrational modes in linear and non-linear polyatomic molecules 13. Use the Boltzmann distribution law to describe energy levels in a molecule and predict the effect on spectroscopic results 14. Given an infrared spectrum, determine the functional groups or coordinate bonds present using correlation charts 15. Given an NMR spectrum, determine the number and type of protons present and the J-J coupling constants for first and second order 16. Given the structure of a compound, predict the number of NMR peaks, chemical shifts, and splitting patterns using electronegativity, symmetry, and hybridization 17. Interpret multi-nuclear NMR spectra of molecules containing 19F, 13C, and 31P by using chemical shifts, reference compounds, and decoupled spectra 18. Given a UV-VIS spectrum, label the transitions occurring based on molecular orbital or crystal field splitting 19. Calculate concentrations or molar extinction coefficients by applying the Beer - Lambert law to a UV-VIS spectrum 20. Analyze the UV-VIS spectrum of a two component mixture to determine individual concentrations 21. Use information from given spectra (IR, UV-VIS, NMR, GC-MS) to determine the structural formula of a compound 22. Use EXCEL spreadsheets to create linear calibration curves, including error analysis 23. Given a mass spectrum, predict the parent compound using fragmentation patterns 24. Determine isotopic information from a high resolution mass spectrum 25. For each spectroscopic method studied, sketch the basic components of the instrument and understand their operation 26. Prepare standards and unknown solutions as required for laboratory analysis and run these samples and use instrumental software to analyze the spectra Textbook Materials Consult the Douglas College Bookstore for the latest required textbooks and materials. Example textbooks and materials may include: • Pavia, Lampman, Kriz, and Vyvyan, Introduction to Spectrosopy, Current Edition, Brooks/Cole • Atkins and De Paula, Physical Chemistry, Custom Version of Current Edition, Freeman • Douglas College, Chemistry 2360 Laboratory Manual • Laboratory Notebook, Safety Goggles, Laboratory Coat • Thomas Engel, Quantum Chemistry and Spectroscopy, Current Edition, Pearson CHEM 1110 (C or better) MATH 1120 (must be completed either prior to or concurrently with this course) • No equivalency courses Course Guidelines Course Transfers Institution Transfer Details Effective Dates Athabasca University (AU) AU CHEM 3XX (3) 2013/09/01 to - Capilano University (CAPU) CAPU CHEM 2XX (4) 2013/09/01 to - College of the Rockies (COTR) COTR CHEM 2XX (3) 2013/09/01 to - Columbia College (COLU) COLU CHEM 2nd (3) 2013/09/01 to - Kwantlen Polytechnic University (KPU) KPU CHEM 2XXX (4) 2013/09/01 to - North Island College (NIC) NIC CHE 2XX (3) 2013/09/01 to - Okanagan College (OC) OC CHEM 2XX (3) 2013/09/01 to - Simon Fraser University (SFU) SFU CHEM 260 (4) 2013/09/01 to - Thompson Rivers University (TRU) TRU CHEM 2X11 (0) & TRU CHEM 2XXX (3) 2013/09/01 to - University of British Columbia - Vancouver (UBCV) UBCV CHEM 2nd (3) 2013/09/01 to - University of Northern BC (UNBC) UNBC CHEM 2XX (4) 2013/09/01 to - University of the Fraser Valley (UFV) UFV CHEM 2XX (3) 2020/01/01 to - University of Victoria (UVIC) UVIC CHEM 213 (1.5) 2013/09/01 to 2020/04/30 University of Victoria (UVIC) UVIC CHEM 2XX (1.5) 2020/05/01 to - Vancouver Island University (VIU) VIU CHEM 213 (3) 2013/09/01 to - Course Offerings Winter 2022 There aren't any scheduled upcoming offerings for this course.
c9ee092153f6326b	ISSN: 2320-2459 Incompleteness of Quantum Theory on EinsteinÃ¢ÂÂs Program Claude Elbaz* Academie Europeenne Interdisciplinaire de Science (A.E.I.S.), Paris, France *Corresponding Author: Claude E Academie Europeenne Interdisciplinaire de Science (A.E.I.S.) Paris, France Tel: 0512-2533812 E-mail: [email protected] Received date: 18/04/2017; Accepted date: 03/05/2017; Published date: 09/05/2017 Adiabatic variations of frequencies lead to electromagnetic interaction constituted by progressive waves. By comparison, the consistent quantum theory, based upon time-like equations, of Dirac for fermions and of Klein- Gordon for bosons, is incomplete since it does not take account of a spacelike function u0(k0r0) to describe massive particle extension. It leads to Dirac’s distribution δ(r0) in the geometrical optics approximation, in order to handle a particle as a singularity within a continuous field, and to energymomentum conservation laws, and to least action law, with determination of the Lagrangian. Einstein’s program, Quantum field theory, Adiabatic invariant, Hidden variables, Wave-particle duality The Standard Model of particles, experimentally validated by the Higgs boson detection, represents the crowning of quantum theory. It is admitted that the whole universe is composed of fundamental particles, at one and the same time for matter and for three out of the four different interactions. They all manifest experimentally either as waves or as particles, justifying that they leans upon a quantum mechanical probabilistic framework. Since it has not still incorporated gravitation, the Standard Model describes only an incomplete aspect of the universe. While Einstein’s contribution for relativity is admitted as essential, it appears as less important for quantum mechanics, even though his Nobel Prize was awarded for his discovery of the first quantum particle in photoelectric effect. More especially as his resolute opposition to the probabilistic framework is well known. For him, the probabilistic experimental behavior of quantum particles, like electrons, proves that the quantum mechanics description is incomplete. The statistical character of the present theory would then have to be a necessary consequence of the incompleteness of the description of the systems in quantum mechanics. Until now, gravitation has resisted to its theoretical quantification. It remains well described by general relativity, based upon a continuous field in a classical framework, [1-5] widely confirmed by numerous experiments, by its theoretical consequences and practical applications. The graviton, as quantum particle mediating gravitation interaction, has not yet been experimentally detected and validated [6,7]. On another hand, quantum field theories of gravity generally break down theoretically before reaching the Planck scale, which determines the limit between the wave and particle behavior of quantum particles [8]. We may conclude that the discrepancy between quantum mechanics and general relativity leans on the wave particle duality, together with the classical deterministic or quantum probabilistic approaches. In order to circumvent these difficulties, the program proposed by Einstein offered an alternating approach founded upon a general energy field. We have two realities: matter and field. …We cannot build physics on the basis of the matter concept alone. But the division into matter and field is, after the recognition of the equivalence of mass and energy, something artificial and not clearly defined. Could we not reject the concept of matter and build a pure field physics? We could regard matter as the regions in space where the field is extremely strong. In this way a new philosophical background could be created… Only field-energy would be left, and the particle would be merely an area of special density of field-energy. In that case one could hope to deduce the concept of the mass-point together with the equations of the motion of the particles from the field e quations - the disturbing dualism would have been removed… One would be compelled to demand that the particles themselves would everywhere be describable as singularity free solutions of the completed field-equations… One could believe that it would be possible to find a new and secure foundation for all physics upon the path which had been so successfully begun by Faraday and Maxwell. From an experimental point of view, the Einstein’s Program has been validated by the International Legal Metrology Organization when it admitted, in one hand, that the light velocity in vacuum was a primary, fundamental constant in physics, with its numerical value strictly fixed, and in other hand, when it choose one particular period an electromagnetic wave frequency as standard for measures of time. As shown in different articles, the Einstein’s Program yields a consistent system for universe description, beside the Standard Model [8-10]. It enlarges our approach, comparably to using both eyes for tridimensional vision, or both ears for stereophonic audition. It is founded upon a scalar field propagating at speed of light c. Matter properties derive from standing waves, while electromagnetic interaction derives from adiabatic variations of frequencies. In the geometrical optics approximation, when very high frequencies are experimentally undetectable, the oscillations are hidden. This holds at one and the same time in classical relativistic and quantum frameworks, yielding their descriptions being incomplete. In this article we propose to show how the Einstein’s program offers a means to remedy this problem, It leads to a space-like Helmoltz equation available to describe particle extension, especially in quantum theory, since the Standard Model is basically founded upon particles, in which the behaviour of massive particles is described only by time-like fundamental equations: of Dirac for fermions and of Klein-Gordon for bosons, or by their Lagrangians in variational equations. Then, particles can be indifferently extended or point-like. In accordance with experiment, they are admitted as point-like. This is necessarily a physical approximation, since their mass-energy density cannot be infinite. The Dirac’s distribution δ(r0), which offers a means to circumvent this difficulty in order to handle a particle as a singularity within a continuous field, derives itself from a space-like Helmoltz equation. The Einstein’s Program We restrict to summarize some equations deduced from Einstein’s program, in order to show how they are related to main equations of quantum mechanics, otherwise widely documented [11]. Kinematical Properties of Standing Fields From the d'Alembertian’s equation describing a scalar field ε propagating at light velocity c Derive two kinds of elementary harmonic solutions with constant frequency ω0, with different kinematical properties. The progressive waves, retarded cos(0t0-k0x0) or advanced cos(ω0t0+k0x0), are in motion with light velocity c=ω0/k0. The standing waves ε0(x0,t0)=u0(k0x000t0)=cos(ω0t0)cos(k0x0), where space and time variables are separated, oscillate locally. They allow to define a system of coordinates at rest (x0,t0). They may be considered as resulting from superposition of progressive waves. cos(ω0t0+ k0x0)+ cos(ω0t0 - k0x0)=2cos(ω0t0)cos(k0x0) (2) When, in a system of reference (x,t), the frequencies of opposite progressive waves are different. cos(ω1t-k1x)+ cos(ω2t + k2x)=2 cos(ωt-βkx)cos(kx-βωt) Where ω=(ω1+ω2)/2=kc, and β=(ω1212). By identification with eqn. (2), they form a standing wave in motion with a speed v=βc=(ω1212)c with frequency ω=(ω12)/2=kc, and ω0=√ω1ω2at rest, defining the Lorentz transformation between the systems of reference(x0,t0) and (x,t), and leading to its whole consequences. It can be shown that the Lorentz transformation, fundamental in special relativity, is specific of c-field standing waves, particularly through the coefficient √(1-β2), which becomes (1 ± β) for progressive waves. The four-dimensional Minkowski’s formalism expresses invariance properties of standing waves at rest, in which the variables of space and time are separated, when they move uniformly with a speed v=βc<c since β is a relative difference [12]. Confirmation is found into invariant quantities obtained from four-quantities, such as coordinates equation, and functions equation or equation. Their space-like or time-like characters are absolute, depending of their refering quantities defined in the rest system. Since the functions u0(k0x0) and ψ00t0) are independent, the frequency ω0 is necessarily constant in: equation (4) The function of space u0(k0x0), obeying the Helmholtz’s equation at rest Δ0u0+k02u0=0, which becomes Δu-(1/c2) (∂2u/∂t2)+k02u=0 in motion. It describes geometric properties of standing waves. It verifies Bessel spherical functions solutions, and particularly its simplest elementary solution, with spherical symmetry, finite at of the references system, equation (5) Dirac’s distribution equation is favored when frequency is tending towards infinityequation In Cartesian system of reference, the central extremum of an extended standing wave determines its standing position x0=r0=0. equation (6) In order to point out the constant frequency of a standing field, we express it as: equation (7) The equations of special and general relativity are based on mass-points, as singularities, moving on trajectories, deriving then directly from geometrical optics approximation. Then, the kinematic properties of standing waves for a scalar field propagating at light velocity c, with constant frequency ω and velocity v, reduce formally to kinematical properties of isolated point-like matter. Dynamical Properties of Standing Fields In order to limit the field ε(ωt,kx) with respect to space and time, which cannot be infinite, one imposes usually boundary conditions exerted by matter [13]. It fixes the wavelength λ through k=2π/λ fixing the frequency ω or as a detector annealing it which is not felicitous from relativistic consistency since space and time operate separately It behaves either as a source: Two progressive waves with different frequencies ω1,ω2 propagating in the same direction at light velocity, give rise to a wave packet propagating in the same direction at light velocity. Its main wave with frequency ω=(ω1+ω2)/2, is modulated by a wave with frequency βω=(ω1-ω2)/2=Δω/2=Δkc/2, wavelength Λ=2π/βk, and period T=Λ/c. Since β<1, the modulation wave acts as an envelope with space and time extensions Δx=Λ/2, Δt=T/2, yielding well known Fourier relations Δx.Δk=2π and Δt.Δω=2π. The boundary conditions for the scalar field ε are represented by the Fourier relations which should supplement d'Alembertian’s eqn. (1) to emphasize that field cannot extend to infinity with respect to space and time. When the difference of frequencies βω=(ω1-ω2)/2=Δω/2<< ω is very small, it can be considered as a perturbation with respect to the main frequency, βω=δω. Monochromatic wave by a frequency Ω(x,t) can be characterized around a constant ω. equation (8) This is also the definition of an adiabatic variation for the frequency. Consequently, all following properties of almost fields arise inside such a process [14]. The necessarily constant frequency of a standing wave must be considered, not as a given data, but rather as the mean value, all over the field, of different varying frequencies Ω(x,t). The perturbation frequencies δΩ(x,t) of modulation waves propagating at light velocity, behave as interactions between main waves, which yield the mean frequency ω to remains practically constant all over the space-time. Such a behavior authorizes mathematically to derive almost fields properties from monochromatic ones, through the variation of constants method (Duhamel principle). Instead of eqn. (8), we express it, as: equation (9) Where products of second order δΩdt≈0 and δK.dx≈0, defined modulo 2π, are neglected at first order of approximation. This is equivalent to incorporate directly the boundary conditions defined by Fourier relations, in almost monochromatic solutions, equation (10) Following eqn. (1), the field ε(x,t) defined by eqn. (9) verifies, equation (11) equation (12) These relations apply to progressive waves for β=±1, to standing waves at rest for β=0 and in motion for β<1, to monochromatic waves for ω and k constant, to almost monochromatic waves for varying Ω(x,t) et K(x,t). They lead to dynamical properties for energy-momentum conservation, and to least action principles, for standing fields and almost standing fields. For a standing wave with constant frequencyω, either at rest or in motion, (12) reduces to: equation (13) Where wμ=(u2,u2v/c)=u0(x0)2(1,v/c)/(1-β2) is a four-dimensional vector. This continuity equation for u2 is formally identical with Newton’s equation continuity for matter-momentum density: equation (14) By transposition, we can then admit that u2 represents the energy density of the standing field. Centre of amplitude is taken into consideration to determine the kinematical behavior of a standing field, with position x0 defined by eqn. (6) The position x0 of the energy density is such that, equation (15) The standing wave energy density u2 is spread in space [15]. It corresponds then to a potential energy density. F=-∇u2=-∇wP=is a density force, and ∂u2v/c2∂t a density momentum, and πν is a four-dimensional force density. In eqn. (15), the vanishing four-dimensional force density tensor πμν of a standing wave, asserts its space stability remains in motion, and that the energy-momentum density four-vector wμ is four-parallel, or directed along the motion velocity v. Eqn. (15) is mathematically equivalent to the least action relation, in which energy density wμ is a four-dimension gradient ∂μa, equation (16) When we transpose the mass density μ=u2/c2, and we take into account the identities ∇P2=2(P.∇)P+2P × (∇ × P) and dP/dt=∂P/∂t+(v.∇)P for c and v constant, after integration with respect to space, we get the equation for matter: equation (17) We retrieve the relativistic Lagrangian of mechanics for free matter Lm=-m0c2√(1-β2). Electromagnetic Interaction The continuity equation of an almost standing wave, expresses the total energy density, W=U2Ω=w+δW, sum of the mean standing wave w and of the interactions δW. Relation eqn. (15) becomes: equation (18) The total density force Πμν for an almost standing wave vanishes. This asserts its stability when it is in motion; Its total energy-momentum density four-vector Wμ is directed along the motion velocity v. However it behaves as a system composed of two sub-systems, the mean standing field with high frequency Ω(x,t) ≈ ω, and the interaction field with lower frequency δΩ(x,t), each one exerting an equal and opposite non vanishing density force πμν=-δΠμν against the other [16]. By difference with the null four-dimensional density force πμν for a standing wave, only the total density force Πμν for an almost standing wave vanishes. In the first case, this asserts the space stability of an isolated moving standing wave, while in the second case, the space stability concerns the whole almost standing wave. It behaves as a system composed of two sub-systems, the mean standing field with high frequency Ω(x,t) ≈ ω, and the interaction field with lower frequency δΩ(x,t), each one exerting an equal and opposite non vanishing density force πμν=-δΠμν against the other. The mean energy-momentum density tensor πμν, no longer vanishes in eqn. (18), as previously in eqn. (15). This comes from the mean energy-momentum density four-vector wμ, which is no longer parallel, because of the opposite density force δμν exerted by the interaction. It appears that an almost standing field behaves as a whole system in motion which can be split in two sub-systems, the mean standing field and the interaction field. Both are moving with velocity v, while exerting each other opposite forces in different directions, including perpendicularly to the velocity v. The perturbation field, arising from local frequency variations δΩ(x,t), introduces orthogonal components in interaction density force and momentum. After generalizing relations eqn. (17) by constants variation method for mass M(x,t)=m ± δM(x,t), we get, equation (19) The density force δΠμν≠0 exerted by the interaction is formally identical with the electromagnetic tensor Fμν=∂μAν- ∂A≠0. We can set them in correspondence δΠμν=eFμ, through a constant invariant charge e, with δM(x,t)=eV(x,t)/c2 and δP(x,t)=eA(x,t)/c. The double sign for mass variation corresponds to the two signs for electric charges, or to emission and absorption of electromagnetic energy by matter. We retrieve the minimum coupling of classical electrodynamics, Pμ(x,t)=pμ+eAμ(x,t)/c, with M(x,t)c2=mc2+eV(x,t), and P(x,t)=p+eA(x,t)/c, where electromagnetic energy exchanged with a particle is very small compared to its own energy eAμ(x,t)/c=δPμ(x,t) << pμ. Electromagnetic interaction is then directly linked to frequencies variations of the field ε. Relation eqn. (19) yields the relativistic Newton’s equation for charged matter with the Lorentz force. dP/dt=-∇m0c2√(1-β2) + e(E+v×H/c). (20) Adiabatic Invariant The relation eqn. (11) leads to first order approximation for an almost standing wave, equation (21) with energy density W=w ± δW=μc2=μc2 ± δμc2, four-dimensional energy density Wν=wν ± δWν, frequency Ω=ω ± δΩ, and four-dimensional frequency Ων=(Ω,Ωv/c), leading to, equation (22) When we take into account the double sign in frequency variation δΩ. The constant I is an adiabatic invariant density [17]. Integrations with respect to space of μ and I densities, lead to relations between four-energy and four-frequency through the adiabatic invariant H, formally identical with the Planck’s constant h. equation (23) For the standing wave corresponding to matter, adiabatic variations of its frequency Ω lead to electromagnetic interaction constituted by progressive waves. The energy of electromagnetic interaction derives from mass variation dE=c2dm. It leans directly upon the wave property of matter: its energy dE=hdν=c2dm derives from variations of matter energy E=hν=mc2. Applications to Quantum Theory Relativistic Foundations The Einstein’s program, based upon a basic scalar field propagating at speed of light, tends toward a theoretical economy by deriving mathematically, and independently of any interpretations, many different fundamental principles of Quantum Theory. The fundamental role of the speed of light in vacuum appeared already in the beginning of Quantum Mechanics, when one noticed that, beyond its agreement with experiment, the non-relativist Schrödinger’s equation, with energy mv2/2, derived as an approximation from the relativist Klein-Gordon’s equation with energy mc2. Nevertheless it remained as foundation to elaborate the theoretical and experimental Copenhagen consistent interpretation: description of a single quantum particle in motion, superposition principle, wave-particle duality behavior with uncertainty principle, Dirac’s relativist equation, the statute of the observer associated with the admitted collapse of the wave function… Nowadays, the more general relativist Quantum Field Theory has introduced some distance with Quantum Mechanics description. For instance, a single quantum particle is no longer considered alone. It is not necessarily permanent experimentally, since it can be created or annihilated. Its experimental point-like behavior is not of prime importance, since it derives as a kind of resonance from a continuous field expressed by partial differential equations. Its mass is not a time-independent constant but varies according to the Feynman process. Its interactions verify gauge theories, with a Lagrangian invariant under continuous local transformations. In addition to the generating time-like ψ function of quantum theory, the Einstein’s program points out the generating role of the space-like amplitude function u. It yields to define the position of a particle as a dynamical variable eqn. (6), either at rest x0 or in x motion with x0→x-vt, to express the continuity eqn. (15) with the energy-momentum conservation laws, and to get the least action law eqn. (16) with determining the Lagrangian eqn. (17). This is not too surprising since the amplitude function, u0 at rest and u in motion with u0 (x0)→u(x-vt), remains always closely linked to the particle displacement. These properties apply both in classical and in quantum domains. Since the Einstein’s program leans on a generating basic c-scalar field, which provides a physical general framework, it draws our attention to some main problems of Quantum Theory such as the point-like behavior of particles, and the problem of iincompleteness of quantum theory with hidden variables, which should ask to be deepened little further. Size of Particles It is admitted that the Standard Model represents the crowning of the Quantum theory. Consequently, the different fundamental component particles, either of matter or of interactions, are usually gathered according their spin, as bosons or fermions, which are a typical quantum property. They are considered as point-like. From a physical point of view, it is obvious that a particle cannot be strictly point-like, since its energy density would be infinite. Since their size is not of prime importance, it does not figure usually beside their mass and electric charge. We may conclude that the description of particles by Quantum theory is not complete. The Einstein program incites us to differentiate the fundamental particles by their relativist properties of motion, according they have a rest mass or not. The distinction is exclusive since particles without mass, like photons and gluons for interactions, move always at speed of light c: it can never be different. On the contrary, particles with mass, like fermions for matter, have a motion velocity v necessarily inferior to light speed c, following the generic relation v=βc=(ω1212)c, in which the frequencies ω12 are hidden in the geometrical optics approximation. As a consequence of the Einstein’s program eqn. (5), a space-like bunched function u0(k0r0), with Compton’s wavelength λ0=h/m0c=hk0/2π characterizes a material quantum particle. It tends towards a Dirac’s distribution u0(k0r0) → δ(r0), without the Planck’s constant, when the very high frequency becomes infinite ω0=k0 →∝, in geometrical optics approximation. The standing wave of the field behaves then as a free classical material particle isolated in space. Such an approach, which links the point-like aspect of particles to experimental interacting or measuring conditions, appears as more physical than the difficult and controversial collapse of the wave function ψ of Quantum Mechanics, since its time-like character is not adapted to describe any space repartition. In quantum domain, in absence of space-like function describing the extension of a quantum particle, its point-like character is implicitly admitted because of the Planck’s constant in the Einstein’s relation E=hν for the photon, and in the de Broglie’s relation E=hν=mc2 for matter. A physical particle cannot be strictly point-like, since its energy density would be infinite. In quantum mechanics, where experimental behaviour is privileged, like in quantum mechanical Young double-slit, energy repartition is approximated by splitting it in two parts, instead of being considered as extended in space. The whole energy is concentrated in a mass-point, while the neglecting part of energy remains extended all around space-time. It behaves then as information which fixes implicitly interactions with space-time boundaries, according to eqns. (5) and (6) approximations [14-16]. For instance, such a split is clearly described by the Dirac’s distribution δ(x0). By integration all over space, mass is totally concentrated at x0 coordinate. As δ(x)=0 for x0≠x0, no mass-energy remains elsewhere. Nevertheless, since centuries, its hidden informative action had intrigued physicists about the least action principle controlling a mass-point behavior, even evoking teleological explanations: how the particle was aware of the far boundaries in order to adjust and minimize its path? Incompleteness of Quantum Theory Einstein considered the probabilistic experimental behaviour of quantum particles, like electrons, as a proof of the incomplete description by quantum mechanics. The statistical character of the present theory would then have to be a necessary consequence of the incompleteness of the description of the systems in quantum mechanics. However, such incompleteness does not rest in the equations of quantum mechanics themselves. From a mathematical point of view they do not need to be modified or supplemented. They are mathematically complete inside the consistent quantum framework. For instance, in Bohm's hidden variable theory, the nonlocal quantum potential Q=-(h22a)/8π2a, which constitutes an implicate hidden order in the guidance of a particle, derives from the non-relativist Schrödinger equation ih∂ψ/2π∂t=-(h22ψ)/82m, inside the solution ψ=a.expi2πS/h. The equations deduced from Einstein’s program show particularly how, and why, quantum mechanics, and more generally quantum theory formalisms, are physically incomplete [15-17]. They do not involve the space-like amplitude function u(r0), which describes a particle extension in its rest system, yielding many fundamental laws. It supplements the consistent quantum framework, based upon time-like equations: of Klein-Gordon for bosons, (from which derives the non-relativist Schrödinger equation), of Dirac for fermions, or of introduced Lagrangian densities for massive particles. As well known, the distinction between time-like and space-like characters is absolute. Consequently, the quantum framework of the Standard Model of particles, which is theoretically consistent and complete, remains unaffected by the Einstein’s program. However, the introduction of an extraneous space-like function for the elementary particles description emphasizes that their admitted size, with zero dimension, is physically only an approximation. This points out a problem which, according to Einstein, merits probably to be more investigated: Above all, however, the reader should be convinced that I fully recognize the very important progress which the statistical quantum theory has brought in physics.... this theory is until now the only one which unites the corpuscular and undulatory dual character of matter in a logically satisfactory fashion; and the (testable) relations, which are contained in it, are, within the natural limits fixed by the indeterminacy-relation, complete. The formal relations which are given in this theory—i,e, its entire mathematical formalism—will probably have to be maintained, in the form of logical inferences, in every useful future theory.
d17d1015d6074914	Quantum mechanics (Redirected from Quantum Mechanics) Printable version \| Disclaimers \| Privacy policy Quantum mechanics, a theory of modern physics formulated in the first half of the twentieth century, successfully describes the behavior of matter on small scales. It explains and quantifies three effects that classical physics cannot account for: • The values of some measurable variables of a system, most notably the total energy of a bounded system, can attain only certain discrete values determined by the system. (The smallest possible jumps in the values of those observables are called "quanta" (Latin quantum, quantity), hence the name quantum mechanics.) • Matter exhibits properties of waves (see wave-particle duality). • Certain pairs of observables, for example the position and momentum of a particle, can never be simultaneously ascertained to arbitrary precision (see Heisenberg's uncertainty principle). Description of the theory In quantum mechanics, all of these are resolved by describing the instantaneous state of a system with a wave function that encodes the probability distributions of all observables. Quantum mechanics makes predictions only about these probability distributions and not about the precise values of observables. The wavelike nature of matter is readily explained as interference effects between probability waves. Many systems that were formerly seen as changing over time (for instance, an electron circling a proton) are now described as static (a proton surrounded by a "probability cloud" describing the likelihood of locating the electron at a specific place). If the probability distributions do change over time, then the Schrödinger equation is used to describe the corresponding evolution of the wave function. Mathematical formulation In the formal mathematical theory, the state of a system is described by an element of a Hilbert space and observables are modeled as self-adjoint operators on this Hilbert space. Given a state and such an operator, the probabilities for the various outcomes of the corresponding observation can be calculated. The time evolution of a system is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the energy observable, plays a prominent role. The details of the mathematical formulation are contained in the article mathematical formulation of quantum mechanics. Quantum mechanics omits the weak nuclear force, the strong nuclear force, gravity and a full treatment of the the electromagnetic force. The principles of quantum mechanics can be applied to the well-established classical field theories. If electromagnetism is quantized, the resulting quantum field theory is called quantum electrodynamics. It is (at least in principle) able to explain the chemical elements and molecules and their properties, as well as interactions of matter and electromagnetic radiation. If the strong nuclear force is quantized, one obtains quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force can be combined, in their quantised forms, and shown to be expressions of one underlying quantum field theory: electroweak theory. The unification of these theories with gravity and hence with general relativity has so far eluded researchers (see Theory of everything). Quantum mechanical explanations for the behavior of transistors and diodes underlie all of modern technology. Quantum mechanics has been used to build lasers and electron microscopes as well as nuclear magnetic resonance imaging, which is also known as magnetic resonance imaging when used in medicine. Computational chemistry includes applied quantum mechanics done on a computer. There are efforts underway to build quantum computers which process vast amounts of data by exploiting the possibility of one system being in several states at once. Philosophical debate Quantum mechanics has provoked a strong philosophical debate. The fundamental problem is that causality and determinism is lost: while the probability distributions evolve according to a well-established deterministic law, the values of the observables themselves do not. Because of this, Albert Einstein held that quantum mechanics must be incomplete. He rejected the now standard Copenhagen interpretation by Niels Bohr which contends that quantum theory describes all there is to know about reality and that the probability statements are irreducible and do not simply reflect our limited knowledge. This interpretation further holds that the act of observation overrides the Schrödinger equation and causes the system to instantaneously change to an eigenstate (the so-called collapse of the wave function). Everett's newer many worlds interpretation holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. While the multiverse is deterministic, we perceive nondeterministic behavior governed by probabilities because we can observe only one "random" universe, with other copies of ourselves observing others. In Everett's interpretation, the act of observation is described by the regular Schrödinger equation and does not require a special treatment. While attempting to derive the correct frequency dependence of the energy emitted by a black body at a certain temperature, Max Planck in 1900 introduced the idea that energy is quantized. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. In 1913, Bohr deduced the spectral lines of the hydrogen atom again by using quantization. Louis de Broglie put forward his theory of matter-as-wave in 1924. Starting in 1925, Heisenberg developed his matrix method, while Schrödinger introduced wave mechanics and the Schrödinger equation. Schrödinger subsequently showed that the two approaches were equivalent. Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation took shape at about the same time. Paul Dirac, in 1928, unified quantum mechanics with special relativity. He also pioneered the use of operator theory, including the influential bra-ket notation. In 1932, John von Neumann formulated the rigorous mathematical basis for quantum mechanics as operator theory. Quantum electrodynamics was fully developed by Feynman, Dyson, Schwinger and Tomonaga starting in the late 1940s, and served as a role model for subsequent quantum field theories. The many worlds interpretation was formulated by Hugh Everett III in 1956. Quantum chromodynamics was proposed in 1964 by Greenberg and Nambu. Some quotations Erwin Schrödinger, speaking of quantum mechanics Niels Henrik David Bohr God does not play dice with the universe. Albert Einstein Richard Feynman Further information:
18558d7141de804e	Problems in Nonlinear Wave Propagation A Walk in Physics from Plasmas to Bose-Einstein Condensates with some Examples of Unifying Themes in Nature Doktorsavhandling, 2001 Waves are a phenomenon that can be found virtually everywhere in nature. A first description of wave propagation can be given in the linear limit, but the nonlinear regime of propagation is of the utmost importance, also in view of possible applications in several scientific fields. In the course of this work, nonlinear wave propagation in physical systems from plasmas interacting with super-intense laser light to Bose-Einstein condensates (BEC) has been investigated, making use of the analogies brought to light by the mathematical modelization of such different systems. In the case of laser-plasma interactions, the main problem is the propagation of electromagnetic waves through a plasma. The nonlinear character is due to the high laser intensity which sets the plasma electrons into relativistic motion and exerts a force strongly perturbing their equilibrium density distribution. This deeply modifies the physics of the propagation leading to effects like self-induced transparency or the generation of plasma-field structures. Self-induced transparency is originated by the relativistic quiver motion of the plasma electrons and allows light to propagate through plasmas with a density so high that light propagation would classically be impossible. We have studied the problem of a threshold for induced transparency via an exact analytical investigation which has led furthermore to an exact description of the structures (electron depletion regions and light filaments) generated in the plasma as a consequence of the interaction, for both high and low density plasmas. The physics behind the generation of these structures can be described, in the weakly nonlinear limit, by the nonlinear Schrödinger equation (NLS), one of the fundamental nonlinear equations of physics. The importance and effectiveness of analytical investigations has been demonstrated by an analysis of the NLS equation generalized to the case of multi-dimensional non conservative systems (Ginzburg-Landau equation) and applied to the description of a scheme for the amplification of laser pulses (the chirped pulse amplification scheme). Furthermore, the same mathematical structure of the NLS equation describes the physics of BEC, systems of bosonic atoms that have undergone a phase transition such that they occupy the same ground state. It is the wave nature of matter which brings to light deep analogies with the nonlinear classical physics of optics and we have made of these analogies a tool for investigating certain aspects of the nature of a condensate. Once more, the mathematical modeling of physical phenomena has revealed new features of the underlying physics. nonlinear Schroedinger equation Bose-Einstein condensates Federica Cattani Chalmers, Institutionen för elektromagnetik Elektroteknik och elektronik Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1775
1ef10c75bca26e60	Thursday, July 28, 2016 The birth of quantum holography—making holograms of single light particles July 19, 2016 Scheme of the experimental setup for measuring holograms of single photons at the Faculty of Physics, University of Warsaw. (The experiment started with a pair of photons with flat wavefronts and perpendicular polarizations. The different …more Until quite recently, creating a hologram of a single photon was believed to be impossible due to fundamental laws of physics. However, scientists at the Faculty of Physics, University of Warsaw, have successfully applied concepts of classical holography to the world of quantum phenomena. A new measurement technique has enabled them to register the first-ever hologram of a single light particle, thereby shedding new light on the foundations of quantum mechanics. Scientists at the Faculty of Physics, University of Warsaw, have created the first ever hologram of a single light particle. The spectacular experiment was reported in the prestigious journal Nature Photonics.The successful registering of the hologram of a single photon heralds a new era of quantum holography, which offers a whole new perspective on quantum phenomena. "We performed a relatively simple experiment to measure and view something incredibly difficult to observe: the shape of wavefronts of a single photon," says Dr. Radoslaw Chrapkiewicz. In standard photography, individual points of an image register light intensity only. In classical holography, the interference phenomenon also registers the phase of the light waves—it is the phase that carries information about the depth of the image. When a hologram is created, a well-described, undisturbed light wave—the reference wave—is superimposed on another wave of the same wavelength but reflected from a three-dimensional object. The peaks and troughs of the two waves are shifted to varying degrees at different points of the image. This results in interference and the phase differences between the two waves create a complex pattern of lines. Such a hologram is then illuminated with a beam of reference light to recreate the spatial structure of wavefronts of the light reflected from the object, and as such, its 3D shape. One might think that a similar mechanism would be observed when the number of creating the two waves were reduced to a minimum—that is, to a single reference photon and a single photon reflected by the object. But that is not the case. The phase of individual photons continues to fluctuate, which makes classical interference with other photons impossible. Since the Warsaw physicists faced a seemingly impossible task, they attempted to tackle the issue differently: Rather than using classical interference of electromagnetic waves, they tried to register in which the wave functions of photons interact. Hologram of a single photon: reconstructed from raw measurements (left) and theoretically predicted (right). Credit: FUW Wave function is a fundamental concept in and the core of its most important principles, the Schrödinger equation. In the hands of a skilled physicist, the function could be compared to putty in the hands of a sculptor. When expertly shaped, it can be used to 'mould' a model of a quantum particle system. Physicists are always trying to learn about the of a particle in a given system, since the square of its modulus represents the distribution of the probability of finding the particle in a particular state, which is highly useful. "All this may sound rather complicated, but in practice, our experiment is simple at its core. Instead of looking at changing light intensity, we look at the changing probability of registering pairs of photons after the quantum interference," explains doctoral student Jachura. Why pairs of photons? A year ago, Chrapkiewicz and Jachura used an innovative camera built at the University of Warsaw to film the behaviour of pairs of distinguishable and non-distinguishable photons entering a . When the photons are distinguishable, their behaviour at the beam splitter is random—one or both photons can be transmitted or reflected. Non-distinguishable photons exhibit quantum interference, which alters their behaviour. They join into pairs and are always transmitted or reflected together. This is known as two-photon interference or the Hong-Ou-Mandel effect. "Following this experiment, we were inspired to ask whether two-photon quantum interference could be used similarly to classical interference in holography in order to use known-state photons to gain further information about unknown-state photons. Our analysis led us to a surprising conclusion: it turned out that when two photons exhibit quantum interference, the course of this interference depends on the shape of their wavefronts," says Dr. Chrapkiewicz. Quantum interference can be observed by registering pairs of photons. The experiment needs to be repeated several times, always with two photons with identical properties. To meet these conditions, each experiment started with a pair of photons with flat wavefronts and perpendicular polarisations; this means that the electrical field of each photon vibrated in a single plane only, and these planes were perpendicular for the two photons. The different polarisation made it possible to separate the photons in a crystal and make one of them 'unknown' by curving their wavefronts using a cylindrical lens. Once the photons were reflected by mirrors, they were directed toward the beam splitter (a calcite crystal). The splitter didn't change the direction of vertically-polarised photons, but it did diverge diplace horizontally polarised photons. In order to make each direction equally probable and to make sure the crystal acted as a beam splitter, the planes of photon polarisation were bent by 45 degrees before the photons entered the splitter. The photons were registered using the state-of-the-art camera designed for the previous experiments. By repeating the measurements several times, the researchers obtained an interference image corresponding to the hologram of the unknown photon viewed from a single point in space. The image was used to fully reconstruct the amplitude and phase of the wave function of the unknown photon. Dr. Radoslaw Chrapkiewicz (right) and doctoral student Michal Jachura at the apparatus for registration of holograms of single photons at the Faculty of Physics, University of Warsaw. Credit: FUW, Grzegorz Krzy?ewski The experiment conducted by the Warsaw physicists is a major step toward improving understanding of the fundamental principles of quantum mechanics. Until now, there has not been a simple experimental method of gaining information about the phase of a photon's wave function. Although quantum mechanics has many applications, and it has been verified many times with a great degree of accuracy over the last century, we are still unable to explain the nature of wave functions—are they simply a handy mathematical tool, or are they something real? "Our experiment is one of the first allowing us to directly observe one of the fundamental parameters of photon's wave function—its phase—bringing us a step closer to understanding what the wave function really is," explains researcher Michal Jachura. The Warsaw physicists used quantum holography to reconstruct wave function of an individual photon. Researchers hope that in the future, they will be able to use a similar method to recreate wave functions of more complex quantum objects, such as certain atoms. Will quantum holography find applications beyond the lab to a similar extent as classical holography? Such existing practical applications include security (holograms are difficult to counterfeit), entertainment, transport (in scanners measuring the dimensions of cargo), microscopic imaging and optical data storing and processing technologies. "It's difficult to answer this question today. All of us—I mean physicists—must first get our heads around this new tool. It's likely that real applications of holography won't appear for a few decades yet, but if there's one thing we can be sure of it's that they will be surprising," summarises Prof. Konrad Banaszek. More information: "Hologram of a Single Photon"; R. Chrapkiewicz, M. Jachura, K. Banaszek, W. Wasilewski; Nature Photonics, DOI: 10.1038/nphoton.2016.129 Read more at: No comments:
30e5c1d35ebb959a	Quantum many-particle dynamics in 1D with matrix product states View the Project on GitHub amilsted/evoMPS Tutorial videos: evoMPS simulates time-evolution (real or imaginary) of one-dimensional many-particle quantum systems using matrix product states (MPS) and the time dependent variational principle (TDVP). It can be used to efficiently find ground states and simulate dynamics. The evoMPS implementation assumes a nearest-neighbour or next-nearest-neighbour Hamiltonian and one of the following situations: It is based on algorithms published by: and available on arxiv.org under arXiv:1103.0936v2. The algorithm for handling localized nonuniformities on infinite chains was developed by: and is detailed in arXiv:1207.0691. For details, see doc/implementation_details.pdf and the source code itself, which I endeavour to annotate thoroughly. evoMPS is implemented in Python using Scipy http://www.scipy.org and benefits from optimized linear algebra libraries being installed (BLAS and LAPACK). For more details, see INSTALL. evoMPS was originally developed as part of an MSc project by Ashley Milsted, supervised by Tobias Osborne at the Institute for Theoretical Physics of Leibniz Universität Hannover http://www.itp.uni-hannover.de/. The evoMPS algorithms are presented as python classes to be used in a script. Some example scripts can be found in the "examples" directory. To run an example script without installing the evoMPS modules, copy it to the base directory first e.g. under Windows:: copy examples\transverse_ising_uniform.py . python transverse_ising_uniform.py Essentially, the user defines a spin chain Hilbert space and a nearest-neighbour Hamiltonian and then carries out a series of small time steps (numerically integrating the "Schrödinger equation" for the MPS parameters):: sim = EvoMPS_TDVP_Uniform(bond_dim, local_hilb_dim, my_hamiltonian) for i in range(max_steps): my_exp_val = sim.expect_1s(my_op) Operators, including the Hamiltonian, are defined as arrays like this:: pauli_z = numpy.array([[1, 0], [0, -1]]) or as python callables (functions) like this:: def pauli_z(s, t): if s == t: return (-1.0)**s return 0 Calculating expectation values or other quantities can be done after each step as desired. Switching between imaginary time evolution (for finding the ground state) and real time evolution is as easy as multiplying the time step size by a factor of i! Please send comments to: ashmilsted at To submit ideas or bug reports, please use the GitHub Issues system http://github.com/amilsted/evoMPS/.
1936ba43e96ff701	You are currently browsing the tag archive for the ‘Larry Bird’ tag. Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and I have just uploaded to the arXiv the paper “Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation“, which we have submitted to Inventiones Mathematicae. This paper concerns the numerically observed phenomenon of weak turbulence for the periodic defocusing cubic non-linear Schrödinger equation -i u_t + \Delta u = \|u\|^2 u (1) in two spatial dimensions, thus u is a function from {\Bbb R} \times {\Bbb T}^2 to {\Bbb C}. This equation has three important conserved quantities: the mass M(u) = M(u(t)) := \int_{{\Bbb T}^2} \|u(t,x)\|^2\ dx the momentum \vec p(u) = \vec p(u(t)) = \int_{{\Bbb T}^2} \hbox{Im}( \nabla u(t,x) \overline{u(t,x)} )\ dx and the energy E(u) = E(u(t)) := \int_{{\Bbb T}^2} \frac{1}{2} \|\nabla u(t,x)\|^2 + \frac{1}{4} \|u(t,x)\|^4\ dx. (These conservation laws, incidentally, are related to the basic symmetries of phase rotation, spatial translation, and time translation, via Noether’s theorem.) Using these conservation laws and some standard PDE technology (specifically, some Strichartz estimates for the periodic Schrödinger equation), one can establish global wellposedness for the initial value problem for this equation in (say) the smooth category; thus for every smooth u_0: {\Bbb T}^2 \to {\Bbb C} there is a unique global smooth solution u: {\Bbb R} \times {\Bbb T}^2 \to {\Bbb C} to (1) with initial data u(0,x) = u_0(x), whose mass, momentum, and energy remain constant for all time. However, the mass, momentum, and energy only control three of the infinitely many degrees of freedom available to a function on the torus, and so the above result does not fully describe the dynamics of solutions over time. In particular, the three conserved quantities inhibit, but do not fully prevent the possibility of a low-to-high frequency cascade, in which the mass, momentum, and energy of the solution remain conserved, but shift to increasingly higher frequencies (or equivalently, to finer spatial scales) as time goes to infinity. This phenomenon has been observed numerically, and is sometimes referred to as weak turbulence (in contrast to strong turbulence, which is similar but happens within a finite time span rather than asymptotically). To illustrate how this can happen, let us normalise the torus as {\Bbb T}^2 = ({\Bbb R}/2\pi {\Bbb Z})^2. A simple example of a frequency cascade would be a scenario in which solution u(t,x) = u(t,x_1,x_2) starts off at a low frequency at time zero, e.g. u(0,x) = A e^{i x_1} for some constant amplitude A, and ends up at a high frequency at a later time T, e.g. u(T,x) = A e^{i N x_1} for some large frequency N. This scenario is consistent with conservation of mass, but not conservation of energy or momentum and thus does not actually occur for solutions to (1). A more complicated example would be a solution supported on two low frequencies at time zero, e.g. u(0,x) = A e^{ix_1} + A e^{-ix_1}, and ends up at two high frequencies later, e.g. u(T,x) = A e^{iNx_1} + A e^{-iNx_1}. This scenario is consistent with conservation of mass and momentum, but not energy. Finally, consider the scenario which starts off at u(0,x) = A e^{i Nx_1} + A e^{iNx_2} and ends up at u(T,x) = A + A e^{i(N x_1 + N x_2)}. This scenario is consistent with all three conservation laws, and exhibits a mild example of a low-to-high frequency cascade, in which the solution starts off at frequency N and ends up with half of its mass at the slightly higher frequency \sqrt{2} N, with the other half of its mass at the zero frequency. More generally, given four frequencies n_1, n_2, n_3, n_4 \in {\Bbb Z}^2 which form the four vertices of a rectangle in order, one can concoct a similar scenario, compatible with all conservation laws, in which the solution starts off at frequencies n_1, n_3 and propagates to frequencies n_2, n_4. One way to measure a frequency cascade quantitatively is to use the Sobolev norms H^s({\Bbb T}^2) for s > 1; roughly speaking, a low-to-high frequency cascade occurs precisely when these Sobolev norms get large. (Note that mass and energy conservation ensure that the H^s({\Bbb T}^2) norms stay bounded for 0 \leq s \leq 1.) For instance, in the cascade from u(0,x) = A e^{i Nx_1} + A e^{iNx_2} to u(T,x) = A + A e^{i(N x_1 + N x_2)}, the H^s({\Bbb T}^2) norm is roughly 2^{1/2} A N^s at time zero and 2^{s/2} A N^s at time T, leading to a slight increase in that norm for s > 1. Numerical evidence then suggests the following Conjecture. (Weak turbulence) There exist smooth solutions u(t,x) to (1) such that \\|u(t)\\|_{H^s({\Bbb T}^2)} goes to infinity as t \to \infty for any s > 1. We were not able to establish this conjecture, but we have the following partial result (“weak weak turbulence”, if you will): Theorem. Given any \varepsilon > 0, K > 0, s > 1, there exists a smooth solution u(t,x) to (1) such that \\|u(0)\\|_{H^s({\Bbb T}^2)} \leq \epsilon and \\|u(T)\\|_{H^s({\Bbb T}^2)} > K for some time T. This is in marked contrast to (1) in one spatial dimension {\Bbb T}, which is completely integrable and has an infinite number of conservation laws beyond the mass, energy, and momentum which serve to keep all H^s({\Bbb T}^2) norms bounded in time. It is also in contrast to the linear Schrödinger equation, in which all Sobolev norms are preserved, and to the non-periodic analogue of (1), which is conjectured to disperse to a linear solution (i.e. to scatter) from any finite mass data (see this earlier post for the current status of that conjecture). Thus our theorem can be viewed as evidence that the 2D periodic cubic NLS does not behave at all like a completely integrable system or a linear solution, even for small data. (An earlier result of Kuksin gives (in our notation) the weaker result that the ratio \\|u(T)\\|_{H^s({\Bbb T}^2)} / \\|u(0)\\|_{H^s({\Bbb T}^2)} can be made arbitrarily large when s > 1, thus showing that large initial data can exhibit movement to higher frequencies; the point of our paper is that we can achieve the same for arbitrarily small data.) Intuitively, the problem is that the torus is compact and so there is no place for the solution to disperse its mass; instead, it must continually interact nonlinearly with itself, which is what eventually causes the weak turbulence. Read the rest of this entry »
aef306369e2ce1b2	San José State University Thayer Watkins Silicon Valley & Tornado Alley The Hartree-Fock Method for Finding Self-Consistent Field Wave Functions for Multi-electron Atoms In 1926 Erwin Schrödinger published his work that provided physicists with a simple way to formulate the quantum dynamics of the electrons in an atom. The Schrödinger Equation is easy to formulate but in all but the simplest cases impossible to solve analytically and not easy to solve numerically. The procedure first involves declaring the Hamiltonian for the system. For example, the Hamiltonian for a helium atom is H = K1 + K2 + V1 + V2 + V12 where K1 and 2 are the kinetic energies of the two electrons. V1 and V2 are the potential energies of the two electrons in the field of the two protons in the nucleus. V12 is the potential energy due to the two electrons with respect to each other. If r1 and r2 are the distances of the two electrons from the nucleus and v1 and v2 are their velocities then the Hamiltonian reduces to H = ½mv1² + ½mv2² −2k/r1² −2k/r1² + k/(\|r1r2\|)² where r1 and r2 are the position vectors of the two electrons with respect to nucleus. The symbol k represents a constant that is the product of the constant for the electrostatic force and the square of the charge of an electron. In the late 1920's Douglas Hartree began trying to find ways to simplify the numerical solution. He formulated the concept of the Self-Consistent Field. In this method the effect on a single electron of the rest of the electrons is assumed to reduce to a central field which is added the central field established by the nucleus of the atom. Starting with an approximation of this central field the wave function of the single electron can be found. This wave function is the used to determine the central field of the other electrons. This procedure is applied iteratively until the solutions converge, or at least do not change by a significant amount. Hartree's procedure involved computing the eigenvectors and their eigenvalues of the discrete version of the self-consistent field version of the Schrödinger equation. The eigenvectors corresponded to the wavefunctions for the electron and the eigenvalues to the negative of their energies. Hartree compared the computed eigenvalues had a good correspondences with the energies of X-rays required to knock the various electrons out of their orbitals; i.e., their ionization energies. Shortly after Hartree published his method in 1928, J.A. Gaunt found that the eigenvalue for an electron could be determined to a close approximation by computing the energy of the atom with the electron and the energy of the ion in which the electron is missing. Hartree assumed that the wave function of a multi-electron atom was the product of the wave functions of the individual electrons. John Slater in 1929 found that theoretically and empirically it was better to take the multi-electron wave function as being the determinant formed from individual electron wave function, which came to be known as the Slater determinant of the system. The Slater determinant automatically produced multi-electron wave function that are anti-symmetric. Vladimir Fock modified Hartree's method so as to obtain on each step wave functions that satisfied the theoretical requirement of a solution to the n electron problem. Tjalling Koopmans published in 1934 a further refinement. He found that in general the spin-orbitals can be be chosen in a way such that a matrix of interaction energies is diagonal and thus the eigenvalues are simply equal the diagonal elements. See Koopmans' Theorem for more on this. In the 1920's there was a competition between the wave mechanics of Werner Heisenberg based on infinite matrices and the quantum mechanics of Erwin Schrödinger based on partial differential equations. Schrödinger showed that the two formulation are equivalent and over time the theory was couched in terms of Schrödinger's formulation but when anyone does numerical computation in effect they are utilizing Heisenberg's wave mechanics with the infinite matrices truncated. For understanding the computations and the approximation involved in solving a physical system a matrix formulation has definite advantages. For that reason a matrix version of a model for a helium atom is given in Matrix Model of Helium. HOME PAGE OF applet-magic HOME PAGE OF Thayer Watkins
0f3e6fb5365db903	Interferometric visibility From Wikipedia, the free encyclopedia Jump to: navigation, search The interferometric visibility (also known as interference visibility and fringe visibility, or just visibility when in context) quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals. Generally, two or more waves are combined and as the phase difference between them varies, the power or intensity (probability or population in quantum mechanics) of the resulting wave oscillates, forming an interference pattern. The pattern may be visible all at once because the phase difference varies as a function of space, as in a 2-slit experiment. Alternately, the phase difference may be manually controlled by the operator, for example by adjusting a vernier knob in an interferometer. The ratio of the size or amplitude[clarification needed] of these oscillations to the sum of the powers of the individual waves is defined as the visibility. The interferometric visibility gives a practical way to measure the coherence of two waves (or one wave with itself). A theoretical definition of the coherence is given by the degree of coherence, using the notion of correlation. Visibility in optics[edit] In linear optical interferometers[clarification needed] (like the Mach-Zehnder interferometer, Michelson interferometer, and Sagnac interferometer), interference manifests itself as intensity oscillations over time or space, also called fringes. Under these circumstances, the interferometric visibility is also known as the "Michelson visibility" [1] or the "fringe visibility." For this type of interference, the sum of the intensities (powers) of the two interfering waves equals the average intensity over a given time or space domain. The visibility is written as:[2] in terms of the amplitude envelope of the oscillating intensity and the average intensity: So it can be rewritten as:[3] where Imax is the maximum intensity of the oscillations and Imin the minimum intensity of the oscillations. If the two optical fields are ideally monochromatic (consist of only single wavelength) point sources of the same polarization, then the predicted visibility will be where and indicate the intensity of the respective wave. Any dissimilarity between the optical fields will decrease the visibility from the ideal. In this sense, the visibility is a measure of the coherence between two optical fields. A theoretical definition for this is given by the degree of coherence. This definition of interference directly applies to the interference of water waves and electric signals. Visibility in a Mach-Zehnder, Michelson or Sagnac interferometer. Visibility is similarly defined in double-slit interference. However, now the max and min vary across the interference pattern. The example shows a visibility of 80% (i.e. 0.8). Visibility in quantum mechanics[edit] Since the Schrödinger equation is a wave equation and all objects can be considered waves in quantum mechanics, interference is ubiquitous. Some examples: Bose–Einstein condensates can exhibit interference fringes. Atomic populations show interference in a Ramsey interferometer. Photons, atoms, electrons, neutrons, and molecules have exhibited interference in double-slit interferometers. Visibility in Hong–Ou–Mandel interference. At large delays the photons do not interfere. At zero delays, the detection of coincident photon pairs is suppressed. See also[edit] 1. ^ 2. ^ [1] 3. ^ [2] External links[edit]
20355d95499a7822	How Do We Perceive Words? If the meaning behind them medium is immaterial, then how do we perceive it? If the meaning behind the media is immaterial, then how do we comprehend it? If you cannot see, hear, feel, taste, or smell the meaning carried by these black symbols that you’re staring at, then why can you comprehend them? How can we comprehend the simplest of words? For example, how could the brain possibly perceive the meaning behind the word three in a way that an abacus or book or a smartphone or a laptop cannot? Or how do we perceive the meaning of the word red in a way that a camera cannot perceive it? How do we perceive the meaning of information? We’ve seen that its immaterial nature is an objective, testable, falsifiable, scientific fact. And yet we don’t know how it is possible for us to perceive the meaning behind the media of our trigonometry textbooks and music videos and fuel gauges? Even if biologists can theorize about how the brain evolved, that is totally and completely irrelevant to the fact that we are able to use our brains to process nonphysical meaning. We will listen to two primary explanations for how we comprehend meaning: that of monism and that of dualism. Monism, the theory held by Naturalists, holds that our mind are matter-in-motion and that nothing nonphysical exists. Dualism, on the other hand, holds that we have physical bodies and nonphysical minds. We’ll look at monism first. After showing how zealously they presuppose materialism, I will summarize three methods they use for explaining how our minds emerge from our brains. The scientific establishment today presupposes that we are our brains and that we will eventually discover, as a recent Scientific American article title put it, “How Matter Becomes Mind”.[i] As Vernon B. Mountcastle (1918-2015), former Professor Emeritus of Neuroscience at Johns Hopkins University, put it in 1998: Few neuroscientists now take a non-naturalist position, and still fewer hold to a principled agnosticism on the mind-brain question. The vast majority believe in physical realism and in the general idea that no nonphysical agent in the universe controls or is controlled by brains. Things mental, indeed minds, are emergent properties of brains.[ii] Now it is not just neuroscientists who take this position. A hundred years ago physicists became obsessively interested in consciousness and the curious appearance of its possible connection with physical events. But after various theories were put forward, the establishment again rested on the presupposition of a materialistic view of the mind. As Brian Greene, professor of physics at Columbia University: Somewhere between the first prokaryotic cells four billion years ago and the human brain’s ninety billion neurons entangled in a network of one hundred trillion synaptic connections, the ability emerged to think and feel, to love and hate, to fear and yearn, to sacrifice and revere, to imagine and create—newfound capacities that would ignite spectacular achievement as well as untold destruction.[iii] Although many protested this view for a while, today it is no longer debated at all. Here is Tufts University Professor Daniel Dennett claiming that this presupposition is bland: How come there are minds? And how is it possible for minds to ask and answer this question? The short answer is that minds evolved and created thinking tools that eventually enabled minds to know how minds evolved, and even to know how these tools enabled them to know what minds are…There is a winding path leading through a jungle of science and philosophy, from the initial bland assumption that we people are physical objects, obeying the laws of physics, to an understanding of our conscious minds.[iv] “Bland assumption”? To the contrary, this is an extremely consequential and profound assumption. It declares that every bit of intuition or feeling or inspiration or motivation or wisdom you have—that those are all, literally, physical phenomena. You plus your personality plus your aspirations equal a 3-pound organ that’s in charge of a body. This debate between materialism and spirituality has everything to do with what we teach our high school students about the meaning of truth, humanity, morality, sexuality, etc. Any scientist who claims to be oblivious to such authority sounds about as convincing as a teenage boy who says he only wants to study the Victoria’s Secret catalog in order to examine the photographers’ use of light and shadow. Indeed, Dr. Richard Lewontin, Evolutionary Biology Professor at Harvard University, makes it very clear that they are using this presupposition to try to usurp God’s authority: Keep in mind that many people who believe in evolution also believe in God and in spirituality. Nevertheless, Naturalists insist it is necessary to presuppose the absence of anything but the material world. They arbitrarily declare this to be a more rational, scientific, objective stance. So then, “armed” with this presupposition, Lewontin can declare that it is “trivially true” that that consciousness and rationality are physiological phenomena: It is trivially true that human cognition has evolved. The human species has evolved from nonhuman ancestors and, if we go back in time far enough, from one-celled organisms swimming in water. Those one-celled organisms certainly did not have human cognition, if they had cognition at all. They did not have a language, they did not decide to create a government, they did not engage in religious worship. Thus it must be that human cognition, like every other characteristic of the human species, has arisen during the continuous course of human evolution.[vi] Trivially true? Well, it is true that the scientific establishment has fully embraced this position. In 2013 the National Institutes for Health launched a public-private research alliance called the BRAIN Initiative (Brain Research through Advancing Innovative Neurotechnologies) for the purpose of developing treatment for mental disorders, brain diseases and brain injuries. Of course that is a wonderful goal and no doubt much wonderful medicine will come from it. Yet the scientists also included in their charter the entirely unnecessary presupposition that the mind emerges from patterns in neural circuitry: “By exploring these patterns of activity, both spatially and temporally, and utilizing simpler systems to learn how circuits function, we can generate a more comprehensive understanding of how the brain produces complex thoughts and behaviors.”[vii] No one better represents this establishment position than Kenneth R. Miller, professor of biology at Brown University and coauthor of a major high school biology textbook. He has served as an expert witness in a couple of high profile court cases about whether Intelligent Design can be taught as an alternative to Darwinism in public schools, so he is very familiar with the debates. He considers materialism and monism to be easy presuppositions on which to base his work. In his book The Human Instinct (which is different from his high school biology textbook) Miller begins his explanation of consciousness this way: Let’s assume the obvious, which is that human consciousness is a product of the workings of our nervous system as it interacts with the rest of the body and with the outside world. In other words, that consciousness is a physiological function in the broadest possible sense. What that means, of course, is that consciousness, like every other human characteristic, is a product of evolution.[viii] First Dennett said it was “bland”. Then Lewontin says it is “trivially true”. And now Miller says it is “obvious”? We are obviously nothing more than our brains? No, there is nothing remotely obvious about that. To the contrary, there might be a good, non-delusional reason that 95% of the planet believes in classical spirituality. No, the only thing that is obvious is that these naturalists are making assumptions. What is not so obvious is how the 3-pound organs in our skulls can create governments and rocket ships and artificial cheeseburgers. Now although Miller is a monist he is also a devout Catholic. That may sound curious, but such stances are not at all uncommon. Many people will fully embrace both religion and materialism at the same time. To take an example from the Bible which nuances Miller’s religious beliefs, it says that many of the religious elite who opposed Jesus of Nazareth were devout Jewish teachers, called Sadducees, who passionately argued that there was no afterlife, nor any such things as spirits or angels.[ix] For them, religion was about politics and culture and the need for a moral authority. Any talk of spirituality simply referred to emotional experiences or perhaps ethical convictions. Regardless, back to the topic of what constitutes “us”, Professor Miller teaches that we can assume materialism to be obviously true: “Consciousness is a process generated by the hugely complex interactions of highly active cells within the brain and associated nervous tissue.”[x] Now these Naturalists are all acutely aware that the alternative to presupposing materialism is nothing less that spirituality. As linguist Noam Chomsky put it: Assuming that we’re organic creatures, and not angels, we have certain fixed capacities which yield the range of abilities that we have—but they impose limits as well…[Thought] is an aspect of matter, just as electrical properties are an aspect of matter.[xi] Well, given this foundation of assumptions, how does the scientific establishment theorize about our ability to perceive meaning? They face quite a dilemma. On the one hand, they are passionately determined to explain how the mind “emerged” from the brain over millions of years of evolution. However, since they are so passionately committed to materialism, they cannot acknowledge that the thing which our minds comprehend—information, the meaning behind the medium—is immaterial. So instead of acknowledging that it is immaterial they give it all kinds of philosophical names and pass the question to philosophy, which then branches into various competing “schools of thought” about both knowledge (epistemology) and reality (ontology). By doing this they invariably skip the question “What are we perceiving?” and instead go goes straight to trying to explain how we perceive it. Nevertheless, the question remains: how to we perceive it and use it. Regardless of what we call information, how does the three-pound organ in your skull perceive and use words in the way that dictionaries and supercomputers cannot? Why do we comprehend chemistry and biology and mathematics and art? Why can we see things that movie cameras cannot see, and hear things smart phones cannot hear, and comprehend words that telecommunication satellites cannot comprehend? They call this the hard problem of consciousness. So what methods do the Naturalists use to address this problem? When Lewontin first wrote the above assertion that it was “trivially true” that our minds are physical things, his editors asked him for a bit more explanation in order to defend against creationists. He replied that that such an explanation was both futile and unnecessary: I must say that the best lesson our readers can learn is to give up the childish notion that everything that is interesting about nature can be understood. History, and evolution is a form of history, simply does not leave sufficient traces, especially when it is the forces that are at issue. Form and even behavior may leave fossil remains, but forces like natural selection do not. It might be interesting to know how cognition (whatever that is) arose and spread and changed, but we cannot know. Tough luck.[xii] So first he said that they needed to keep God’s foot out of the door by presupposing materialism “no matter how counter-intuitive, no matter how mystifying to the uninitiated.” Then he said that if we can’t understand the origins of consciousness, the reason might be that we can’t understand the origins of consciousness. Sixteen years after he gave this response to his editors, he joined Chomsky and six other scientists to acknowledge that their bland, trivial, obvious presupposition cannot even begin to provide a foundation for understanding how humans developed the ability to comprehend words and sentences and paragraphs: Now in saying that the answers may remain mired in mystery, they were still talking about how our linguistic abilities evolved, not about how the brain actually produces them. However, it did not take long for scientists to extend the argument to cover the later. In 2016, two years after the above quote, Chomsky said that we may simply not have the cognitive ability to understand cognition, especially in regard to our ability to use language creatively—i.e. to author new plans and ideas: There is interesting work on precepts for language use under particular conditions—notably intent to be informative, as in neo-Gricean pragmatics—but it is not at all clear how far this extends to the normal use of language, and in any event, it does not approach the Cartesian questions of creative use, which remains as much of a mystery now as it did centuries ago, and may turn out to be one of those ultimate secrets that ever will remain in obscurity, impenetrable to human intelligence.[xiv] Now I have no idea what “neo-Gricean pragmatics are”, but do not let that distract you from what he is saying. “Cartesian questions of creative use” refers to the questions that René Descartes (famous for saying, “I think, therefore I am”) asked about our minds’ ability to be creative and to exercise free will. Chomsky is saying we may not ever be able to understand it. He compares the revolutionary notion of physiological consciousness to the Isaac Newton’s revolutionary theories of gravity. Just as Newton’s discoveries pointed to deep mysteries that scientists could not understand at the time (not that materialists can even begin to explain the verbal laws of gravity now), so also we should be patient with things we cannot understand. In writing the forward to Chomsky’s book, Akeel Bilgrami, professor of philosophy at Columbia University, emphasized the importance of this conclusion. “It is a very important part of this methodological picture that we should learn to relax with the fact of our cognitive limits and the ‘mysteries’ that they inevitably force us to acknowledge.”[xv] In other words, even though we are faced with a profound, inexplicable mystery, we should relax and trust in our presuppositions anyway. “Thus limits on our cognition are inevitable for a variety of reasons,” Bilgrami writes, “chief among which is the taking seriously of the sheer fact that we are biological creatures.”[xvi] Just let that argument sink in for a moment. Our presuppositions lead to something inexplicable, but we should relax: that inexplicability inevitably rests on the fact that the presuppositions are true. That is a textbook example of blind faith—insisting that something is true because it is true. Given such a stubborn mindset, would you be surprised if Naturalists encourage everyone else to celebrate their own presuppositions. Listen to Michael Shermer, executive director of the Skeptics Society and a former columnist for Scientific American, explain how to handle the debate between Creationism and Naturalism: Shermer is saying that if you want to believe in God, you should more or less do it by blind faith—that is to say, with presuppositions. Religion, as he defines it, should not need any reason or evidence. In fact, he strongly implies that if your religious faith is not blind, then it is weak. So religious believers should start (like the Naturalists do) with assumptions, and they should not be interested in finding any evidence for them—especially scientific evidence. Instead, they should just embrace them as truth. Don’t worry about how counter-intuitive they might be, or how mystifying they might sound to the uninitiated. (For that matter, you can still be, like Brown University’s Professor Miller, a materialist.) Of course, they won’t call their stance blind faith. So what do they call it? The Naturalists say that we are operating on instinct. In 1848 Charles Darwin wrote to his friend, John Henslow, a Brittish priest, botanist, and geologist, saying that: “I believe there exists, and I feel within me, an instinct for the truth, or knowledge or discovery, of something of the same nature as the instinct of virtue, and that our having such an instinct is reason enough for scientific researches without any practical results ever ensuing from them.”[xviii] Now when he calls the pursuit of truth and virtue “instinctive”, he seems to be speaking figuratively and not necessarily proposing a scientific theory. His modern-day disciples, however, have concluded that our rational, linguistic, and mathematical abilities are all instinctive. Just consider the titles of some of the more robust books on the subject: • The Language Instinct by psychologist Stephen Pinker • The Number Sense by neuroscientist Stanislas Dehaene • The Math Instinct by mathematician Keith Devlin • The Human Instinct by biologist Kenneth Miller • The Consciousness Instinct by cognitive psychologist Michael Gazzaniga They all make more or less the same argument, so I will only quote the first one here and summarize some of the others later. So here is Dr. Stephen Pinker in 1994: Language is not a cultural artifact that we learn the way we learn to tell time or how the federal government works. Instead, it is a distinct piece of the biological makeup of our brains. Language is a complex, specialized skill, which develops in the child spontaneously, without conscious effort or formal instruction, is deployed without awareness of its underlying logic, is qualitatively the same in every individual, and is distinct from more general abilities to process information or behave intelligently. For these reasons some cognitive scientists have described language as a psychological faculty, a mental organ, a neural system, and a computational module. But I prefer the admittedly quaint term “instinct.” It conveys the idea that people know how to talk in more or less the sense that spiders know how to spin webs. Web-spinning was not invented by some unsung spider genius and does not depend on having had the right education or on having an aptitude for architecture or the construction trades. Rather, spiders spin spider webs because they have spider brains, which give them the urge to spin and the competence to succeed.[xix] Pinker is a great writer, and the research that leads to this conclusion is fascinating. Our ability to do language comes as effortlessly as our ability to pull our hand away from fire or our ability to drink water when we are thirsty. “The crux of the argument is that complex language is universal because children actually reinvent it, generation after generation—not because they are taught, not because they are generally smart, not because it is useful to them, but because they just can’t help it.”[xx] Again, it is wonderful to hear the scientific research that backs up that last statement. Regardless, for a materialist to call our linguistic ability instinctive is, at its best, not a scientific explanation: we can do language because we are programmed to do language? But it’s not just a redundant statement. Indeed, it does not actually make any sense…at all. For an instinct is something we do automatically, without thinking, because it is programmed into us. Beavers build dams and spiders build webs for the same reason that antilock brakes kick in when you slam on them: that’s what they are programmed to do. But it goes way beyond begging the question to use that as an explanation for how the brain comprehends words and sentences and soliloquies. The fact is that you cannot communicate without thinking. You cannot, for example, comprehend this sentence if you’re on autopilot. (And when you are on auto-pilot, you’re most assuredly thinking about something else!) Nor can you comprehend the meaning of a sentence like “What’s the square root of nine million?” without concentrating your mind just a bit. Smart phones and supercomputers can simulate such calculations instantly because that is what they are programmed to do. But they don’t perceive the meaning of those calculations any more than my colon comprehends the meaning of the Krebs cycle. Our minds, however, can perceive the meaning. Why? Why are we able to both perceive and use words in such creative and analytical ways? How do we do it? How do children do it? Pinker’s book is titled The Language Instinct, but saying that we communicate instinctively makes about as much sense as saying we can think without thinking. Yet the argument has been fully embraced by Naturalists. It has become popular for them to compare our mind’s emergence from our brains to an ant colony’s emergence from a bunch of ants. So although I will summarize some of those other books’ arguments later, I want to bring another one into the discussion here in order to explore that analogy. David Eagleman teaches as an adjunct professor in the department of Psychiatry & Behavioral Sciences at Stanford University, and also serves as the director of the Center for Science and Law. And he is CEO of NeoSensory, a company that develops devices for sensory substitution. He acknowledges that we still don’t know how our brains perceive meaning as a result of emergence, but in his book The Brain he says that he confident that we will eventually figure it out. We know a lot about the mechanics of neurons and networks and brain regions—but we don’t know why all those signals coursing around in there mean anything to us. How can the matter of our brains cause us to care about anything? The meaning problem is not yet solved. But here’s what I think we can say: the meaning of something to you is all about your webs of associations, based on the whole history of your life experiences.[xxi] Eagleman explains better than anyone else how incredibly complex this web is in our brains. (The Society of Neuroscience named him science educator of the year in 2012, and he later created the PBS documentary series The Brain with David Eagleman.) He says that if we wanted to record a high-resolution architecture of a single human brain, we would need a zettabyte of capacity—the same size as all the digital content on planet earth right now. Because it’s not just the neurons that are at work, but also chemical processes and protein changes. “The alchemy of thought, of feeling, of awareness—this emerges from quadrillions of interactions between brain cells every second: the release of chemicals, the changes in the shapes of proteins, the traveling waves of electrical activity down the axons of neurons.”[xxii] With such mind-numbing complexity, surely consciousness could emerge, right? Eagleman says emergence refers to how something completely new can arise out of different parts. It doesn’t exist in the individual parts, but it does exist in the whole, just as a thriving, competitive ant colony can emerge from thousands of ants even though the individual insects do not have a clue and are only reacting to their immediate surroundings. What is key is the interaction between the ants. And so it goes with the brain. A neuron is simply a specialized cell, just like other cells in your body, but with some specializations that allow it to grow processes and propagate electrical signals. Like an ant, an individual brain cell just runs its local program its whole life, carrying electrical signals along its membrane, spitting out neurotransmitters when the time comes for it, and being spat upon by the neurotransmission of other cells. That’s it. It lives in darkness. Each neuron spends its life embedded in a network of other cells, simply responding to signals. It doesn’t know if it’s involved in moving your eyes to read Shakespeare, or moving your hands to play Beethoven. It doesn’t know about you. Although your goals, intentions, and abilities are completely dependent on the existence of these little neurons, they live on a smaller scale, with no awareness of the thing they have come together to build.[1] Although that is an interesting analogy, it actually only address issues of biology and neuroscience, not necessarily consciousness. That is to say that it is operating on the assumption that if we can explain the brain then we can explain consciousness. Apart from that presupposition, the analogy actually works just as well in the opposite direction—for explaining why the brain does not need a central command of consciousness. Just as the individual ants are only doing what they are programmed to do, so also the colony as a whole is only doing what it is programmed to do—just as communication satellites and mars robots are only doing what they are programmed to do. None of these things are aware that they’ve been programmed, and yet the truly mysterious part of consciousness is that we are aware of it. We are aware that we are responding to our environment. After all his interesting analogies and explanations, Eagleman doesn’t even try to explain how we can perceive and integrate information. Let’s be more precise: Eagleman’s explanation mistakes the medium of information with the meaning of information. Although it’s plausible to argue that we use our neurons to process information, that is no different, in principle, from how we might use an abacus to process information. No matter how many gazillion neurons we tie together, they are still just the mediums for information. So even if, for the sake of argument, we allowed for the Naturalists’ theory that the brain emerged in all its complexity, that still actually says precisely zero about consciousness. They still have to depend 100 percent on their arbitrary presuppositions. Everywhere you look you can find systems with emergent properties. No single hunk of metal on an airplane has the property of flight, but when you arrange the pieces in the right way, flight emerges. Pieces and parts of a system can be individually quite simple. It’s all about their interaction. In many cases, the parts themselves are replaceable. What is required for consciousness? Although the theoretical details are not yet worked out, the mind seems to emerge from the interaction of the billions of pieces and parts of the brain. This leads to a fundamental question: can a mind emerge from anything with lots of interacting parts?[2] This is another textbook example of blind faith, only now it is hiding behind a bunch of complex imagery. It still requires pure presupposition to argue that consciousness will somehow, some way, instinctively, intrinsically, innately emerge from such complexity. Well, if we don’t have the cognitive ability to comprehend cognition, and we just explain it as being an instinct that emerges from billions of neurons after billions of years, the next thing to do is wax poetical and philosophical. Dr. Alan Jasanoff is an associate investigator of the McGovern Institute for Brain Research at MIT, where he is a professor of biological engineering with joint appointments in the departments of brain and cognitive sciences and of nuclear science and engineering. In the age of neuroscience, we can doubt neither the life of our minds nor the central role of our brains in it. But at the same time, we cannot doubt that external forces extend their fingers into the remotest regions of our brains, feeding our thoughts with a continuous influx of sensory input from which it is impossible to hide. We also cannot deny that each of our acts is guided by the minute contours of our surroundings, from the shapes of the door handles we use to the social structures we participate in. Science teaches us that the nervous system is completely integrated into these surroundings, composed of the same substances and subject to the same laws of cause and effect that reign at large—and that our biology-based minds are the products of this synthesis. Our brains are not mysterious beacons, glowing with inner radiance against a dark void. Instead, they are organic prisms that refract the light of the universe back into itself.[xxiii] That quote comes from Jasanoff’s book, The Biological Mind, the main point of which is that our brains are intimately connected with our bodies and our environment. Instead of saying that we are our brains, Jasanoff says that we are our bodies interacting with our environment. Regardless, when it comes to explaining what our minds are, although Jasanoff has done some wonderfully interesting research, the very best he can do is to say that they are “organic prisms that refract the light of the universe back into itself.” As beautiful as such analogies are, they nothing more than a poetic way of stating naturalism’s presupposition: the mind is a “biology-based” thing. They have not offered one scintilla of scientific data with which to back up this presupposition, much less a coherent theory of how the brain comprehends information. Yes, the brain is complex, no one questions that. What we question is how the brain comprehends immaterial phenomena—how it could be said to comprehend anything at all. Danielle S. Bassett, an associate professor in the department of bioengineering at the University of Pennsylvania, and Max Bertolero is a postdoctoral fellow in Bassett’s Complex Systems Group, recently wrote an article for Scientific American titled, “How Matter Becomes Mind”. They start with the presupposition: “In the most fundamental sense, what the brain is—and thus who we are as conscious beings—is, in fact, defined by a sprawling network of 100 billion neurons with at least 100 trillion connecting points, or synapses.”[xxiv] Throughout their article, which is very interesting and well written, they repeatedly compare consciousness to music. Put simply, your thoughts, feelings, quirks, flaws and mental strengths are all encoded by the specific organization of the brain as a unified integrated network. In sum, it is the music that your brain plays that makes you.[xxv] That’s a nice analogy. But wait, what exactly is music? Although we are most familiar with music being translated as sound waves, it can just as easily be translated onto paper. For example, Beethoven first recorded his Ninth Symphony as black symbols on paper…when he was deaf! But very few people can appreciate it in any way simply by reading it. Instead, we need it translated from a pattern of black symbols into a pattern of sound waves. But then who has the time or the money to go to a concert and listen to a hundred musicians do that?! Most of us actually need it translated from that pattern of sound waves into either a pattern of dents on a DVD or into a pattern of electromagnetic waves that we can download wirelessly onto our smartphones. Then we can use our car’s computers to translate it back into sound waves. So what exactly is that “it” that is being translating? What do the stack of paper, the sound waves, the DVD, and the electromagnetic waves all have in common? They don’t have any physical qualities in common, so whatever they do have in common is intangible. That intangible nature of music is why a DVD player doesn’t comprehend the meaning of Beethoven’s Ninth Symphony any more than a book comprehends the meaning of Shakespeare’s 9th Sonnet, right? In fact, whatever it is that does perceive music must likewise be nonphysical (i.e. you are a soul) and so you must be using your brain, in principle, the same way that you use pen and paper or a smart phone or a car. How is that possible? If we’re asking how it’s possible that Bassett and Bertolero made such a self-defeating analogy, it’s because, like so many others, they are mistaking the medium for information—in this case, music—for the meaning of information. And if we consider that music, like mathematics and language, is an immaterial phenomenon, then it again becomes clear that taking both information and our perception of information for granted is indistinguishable from taking spirituality for granted. But if we’re asking how a nonphysical mind could use a physical brain, then we should now consider the dualistic explanation of the mind-over-matter mystery. Just consider the possibility that our minds are, like information, immaterial. According to this view we use our physical brains, in principle, the same way that we use our hands and our laptops and our cars. Although it’s called dualism we should actually recognize three distinct phenomena—physical media, nonphysical meaning, and the nonphysical minds that can perceive and author meaning. But for now we will simply stick with the term dualism to refer to the physical body and the nonphysical mind. Let’s go back to square one and see if this idea is plausible and coherent. On the one hand, although the explanation comes from quantum mechanics, it is really quite simple. Instead of asking how the mind directs matter, we want to ask when. After all, we know—to the extent that we know anything at all—that we do in fact direct our bodies, and that we can use our bodies to direct everything from lawn mowers to Mars robots. So when exactly do our minds do the controlling? Well physicists discovered that before any quantum particle (the smallest physical thing we know about) materializes from a wave—quite literally materializes—there is what is called the collapse of the wave function. The wave function is a complex mathematical equation that gives the probability of finding a quantum particle in any particular location. And the key to understanding this explanation—the party of the mystery that so baffled scientists from the day they first discovered until now—is the word probability. The outcome of the wave function collapse is unpredictable and so subject to the whims of the scientists doing the experimenting. Their conscious decisions affected the materialization of matter. As one of the 20th century’s leading physicists, Henry Stapp, put it: Heisenberg’s discovery was that the process of observation—whereby an observer comes to consciously know the numerical value of a material property of an observed system—cannot be understood within the framework of materialist classical mechanics. A non-classical process is needed. This process does not construct mind out of matter, or reduce mind to matter. Instead, it explains, in mathematical terms, how a person’s immaterial conscious mind interacts with that person’s material brain.[xxvi] Because the wave function equations are about probability and thus unpredictable, that means that prior to the materialization of an electron from a wave, the equation/sentence must be completed. When that sentence is finalized the wave collapses into a particle. So when scientists realized that their decisions can affect the outcome of such equations, those decisions themselves became paramount. Yet decisions, simply put, are not physical phenomena. Stapp says that in the beginning, Heisenberg and his colleagues were completely baffled by these results, but it didn’t even occur to them to challenge the prevailing materialistic worldview. Instead, they simply identified a series of principles which would guide them in their experimentation. Only later did they realize that their conclusions demanded a paradigm shift in physics. The classical, materialistic view of the world was being replaced by something radically different. Quantum mechanics accounts with fantastic accuracy for the empirical data both old and new. The core difference between the two theories is that in the earlier classical theory all causal effects in the world of matter are reducible to the action of matter upon matter, whereas in the new theory our conscious intentions and mental efforts play an essential and irreducible causal role in the determination of the evolving material properties of the physically described world. Thus the new theory elevates our acts of conscious observation from causally impotent witnesses of a flow of physical events determined by material processes alone to irreducible mental inputs into the determination of the future of an evolving psycho-physical universe.[xxvii] Decisions can cause the materialization of matter? Yes. First, we should recognize that sentences precede everything. Just as you can’t have a pizza without a recipe or a building without a blueprint or a cat without a DNA “blueprint”, so also you can’t have a particle without a sentence. And with our decisions we can effectively author some of those sentences. That doesn’t mean that we have to be able to solve wave functions in order to use our brains any more than we have to be able to write computer code in order to use our computers. It means that just as scientists can make decisions in the laboratory that cause quantum particles to materialize, so also we can make decisions that can direct our brains. It’s just that simple. On the other hand, those wave functions are difficult to comprehend. Myself, I comprehend them about as well as I comprehend Egyptian hieroglyphics. And although I like to call them sentences whose main verb is equal, if you wrote one of them out in plain English—which I will not try to do—it would look more like a paragraph. You’ve got to study a whole lot of math before you can begin to understand them, similar to how you would have to study a whole lot of Spanish before you could understand a Mexican news show. Stapp, now age 91, is one of the physicists who first learned to read and write them. He worked closely with such twentieth century giants Werner Heisenberg, Wolfgang Pauli, and John Wheeler. He has published many papers pertaining to quantum mechanics’ non-local aspects, which reveal that an object can be moved or affected without being physically touched. He says that no physicist can deny the overwhelming empirical evidence that scientists have found for such faster-than-light action-at-a-distance. When he says that the classical theory had reduced the world to “the action of matter upon matter”, he means that prior to quantum mechanics physicists believed that everything that happened in the universe was, in principle, as predictable as billiard balls. If you are good at billiards, then you know exactly how to cause the balls hit each other so that you put one in a pocket. In similar fashion, they believed that if you were really good at physics then you could predict exactly what the billiards player himself will do. What quantum mechanics revealed was that the decision of the pool player was, in effect, quite literally unpredictable. He has free will. Although many Naturalists want to believe in free will, none of them can even begin to articulate a theory as to how a materialistic free will is possible—unless, of course, they resort to a whole lot of esoteric philosophical jargon. By contrast, Stapp says quantum mechanics reveals exactly, measurably, how an immaterial mind can cause small-scale changes in the brain that can lead to large-scale changes in the world. It is exactly this problem of the connection between physically described small-scale properties and directly experienced large-scale properties that orthodox quantum theory successfully resolves. To ignore this solution, and cling to the false precepts of classical mechanics that leave mind and consciousness completely out of the causal loop, seems to be totally irrational. What fascination with the weird and the incredible impels philosophers to adhere, on the one hand, to a known-to-be-false physical theory that implies that all of our experiences of our thoughts influencing our actions are illusions, and to reject, on the other hand, the offerings of its successor, which naturally produces an image of ourselves that is fully concordant with our normal intuitions, and can explain how bodily behavior can be influenced by felt evaluations that emerge from an aspect of reality that is not adequately conceptualized in terms of the mechanistic notion of bouncing billiard balls?[xxviii] Thus quantum mechanics confirms what we intuitively know to be true: we have free will in choosing what to do with our bodies, whether that choice is as simple as raising your hand or as complex as designing and launching a rocket ship to mars. If we allow for the presence of a soul—with genuinely subjective experiences (as opposed to calling them illusions or hallucinations) and free will and intentionality, etc.—then all the scientific facts fit together beautifully. Stapp says there is nothing goofy or weird, not to mention unscientific, about figuring an immaterial mind into the equations—literally, part of the equations. Quantum mechanics thereby provides a rational science-based escape from the philosophical, metaphysical, moral, and explanatory dead ends that are the rational consequences of the prevailing entrenched and stoutly defended in practice—although known to be basically false in principle—classical materialistic conception of the world and our place within it.[xxix] Again, this was the original, most straightforward explanation of the facts. In one of the first quantum mechanics textbooks, written in 1932, the Hungarian mathematical physicist John von Neumann explained the already massively confirmed conclusion that wave function collapse happened through the intervention of an observer rather than through static physical laws. Furthermore, it was already clear that the observer (i.e. the scientist who was doing the experiment) was “a new entity relative to the physical environment”. In the 21st century we don’t hear scientists use this sort of language, but Neumann was simply explaining what the data revealed: This next point might sound odder still, but it is interesting to hear how these scientists were processing what they learned the laboratory. Neumann went on to explain that the boundary between the observer (“the new entity relative to the physical environment”—i.e. a soul) and the observed physical system was arbitrary, but that the observer was located within a scientist’s physical body. It must be possible to describe the extra-physical process of the subjective perception…That is, we must always divide the world into two parts, the one being the observed system, the other the observer. In the former, we can follow up all physical processes (in principle at least) arbitrarily precisely. In the latter, this is meaningless. The boundary between the two is arbitrary to a very large extent…but this does not change the fact that in each method of description the boundary must be put somewhere, if the method is not to proceed vacuously, i.e., if a comparison with experiment is to be possible. Indeed experience only makes statements of this type: an observer has made a certain (subjective) observation; and never any like this: a physical quantity has a certain value.[xxxi] Now the original, orthodox explanation is called the Copenhagen Interpretation because it was largely outlined by German physicist Werner Heisenberg and Danish physicist Niels Bohr at the Niels Bohr Institute for Theoretical Physics at the University of Copenhagen in the 1920s. Of course many other physicists, including Albert Einstein, Max Born, Erwin Shrödinger, and John von Neumann, contributed to the work. The Copenhagen Interpretation does not explicitly state, as von Neumann did in his textbook, that the process of subjective perception is “extra-physical”. Instead it simply says that the act of measurement—the act of a physicist deciding to ask a question or, literally, looking for a particle in a laboratory—causes a set of statistically probable answers to reduce (“collapse”) down to one answer. It states that physical systems do not have definite, measurable properties prior to being measured. But the implication that the observer/measurer is extra-physical—that implication is unavoidable regardless of whether it is articulated. And so the Copenhagen Interpretation has to be rejected by Naturalists. It is still often taught in universities, but just as Naturalists can avoid talking about the nonphysical nature of information, so also they can avoid talking about the nonphysical nature of the observer. Instead, they simply call this mystery “the measurement problem”. Stapp emphasizes that Neumann and his colleagues were drawing conclusions without regard to any presuppositions or any “a priori adherence to material causes”. Let me repeat a quote from him that I gave when we explored the “Who?” question: “The strangle-hold of materialism was broken simply by the need to accommodate the empirical data of atomic physics, but the ontological ramifications went far deeper, into the issue of our own human nature and the power of our thoughts to influence our psycho-physical future.”[xxxii] Why don’t students really hear about any of this today? It’s all so very interesting. Why not tell them about it? The answer to that question might be exemplified by Philip Ball, an editor for the journal Nature and a columnist for Chemistry World, who wrote a fantastic book on quantum physics titled Beyond Weird. In one part of the book he explains the Copenhagen Interpretation and then reviews several of the competing ways to deal with the measurement problem. But he only touches briefly on this first one—the one that Neumann and Stapp wrote about—which he calls “mind-induced collapse”. He just summarizes it and then dismisses it. Why? In particular, mind-induced collapse seems to demand that we attribute to the mind some feature distinct from the rest of reality: to make mind a non-physical entity that does not obey the Schrödinger equation. How else could it do something to quantum processes that nothing else can? Perhaps most problematically of all, if wavefunction collapse depends on the intervention of a conscious being, what happened before intelligent life evolved on our planet? Did it then develop in some concatenation of quantum superpositions?[xxxiii] Ball dismissed this explanation because (1) it requires an immaterial mind and (2) it completely contradicts evolutionary theory by making the whole universe contingent upon the existence of (a) conscious mind(s). Being a good Naturalist, he cannot allow for either of those possibilities. None of them can. (It makes no difference that, as Chomsky, Lewontin, and six other scientists concluded as late as 2014, “the most fundamental questions about the origins and evolution of our linguistic capacity remain as mysterious as ever.”) So what do they do instead? What do they teach students? Well, today there are a couple of dozen competing interpretations of quantum mechanics—competing theories about how to solve the measurement problem—including, theories about how it produces consciousness materialistically. So it is very easy to quickly get lost and overwhelmed. But we need to understand that the original, orthodox, “mind-induced collapse” that Neumann and Stapp wrote about is not a theory of how quantum events produce consciousness materialistically, but actually just the opposite—how conscious decisions can produce quantum events. Just to be clear, let me repeat that: materialists have offered several theories about how quantum mechanics might create consciousness, but the original orthodox theory explained how an immaterial/nonphysical mind can precede and influence quantum events. Decisions made by scientists in the laboratory literally caused the unpredictable collapse of the wave function, so that an electron materialized. Although Naturalists cannot tolerate this explanation since it annihilates their presuppositions, there are many scientists who buck the establishment, arguing that mind-induced-collapse provides the most—if not the only—coherent explanation of the data. As Stephen M. Barr, professor in the Department of Physics and Astronomy at the University of Delaware explained, the original orthodox interpretation is still entirely plausible and realistic: In the opinion of many physicistsincluding such great figures in twentieth-century physics as Eugene Wigner and Rudolf Peierls—the fundamental principles of quantum theory are inconsistent with the materialist view of the human mind. Quantum theory, in its traditional, or “standard,” or “orthodox” formulation, treats “observers” as being on a different plane from the physical systems that they observe. A careful analysis of the logical structure of quantum theory suggests that for quantum theory to make sense it has to posit the existence of observers who lie, at least in part, outside of the description provided by physics. This claim is controversial. There have been various attempts made to avoid this conclusion, either by radical reinterpretations of quantum theory (such as the so-called “many-worlds interpretation”) or by changing quantum theory in some way. But the argument against materialism based on quantum theory is a strong one, and has certainly not been refuted. The line of argument is rather subtle. It is also not well- known, even among most practicing physicists. But, if it is correct, it would be the most important philosophical implication to come from any scientific discovery.[xxxiv] The work done by Eugene Wigner (1902-1995) that Barr refers to was more experimental evidence that made it clear that the scientists themselves were not simply measuring quantum events but somehow causing the events to materialize. Literally. As Wigner, a Hungarian-American theoretical physicist who received the Nobel Prize in Physics in 1963, put it, “It follows that the being with a consciousness must have a different role in quantum mechanics than the inanimate measuring device.”[xxxv] Wigner is emphasizing that the experiments not only reveal the presence of an immaterial mind but, to be more precise, an immaterial free will. That is the only way to articulate the causal gap between the questions that the scientists asked in the laboratory and the answers which experiments produced. They found not only that future physical actions were truly unpredictable but that the scientists themselves were the cause of that unpredictability. As simple as this discovery might sound, it pointed to a radical paradigm shift into our view of the physical world: our decisions are entirely free, in the most profound sense of that word, and they have a permanent impact upon future events. As Nobel Prize-winning Dutch Physicist Gerard t’Hooft put it: Indeed, when one attempts to construct models that visualize what might be going on in a quantum mechanical process, one ﬁnds that deterministic interpretations usually lead to predictions that would obey his inequalities, while it is well understood that quantum mechanical predictions violate them. In attempts to get into grips with this situation, and to derive its consequences for deterministic theories, the concept of “free will” was introduced. Basically, it assumes that any ‘observer’ has the freedom, at all times and all places, to choose, at will, what variables to observe and measure.[xxxvi] So although wave functions collapse all the time, the point is that scientists discovered that they themselves can choose to cause the collapse simply by asking questions. Now some have tried to disprove the experimental evidence for this by arguing that machines or animals can make the decision to cause the collapse. But their line of reasoning would also conclude that calculators know trigonometry, that movie cameras can see, that cranes lift up heavy objects, that oxen plow fields and police dogs search for drugs, that spiders comprehend geometry and beavers comprehend engineering, that ants comprehend not only air-conditioning systems but also colonial government, etc., etc. Yet so far as we know, humans are not only the only ones who can comprehend information and use it creatively—which includes using animals and machines in our projects and experiments—but also the only ones who can ask the questions in the laboratory. And this is observable, testable, falsifiable evidence that we have free will. This is all such excellent news. And yet the establishment simply cannot set their presuppositions aside. Stapp says the modern obsession with materialism shows a reckless, stubborn disregard for empirical evidence. But after what we have read, that should come as no surprise. If naturalists refuse to even acknowledge the immaterial nature of information, how much easier is it for them to ignore physicists’ more demanding discovery of a direct link between consciousness and our physical actions and behavior? Stapp is aghast: Given this recognized major importance of the mind-brain problem, you might think that the most up-to-date, powerful, and appropriate scientific theories would be brought to bear upon it. But just the opposite is true! Most neuro-scientific studies of this problem are based on the precepts of nineteenth century classical physics, which are known to be fundamentally false. Most neuroscientists follow the recommendation of DNA co-discoverer Francis Crick, and steadfastly pursue what philosopher of science Sir Karl Popper called “Promissory Materialism”.[xxxvii] Now remember that Ball, as a good representative the naturalistic view, had given two reasons for dismissing this original conclusion. The first was that it requires an immaterial mind. As the above quotes show, many scientists agree that an immaterial mind is indeed what the data reveal. Ball’s second reason for dismissing mind-induced collapse was that it contradicts evolutionary theory by making the whole universe contingent upon (a) conscious mind(s). That is to say that, as Eric Holloway, a fellow at the Walter Bradley Center for Natural & Artificial Intelligence, explains, the conclusions point directly to a Creator: And herein lies the rub. If human observers are necessary for physical final causality to occur, how do humans come to have the capability in the first place? This question points to a yet even higher source of final causality that extends beyond the human realm, and is responsible for the final causality that humans exhibit. Thus, these quantum physicists are showing that—far from final causality being a minor physical phenomena that can be explained away with an experiment—our entire universe is imbued with final causality within its very fabric and this final causality must come from some source beyond the universe.[xxxviii] So if we simply examine the facts, they point to a conscious, immaterial, intelligent Author for the entire universe. Stapp says that is the very straightforward extension of the interpretation of the data that Heisenberg and his colleagues discovered. This situation is concordant with the idea of a powerful God that creates the universe and its laws to get things started, but then bequeaths part of this power to beings created in his own image, at least with regard to their power to make physically efficacious decisions on the basis of reasons and evaluations. I see no way for contemporary science to disprove, or even render highly unlikely, this religious interpretation of quantum theory, or to provide strong evidence in support of an alternative picture of the nature of these ‘free choices’. These choices seem to be rooted in reasons that are rooted in feelings pertaining to value or worth. Thus it can be argued that quantum theory provides an opening for an idea of nature and of our role within it that is in general accord with certain religious concepts, but that, by contrast, is quite incompatible with the precepts of mechanistic deterministic classical physics. Thus the replacement of classical mechanics by quantum mechanics opens the door to religious possibilities that formerly were rationally excluded. This conception of nature, in which the consequences of our choices enter not only directly in our immediate neighborhood but also indirectly and immediately in far-flung places, alters the image of the human being relative to the one spawned by classical physics. It changes this image in a way that must tend to reduce a sense of powerlessness, separateness, and isolation, and to enhance the sense of responsibility and of belonging. Each person who understands him-or herself in this way, as a spark of the divine, with some small part of the divine power, integrally interwoven into the process of the creation of the psycho-physical universe, will be encouraged to participate in the process of plumbing the potentialities of, and shaping the form of, the unfolding quantum reality that it is his or her birthright to help create.[xxxix] Now as to our place in the world, so far as we know it is confined to our bodies. So what would a disembodied mind/soul be able to do? Where would it exist? I am quite sure that I definitely will not try to speculate on such matters. We have these profoundly complex, amazingly powerful brains—organs which the materialists have done amazing work in studying and explaining—that we can use to put men on the moon and to compose symphonies and to compose chili. But after our bodies die? That’s an excellent question. [1] IBID, 213-214. [2] IBID, 214-215. [i] “How Matter Becomes Mind,” by Max Bertolero and Danielle Bassett. Scientific American, July 2019, Volume 320 Number 6. Pp. 26-33. [ii] Vernon B. Mountcastle, “Brain Science at the Century’s Ebb”, Daedalus Vol. 127, No. 2, The Brain (Spring, 1998), pp. 1-36. (The MIT Press on behalf of American Academy of Arts & Sciences) 1. ( [iii] Brian Greene, Until the End of Time (New York: Alfred A. Knopf, 2020), Kindle Edition, location 2004. [iv] Daniel Dennett, From Bacteria to Bach and Back (New York: W.W. Norton & Company, 2018) Kindle Location 184-195. [v] Richard Lewontin, “Billions and Billions of Demons,” a review of The Demon-Haunted World (by Carl Sagan, 1997), The New York Review of Books, January 9, 1997, 31. [vi] Richard Lewontin, “The Evolution of Cognition: Questions We Will Never Answer,” An Invitation to Cognitive Science, Volume 4, edited by Daniel N. Osherson, Don Scarborough, Saul Sternberg (Cambridge, MA: The MIT Press, 1998) 108. [viii] Kenneth R. Miller, The Human Instinct (New York: Simon & Schuster, 2018), 150. [ix] Matthew 22:23; Mark 12:18; Luke 20:27; Acts 23:8 [x] IBID 168. [xiii] Marc D. Hauser, Charles Yang, Robert C. Berwick, Ian Tattersall, Michael J. Ryan, Jeffrey Watumull, Noam Chomsky, and Richard C. Lewontin, “The mystery of language evolution,” 2014. ( [xiv] Noam Chomsky, What Kind of Creatures Are We? (New York: Columbia University Press, 2016) 128. [xv] Noam Chomsky, What Kind of Creatures Are We? (New York: Columbia University Press, 2016) Kindle Location 172. [xvi] Noam Chomsky, What Kind of Creatures Are We? (New York: Columbia University Press, 2016) Kindle Location 221 [xvii] Michael Shermer, Why People Believe Weird Things (New York: St. Martin’s Griffin, 2002), 135. [xviii] Charles Darwin, The Correspondence of Charles Darwin, Vol. 4. (1847-50), Frederick Burkhardt and Sydney Smith, editors (London: Cambridge University Press, 1989); [xix] Steven Pinker, The Language Instinct (New York: HarperCollins, 1994), 4-5. [xx] IBID 20. [xxi] David Eagleman, The Brain (New York: Pantheon Books, 2017), 35. [xxii] IBID 201. [xxiii] Alan Jasanoff, The Biological Mind (New York: Basic Books, 2018), 170. [xxiv] Max Bertolero and Danielle Bassett, “How Matter Becomes Mind,” Scientific American, July 2019, Volume 320 Number 6. (pp. 28) [xxv] IBID 32. [xxvi] Henry Stapp, Quantum Theory and Free Will (Springer International Publishing, 2017), Kindle Locations 339-342. [xxvii] Henry Stapp, Quantum Theory and Free Will (Springer International Publishing, Kindle Edition, 2017), Kindle Locations 47-52. Physicists talk about time evolution, which is when a wave collapses at a particular configuration of space (the most “fit” configuration) to form a particle. [xxviii] Henry Stapp, “Minds and Values in the Quantum Universe,” in Information and the Nature of Reality, ed. by Paul Davies and Niels Henrik Gregersen (New York: Cambridge University Press, 2010), 108. [xxix] Henry Stapp, Quantum Theory and Free Will (Springer International Publishing, 2017), Kindle Locations 1706-1708). [xxx] John von Neumann, Mathematical Foundations of Quantum Mechanics, published 1932, translated from the German edition by Robert T. Beyer in 1949 (Princeton, NJ: Princeton University Press, 1983), 418. [xxxi] IBID 419-420. [xxxii] IBID 759-761. [xxxiii] Philip Ball, Beyond Weird: Why everything you thought you knew about quantum physics is different (Chicago, IL: The University of Chicago Press, 2018), 118. [xxxiv] Stephen M. Barr, Modern Physics and Ancient Faith (Notre Dame: University of Notre Dame Press, 2003), 27-28. [xxxv] Eugene Wigner, “Remarks on the Mind-Body Question”, Eugene Wigner, in John Wheeler and Wojciech Hubert Zurek, Quantum Theory and Measurement (Princeton: Princeton University Press, 1983), p. 180. [xxxvi] Gerard t’Hooft, “On the Free-Will Postulate in Quantum Mechanics”, arXiv (January 15, 2007) [xxxvii] Henry Stapp, Quantum Theory and Free Will (Springer International Publishing, 2017), Kindle Locations 870-874. Recent Posts
d228e372c68d78f6	Adiabatic and diabatic responses of H2+ to an intense femtosecond laser pulse: Dynamics of the electronic and nuclear wave packet Isao Kawata, Hirohiko Kono, Yuichi Fujimura Research output: Contribution to journalArticle 141 Citations (Scopus) We investigate the quantal dynamics of the electronic and nuclear wave packet of H2+ in strong femtosecond pulses (≥ 1014W/cm2). A highly accurate method which employs a generalized cylindrical coordinate system is developed to solve the time-dependent Schrödinger equation for a realistic three-dimensional (3D) model Hamiltonian of H2+. The nuclear motion is restricted to the polarization direction z of the laser electric field E(t). Two electronic coordinates z and ρ and the internuclear distance R are treated quantum mechanically without using the Born-Oppenheimer approximation. As the 3D packet pumped onto 1 σu moves toward larger internuclear distances, the response to an intense laser field switches from the adiabatic one to the diabatic one; i.e., electron density transfers from a well associated with a nucleus to the other well every half optical cycle, following which interwell electron transfer is suppressed. As a result, the electron density is asymmetrically distributed between the two wells. Correlations between the electronic and nuclear motions extracted from the dynamics starting from 1 σu can be clearly visualized on the time-dependent "effective" 2D surface obtained by fixing ρ in the total potential. The 2D potential has an ascending and descending valley along z = ±R/2 which change places with each other every half cycle. In the adiabatic regime, the packet starting from 1 σu stays in the ascending valley, which results in the slowdown of dissociative motion. In the diabatic regime, the dissociating packet localized in a valley gains almost no extra kinetic energy because it moves on the descending and ascending valleys alternately. Results of the 3D simulation are also analyzed by using the phase-adiabatic states \|1〉 and \|2〉 that are adiabatically connected with the two states 1 σg and 1 σu as E(t) changes. The states \|1〉 and \|2〉 are nearly localized in the descending and the ascending valley, respectively. In the intermediate regime, both \|1〉 and \|2〉 are populated because of nonadiabatic transitions. The interference between them can occur not only at adiabatic energy crossing points but also near a local maximum or minimum of E(t). The latter type of interference results in ultrafast interwell electron transfer within a half cycle. By projecting the wave packet onto \|1〉 and \|2〉, we obtain the populations of \|1〉 and \|2〉, P1 and P2, which undergo losses due to ionization. The two-state picture is validated by the fact that all the intermediates in other adiabatic states than \|1〉 and \|2〉 are eventually ionized. While E(t) is near a local maximum, P2 decreases but P1 is nearly constant. We prove from this type of reduction in P2 that ionization occurs mainly from the upper state \|2〉 (the ascending well). Ionization is enhanced irrespective of the dissociative motion, whenever P2 is large and the barriers are low enough for the electron to tunnel from the ascending well. The effects of the packet's width and speed on ionization are discussed. Original languageEnglish Pages (from-to)11152-11165 Number of pages14 JournalJournal of Chemical Physics Issue number23 Publication statusPublished - 1999 Jun 15 ASJC Scopus subject areas • Physics and Astronomy(all) • Physical and Theoretical Chemistry Fingerprint Dive into the research topics of 'Adiabatic and diabatic responses of H<sub>2</sub><sup>+</sup> to an intense femtosecond laser pulse: Dynamics of the electronic and nuclear wave packet'. Together they form a unique fingerprint. • Cite this
70b0c9efc80e98cb	01206nas a2200133 4500008004300000245007600043210006900119260001300188520077600201100002100977700001900998700001901017856003601036 2003 en_Ud 00aEffective dynamics for Bloch electrons: Peierls substitution and beyond0 aEffective dynamics for Bloch electrons Peierls substitution and bSpringer3 aWe consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\\\\phi(\\\\epsi x)$, and vector potential $A(\\\\epsi x)$, with $x \\\\in \\\\R^d$ and $\\\\epsi \\\\ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\\\\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\\\\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/304001403nas a2200133 4500008004300000245004000043210003900083260002400122520102800146100002101174700001901195700001901214856003601233 2003 en_Ud 00aSpace-adiabatic perturbation theory0 aSpaceadiabatic perturbation theory bInternational Press3 aWe study approximate solutions to the Schr\\\\\\\"odinger equation $i\\\\epsi\\\\partial\\\\psi_t(x)/\\\\partial t = H(x,-i\\\\epsi\\\\nabla_x) \\\\psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\\\\Hi_{\\\\rm f}$ of fast ``internal\\\'\\\' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \\\\cite{NS} we prove that interband transitions are suppressed to any order in $\\\\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\\\\mathbb{R}^d,\\\\Hi _{\\\\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/304101131nas a2200133 4500008004100000245006000041210005900101260003000160520071200190100002100902700001900923700001900942856003600961 2002 en d00aSpace-adiabatic perturbation theory in quantum dynamics0 aSpaceadiabatic perturbation theory in quantum dynamics bAmerican Physical Society3 aA systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/5985

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