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f1fb124e1e9ca823	Discussing the Turing Paradox HABIB UNIVERSITY: There is little doubt that Quantum Mechanics will reshape the face of the planet and humanity in the coming decades. The current hardware technology is reaching its limit in exploiting Moore’s law to make more compact and powerful computation devices. As boundaries of knowledge are being pushed further, the need to have more powerful computers is immense. Quantum Computing technology is rapidly developing as the future of computation, however, a real life Quantum Computer is far from replacing our current computers. Dr. Adam Zaman Chaudhry’s talk on ‘The Effect of Repeated Measurements in Quantum Systems’, here at Habib University’s Soorty Lecture Theater, turned into an intensely interactive session as professionals and students from around Karachi took the opportunity to gain a deeper understanding of the world of Quantum Computing. Organized by Habib University’s School of Science & Engineering (SSE) as part of the SSE Public Lecture Series, the talk attracted a large audience of students from both Habib University and other educational institutions across Karachi. Computation in Quantum Computers involves measuring the state of an underlying quantum system. A quantum system simultaneously exists in multiple states. When a measurement is made on the system, it assumes one of the possible states available by collapsing the wave function of the system. If the system is described at any point in time, it can be described at any other point using the time evolution operator in the Schrödinger equation. Rapid measurements on the quantum systems freezes the evolution of the system in time due to repeated collapse of the wave function; this is called the Quantum Zeno Effect (QZE). However, if no measurements are made then the states of the system rapidly evolve at an accelerating rate; this is called the Quantum Anti-Zeno Effect (QAZE). Such rapid evolution of a quantum system can threaten the coherence between the states of the system. Thus both these effects are important parameters in developing computation techniques in Quantum Computers. Both Zeno and Anti-Zeno effects have been studied in a wide variety of quantum systems. In his talk, Dr. Zaman spoke about the basics of the concept of measurement in Quantum Mechanics. He further went on to elaborate on the Zeno and Anti-Zeno Effect. He explained how the decay time of quantum systems has always been studied in context of particular quantum systems like superconducting qubits, nano-mechanical oscillators, Josephson junction etc. He then further explained his work in which, for the first time, these effects have been studied in a general framework which is applicable to weakly interacting quantum systems. He concluded his talk by elaborating on his plans to study this problem in a more general way by extending the formalism to strongly interacting systems. Comments are closed.
e9dadcb59fc5708a	Chapter 5. Quantum chemistry in Molecular Modeling 5.5 Energy calculations Ab initio calculations give the absolute energy of the system of fixed nuclei and moving electrons. These are large numbers, for example for cyclohexane the HF energy with the 6-31G* basis set is -234.2080071 a.u., which is equal to 146967.86 kcal/mol. Thus, the chemically significant energy quantities of a few kcal/mol are very much smaller than the computed quantity, and high accuracy is required. The absolute energy is not a directly useful quantity. It can however be used to calculate the Heat of Formation with a reasonable accuracy. According to G1 and G2 theories [14, 15] the molecular structure and vibrational frequencies are first determined at the HF/6-31G* level. The frequencies are used to calculate the zero-point energy. Then, the geometry is further optimized at the MP2 level. Subsequently, basis set effects and correlation energies are calculated at various levels of theory, to allow an extrapolation (using small empirical contributions !) to the limits of full CI and the Hartree-Fock limit, that is to the complete Schrödinger equation for the motionless molecule. Finally, the zero-point vibrational energy is added. This procedure can account for heats of formation with an accuracy of < 2 kcal/mol, which rivals the quality of experimental data. Other authors [16] calculate the heat of formation based on the 6-31G* calculation and bond increments, similar to the way MM2 deals with this. This is a much less elaborate procedure than the G1 and G2 theories, but it is essentially empirical. The empirical corrections needed in G1 and G2 are of a very "mild" kind, they are not related to the structure of the species, but only depend on the number of electrons. The isodesmic reaction approach allows a fairly accurate calculation of the heat of reactions, even at the HF level. Isodesmic reactions are defined as transformations in which the numbers of bonds of each formal type are conserved, and only the relationships among the bonds are altered [4, 6]. For example : CH4 + CH3CH2OH --> CH3CH3 + CH3OH (1) CF4 + 3 CH4 --> 4 CH3F (2) Energy changes (kcal/mol) for these two reactions are : STO-3G 3-21G 6-31G*//STO-3G experimental deltaE (1) 2.6 4.8 4.1 5.0 (5.7) deltaE (2) 53.5 62.4 49.6 49.3 (52.8) (The experimental numbers in parentheses are without correction for zero-point energy changes). [4] W.J. Hehre, L. Radom, P. von R. Schleyer and J.A. Pople Ab initio molecular orbital theory, Wiley, 1986 [6] Spartan User's Guide, version 3.0, Wavefunction, Inc., 1993. [14] Pople, J.A.; Head-Gordon, M.; Fox, D.J.; Raghavachari, K.; Curtiss, L.A., J. Chem. Phys., 1989, 90, 5622 - 5629. [15] Curtiss, L.A.; Raghavachari, K.; Trucks, G.W.; Pople, J.A., J. Chem. Phys., 1991, 94, 7221. Next paragraph, 5.6 Quality of ab initio results Previous paragraph 5.4 Limitations of the HF method; Electron correlation. Chapter 5 MM Syllabus 1995 MODIFIED November 8, 1995 Fred Brouwer, Lab. of Organic Chemistry, University of Amsterdam.
7ab95b8f9aafb425	Subscribe to our newsletter: Physics to Go! Part 1 iPhone / iPad • Entertainment • Education I want this app Download from AppStore Physics to Go! Part 1   Interactive Quantum Mechanics These are the basic controls of the app: home page -    set the range of the x-value for which the Schrödinger equation is calculated (box size) with xmin, xmax   set basic step size for the time evolution   set the number of points of the wave function to be calculated. A larger number slows down the calculation and uses up more memory, but can increase the accuracy of the calculation.    pick from a list of predefined potentials or choose "user defined" and type a formula into the potential text field like, e.g., x*x .  Pressing "Graph" brings you to the plotting window. The graph displays the potential, the wave function (blue real part, red imaginary part), and the energy of the system (indicated as white bar). The left y axis denotes the values of the potential and energy, the right one corresponds to the value of the wave function. "PLOT!” starts the plotting. With the +/- stepper you can choose to calculate the ground state or one of the first nine excited states. Note that in order to get the correct, e.g., third excited state, you first have to calculate the ground, first and second excited state in this sequence. You can choose between real time solution of the Schrödinger equation or imaginary time, which relaxes the state to the ground, or chosen excited, state. As you might observe, in a real time calculation, after a while the wave function tends to relax into these states as well. This is essentially due to an artefact of the numerical procedure for solving the equations, effectively introducing a small imaginary part to the time. In principle, with more effort this could/should be done better. A few remarks about the underlying physics. The app solves the time-dependent Schrödinger equation, working best for imaginary time. Essentially, in the latter case it solves for the eigenstates and the corresponding eigenvalues of the energy of the system. For more on the math, interpretation, … there are infinitely many documents on the web, just look at the entry for the Schrödinger equation in Wikipedia or any other sites. Try this with the app! You’ll observe some deviations, which is due to the imperfect numerical scheme and sometimes also because the calculation is done in a finite box in the x direction, setting the wave function to zero at the edges of the box. In spite of the app's shortcomings, it should still be quite entertaining and also instructive to play around with the app. For instance, look at the gaussian potential. You see that the excited states have energies higher than the potential. Thus, without finite box, the particle could not be kept in the potential well anymore but would escape. If you construct a very narrow potential well, this could even happen with the ground state of the system. Then, no state can be localised by the potential (although a classical particle at minimum energy would just sit at the lowest point of the potential well). This is a striking example of the effect of the quantum zero point energy. Thus, if you squeeze some solid states, which usually would form a crystal, more and more, going to very high densities, the potential well might narrow so that the ground state energy is above the potential, transforming the substance back again to a liquid at very high densities due to the zero-point motion even if the temperature of the system is at zero degree Kelvin (this might happen with lattices of atomic nuclei in the outer regions of neutron stars, for example)! You can also observe, how the wave function of the particle leaks into x-ranges, which a classical particle could not access, and might even connect two separate potential wells (as in the double-well example), which is a case of the famous quantum tunnelling through barriers that could not be traversed in classical physics. Enjoy the app!
fa88af8e75dc4901	Tag Archives: mechanics What If Everyone JUMPED At Once? Published on Aug 18, 2012 Follow Michael Stevens http://www.twitter.com/tweetsauce Tell Geek and Sundry that Vsauce says “hi!” and SUB!:http://www.youtube.com/watch?v=6-Wef0… music courtesy of http://www.SoundCloud.com/JakeChudnow Thanks to: For helping me with this video at Summer in the City!! theINENIvlogs behind the scenes Summer in the City vid with me:http://www.youtube.com/watch?v=F_Rd35… Why videos views freeze in the 300s on YouTube (sub to this channel FTW, seriously): http://www.youtube.com/watch?v=oIkhga… Japan Earthquake and Earth’s rotation:http://en.wikipedia.org/wiki/2011_T%C… All people in one place LIVING:http://persquaremile.com/2011/01/18/i… shoulder-to-shoulder: http://news.nationalgeographic.com/ne… BBC Jump Video: http://www.bbc.co.uk/learningzone/cli… SCALE OF UNIVERSE AWESOME: http://htwins.net/scale2/ STRAIGHTDOPE article on a jump:http://www.straightdope.com/columns/r… Dot Physics on the jump: http://www.wired.com/wiredscience/201… Interactive scale of the universe: http://scaleofuniverse.com/ Decimate: http://www.etymonline.com/index.php?a… Dunbar’s Number: http://en.wikipedia.org/wiki/Dunbar&#… NPR story on Dunbar’s number:http://www.npr.org/2011/06/04/1367233… Life Expectency: http://en.wikipedia.org/wiki/Life_exp… How many people you meet in your life:http://eclectic24.wordpress.com/2009/… Newton’s Third Law: http://www.physicsclassroom.com/class… Stable Orbits Published on May 3, 2012 If gravity is so attractive, why doesn’t the earth just crash into the sun? Or the moon into the earth? The answer: Stable Orbits hyperbolic funnel video: http://bit.ly/r5xhng And facebook – http://facebook.com/minutephysics And twitter – @minutephysics floating pyramids Physics of Floating Pyramids 1: Motor radial de un avión 2: Distribución oval 3: Principio de la máquina de coser 4: Movimiento de Cruz de Malta – de la mano del segundero, que controla al reloj 5: Mecanismo de cambio de velocidades (automóvil) 6: Junta universal para velocidad constante automática 7: Sistema de carga de proyectiles 8: Motor giratorio – motor de combustión interna, el calor y no el movimiento del pistón, causa el movimiento giratorio 9: Motor en línea – cilindros alineados en forma paralela Poisson bracket mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. It places mechanics and dynamics in the context of coordinate-transformations: specifically in coordinate planes such as canonical position/momentum, or canonical-position/canonical transformation. (A so-called “canonical transformation” is a function of the canonical position and momentum satisfying certain Poisson-bracket relations). Note that one example of a canonical transformation is the Hamiltonian itself: H = H(q,p;t)\,. Namely: the Hamiltonian-canonical-transformation transforms canonical position/momenta into the conserved (constant-of-time-integration) quantity known as “energy”. In a more general sense: the Poisson bracket is used to define a Poisson algebra, of which the Poisson manifolds are a special case. These are all named in honour of Siméon-Denis Poisson. Quantum particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a heavy box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never “sit still”. Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes. The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. This means that the observable properties of the particle (such as its energy and position) are related to the mass of the particle and the width of the well by simple mathematical expressions. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. See also: the history of quantum mechanics. One-dimensional solution In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction.[3] The wavefunction ψ(x,t) can be found by solving the Schrödinger equationfor the system where \hbar is the reduced Planck constantm is the mass of the particle, i is the imaginary unit and t is time. \psi(x,t) = [A \sin(kx) + B \cos(kx)]\mathrm{e}^{-\mathrm{i}\omega t},\; where A and B are arbitrary complex numbers. The frequency of the oscillations through space and time are given by the wavenumber k and the angular frequency ω respectively. These are both related to the total energy of the particle by the expression E = \hbar\omega = \frac{\hbar^2 k^2}{2m}, The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by P(x,t) = \| ψ(x,t) \| 2. The wavefunction must therefore vanish everywhere beyond the edges of the box.[1][4] Also, the amplitude of the wavefunction may not “jump” abruptly from one point to the next.[1] These two conditions are only satisfied by wavefunctions with the form \psi_n(x,t) = \begin{cases} A \sin(k_n x)\mathrm{e}^{-\mathrm{i}\omega_n t}, & 0 < x < L,\\ 0, & \text{otherwise,} \end{cases} where n is a positive, whole number. The wavenumber is restricted to certain, specific values given by[5] k_n = \frac{n \pi}{L}, \quad \mathrm{where} \quad n = \{1,2,3,4,\ldots\}, where L is the size of the box.[7] Negative values of n are neglected, since they give wavefunctions identical to the positive n solutions except for a physically unimportant sign change.[6] Finally, the unknown constant A may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that \left\| A \right\| = \sqrt{\frac{2 }{L}}. Thus, A may be any complex number with absolute value √(2/L); these different values of A yield the same physical state, so A = √(2/L) can be selected to simplify. Energy levels E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}. The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionless at the bottom of the box. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.[9] Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.[8] The kinetic energy of a particle is given by Ep2 / (2m), and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.[8] Spatial location In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as P(x) = \| ψ(x) \| 2. For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by P_n(x) = \begin{cases} \frac{2 }{L}\sin^2\left(\frac{n\pi x}{L}\right); & 0 < x < L \\ 0; & \text{otherwise}. \end{cases} Thus, for any value of n greater than one, there are regions within the box for which P(x) = 0, indicating that spatial nodes exist at which the particle cannot be found. \langle x \rangle = \int_{-\infty}^{\infty} \psi^(x) x \psi(x)\,\mathrm{d}x. For the particle in a box, it can be shown that the average position is always \langle x \rangle = L/2, regardless of the state of the particle. In other words, the average position at which a particle in a box may be detected is exactly in the center of the quantum well; in agreement with a classical system. Higher-dimensional boxes If a particle is trapped in a two-dimensional box, it may freely move in the x and y-directions, between barriers separated by lengths Lx andLy respectively. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies are given respectively by \psi_{n_x,n_y} = \sqrt{\frac{4}{L_x L_y}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y\right), E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m}, where the two-dimensional wavevector is given by \mathbf{k_{n_x,n_y}} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}. For a three dimensional box, the solutions are \psi_{n_x,n_y,n_z} = \sqrt{\frac{8}{L_x L_y L_z}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y \right) \sin \left( k_{n_z} z \right), E_{n_x,n_y,n_z} = \frac{\hbar^2 k_{n_x,n_y,n_z}^2}{2m}, where the three-dimensional wavevector is given by \mathbf{k_{n_x,n_y,n_z}} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} + k_{n_z}\mathbf{\hat{z}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}} + \frac{n_z \pi }{L_z} \mathbf{\hat{z}}. An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. LxLy), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with nx = 2,ny = 1 has the same energy as the wavefunction with nx = 1,ny = 2. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation. Modern Quantum Mechanics (2nd Edition) Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3 Introduction to Quantum Mechanics (2nd Edition) Introductory Quantum Mechanics (4th Edition) Stark effect The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects. The first-order effect is linear in the applied electric field, while the second-order effect is quadratic in the field. The Stark effect is responsible for the pressure broadening (Stark broadening) of spectral lines by charged particles. When the split/shifted lines appear in absorption, the effect is called the inverse Stark effect. The Stark effect is the electric analogue of the Zeeman effect where a spectral line is split into several components due to the presence of a magnetic field. The Stark effect can be explained with fully quantum mechanical approaches, but it has also been a fertile testing ground for semiclassical methods. Classical electrostatics The Stark effect originates from the interaction between a charge distribution (atom or molecule) and an external electric field. Before turning to quantum mechanics we describe the interaction classically and consider a continuous charge distribution ρ(r). If this charge distribution is non-polarizable its interaction energy with an external electrostatic potential V(r) is E_{\mathrm{int}} = \int \rho(\mathbf{r}) V(\mathbf{r}) d\mathbf{r}.\, If the electric field is of macroscopic origin and the charge distribution is microscopic, it is reasonable to assume that the electric field is uniform over the charge distribution. That is, V is given by a two-term Taylor expansion, V(\mathbf{r}) = V(\mathbf{0}) - \sum_{i=1}^3 r_i F_i \quad \hbox{with the electric field:}\quad F_i \equiv -\left. \left(\frac{\partial V}{\partial r_i} \right)\right\|_{\mathbf{0}}, where we took the origin 0 somewhere within ρ. Setting V(\mathbf{0}) as the zero energy, the interaction becomes E_{\mathrm{int}} = - \sum_{i=1}^3 F_i \int \rho(\mathbf{r}) r_i d\mathbf{r} \equiv - \sum_{i=1}^3 F_i \mu_i = - \mathbf{F}\cdot \boldsymbol{\mu}. Here we have introduced the dipole moment μ of ρ as an integral over the charge distribution. In case ρ consists of N point charges qj this definition becomes a sum \boldsymbol{\mu} \equiv \sum_{j=1}^N q_j \mathbf{r}_j. Perturbation theory Turning now to quantum mechanics we see an atom or a molecule as a collection of point charges (electrons and nuclei), so that the second definition of the dipole applies. The interaction of atom or molecule with a uniform external field is described by the operator V_{\mathrm{int}} = - \mathbf{F}\cdot \boldsymbol{\mu}. This operator is used as a perturbation in first- and second-order perturbation theory to account for the first- and second-order Stark effect. First order Let the unperturbed atom or molecule be in a g-fold degenerate state with orthonormal zeroth-order state functions \psi^0_1, \ldots, \psi^0_g . (Non-degeneracy is the special case g = 1). According to perturbation theory the first-order energies are the eigenvalues of the gg matrix with general element (\mathbf{V}_{\mathrm{int}})_{kl} = \langle \psi^0_k \| V_{\mathrm{int}} \| \psi^0_l \rangle = -\mathbf{F}\cdot \langle \psi^0_k \| \boldsymbol{\mu} \| \psi^0_l \rangle, \qquad k,l=1,\ldots, g. If g = 1 (as is often the case for electronic states of molecules) the first-order energy becomes proportional to the expectation (average) value of the dipole operator \boldsymbol{\mu}, E^{(1)} = -\mathbf{F}\cdot \langle \psi^0_1 \| \boldsymbol{\mu} \| \psi^0_1 \rangle = -\mathbf{F}\cdot \langle \boldsymbol{\mu} \rangle. Because a dipole moment is a polar vector, the diagonal elements of the perturbation matrix Vint vanish for systems with an inversion center (such as atoms). Molecules with an inversion center in a non-degenerate electronic state do not have a (permanent) dipole and hence do not show a linear Stark effect. In order to obtain a non-zero matrix Vint for systems with an inversion center it is necessary that some of the unperturbed functions \psi^0_i have opposite parity (obtain plus and minus under inversion), because only functions of opposite parity give non-vanishing matrix elements. Degenerate zeroth-order states of opposite parity occur for excited hydrogen-like (one-electron) atoms. Such atoms have the principal quantum number n among their quantum numbers. The excited state of hydrogen-like atoms with principal quantum number n is n2-fold degenerate and n^2 = \sum_{\ell=0}^{n-1} (2 \ell + 1), where \ell is the azimuthal (angular momentum) quantum number. For instance, the excited n = 4 state contains the following \ell states, 16 = 1 + 3 + 5 +7 \;\; \Longrightarrow\;\; n=4\;\hbox{contains}\; s\oplus p\oplus d\oplus f. The one-electron states with even \ell are even under parity, while those with odd \ell are odd under parity. Hence hydrogen-like atoms with n>1 show first-order Stark effect. The first-order Stark effect occurs in rotational transitions of symmetric top molecules (but not for linear and asymmetric molecules). In first approximation a molecule may be seen as a rigid rotor. A symmetric top rigid rotor has the unperturbed eigenstates \|JKM \rangle = (D^J_{MK})^ \quad\mathrm{with}\quad M,K= -J,-J+1,\dots,J with 2(2J+1)-fold degenerate energy for \|K\| > 0 and (2J+1)-fold degenerate energy for K=0. Here DJMK is an element of the Wigner D-matrix. The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent electric dipole moment of the symmetric top molecule. Second order As stated, the quadratic Stark effect is described by second-order perturbation theory. The zeroth-order problems H^{(0)} \psi^0_k = E^{(0)}_k \psi^0_k, \quad k=0,1, \ldots, \quad E^{(0)}_0 < E^{(0)}_1 \le E^{(0)}_2, \dots are assumed to be solved. It is usual to assume that the zeroth-order state to be perturbed is non-degenerate. If we take the ground state as the non-degenerate state under consideration (for hydrogen-like atoms: n = 1), perturbation theory gives E^{(2)} = \sum_{k>0} \frac{\langle \psi^0_0 \| V_\mathrm{int} \| \psi^0_k \rangle \langle \psi^0_k \| V_\mathrm{int} \| \psi^0_0 \rangle}{E^{(0)}_0 - E^{(0)}_k} =- \frac{1}{2} \sum_{i,j=1}^3 F_i \alpha_{ij} F_j with the components of the polarizability tensor α defined by \alpha_{ij}\equiv -2\sum_{k>0} \frac{\langle \psi^0_0 \| \mu_i \| \psi^0_k \rangle \langle \psi^0_k \| \mu_j \| \psi^0_0\rangle}{E^{(0)}_0 - E^{(0)}_k}. The energy E(2) gives the quadratic Stark effect. Because of their spherical symmetry the polarizability tensor of atoms is isotropic, \alpha_{ij} = \alpha_0 \delta_{ij} \Longrightarrow E^{(2)} = -\frac{1}{2} \alpha_0 F^2, which is the quadratic Stark shift for atoms. For many molecules this expression is not too bad an approximation, because molecular tensors are often reasonably isotropic. The perturbative treatment of the Stark effect has some problems. In the presence of an electric field, states of atoms and molecules that were previously bound (square-integrable), become formally (non-square-integrable) resonances of finite width. These resonances may decay in finite time via field ionization. For low lying states and not too strong fields the decay times are so long, however, that for all practical purposes the system can be regarded as bound. For highly excited states and/or very strong fields ionization may have to be accounted for. (See also the article on the Rydberg atom). Modern Quantum Mechanics (2nd Edition) The Theory of Atomic Spectra
16522523e8309eff	måndag 30 september 2013 Thomas Stocker's Defense of IPCC Climate Models at Variance with Observations Global temperature increased according to observations in the periods 1920 - 40 and 1978 - 1996 with about 0.35 C, while the temperature was slightly decreasing in the periods 1880 - 1920, 1940 - 1978, 1997 - 2013. The main argument used by IPCC co-chairman Thomas Stocker when defending the climate models of IPCC AR5 predicting steady warming, was that the 17 year period 1997 - 2013 with no warming was too short to allow any conclusion that CO2 forcing was too small to be observed, while the 18 year period 1978 - 1996 with warming was long enough be able to attribute with 95% likelihood the warming to CO2 forcing. Stocker clarified the argument by adding that a period of 30 years would be required to be able to detect a trend in global cooling. PS1 The temperature curves for the periods 1920 - 1950 and 1973 - 2013 are very similar with first warming and then slight cooling, only shifted with a steady rise of about 0.5 C/century after the Little Ice Age. The rise 1920 - 1940 is not attributed to CO2 by IPCC while the rise 1976 - 1996 is. The logic is missing. PS2 The above graph produced by IPCC appears to present lower than actual temperatures before 1960 thus enhancing the warming thereafter. Will Skeptics Now Be Able to Unite? As the IPCC along with its politicized scientific apparatus now sinks into the Deep Ocean, it is natural to ask if skeptics of different brands, from IPCC refugees over lukewarmers to socalled deniers, will now be able to unite instead of beating each other with secteristic fervor? In particular, I could ask if the ban of my writings on some skeptics blogs, because of my questioning of the reality of a Holy Sky Spirit of Back Radiation or DLR (Downwelling Longwave Radiation), can now be lifted? In particular, can the lack of global warming since 1997 under steadily rising CO2 levels, be viewed as evidence of non-existence of radiative forcing as an effect of DLR from a Holy Sky Spirit? Is DLR fictional in the same sense as the Holy Spirit, when confronted with observed realities? PS One of the blogs where I have been banned is Roy Spencer's because of my insistence that back radiation is non-physical and that the starting point of 1 C warming from doubled CO2 is a definition based on a simple algebraic relation (Stefan-Boltzmann's law), which does not have any real meaning for the complex system of global climate. Roy sums up the basic physics supposedly carrying climate modeling as follows: • It is sometimes said that climate models are built upon physical first principles, as immutable as the force of gravity or conservation of energy (which are, indeed, included in the models). But this is a half-truth, at best, spoken by people who either don’t know any better or are outright lying. • The most physically sound portion of global warming predictions is that adding CO2 to the atmosphere causes about a 1% energy imbalance in the system (energy imbalances are what cause temperature to change), and if nothing else but the temperature changes, there would only be about a 1 deg. C warming response to a doubling of atmospheric CO2 (we aren’t even 50% of the way to doubling). • But this is where the reasonably sound, physical first principles end (if you can even call them that since the 1 deg estimate is a theoretical calculation, anyway). Roy thus appears to question the 1 C and so we agree on this point. The red card must then result from back radiation. söndag 29 september 2013 Judith Curry: From Sick to Healthy Climate Science Judith Curry has gone a long way from supporter to opponent of the CO2 global warming science of IPCC, by realizing that the science of IPCC is sick and therefore has to be eliminated to allow healthy climate science to develop: • We need to put down the IPCC as soon as possible – not to protect the patient who seems to be thriving in its own little cocoon, but for the sake of the rest of us whom it is trying to infect with its disease. Since 97% of institutions and people of climate science reportedly have been infected by IPCC, Judith is asking for a revolution with a small group of healthy skeptics leading climate science into the future. Interesting perspectives... PS1 Judith started her transformation from alarmist to heretic by suddenly realizing that "back radiation" as the basis of greenhouse gas alarmism, is non-physical, which was also my door to skepticism. PS2 Judith makes the same analysis as Pointman: • We’ve just witnessed the embarrassing and public humiliation of climate science as a field of honest scientific endeavour. It has lost all claim to be taken seriously and is now tarred with the same pathological science brush that aberrations like Lysenkoism or Eugenics were. It’s now up to the non-activist scientists in the field, who’ve stayed silent for far too long, to save it from that fate by speaking out and reclaiming their field from fanatics posing as scientists. As Elvis said, it’s now or never. PS3 Judy's death sentence has now been printed in Financial Post. fredag 27 september 2013 IPCC Follows Warming into the Deep Ocean tisdag 24 september 2013 The Funeral of IPCC: Too Strong Response to Greenhouse-Gas Forcing The leaked IPCC AR5 Summary for Policymakers tells the world and its leaders that climate models tuned to the observed warming 1970 - 1998, do not fit with the observed lack of warming 1998 - 2013 • There is very high confidence that models reproduce the more rapid warming in the second half of the 20th century. IPCC thus admits that climate models are constructed to have • too strong a response to increasing greenhouse-gas forcing, and are unable to capture • unpredictable climate variability. This must be the end of IPCC since IPCC was formed on the sole doctrine of strong a response of green-house gas (CO2) forcing in climate model predictions. Since IPCC was born in Stockholm from the mind of the Swedish meteorologist Bert Bolin, it is fully logical that the funeral of IPCC now takes place in Stockholm along with the Bert Bolin Center for Climate Research. PS Concerning climate predictions recall the prediction I made in 2009. torsdag 19 september 2013 Royal Swedish Academy of Sciences Platform for IPCC IPCC announces to release its 5th Assessment Report on September 27 on a platform offered by The Royal Swedish Academy of Sciences under the title Climate Change: the state of the science: • On 27 September 2013, IPCC’s Working Group I releases the Summary for Policymakers of the first part of the IPCC’s Fifth Assessment Report, Climate Change 2013: the Physical Science Basis. This is the first event at which the Working Group I Co-Chairs present the findings of the newly approved report to the general public. This historic event expresses the historically strong bond between IPCC and the Royal Swedish Academy and Swedish climate science, with IPCC determining the state of science and the Academy acting as platform for the political CO2 alarmism of IPCC. If IPCC falls so will the Academy. PS A key finding to be reported is: It's come to this: "The heat is still coming in, but it appears to have gone into the deep ocean and, frustratingly, we do not have the instruments to measure there" Climate change: IPCC cites global temperature rise over last century \| Environment \| The Observer tisdag 17 september 2013 Staggering Consequences of One IPCC Graph Ross McKitrick comments on the reality facing the 5th Assessment Report of IPCC to be presented September 23 - 26 in Stockholm in IPCC Models Getting Mushy (Financial Post): • Everything you need to know about the dilemma the IPCC faces is summed up in one remarkable graph. • Models predict one thing and the data show another. • There is a high probability we will witness the crackup of one of the most influential scientific paradigms of the 20th century, and the implications for policy and global politics could be staggering. The message is the same as that Richard Feynman sent to his physics colleagues: Even if the climate theory of IPCC is ugly and IPCC lead authors have limited smartness, if it does not agree with observation, it is wrong. The graph shows that IPCC climate theory does not agree with observation. The implications for policy and global politics will be staggering. PS Roy Spencer draws the same conclusion in Turning Point for IPCC and Humanity: lördag 14 september 2013 Quantum Mechanics as Smoothed Particle Mechanics As a follow up of the ideas in the sequence of posts on Quantum Contradictions I sketch here an approach to quantum mechanics as a form of smoothed particle mechanics which allows a deterministic physical interpretation and is computable, thus without the difficulties of the standard multidimensional wave function, which according to Nobel Laureate Walter Kohn is not a legitimate scientific concept. Simulations using this approach are under way. tisdag 10 september 2013 The Crisis in Modern Physics: Too Complicated The last sequence of posts on Quantum Contradictions 1 - 20 gives examples of the crisis in modern physics recently described by the Perimeter Institute Director Neil Turok as follows: The crisis in modern physics resulting from the confusion of modern physicists originates from the statistical mechanics of Boltzmann used by Planck in a desperate attempt to explain blackbody radiation as statistics of quanta, which led to the quantum mechanics of Bohr and Heisenberg based on atomistic roulettes without casusality and physical reality. But blackbody radiation can be explained without statistics in a classical model subject to finite precision computation as exposed on Computational Blackbody Radiation, which is simple and therefore possibly correct in the spirit of the above. måndag 9 september 2013 Quantum Contradictions 20: Averaged Hartree Model between Scylla and Charybdis The present sequence of posts Quantum Contradictions 1 - 20 with the key posts 6, 9 and 12 focussing on Helium (and two-electron ions) lead to a modified Hartree model for the ground state as a system of single electron wave functions defined by minimization of the total energy as the sum of (i) kernel potential energy, (i) inter-electron potential energy and (iii) kinetic energy, where the electrons keep individuality defined by individual presence in space as expressed by the single electron wave functions as concerns (i) and (ii), while the kinetic energy is computed after angular averaging as an expression of lack of individuality. This model gives according to 12 a ground state energy of Helium of - 2.918 for spherical wave functions with polar decentration, to be compared with the observed - 2.903. Not bad. In this model electrons thus keep individuality as concerns potential energies but lack individuality as concerns kinetic energy as a result of polar averaging. Helium is thus described by two electronic wave functions, defined on 3-dimensional space, of the form: • $\psi_1(r,\theta )^2 = (1 + \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$, • $\psi_2(r,\theta )^2 = (1 - \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$, where $\alpha$ and $\beta$ are positive parameters determined by total energy minimization. This corresponds to a configuration with the two electrons being (more or less) separated, with electron 1 shifted towards the North pole of a spherical atom and electron 2 towards the South pole. The kinetic energy is computed after polar averaging or summation of $\psi_1$ and $\psi_2$. We compare with the full wave function $\psi (r_1,\theta_1,\phi_1, r_2,\theta_2, \phi_2)$ satisfying Schrödinger's linear wave equation in 6 spatial dimension, where the electrons have lost all individuality as being indistinguishable and the wave function is given a statistical meaning. We have understood that this model is unphysical and should not be used. The averaged Hartree model is a system in 3 dimensions and as such can be given a physical meaning without statistics, while the angular averaging removes the observed unphysical nature of the original Hartree model as a classical electron cloud (or Bohr) model of the atom. The averaged Hartree model thus can be seen as a semi-classical physical model obtained by angular averaging in a classical model, instead of the full statistics of the full quantum model necessarily introducing non-physical aspects. The averaged Hartree model steers between the Scylla of a classical model with full electronic individuality, which does not seem to describe the atomistic world, and the Charybdis of a full multidimensional quantum model with no electronic individuality, which is an unphysical model loaded with contradictions. PS For the hydrogen ion H- with two electrons surrounding a +1 kernel, we obtain similarly a ground state energy of - 0.531 to be compared with observed - 0.528, and with - 0.500 for Hydrogen, thus indicating that H- is a stable configuration. torsdag 5 september 2013 Quantum Contradictions 19: Summary Here is a summary of contradictions of text book quantum mechanics: 1. The multidimensional wave function, as the solution of the multidimensional linear Schrödinger's equation as the basic mathematical model of quantum mechanics, is not a legitimate scientific concept according to Nobel Laureate Walter Kohn, because it cannot be solved analytically nor computationally. 2. It follows that the linear multidimensional Schrödinger equation, which is an ad hoc model invented by Schrödinger and canonized in its Copenhagen Interpretation by Bohr, Born, Dirac and Heisenberg (and then abandoned by Schrödinger), should be removed from text books. 3. Doing so eliminates the need of inventing the microscopic roulettes of the Copenhagen Interpretation in order to give the multidimensional wave function at least some physical meaning, roulettes which violate basic physical principles of causality and reality and therefore were never accepted by Einstein and Schrödinger despite strong pressure from the physics community to confess to statistics. 4. With the linear multidimensional Schrödinger equation and its statistics put into the wardrobe of scientific horrors, focus can instead be put on developing non-linear three-dimensional deterministic equations describing the atomistic world formed by interaction of positive kernels and negative electrons such as Hartree and density functional models. The challenge is then to e.g. explain the (shell structure) of the periodic table from such a model. 5. A tentative such model will be described in final post of this series. tisdag 3 september 2013 Quantum Contradictions 18: Heisenberg The text book canon of quantum mechanics was formed by Bohr and Heisenberg in the 1920s and was named the Copenhagen Interpretation (CI) by Heisenberg in the 1950s. Let us seek the origin and motivation behind the Copenhagen Interpretation in Heisenberg's confessional treatise The Physicist's Concept of Nature. We find the following basic beliefs of Heisenberg: • Even in the ancient atomic theory of Democritus and Leucippus it was assumed that large-scale processes were the results of many irregular processes on a small scale. • Thus we always use concepts which describe behaviour on the large scale without in the least bothering about the individual processes that take place on the small scale. • Now, if the processes which we can observe with our senses are thought to arise out of the inter-actions of many small individual processes, we must conclude that all natural laws may be considered to be only statistical laws. • Thus it is contended that while it is possible to look upon natural processes either as determinedby laws, or else as running their course without any order whatever, we cannot form any picture of processes obeying statistical laws. • Planck, in his work on the theory of radiation, had originally encountered an element of uncertainty in radiation phenomena. He had shown that a radiating atom does not deliver up its energy continuously, but discreetly in bundles. This assumption of a discontinuous and pulse-like transfer of energy, like every other notion of atomic theory, leads us once more to the idea that the emission of radiation is a statistical phenomenon. • However, it took two and a half decades before it became clear that quantum theory actually forces us to formulate these laws precisely as statistical laws and to depart radically from determinism. • With the mathematical formulation of quantum-theoretical laws pure determinism had to be abandoned. We understand that Heisenberg believed that the microscopic atomic world is a roulette world of non-deterministic processes for which we cannot form any pictures but we anyway have to believe obey statistical laws. But atomic roulettes require microscopics upon microscopics, since a roulette is not a simple pendulum but a complex mechanical device, which leads to reduction in absurdum and thus a logical deadlock. This was understood and voiced by Schrödinger and Einstein but Bohr and Heisenberg could scream louder and took the game despite shaky shady arguments. But a scientific deadlock is a deadlock and so a new direction away from the quagmire of microscopic statistics must be found.
4f5fcef0d17fb0f6	Simple proof QM implies many worlds don't exist Let's take an electron and measure its spin component $j_z$ via the Stern-Gerlach apparatus i.e. via a magnetic field. The initial state of the electron is prepared to be "up" with respect to a particular tilted axis – every state of the spin in 3 dimensions is "up" with respect to a semi-axis – so that we have\[ \ket\psi = 0.6 \ket{\rm up} + 0.8 \ket{\rm down}. \] So the electron will have a 36% chance to have the spin "up" and 64% chance to have the spin "down". Note that it's not just the absolute values of the amplitudes that matter. The relative phase matters, too. If we changed the relative phase of the two terms by the factor of $\exp(i\alpha)$, it would mean that the axis with respect to which the electron is polarized "up" would rotate by the angle $\alpha$. Such a rotation may be inconsequential for our measurement of $j_z$ but it would matter for the measurement of all other components of the spin. Now, let's ask the key MWI question: will there be an electron with spin "up" as well as an electron with spin "down"? The MWI proponents say "Yes". They imagine that different possibilities "really occur" in different universes, and so on. So this is the main question that decides about the validity of the MWI. Stupid monkeys are obsessed by questions whether MWI and other things are "not even wrong", "politically correct", "obeying Occam's razor", "pretty", and all such irrational adjectives, but no one seems to care about the question whether it is scientifically false or true. Quantum mechanics offers a universal rule to answer all Yes/No questions that have any physical meaning, that are in principle observable. For the given question, we identify the projection operator $P$, i.e. a Hermitian operator $P=P^\dagger$ obeying $P^2=P$ (which is why its eigenvalue have to obey $p^2=p$ as well and they must belong to the set $\{0,1\}$ i.e. {No, Yes}). The expectation value\[ {\rm Prob} = \bra \psi P \ket \psi \] is interpreted as the probability that the answer is Yes. Quantum mechanics doesn't allow us to predict anything else than probabilities. So there's always some uncertainty about the answer to the question. The only exceptions are projection operators whose expectation values are equal to $0$ or $1$: these values correspond to "certainly No" or "certainly Yes" and there's no uncertainty left. We will see that the "key question of MWI" is of this sort. The projection operator for a question "A and B" is constructed as\[ P = P_A \cdot P_B. \] When it comes to operators, "and" is multiplication. That's why Logical AND i.e. conjunction is also known as "binary multiplication". And that's also why the probabilities of two independent questions' having answers "Yes" is equal to the product of probabilities. Fine, what are $P_A$ and $P_B$? They are projection operators on the subspaces for which the answers to questions A and B are "Yes". In particular, we have\[ P_A = \ket{\rm up}\bra{\rm up}, \quad P_B=\ket{\rm down}\bra{\rm down}. \] They're projection operators on the "up" and "down" states of the electron, respectively. There are just no other states in the Hilbert space for which the statement "there is an isolated electron with the spin up" or similarly "...down" would be valid. Now,\[ \braket{\rm up}{\rm down} = 0 \] and therefore\[ P = P_A P_B = \ket{\rm up}\bra{\rm up}\cdot \ket{\rm down}\bra{\rm down} = 0. \] Therefore, the probability that there will be both an electron "up" and an electron "down" is\[ \bra\psi P \ket \psi = \bra \psi 0 \ket\psi = 0 \braket\psi\psi = 0. \] I've written the derivation really, really slowly so that at least 10% of the stupid monkeys have a chance to follow it. At any rate, we may prove that the probability that the electron exists in both mutually exclusive states simultaneously is zero. It can't happen. The derivation is identical for any other mutually excluding alternative properties of any physical system. Note that the operators $P_A,P_B$ commute with one another, i.e. $P_A P_B=P_B P_A=0$, which means that both questions may have an answer at the same moment (the uncertainty principle adds no extra hassle). That allows us to avoid some discussions. The simple conclusion is that there aren't many worlds. QED. Get used to it, monkeys. ;-) Let me now spend some time by discussing how indefensible various "loopholes" would be and why there are many other ways to see that the answer to the question "Are there many worlds?" had to be "No". And I want to mention several likely fundamental and rudimentary errors that prevent MWI advocates from deriving the right answer to this simple question and from seeing that this is truly a kindergarten stuff and not something that they should be confused by for days, weeks, months, years, decades, or centuries. First, let me discuss the interpretation of the "plus" sign. As I already suggested, it's important to distinguish addition and multiplication. (If you don't know what multiplication is, watch 0:40-0:45 Miss USA on maths.) The key fact is that the wave function composed of several mutually exclusive pieces such as\[ \] has a plus sign that roughly means "OR", not "AND" as many people apparently think. When we care about the $j_z$ component of the spin, the formula above says that the state $\ket\psi$ allows the electron to be either "up" OR "down". It doesn't say that there is both a spin "up" AND a spin "down". If we need to say "AND" in quantum mechanics, either "one proposition/question AND another proposition/question" (as discussed with the $P=P_A P_B$ relationship above) or "one object added on top of another object", we need multiplication, not addition. For the case of the two propositions, we have already discussed an example, the $P=P_A P_B$ relationship above. If we discussed physical systems composed of several pieces, e.g. a group of 2 apples and a group of 3 apples, we would need another kind of a product, the tensor product,\[ \ket{\text{5 apples}} = \ket{\text{2 apples here}} \otimes \ket{\text{3 apples there}}. \] The matrix elements extracted from similar "tensor products" are products of the matrix elements for the individual subsystems and the same thing therefore holds for the probabilities, too. Some people may be thinking that it almost looks like I am suggesting that the MWI advocates are complete idiots with the IQ of a retarded third-grader because they can't distinguish addition from multiplication. The reason why it looks so is that this is exactly what I am trying to say. In fact, it's pretty obvious that my attempts to say such a thing are successful and I am actually saying this thing. ;-) Why is there so much confusion about the meaning of addition and multiplication here? Because people with common sense – as it evolved for millions of years – and no genuine knowledge of the pillars of modern physics (which includes the MWI advocates) always think in terms of objects, e.g. apples. So when you're adding two apples and three apples, place the two groups next to one another, you're adding apples. Similar addition more or less applies to lengths of sticks, momenta and other conserved quantities, and even quantities such as voltages, currents, charges, and many others. But this "combining objects that exist simultaneously is addition" is fundamentally and completely wrong for wave functions in quantum mechanics. In quantum mechanics, addition of wave functions or density matrices roughly corresponds to "OR", not "AND", and "AND" must be expressed by multiplication. How can we understand the origin of this flagrant difference between the classical thinking and quantum mechanics? For propositions and their probabilities (expectation values of the projection operators), addition is simply not "AND", addition is "OR". The right mathematical expression for "AND" is another operation, namely multiplication rather than addition. An MWI advocate could start to spread fog. It may be debatable which one it is, the difference between "AND" and "OR" isn't that important, anyway, and it may be up to centennial deep philosophical discussions which way it goes. Well, all these statements are pure rubbish. There isn't any ambiguity, confusion, or room for modifications. Addition and multiplication are completely different operations so you should better not confuse them. The right theory that is tested is the theory that says the same thing about the interpretation of addition and multiplication as I did. Be sure that if you modify its rules, the rules of quantum mechanics, by randomly replacing addition by multiplication and vice versa at various places, you will get a completely, qualitatively different theory that will yield a totally different description of the reality and it will disagree with almost all observations, including some extremely elementary ones. There just isn't any room for confusions and debates. Just like a 7-year-old schoolkid who invents arrogant excuses why she cannot learn the difference between the addition and multiplication (note that I am politically correct so I sometimes include "she" in similar sentences, especially if it increases the degree of realism), the MWI proponents should be given a failing grade and should be spanked. Be sure that any "technical" modification of my proof that there aren't many worlds will damage the theory so that it will become totally incompatible with the experimental tests. For example, if you suggested that the projection operator for "A and B" should be $P_A+P_B$ rather than $P_A P_B$, you will easily find out that the same rule used for any experimentally testable situation will lead to wrong predictions. In fact, pure thinking is enough to see that "AND" must be expressed by the product of the projection operators and not the sum. Using charge conservation to prove there aren't many worlds The fact that one electron can't suddenly be split to two electrons so that it would be both "here" and "there" may also be derived from charge conservation, angular momentum conservation, mass conservation, or other conservation laws. In quantum mechanics, such laws still hold. If the initial state $\ket\psi$ is an eigenstate of the electric charge operator $Q$,\[ Q\ket\psi = q\ket\psi, \] then, because $QH=HQ$ i.e. the charge is conserved i.e. the symmetry generated by it is a symmetry of the Hamiltonian i.e. of the laws of physics, the final state will obey the same relationship with the same value of $q$. But if there were an electron on both places, the electric charge could be shown to be doubled and different than the original one. That would conflict with the conservation law. Inflating the Hilbert space along the way Some people could say that my derivations are missing the point that there is an "Everett multiverse". I should have increased the size of the Hilbert space before the measurement etc. There are many wrong things about such a potential objection. First, the constancy of the dimension of the Hilbert space is a mathematical necessity. Especially because some MWI proponents including Brian Greene say that they want to be led by the most natural interpretation of the equations of quantum mechanics, it's totally indefensible to actually change the dimension of the Hilbert space along the way. It's surely not what quantum mechanics tells us to do. In fact, one may easily show that such a proliferation of the degrees of freedom couldn't lead to an internally consistent theory. It may be explained in many ways, e.g. by the quantum xerox no-go theorem. There can't be any evolution of a state in ${\mathcal H}$ to a state in a larger Hilbert space such as ${\mathcal H}\otimes {\mathcal H}$ because the evolution of the state vector in quantum mechanics is linear while the map \[ \] is not linear; it is bilinear or quadratic. If $\ket x$ and $\ket y$ were evolving to $\ket{xx}$ and $\ket{yy}$, respectively, then linearity would dictate that $\ket{x+y}$ evolves to $\ket{xx+yy}$ while the universal squaring formula would say that it should evolve to $\ket{(x+y)^2}=\ket{xx+xy+yx+yy}$. These are different ket vectors on the larger Hilbert space because there are extra mixed terms. At any rate, it's a contradiction: in a quantum world, there can't be any gadget that creates two exact copies out of the arbitrary initial state. Another problem with the objection is that I actually haven't done any assumption about the non-existence of the "Everett multiverse". For example, in the fast "charge conservation" proof, $Q$ could have meant the total electric charge in "all branches" of the world you could ever hypothesize. Clearly, if the number of worlds is being multiplied, the charge won't be conserved. That will be a problem because the symmetry generated by $Q$ won't be a symmetry of the laws that control the "Everett multiverse" anymore. It won't be able to be exact at a fundamental level, you won't be able to use it to constrain the laws of physics, and so on. This "demise" will be fate of all the symmetries in physics (translations, rotations, Lorentz boost, parity, etc.) because all symmetries are related to a conservation law. One more problem with the "splitting of the Universes along the way" is that there can't possibly exist any justifiable rule about "when this splitting takes place". There aren't any sharp qualitative boundaries between phenomena in Nature. It's clear that there can't be any splitting during a sensitive interference experiment – because such an "elephant in china" converting the fuzzy quantum information into the classical one would surely destroy the interference pattern. The problem is that in principle, we may say the same thing about 2 particles, 3 particles, 100 particles, or $10^{26}$ particles. In principle, the interference pattern involving an arbitrarily large system may be measured so the Universe is just not allowed to "split" into possibilities where different classical outcomes are realized because such a splitting would make the "reinterference" permanently impossible while it is arguably always possible in principle. In practice, there's a lot of irreversibility, "decoherence", but this process always depends on our inability to manipulate with the elementary building blocks of information too finely. Decoherence is an emergent phenomenon and it isn't sharp, either. There is no point during the decoherence process when you could say "now it's the right time for the universes to split into many worlds". Decoherence is just a continuous process in which the off-diagonal elements of the density matrix gradually decrease. They decrease faster and faster but they're never "quite" zero. Shannon told us that Brian Greene thinks that he and your humble correspondent have a "little disagreement" about a physics question. ;-) The little disagreement is about the existence of a paradigm shift in the 20th century science that would invalidate the previous framework of classical physics. I am sure it has happened in the 1920s; Brian Greene thinks that it hasn't happened so it is still possible to think about Nature in the "realist" way. Of course, I could also be saying it is a little disagreement, I have also been taught how to be diplomatic, polite, hypocritical, and dishonest. But I just don't think it's right to behave in this way. The disagreement is clearly about a major question, about the very existence of modern physics as something that is outside the box of classical physics. Brian Greene is really denying the existence of quantum mechanics; instead, he is suggesting that what we need are new theories (e.g. nonlocal ones or multiverse ones) within classical physics (although he and others prefer more obscure ways to describe the very same thing, ways that make the naked Emperor's new clothes look more fashionable and decent). Bohr et al. always used legitimate, official, and transparent channels to discuss similar physics questions – e.g. in the Bohr-Einstein debates – and it is the MWI advocates who are using non-standard channels such as popular books to spread misconceptions. Equally importantly, the "universal validity of the laws for small and large objects" is an important consideration, indeed. But it unambiguously says that MWI is wrong and QM as understood by Bohr et al. and the followers – modern physicists – is the only plausible right answer. I have already mentioned why it is so. There just can't be any splitting of the worlds when one quantum particle is coherently and peacefully propagating through an experimental apparatus. The same comment applies to 2 or 3 particles so if we're using the laws of physics coherently for small as well as large systems, there can't ever be any "splitting of the Universes". An impressive song about the Higgs, a new genre of music. There is one more aspect of the unity that could be violated by the MWI advocates to defend the indefensible. They could say that the question "is there an electron here as well as an electron there", the question whose probability we calculated to be zero, shouldn't be answered by the rules of quantum mechanics i.e. by identifying the right projection operator and by computing its expectation value (interpreted as the probability of "Yes"). They could say that this is a question "above the system" that should be answered by some philosophical dogmas. But that's not how physics works or should work. Quantum mechanics has a way to answer all physically meaningful i.e. in principle observable questions and it is the same way for all the questions. In fact, there is nothing unusual about asking whether there are electrons at two places. This is the kind of questions that all of physics is composed of. If you were free (or even eager) to abandon your standardized theory and methodology to answer such questions and if you switched to some metaphysical dogmas just because this question about the many worlds is "ideologically sensitive", it would prove that the theory you may still be using for other questions isn't something you are taking seriously, isn't something to answer really important questions in physics. It would surely show that you have double standards and the technical theory you're using isn't universal and uniformly applicable because you often replace it by metaphysical dogmas. Your attitude would be completely analogous to the attitude of a fundamentalist Christian physicist who just chooses to believe that Jesus Christ could walk on the sea because the laws of gravity and hydrodynamics didn't have to apply and the non-nuclear conservation of carbon atoms could have been invalidated when he was converting water into wine. And I don't mention many other Jesus' hypothetical crimes against the laws of physics that such a physicist could be eager to overlook for political reasons. ;-) The MWI advocates prefer metaphysical dogmas and their naive classical intuition over the standardized quantum mechanical "shut up and calculate" approach to answer such questions about the electron on two places (or pretty much any other question in physics) because they haven't started to think in the quantum way yet. To think in the quantum way is to be deciding about the validity of propositions (or the probabilities of their being valid) and the procedure is always the same. One constructs the projection operator related to the proposition and calculates its expectation value in the quantum state. It's the probability and if the result is $0$ or $1$, we may be certain that the answer is "No" or "Yes", respectively. (The detailed arguments or calculations may proceed differently and avoid concepts such as "projection operators" but they must still agree with the general rules of quantum mechanics.) When we follow this totally universal quantum procedure – valid for questions about microscopic systems as well as macroscopic systems – carefully and rigorously, we will find out that quantum mechanics as it stands, in the same Copenhagen form as it has been known since the 1920s, answers all questions, including those that "look philosophically tainted", correctly i.e. in agreement with the experiments. Sidney Coleman gave many examples in his lecture Quantum Mechanics In Your Face. For example, it's often vaguely suggested by the MWI champions and other "Copenhagen deniers" that the experimenter could feel "both outcomes at the same moment". However, by the correct quantum procedure whose essence is absolutely identical to my discussion of the two positions of the electron at the beginning, we may actually find the answer to the question "whether the experimenter feels both outcomes at the same moment". We will convert the proposition to a projection operator, it has the form $P=P_AP_B$ again, and because its expectation value is zero for totally analogous reasons as those at the top, it follows that according to quantum mechanics, the experimenter doesn't perceive both outcomes at the same moment. This is a completely physical question, not a metaphysical one, and quantum mechanics allows one to calculate the answer. It's just not the answer that the anti-Copenhagen bigots would like to see. Quantum mechanics doesn't predict "unambiguously" which of the outcomes will be perceived by the experimenter (spin is "up" or "down"?) but this uncertainty is something totally different than saying that he will perceive two outcomes. The number of outcomes he will perceive may be calculated unambiguously by the standard rules of quantum mechanics and the number is one. There is no room for "two worlds" or "two perceptions at the same moment". Which outcome will be felt has probabilities strictly between 0 and 100 percent so the answer isn't unequivocal. When the MWI-like folks are discussing these matters, they are constantly making lots of other totally rudimentary errors – and perhaps "deliberate errors" – aside from the confusion of addition and multiplication I mentioned above. A frequent one is to totally forget or deny that quantum mechanics predicts and remembers correlations (in their most general form known as entanglement) between any pairs, triplets, or larger groups of degrees of freedom and properties that may co-exist in the real world. For example, Coleman mentioned the cloud chamber example by Nevill Mott. A particle leaves the source in the cloud chamber. It is in the $s$-wave: its wave function is spherically symmetric so it has the same chance to move to each direction. So why does it create a straight line of bubbles in one direction rather than a spherically symmetric array of bubbles? Again, this may be interpreted as some super-deep metaphysical question that goes well beyond quantum mechanics and the Copenhagen interpretation may be claimed to be incapable of answering such questions. Except that there is nothing hard or metaphysical about this question at all. It is completely physical, quantum mechanics allows us to answer it using a very simple calculation, and the answer is right. There will be a straight line of bubbles because one may prove that due to some demonstrable entanglement between properties of the supersaturated water or alcohol at various points that the propagation of the charged particle causes, the direction of two newly created bubbles as seen from the source is always essentially the same. (One may prove that the charged particle only creates bubbles in a small region around its location; and one may prove that the position of the charged particle goes like $\vec x = \vec p \cdot t / m$ where the momentum $\vec p$ is essentially conserved. That's enough to see that the bubbles will be aligned.) So again, while quantum mechanics gives ambiguous predictions about the direction in which the "bubbly path" will be seen – all directions are equally likely – it actually does unambiguously predict that the bubbles will have a linear shape, they will only emerge along a straight semi-infinite track. There is absolutely no inconsistency between these two assertions. Any wrong idea that QM has to predict that the distribution of the bubbles is spherically symmetric boils down to a trivial error: the omission of the fact that the existence or absence of bubbles at a point is correlated with the existence or absence of bubbles at other points. In fact, the correlation is so tight that for each semi-infinite line, there are either bubbles everywhere along the line or there are no bubbles on it. And there is only one semi-infinite line. As I said many times, the people who have trouble with proper, i.e. Copenhagen or neo-Copenhagen laws of quantum mechanics, are always "eager" to simplify the quantum rules of the game prematurely and convert the situation to some "real physical object" way too early (well, one should really never do so but if one does it too early it may be more damaging). But Nature never does such mistakes. It remembers the wave function which knows about all the possible correlations between all the degrees of freedom, which knows about all the relative phases because they could matter, and only when an observable question has to be answered, it just calculates the right answer. The right calculation looks very different than any kind of reasoning in a classical world but it isn't too hard; it's really straightforward and in all situations in which classical physics used to work, it still gives the same answer (with tiny corrections). When the initial wave function for the charged particle in a cloud chamber is spherically symmetric, it doesn't imply that spherically asymmetric configurations of the bubbles at the end are forbidden i.e. predicted to have vanishing probabilities. On the contrary, we may prove (the right verb really is "calculate" because the proof boils down to the calculation of an expectation value of a projection operator) that the distribution of the bubbles will be spherically asymmetric – a semi-infinite line in a direction. There is no contradiction because the initial wave function isn't a real object such as a classical field, stupid. It's a quantum-generalized probabilistic distribution. A spherically symmetric probabilistic distribution (on a sphere) doesn't mean that the actual objects such as the particles (or, later, the bubbles they will create) are spherically symmetric. Instead, it means that the probability that the objects are found in one direction is the same as it is for another direction. But because the particle may be shown to be in a direction, we know that the actual measurements of positions will inevitably be spherically asymmetric. And that's the memo. Add to Digg this Add to reddit snail feedback (98) : reader Shannon said... I now fully understand your frustration when translating The Hidden Reality. Brian Greene seems to defend this Many Worlds approach saying it is "the most conservative framework for defining Quantum Physics". reader victor said... Have you seen this hilarious video? A Capella Science - Rolling in the Higgs (Adele Parody) reader Luboš Motl said... Right, Shannon. It's about 30 pages of this stuff that keeps on going and it's repeated and repeated and everything is upside down. From the viewpoint of the history, I find it demoralizing and demotivating. Be smart and lucky enough to be one of 3-5 key people who realize the most important revolution of the 20th century science. It ain't easy. Is it worth it? Almost 90 years later, there will still be "mainstream" books published that you haven't really discovered anything, just muddled the waters, and you were a thug who was bullying brave original thinkers (proper word: crackpots of your age), and your theory is unable to do all the things (that it is actually totally able to do), and the original thinker's theory is surely better and more unified (although it isn't even well-defined, it doesn't describe anything correctly whatsoever, and is nothing else than defense of the intellectual inability, laziness, and dogmatism). Imagine you discover the most important thing of the 21st century now, looking at the Universe from a much more far-reaching, accurate, abstract, and unified perspective. Some people won't be able to get it so you will explain them why it's wrong. In 2100, there will be popular books sold to millions of people that you were a bully who used political tricks to modify inconvenient theses, and so on. It's terrible. I don't really know whether, if history is a good guide, I would like to discover the 21st-century counterpart of quantum mechanics. The hassle - not just hassle in your life but apparently also hassle in your "after-life" - may be just too intense. The "conservative" label is particularly silly for the MWI, indeed. Speculations about splitting worlds according to ad hoc rules no one has really meaningfully formulated, ever, because it's not really possible, are the least conservative thing one may imagine. Quantum mechanics is radical but it also preserves the basic scheme and collection of observables and their properties in physics more or less without change. One could say that quantum mechanics only differs from classical physics by having xp-px = i.hbar instead of xp-px=0. The commutator is just a little bit different, a tiny number called Planck's constant times i which must be there because the commutator is anti-Hermitian, and that's it. It's a very modest deformation of the particular laws of physics for the particular classical system we used to have (the classical limit of the quantum theory); one must just learn what it means to work with a theory where xp-px isn't zero but the classical physicists could still be told they were "approximately right" in the observable sense of approximations because their assumption xp-px=0 had an "approximately right results". Adding infinitely many randomly and vaguely fucking and reproducing universes just in order to deny that xp-px isn't zero i.e. x and p (or any observable pair) can't have well-defined properties at the same moment is just the maximally ad hoc, uncontrollable, "progressive", degenerative, irrational thing one can do. Such a framework isn't "approximately equal" to any previous theory, it's not building on anything. It's very clear that it's just a messy attempt to fake the correct theory by some unjustifiable building blocks. reader Luboš Motl said... Thanks, Victor, it's really impressive. The music genre is just like the Yellow Sisters but it also has the extra Higgs bonus points. ;-) reader Gene said... I have never read a clearer explanation of the basic idea of QM, Lubos. Those folks who claim that Schroedinger's cat is half alive and half dead need only to read the + sign as "or". It's so very simple. Regarding your small disagreement with Greene, I recall Sheldon's famous retort regarding LQG, "Small disagreement!! The Pope and Galileo had a small disagreement!". reader rsala said... Hahaha I knew you couldn't leave that mumbo jumbo on Scott's blog go unanswered. Yesterday I skimmed though all 100+ comments and was disappointed not to find yours ... does Scott have you on his "ignore" list? reader Dilaton said... Ha ha, that rocks :-))) ! And now I want to hear this particular parody of "Bohemian Rhapsody" they announced in the last few seconds, LOL :-D reader Luboš Motl said... Thanks, Gene, for your professionally loaded synergy. The quote from TBBT is of course a memorable one, LOL. ;-) reader Fred said... I find this article curious. I'm a believer in the Everett-Wheeler interpretation, and yet I agree with all the physics you describe here. And I don't recognise much of the version of EWI you present. Odd... The EWI is really about the quantum state of the observer, rather than the system being observed. The 'Copenhagen' view it is set up in opposition to is that you have a quantum system obeying quantum physics which is observed by a classical observer obeying classical physics - the transition from one to the other occuring through these projection operators and wavefunction collapse. The EWI view is that the system observed and the observer are both quantum systems, and that the process of observation is simply a temporary coupling of the two systems. It's similar to the way coupled oscillators give rise to normal modes. When coupling is introduced, the modes of vibration of the components become correlated, and turn out to be the eigenvectors of the interaction matrix. If one system starts in a superposition of eigenstates, and is observed by (interacts with) another system, the observer enters a superposition of correlated eigenstates, each state corresponding to an observer observing a particular state. EWI is 'conservative' in the sense that it simply extends without modification the postulates of quantum physics to the observer. It agrees absolutely regarding the physics of the system being observed. The 'many worlds' label comes from an attempt to explain to the lay public what such a quantum observer would experience. Because the different eigenstates are orthogonal, they don't interact, and thus each eigenstate of the superposition would be unaware of all the others. It would be as if there were multiple observers in multiple worlds, each seeing one outcome. Thus, if you want to prove EWI wrong, you need to show how conventional QM handles the quantum state of observers without them ever entering superpositions. Or to say how else you would interpret the experience of a quantum observer in a superposed state. reader Luboš Motl said... I haven't tried it for quite some time whether my comments would be censored now. The odds are always O(50%) for all such left-wing blogs. ;-) That's high enough a risk not to waste time. After all, my blog has a somewhat larger traffic than his blog so I am sure that a blog entry here is more visible than a comment at his blog. reader Neuman said... Song about higgs is not completely new genre. Bjork did the whole album out of vocals only. reader Jonathan Cohen said... It seems to me that you only proved that the spin cannot be both up and down in the same world, but a multiple-world theory would say that the up and down occur in different worlds. So if we let Ua mean up in world A, and Db mean down in world B, then P(Ua) = P(Db) = P(Ua and Db). Of course P(Ua and Da) = 0. Similarly with charge conservation, the MWI is that the conservation law is that the charge is equal across worlds, not that the sum across worlds is equal to one world. reader Luboš Motl said... Sorry, Jonathan, I haven't made any assumption about the number of worlds so it applies to any number of "components". It is also complete nonsense that the charge conservation could mean something else than the conservation of the total charge - of the sum of all the contributions. If there exist many components of the universe, in the usual sense of the word "exist", then the total charge is indisputably the sum over them. The charge can't really mean anything else. The U(1) symmetry generated by the charge has to transform all the charged fields in all components of the Universe so the value of the charge as the quantity has to be the sum. There is no way to escape these facts. The only thing you can achieve by denying these facts is that you completely misunderstand Noether's relationship between charges and symmetries, too. reader tms said... My impression was that Everett was groping towards the idea of decoherence but not really getting there in a comprehensible way. i.e. you don't need a "special rule" to "collapse" the wave function. Bohr et al. might never have believed that, but perhaps lesser minds who came after them did. reader Fred said... "I haven't presented "EWI" because I've never heard any theory that was claiming to be called "EWI". This is a totally bizarre term." That's the Everett-Wheeler Interpretation, which is the correct name for it. 'Many worlds' is a misnomer. "It's good that you explicitly say that "EWI" treats observers differently than the "observed system" because this was one of my primary "accusations", one that disagrees" I haven't said that. Observers and the observed are treated identically, and observation as a physical process is no different to any other sort of coupling or interaction. "The Copenhagen school was talking about observers in isolation but it never meant that such large bound states didn't follow the laws of quantum mechanics" No, it was an attempt to invent laws of quantum mechanics that would explain why a quantum reality appeared to observers to be classical. This was the 'wavefunction collapse'. The idea was that the physics proceeded according to reversible QM until it was 'observed', at which point it would collapse down to a single eigenstate according to some projection. But it was unclear what physically distinguished 'observation' from any other sort of interaction of systems, and all sorts of crazy ideas related to consciousness, gravity, size, complexity, and thermodynamics have been proposed. Everett's thesis was to simply say these devices were unecessary. If a quantum observer interacted with a quantum system, it would enter a superposition of mutually non-interacting states, each corresponding to the observation of one outcome. That is to say, unitary-evolution, reversible QM already fully explained our classical experience with no need to posit a non-reversible 'collapse'. Everett's physics had absolutely no new content - it was simply (a subset of) ordinary and already widely-accepted QM - it just applied it to the observer problem and found there was nothing difficult to explain. "Quantum states always enter superpositions. In the cloud chamber example, the whole system evolves into a superposition of states of a charged particle and bubbles in the same direction - superposition over all directions." Agreed. But the question is about the quantum state of the observer who observes the particle, which is a spherically-symmetric superposition of observers each seeing a particle in one direction. Having explained carefully that the issue was over the quantum state of the observer, and whether it was a superposition, I'm surprised to see you again point to the state of the system being observed. You need to address the question of whether the observer of the decaying particle, or Schrodinger's cat enters a superposition of observer states. What is the quantum state of the detection apparatus? Of the scientist? Can they ever be in superposition? reader Luke Lea said... Dear Lubos, Even though I couldn't say exactly why, as a layman I intuitively felt the many worlds interpretation was inherently absurd the moment I heard. It struck me as the sort of thing a not-to-bright nerdy science-fiction fantasist smarty-pants would go for, like the idea of a technological singularity. reader Dilaton said... If I remember it correctly MWI is used to explain the different "worlds" in "Sliders" ... :-D reader JC said... Jonathan is absolutely right. You should read his point again slowly and try to understand it. reader Luboš Motl said... All these things are completely untrue. You and your fellow many-world cranks may be using the term "Everett-Wheeler Interpretation" but it is a nonsensical term because Wheeler hasn't contributed anything to it aside from the deliberately obscuring title "relative state". It's nonsense that Bohr et al. wanted to "explain why things look classical" by assuming a demarcation line. And it's nonsense that the Copenhagen school ever assumed any "collapse". I have addressed this question about 500 times already, and so did Heisenberg, Bohr, Dirac, and others 85 years earlier. Yes, states always evolve into very general complex linear superpositions of any basis vectors one may choose. As I predicted, you would still be unable to notice in your new comment, and I just added you on the blacklist because your degree of idiocy is something I just won't suffer through again. reader Jonathan Cohen said... The whole point of MWI is to redefine what exists means. Also, by stating P⟨up\|down⟩=0 you are assuming one world, which I assume you are calling component. I've never seen Noether's theorem reformulated to work for multiple worlds, but you definitely can't just take the one world formulation and apply it as is to a multiple world situation. Whether MWI is true or not is different from correctly representing the concept mathematically. My own opinion is that whether MWI is true or not is not knowable by us, if it has been correctly formulated. A misformulated version of it will definitely be false. reader Luboš Motl said... The fact that lesser minds - such as Everett himself - said lot of rubbish after 1925 is surely not a good reason to deny that the foundations of quantum mechanics, the framework of modern physics, was built correctly by Bohr, Heisenberg, Dirac, Pauli, and a few others, is it? reader Luboš Motl said... The whole point of MWI is to redefine what exists means. The only problem is that no one has ever seen such a point. You may dream about "redefining the existence" except that there isn't any "new" definition of existence. It's just a sequence of lies. If you disagree, could you tell me what your new definition and the new derivation of the dynamics of charge etc. is supposed to be? To offer a legitimate counter-evidence, you will have to rediscuss all these elementary derivations I did and offer your "alternative" derivation of all the experimentally tested conclusions for which the MWI discussion is inequivalent to proper quantum mechanics. You know very well that no such an alternative theory may exist. It's very obvious that there can't be any "intermediate" or "third way" of existence that would allow one to have one's cake and eat it, too. It doesn't matter whether the spin exists in another component of the Universe or not. If there is an extra electron anywhere, it carries an electric charge and the conserved quantity has to be the sum of the charges, and similarly for other quantities. Whether the universe is connected or not is an absolute detail, an irrelevant technicality. There isn't any "different kind of existence" in which the conserved quantities wouldn't have contributions from all pieces of the Universe. Also, it's not true that I was assuming one world for the orthogonality. The states are orthogonal whenever they're different-eigenvalue eigenstates of a Hermitian operator, e.g. J_z in this case; it's a trivial one-line proof in linear algebra. I don't need to make any assumption about the number of components in the Universe, it may be anything you want but the directly experimentally measured value is 1. reader a name said... you are still stupid, accept it. reader agenore said... paradox: if the MWI were true there would be a universe where instead the MWI is not true. conclusion: MWI is interpretation is wrong! :-) sorry for my horrible english. reader Vlad said... Luboš,I wonder if you saw following paper: reader wfoster said... As ignorant as I am, I still venture to wonder, Are there any physically meaningful tests of the MWI? I note the citations in wiki: Are these authors presenting worthy suggestions? reader Werdna said... Well, Lubos has argued persuasively that MWI can't be correct, but I don't think this reason can have anything to do with it. if MWI were correct, there couldn't be any universe in which it was not correct. What you are saying is essentially that in MWI any imaginable universe exists - this is not true which is I guess a minor nice thing that one can say about MWI - and that, there must be a universe in which multiple universes do not exist. I'm sorry but what? There is an interesting philosophical problem with MWI that does make me uncomfortable with the idea, Lubos's reasons aside: If all probabilities are somehow realized in "other worlds" there must be some worlds in which every event is an unlikely outcome. In such universes, disturbingly enough, quantum mechanics looks wrong from a statistical point of view: It makes predictions about how likely certain things are to happen, which would appear wrong, purely by chance, in a large number of universes. Not only is the idea that as a scientist you could find yourself in a universe where the correct theory makes unreliable predictions abominable, one could make the argument that it is very unlikely we happen to be in a universe where quantum mechanics looks good-which might lead into anthropic arguments... reader jitter said... In Wiki they link the plus sign to a "disjoint union" Is this your meaning or am I on the wrong track. reader Rehbock said... That one should need prove a theory false although it has no observable consequence and makes no computational advances is still another objection. The burden is on the mwi folk not on those who have so often proven that qm works. Mwi in all formulations add no new useful prediction. If one could not -although you demonstrate one can -demolish it, it would at best be a philosophical construct. If Fred, or anyone can point us to a consistent modification on Noether that would conserve any property such that we could at least retain the explanatory power of symmetry then only would Mwi even rise to not even wrong stature. reader Werdna said... It's a simple statement about how to calculate probabilities. The probability that A or B occurs if the two events are mutually exclusive, is the sum of their probabilities P(A) +P(B) (if they aren't mutually exclusive you must subtract their joint probability (ie "and")). If you want to calculate the the probability that both events occur, well, actually they are mutually exclusive so their joint probability should be zero. But if they are independent it would be the product of their probabilities. reader Peter F. said... This URL is in response to the smoking monkey! reader Carbone said... How is MWI explained in The Hidden Reality? I have some layman questions and I have a hard time googling any layman explanations, e.g., if the electron will have a 36% chance to have the spin "up" and 64% chance to have the spin "down" and both happen how exactly do the probabilities manifest themselves in this deterministic situation? reader Gene said... So, your own opinion is that the truth of MWI is not knowable by us. What the shit does that mean? Again, physics is not about existence; it is about what we can observe! Jesus! reader Shannon said... The prejudice might get worse and worse as time passes but Nature always recognizes the truth. The bad thing though is that humanity will waste a lot of time with these MWI and other crackpots theories. That's the frustrating part. Still I'm here "interneting" with Dr Motl and just last month I spoke with Brian Greene. It feels like I'm close to the Gods'Fight. Cool :-) reader Dilaton said... reader Old Wolf said... Sorry, I have to disagree with the and/or discussion. Schrodinger's cat isn't "alive or dead". That would be a hidden variables "interpretation". The point of the cat experiment is that sentences like "the cat " don't make sense, QM only gives probabilities for the result of measurements. In the case of "0.6\|up> + 0.8\|down>" , we could say: "The result of a measurement will be either \|up> OR \|down>", but we could just as validly say "The eigenvectors are \|up> AND \|down>". This is just a language issue, not an issue of understanding. reader Old Wolf said... Hi Lubos You speak as if an MWI "split" happens when a photon hits a half-silvered mirror, for example . That's not what the theory says though; a "split" may only happen at the same time that an observation is made. Quantum states in linear superposition still exist in MWI and the mathematics is the same. Bohr said that the wavefunction collapses and the state is now an eigenstate of the measurement just made. MWI says that the universe splits into copies when the measurement is made, with one copy for each eigenstate that had a non-zero probability of being found. The issue of the split maybe occurring over a short period of time, is the same as the issue of Copenhagen wavefunction collapse maybe taking some time. Of course you may say that decoherence is superior to both of these views, and you may be right. However, Bohr didn't know about decoherence , he rejected MWI in favour of Copenhagen. So, there is no issue with "split states being unable to reinterfere with themself" or whatever. In the case of a Mach-Zend interferometer, for example, if the probability of detector B triggering is 0%, then MWI says that there is no splitting of universes, in this particular case MWI is just identical to Copenhagen. Also I am not sure why you bring up charge conservation etc. Each individual universe follows the laws of physics, including charge conservation. If the universe splits then there's twice as many electrons and twice as many protons. The Hilbert space that a wavefunction lives on, is something that is just within one universe. When a split occurs, each universe gets its own Hilbert space, the vectors in one space no longer have anything to do with the other space. The different universes, by definition, cannot interact with each other ever again. NB. I don't adhere to MWI personally, but the thing you refute in this post is not what Everett and his fans do adhere to. reader Gordon Wilson said... Jorge Luis Borges, the famous Argentinian short story writer, wrote the story, "The Garden of the Forking Paths" in 1941, which suggested the idea of many worlds. David Deutsch has been championing this idea, and, in his popular book, "The Fabric of Reality" and in some papers, claims to have "proved" it using the double slit experiment. He claims that the interference results from passing single photons through a double slit can only be explained by MWI where "shadow photons" from the many worlds are interacting with the photon in this world.-----really bizarre "proof". It is sort of proof by incredulity or lack of imagination. I must admit that many worlds is a fun science fiction concept, but isn't plausible. Also, the brains entertaining it seriously really don't compare with the QM founders'---Dirac's 1930 book is extremely clear, and Heisenberg's famous paper is complete magic, if very hard for me to follow. reader Yuri said... reader David said... So you both agree that observers themselves can enter into superpositions? (that's pretty much how I understand MWI, that I myself am in some way in a quantum superposition) If so, it's not obvious to me what you're actually disagreeing over other than weather you like the words "many worlds interpretation"... reader Luboš Motl said... Those people never clearly say what they actually believe. A vast majority of them never clearly says whether superpositions of macroscopically different states are legitimate states and the minority is as split as the world can never get. ;-) Of course that my answer is Yes, it's the superposition principle. The reason why such superpositions aren't familiar from the "classical" perceptions is explained by decoherence but there is nothing fundamentally wrong about these superpositions. reader Luboš Motl said... Dear Old Wolf, one could perhaps agree that it is a language issue but your claims about the preferred language are still completely wrong. The sentence "the eigenvectors are up and down" is valid but it has nothing to do with the state vector "psi" itself so it is not equivalent to my original sentence. It only describes a priori possible choices. The sentence I mentioned was meant to only describe possible states whose coefficients are nonzero so it carries some information. Your sentence carries none. There are no "hidden variables" in the sentence "cat is dead or alive". It's just an ordinary logical statement using the conjunction OR. You surely don't want to prevent physicists from using the word "OR", do you? I assure you that physics or any science would be impossible without words like "OR". It's also untrue that statements "observable XY has value xy" may be meaningless. All such propositions are valid in general, by the basic rules of quantum mechanics. Histories constructed out of such sentences may fail to be "inconsistent" if I use the Gell-Mann-Hartle terminology but one surely can't ban any of these sentences at the level of general quantum mechanics, before the dynamics is considered. So it's not really "just" language. You misunderstand the physics, too. reader Luboš Motl said... Dear Old Wolf, when you say that there's a "split" after the measurement, you are back to the question "what is a measurement" (who has enough consciousness or whatever to be allowed to measure: now the same agents have the right to split the worlds!), the same question that was claimed to motivate the whole interpretation because the Copenhagen interpretation is said not to answer such questions. In reality, there can't of course be any splitting of the world and I have proved so. So it's not true that my description isn't relevant for this question. Some people just don't want to hear things that prove that they believe in wrong things. In the Copenhagen interpretation, the "measurement" is a somewhat arbitrarily defined threshold/event after which one may treat the information using the laws of classical physics. So one may talk about a "measurement" at a point behind which classical physics becomes an OK approximation, or later than that, but not before that. It's very important that it's a phenomenological theory only; nothing qualitative is actually changing about the world at the "moment of measurement". There isn't any "moment of measurement". Bohr has never said that the "wave function collapses". All of your comment is just pure bullshit. It's impossible to react to every sentence written by everyone who writes complete bullshit about quantum mechanics, about the way how the world actually works, about the mathematical possibilities how it could work, and about the history of science. All these things are being distorted, rewritten, rotated upside down, and you're a part of the problem, too. reader Eugene S said... I read The Fabric of Reality, too, when it came out. Deutsch also wrote that quantum computing, when it comes, will owe its power to massively parallel computations being performed on the qubits in many universes simultaneously that differ only by a smidgen. I guess the computer operators in our neighbor universes work for us... or we work for them :) Deutsch is near the top of my wishlist of people who I would like to see publish a guest blog at TRF, but ONLY if they stay around for the discussion afterward :) Borges was a genius writer, he surely mined theoretical physics for inspiration. reader Luboš Motl said... The Hidden Reality refers to the Garden of the Forking Paths, too, just mentioning it was Brian Greene's favorite literature on related topics. But one could seriously claim that if the "splitting worlds from quantum mechanics" is a legitimate insight in physics, it wasn't done by Everett for the first time but by this book. Everett hasn't made it more meaningful in any detectable sense. He just subtracted the references that make it obvious it is science fiction, and he only removed them because he was advised by Wheeler. reader Mitchell said... But you have to remember that this is the Copenhagen interpretation; "that observer is in a superposition" doesn't mean "there are two copies of that observer and they are in different states". It means "the wavefunction we should use to make predictions about the physical properties of that observer, is the mathematical sum of several other possible wavefunctions". That is always trivially true, as a mathematical statement; what people would be interested in, is if coherent superpositions of "macroscopically distinct" wavefunctions were ever needed to describe "observers" in the real world. Decoherence prevents this from happening; but as quantum computers become more advanced, we will get closer to having superpositions of observer-like cognitive processes. reader Shannon said... The danger would be a huge waste of time (like a few centuries)... like religions did in the past (and still do in some part of the world). This would throw our civilisation into the dark ages of science. We are living exciting times with the LHC, space probes etc... I hope physicists won't waste it. They have a huge responsibility on humanity's evolution path. reader Dilaton said... Yeah that is right. I am worried too about the fact that even though, due to the advanced knowledge and technologies we have now, we could learn a lot about deep fundamental questions (probably even in the not so far future) this great chance could be gamed away ... I recently had a serious word with my colleague who borrowed "Vom Urknall zum Durchknall" to me, in order to tell him what I really think about it and the author, his misbehavior in Munich, etc ... :-D. I the course of our lively discussion I learned that he belongs to the sourballs who are of the opinion that it is completely legitimate to cut fundamental physics since it is not important compared to the "real world" problems humanity faces at present etc etc ... His office mate (who I considered to be quite a nice and friendly guy too) is even worse; if he had the power to do it, he would probably turn off the LHC immediately to "save energy" for example, and abolish fundamental physics to save the money for something that is "more useful to humanity". Boy how did this discussion upset me since I never thought that these two colleagues are among such hardcore sourballs :-(((. At the moment, I hardly manage to look them in the eyes without turning into an angry shadron again when we meet accidentally in our corridor, our small kitchen, or at our weekly group meetings, ect ... Happily the director of our institute is more reasonable: The day after the higgs-independence day he explicitely pointed out the discovery of the higgs as very important, we all should know about it and we should all learn how the higgs mechanism works by ourself if we do not know it already :-))). The second part of his speech I since then use as a pretext to read TRF even at work ... :-P :-D :-) reader Dilaton said... Thanks Eugene :-) Your translation sounds very nice in my ears, good for Focus Magazine to talk to real physicists instead of trolls ;-). Now after all Prof. Lüst seems to have friends in the media. Maybe Germany has to improve its image after the horrible appearance of our own local troll king in Munich, which leads the science journalists to pull themself together ... :-D Hm, Nicolai wants to directly quantize spacetime ... is he a LQG theorist ? The original German article with the pretty picture I will read as a nice and comforting bedtime story to sleep well and peacefully :-) reader Gordon Wilson said... "The whole point of MWI is to redefine what exists means." That reminds me of Clinton's famous quote " It depends on what the meaning of the word "is" is." reader Shannon said... Thanks for the translation Eugene. reader Shannon said... Funny how these people all look alike to us ;-D (in TBBT Raj's dad confused Leonard with Sheldon and says "oh sorry, you all look alike to us"). reader Luboš Motl said... LOL, a fun scene. But I have some problems to believe it's genuine and there's this symmetry between the subraces as presented by TBBT. I may be wrong but I surely do believe that the diversity of the appearances between the whites is larger than for other races. Or is it really just because we're not optimized to distinguish other races finely enough and they're similar unequipped by the resolution for the whites in a symmetric way? reader Shannon said... You're right. Whites have different hair colours, eyes colours, complexion. Asians and blacks don't have those "striking" differences. reader George Christodoulides said... european populations have the smallest genetical variety by far than any of the other mentioned "groups". hair colours yes, there is a big variety dna, no there isn't. reader Rezso said... Dear Lubos, I think that your derivation doesn't show that the MWI is incorrect, because my interpretation of the projectors is a little bit different than yours. In your proof P_A=\|up><observes down\| and not the pure state you wrote. ( But your derivation doesn't really use the state, that's why I left this to the end. ) What do you think about my position? reader anon said... I think Lubos has explained this very well, especially with his reference to Mott as well as his proofs. Fundamentally the confusion on this point resides in people's adherence to a notion of objective reality, which must be abandoned in the quantum world. As pointed out we have to think in terms of probabilities. Is there some probability that if you made a different decision in your life you could have been the next einstein? Certainly there is and that has to be factored into the evolving wave function, but does that mean there is some version of yourself leading the life of fame and fortune? No, and that's the point. You AND your doppelganger can not be the outcome of an observation. It has to be you OR your doppleganger. The use of superposition by MWI advocates is misleading. All the superposition preserves is the indeterminancy of an unobserved state. However, the state is NOT you AND your doppelganger. A superposed state is unique in itself. It is a state between mutual exclusive possibilities. This is very important because such states do not have classical analogs. MWI proponents want to change the what exist means before they understand what it means now. reader David said... To me, the many worlds interpretation means something like: "The entire universe can be modeled as a quantum system, and the outcomes of experiments can be predicted probabilistically by analyzing the evolving correlations/entanglements". ("Many worlds" comes from the fact that in this model, the whole universe is quantum, and therefore in a superposition of mutually exclusive states.) This seems to be what Fred's saying, and to me the question of weather MWI is valid is basically weather that's true or false. Can you derive the Born rule just by looking at the whole universe's state vector? Can you make a toy model of a quantum mechanical universe containing a scientist measuring the spin of an electron and conclude "he's got a 60% chance of seeing an UP", for example? That's the question I'm curious about. reader Luboš Motl said... Well, I would probably call this paragraph "a few words, far from complete words, describing quantum mechanics in its Copenhagen interpretation. This difference in our wording isn't just terminology, it's about the credit and rewriting of the history of physics because your terminology suggests that there is something in your sentence that the Copenhagen school didn't discover. There's nothing of the sort. So if one strips everything that is demonstrably wrong about "MWI" and anything ever connected with MWI, one is back to the Copenhagen quantum mechanics and a movement trying to deny that it was these men who actually made the revolution in the foundations of physics. reader Old Wolf said... I do understand the physics, but we are disagreeing on which English words to use to describe it. Would you also say that in the double-slit experiment, the electron goes through one slit or the other? It's more commonly said that it goes through one slit and the other (i.e. both slits). reader Luboš Motl said... No, we are disagreeing on the substance. The propositions "A and B" and "A or B" are two completely different, inequivalent statements. The statement "the electron exists in slit A or slit B" is valid - with the disclaimer that one must avoid the wrong classical preconception that an objective answer exists. But the statement "an electron exists in the region of slit A and an electron exists in the region of slit B" is just wrong. It may be more common but it's wrong. You can't learn physics or logic or maths properly by choosing "more common" answers and sentences. You must choose more correct ones. reader David said... Right - fair enough. So according to you, people like me and Fred are copenhagen advocates, and according to us, you're a many worlds advocate. Well as long as people manage to communicate eventually... Is this maybe not what Brian Greene was referring to all along though? I mean I haven't read the book in question, but I imagine this was what he was angling at, maybe while introducing some slightly dodgy analogies to communicate with a lay audience...... I watched that Sidney Coleman lecture last night actually, and it seemed like he'd got most of the way to making the Born rule come from all the other postulates. Which was quite cool actually... This is doable right? Feel like doing a post on it? reader Luboš Motl said... I have no idea what you're talking about. Everything is upside down. I assure you that I have never been a MWI advocate according to myself and you have never grown enough to become a Copenhagen advocate. Still, the claim that outcomes - with most of the information stored in entanglement/correlation - is predicted probabilistically is what the Copenhagen school brought to the world. I have done dozens of posts on what you're saying, discussed a lecture on this topic by Sidney Coleman, and this blog post was another one. But you have probably missed everything. Probably deliberately so. reader Mikael said... Dear Lubos, of course that + does not mean exactly OR in quantum mechanics because of interference. But for cats it becomes a good enough approximation of OR because of interference. reader David said... I just liked the way Sidney Coleman basically explained why an observer would get random results when measuring an electron's spin, without referring to the Born rule at all. He explained reduction of the wavefunction very clearly as well. It's something I'd never seen before, and I found it very interesting. It kind of seemed like you could pretty much the same argument to just totally junk the Born rule and derive it from the other postulates, but I've never seen the full derivation. Not only that, but I've heard people claim that such a derivation is impossible. What's your position on this? Btw I notice Coleman cited Everett in that Quantum Mechanics in your Face lecture.... grin reader Luboš Motl said... You're just an irrational asshole - sorry if you don't like my terminology: I just watched about 10+ additional episodes of Bullshit of Penn and Teller, wonderful. What should I do with your junk? All your opinions, priorities, interpretations, methods are just junk. The Born rule is nothing else than the rule that QM predicts the probabilities and they're equal to \|c_i\|^2 where c_i is the complex coefficient of a decomposition of the wave function to a basis of eigenstates. If one uses modern i.e. quantum physics, every meaningful question may be reduced to a question about eigenvalues of observables and every such a question may be answered by a calculation of the amplitudes followed by the Born rule. It's the most important fundamental pillar of all of modern science. The Born rule is exactly true, it is fundamental, it has earned Max Born a well-deserved Nobel prize, and you as well as everyone else who wants to "junk it" is a deranged scumbag and idiotic fucked-up asshole. Whether one may "derive it" from other postulates is completely irrelevant. One surely has to start with some postulates that are at least as strong and far-reaching as the Born rule. The previous sentence is a tautology; the greater strength of the other hypothetical postulates clearly follows from the fact that the Born rule can be derived from it. Your idiotic ad hominem comments about Coleman are completely distasteful, too. I am sure that there isn't an iota of difference between my comments on QM and his comments, I know that he mentioned Everett and so did I, and I also know that he credits the Copenhagen founding fathers with the discovery of quantum mechanics and all the foundations needed to what he has ever said about the inner workings of quantum mechanics. reader Andrzej Czechowski said... Many World Interpretation (as I understand it) assumes in effect that each path under the path integral correspond to a separate reality. Inserting unity in the form 1=Sum \|><\| is interpreted as a sum over different realities. So your argument misses the point, I think. The measuring instrument cannot be simultaneously in the states \|1> and \|2> (Landau) but Everett assumes it is (in different realities). I am sorry, I would be only to happy to see the MWI disproved. reader Gene said... Is Fred the same Fred that did silly heat experiments a while back? If so, he doesn’t know a whit of physics. reader Bogs_Dollocks said... That sums it up. reader DonGateley said... If I understand you, you are simply removing the process of collapse entirely from the theory (and from the world.) Every interaction results in a superposition, characterized by quantum mechanics, that continues to propagate. End of story. There is a single universe that is much more complex. The idea of collapse is a crutch to try to explain what we think we experience and is the root of all of the confusion about what an observer is and what constitutes an observation. That is how it was explained to me a long time ago in a private conversation with a mathematical physicist of some renown and it seemed a wonderful simplification with an enormously expanded view. I don't think Lubos' attack is relevant to that view. I stand ready to be eviscerated, even blacklisted. :-) reader Luboš Motl said... Andrzej, sloppiness is an important part of this philosophically prejudiced demagogy. The formula 1=Sum \|› ‹\| must be interpreted as the sum over possible realities, not actual realities. There is absolutely no ambiguity about this statement - it may be directly measured. The interpretation of the sum above is exactly the same as int dp dq / 2.pi.hbar the integral over the phase space in classical physics. This is not just an analogy; in the classical limit, the sum above reduces to the integral over the phase space. Now, the individual points of the phase space are clearly not realized simultaneously - the phase space is the complete set of possible states in which the physical system may be but the number of states in which it actually is is demonstrably equal to one. The fact that the squared amplitudes \|c\|^2 are probabilities isn't one of dozens of possible speculations; it is a completely directly observed experimental fact. We may just associate the wave function with a particular experimental situation and if we measure once, we don't see any wave function in the experiment because the wave function isn't observable - both in the linguistic and technical sense. If we measure many times, we see the probabilistic distributions related to the wave function in the usual QM way, thus proving that the wave function is a semi-finished product for (all) probability distributions. This is not a random guess; it is a claim that may be directly observed in experiments. Maybe you would first have to open your eyes. reader Luboš Motl said... There has never been any collapse in the proper, Copenhagen (and physically equivalent, probability-based instrumental) interpretation of quantum mechanics. It has always been a crutch, a deeply misleading popular metaphor for newspapers that kind of influenced even those who shouldn't have been influenced and who should know better. The collapse isn't an actual process; it is just our simplification of our own knowledge, the arbitrary moment of time after which we may just use the conditional probabilities assuming the observed facts and forget about the result of the probability distribution defined for values different from those that were already made known to us. reader Deadlyelder said... This is without a doubt the most clearer and simple explanation of the basic idea of QM. I agree that MW theories are hogwash. To start with, how are many world to be tested? Trusting many worlds would be like trusting a person who claims to have been given so called divine message in sleep.... Looking forward for more of your posts reader DonGateley said... I would say that it is rather just a simplification of our perspective to something we can grasp. That no such moment really exists, just a universe of propagating and expanding superpositions due to prior interactions and leading through subsequent interaction to more of the same. This is the sense in which there are "many worlds." What I perceive as "I" is purely historical and is just a locus through it among which there are many ("many" being an enormous understatement.) Out of all this our perception of singular events and their probability is mysterious and in some deep way relates to measure, which is the real domain of quantum mechanics. Back to the armchair where I belong. Thanks for giving me the floor for a moment. reader Luboš Motl said... Dear Don, fine, if this simplification or any other simplification helps anyone, he may use it to improve his life. But he shouldn't call it science. Science isn't about simplification at any cost; science only allows simplification as long as the theory remains accurate as a description of all the known and relevant phenomena. One may simplify the explanation of sex to children by telling them that babies are brought by storks. Such a simplification helps but it is not valid science. There is no genuine stork that is bringing babies - babies are born without any birds whatsoever (except for the bird that the Czech readers are thinking about now) - and in the very same sense, there is no collapse, no many worlds, no hidden variables, and so on. Quantum mechanics just predicts probabilities of outcomes directly, without any of these intermediate storks, and the assumption that there exists one of these storks leads to a direct conflict with experiments as long as one looks at the experiments comprehensively enough. reader Chris Walsh said... This is more a technical point I guess and doesn't change the content of the article but I was wondering if since the "multiplications" were direct products are the sums really direct sums? My intuitive interpretation of the two types of products as applied to probabilities in general is that states are like marbles in bags. If you have a product of states then it's like one state is a marble (not necessarily a specific marble) from bag 1 and the other is from bag 2 so you take the direct product of the states to symbolize the fact that the probability to have the state s1(x)s2 is the product of the probability to have s1 and the probability to have s2 whereas for a sum of states you're really saying that the two states are two marbles from the same bag so obviously when you draw you only get one or the other. reader Luboš Motl said... I hope I understand you well in which case you're right and it's important. Two marbles have states that live in the tensor product of the single-marble Hilbert spaces. It's important that the tensor product isn't the direct sum. The dimension of the tensor product is d1d2 where d1,d2 are the dimensions of the single-marble Hilbert spaces. On the other hand, the dimension of the direct sum is d1+d2. The direct sum of two linear spaces may also be "geometrically" described as the Cartesian product* of the two individual linear spaces. But this Cartesian product is really just a sum. What one needs for two marbles is the tensor product of the Hilbert spaces and the probabilities for conditions "marble 1 does something and marble 2 does something else" reduces to the product of the two individual probabilities if the marbles are unentagled. And yes, if one talks about one marble e.g. in the double slit experiment, its having more possibilities where it can be corresponds to extending the spaces as direct sum. So if a single marble can sit in one of 15 red holes or 4 blue holes added later, it may sit in one of 19 holes and the Hilbert space is the direct sum of the original 15-state and 4-state Hilbert spaces. The marble is in one of the 15+4 holes, so it's either in the red holes OR the blue holes. This "OR" is what corresponds to the direct summation of Hilbert spaces. It doesn't increase the number of objects; it only increases the number of mutually excluding states/properties that the objects may have. reader Rezso said... Sorry, I don't know what happened with my previous post. Basically, I wanted to write two things: 1) Since the projectors are related to the observations of an observer, P_AP_B=0 only means that the same observer cannot observe the electron in both states simultaneously. In the MWI, there are two different observers in two different worlds who observe the different states, so there is no contradiction. 2) I think that the MWI only works after decoherence, so the MW state should be described by a density operator, where the offdiagonal terms are zero. If the decoherence isn't complete (so there are small offdiagonal terms), than the MWI is only an approximate picture. reader Luboš Motl said... Dear Rezso, you probably used "smaller than" and "greater than" which you shouldn't in a partly HTML-enabled comment editor. 1) Your attempt to escape the unescapable by "restricting it to an observer in one world" would only be justifiable if you could also create the corresponding mathematical objects that would describe whether a meta-observer in the whole system of worlds may see electrons in both spin states - anywhere. If it is in principle impossible to talk about the observations in the whole "MWI multiverse", then the MWI multiverse obviously doesn't exist. Needless to say, such an attempt will fail because it's exactly equivalent to the previous problem with the word "observer" replaced by "meta-observer". There can't be any operator that expresses the existence of objects or their properties in the "multi-world" for the reasons I have already demonstrated and your newest "excuse" is just a terminological sleight-of-hand that tries to redefine the "observer" in such a way that the "multi-world" becomes inaccessible in principle. At any rate, if done correctly, the argument leads to an inescapable conclusion: the other worlds can't exist. 2) Decoherence is a meaningful theory that can be explained and verified by well-defined mathematical formulae but MWI is not. There isn't any "MWI after decoherence". MWI is a philosophical prejudice that was promoted decades before* decoherence was discovered and it is not really equivalent to decoherence, either. And decoherence doesn't produce any "multiple worlds". It explains why some bases of states are more observable in the real complex world than others. You are talking about a non-existent combination of MWI (which is an ill-defined piece of rubbish) and decoherence (which is a homework exercise and totally indisputable consequence of ordinary quantum mechanics applied to states in which an interesting system is treated separately from the uncontrollable environment). reader Rezso said... Dear Lubos, thank you for your answers. 1) I completely agree that we should not introduce extra structure for meta-observers, because in the MWI, we would like to describe the measurement process without the measurement axiom and without adding any extra structure to conventional unitary QM. 2) Do you think that decoherence alone can solve the measurement problem completely? Surely, it can explain why macroscopic objects behave in a classical way. And it is a physical process, no one can deny it's existence. But is it the final answer? The result of decoherence theory is a density operator for the system (after the environment is traced out). The probabilistic interpretation of this object should be put in by hand. For example, if you take a look at this article:, Joos concludes that decoherence is not the final answer and that there are only two possibilities for the good interpretation (if hidden variable theories such as pilot wave theory are excluded): 1) we should modify the Schrödinger equation to get a real, objective collapse 2) we should use the MWI. My opinion: I dislike option 1), because the Schrödinger-equation is equvialent to unitary time evolution, so even a small modification would lead to a completely different philosophy behind the equation. So, if I were forced to choose I would clearly go with option 2). What do you think? You clearly dislike option 2) but I suspect that you will say that you dislike option 1) too. reader Rezso said... Dear Lubos, You wrote: "Also, I feel very uneasy about your sentence "the probabilistic interpretation of [density matrix] should be put by hand". Which hand? What is the sentence supposed to mean except for trying to spread some irrational and totally unjustified doubts by some rhetorical tricks? The density matrix is by definition the quantum version of the probability distributions on the phase space, so of course that it has a probabilistic interpretation, by definition." The meaning of my sentence was the following. In modern decoherence theories, you start with the wavefunction of the system+environment, build an orthogonal projector from it, and then, you trace over the environment to obtain the density operator of the system. So, there are no probabilities in the definition! First, the decoherence term in the master equation has to kill the offdiagonal matrix elements (in the preferred basis). After that, the remaining diagonal matrix elements can be interpreted as classical probabilities. What I wanted to say above is that this is a new assumption which is needed to connect the theory with experiments. This is why some people think that there is something more in the measurement problem. Of course, one can argue with this analysis. So now, I'm going to argue with myself. :) One can say that the properties of the density operator make a probabilistic interpretation natural. Hermicity means that the diagonal matrix elements are real, Tr=1 means that their sum is 1, and positivity means that all of them are positive. So a probabilistic interpretation is natural. Oh no, today it seems like that I have convinced myself that my previous argument was wrong. :S reader Luboš Motl said... You say "So, there are no probabilities in the definition!". That's a highly bizarre assertion. Whenever you want to interpret the calculations physically, you need to use the word probability because it's the only valid interpretation of the matrix elements of the density matrix, of the expectation values of projection operators, and so on. No, there is absolutely no "new assumption" in decoherence. Decoherence is just ordinary quantum mechanics applied to a particular kind of questions about the co-existence of a system with its environment. The interpretations of all objects such as density matrix are exactly the same as they always are in quantum mechanics. The probabilistic interpretation is not only natural but it's also one that may directly derived from observations and the only one that allows the theory to reduce to the previously known classical limits. reader James Gallagher said... Hi Lubos, I highly admire your never-ending defence, against all-comers, of the probabilistic interpretation and encourage you to never stop, it's clearly how nature is Problem is, it's a a mix of psychological non-acceptance and miniscule logical loopholes (eg MWI, super-determinism, crazy godlike pilot waves) that allow the fretting deniers of nature's randomness a corner to fight from, and while you deal superbly well with the logical arguments, you'll never solve the psychiatry problems. FWIW, I agree with YOU. reader Rezso said... Dear Lubos, okay, you convinced me that you are right. The probabilistic interpretation of the density operator follows naturally from it's mathematics, so nothing more (like MWI or something else) is needed, decoherence alone solves the measurement problem. But I still maintain that the fundamental definition of the density operator should use the partial trace and not the classical probabilities. And this is a difference between decoherence theory and ordinary QM (=Copenhagen Interpretation). In ordinary QM, the construction of the theory goes in the following order: 1) Wavefunction, unitary time evolution 2a) Measurement axiom, wavefunction collapse, classical probabilities 3a) Density operator is defined from the probabilities and from the corresponding collapsed wavefunctions But the decoherence motivated construction of QM goes as: 1) Wavefunction, unitary time evolution 2b) System+Environment, density operator is defined by a partial trace 3b) Classical probabilities emerge from the density operator after decoherence is complete So, I want to say that 2b) is a better definition for the density operator than 3a), because 3a) relies on the ad hoc wavefunction collapse rule, while 2b) doesn't. Do you agree with me in this? reader Shooter said... I want to know what you think of this. Do you think interstellar travel is possible, or science fiction? Hearing from a physicist would be great. reader Luboš Motl said... "What I think is that your position is just a linguistically powered rubbish that can't be given any interpretation that makes sense and you are just wasting my time. In an experiment with one electron, "electron exists with spin up" is exactly the same proposition as "electron has spin up". Trying to create any doubts about this is totally irrational. Also, if you used the MWI philosophy to arbitrarily insert existential quantifiers ("there exists a universe in which") in front of all propositions, you would totally screw all rules of logic about the propositions. You can't just add quantifiers without totally changing the logic. In particular, "electron has spin up" is the exact negation of "electron has spin down" but "there exists a universe with electron up" isn't complementary to "there exists a universe with electron down", especially because both propositions would almost certainly be "true" in an MWI. So this is experimentally excluded because we know that they're negations of each other. In such comments, I see that any discussion is totally hopeless after the first sentence. You say that we have different interpretations of projection operators. Holy fuck. How can you have a different interpretation of a projection operator? It is a very elementary object in principle, both mathematically and physically, and there is only one interpretation that is consistent with observations as well as logic and it's the interpretation of QM. The interpretation is that a projection operator is P obeying P^2 = P, we also want P^dagger=P, that is identified with the observable having No/Yes i.e. 0/1 eigenvalues answering to a question - namely the question Is the physical system in a state inside the lambda=1 eigenvalue (of P) subspace of the Hilbert space? The expectation value of this P in a pure state or Tr(P.rho) is the probability that the proposition holds. That's it. What the fuck is your interpretation? You're always promising some other interpretation but there isn't any. Crackpots like you are talking too much. In reality, the MWI babblers haven't even decided whether projection operators play any role in MWI at all. The reason they haven't decided is that none of the two answers makes any sense and they know it. At any rate, I am waiting for your prescription how to use projection operators to make the calculations in your non-Copenhagen framework, the MWI counterpart of my paragraph two paragraphs above. Before you actually have something of the sort, could you please kindly shut up and stop these meaningless tirades that only show one thing, namely that you're never willing to learn anything and you prefer to spit tons of this vague nonsensical mud over the Internet? reader Rezso said... Dear Lubos, the comment you just replied to was my oldest comment in the thread. But it was broken, and I only removed "smaller than" and "greater than" symbols from it to make it work. "In such comments, I usually see that any discussion is totally hopeless after the first sentence." Actually, you already convinced me that you are right and the MWI is an incorrect interpretation. :) "Also, if you used the MWI philosophy to arbitrarily insert existential propositions, you would totally screw all rules of logic about the propositions. You can't just add quantifiers without totally changing the logic." Yes, you are right. I haven't thought about this, when I wrote my old post. But the old Copenhagen interpretation is incorrect too because it uses an ad hoc wavefunction collapse rule and the preferred basis is chosen by hand. Decoherence theory is the correct interpretation of quantum measurements, where classical probabilities naturally emerge from the density operator after decoherence is complete. But the wavefunction never really collapses. Previously, I thought that the decoherence interpretation can be naturally merged with the MWI, but you convinced me that I was wrong. reader anon said... Your calculations and logic are both wrong. You're simply calculating the probability that electrons within the same time-line would exist in multiple states at once to be 0, which would obviously be true. Within the same time-line the probability of existing in different states would be 0, but that doesn't have anything to do with the MWI. All interpretations of the double-slit experiment use the same mathematics and quantum theory, the results of double-slit experiment are just interpreted differently. In MWI there is no collapse, you see the electron that exists in your time-line. So your disproof is all wrong. I'm certain that in the future multiple time-lines will be experimentally proven, though I'm not sure about the MWI in particular. reader Luboš Motl said... By a timeline, you probably meant a world line, right? ;-) Show me your "correct MWI" calculation of the same trivial thing but before you do so, please accept the fact that your comment is just a laymen's rubbish. reader Eugene S said... Very nice find, Victor. Also nice (in my opinion): Newton's First Law, by the Number Sixes and the Double Slit Experiment by Future Management Agency. Both of them Creative Commons licenses so you can't beat the price :D reader notallama said... As far as the proponents of many worlds go, it's not the laymen and self-proclaimed experts and prophets who just can't get their minds around the wave function being a subjective probability distribution because it mathematically looks like a classical wave for one particle, preferably spinless, that bother me. Most of those people are on the level of the Flat Earth Society. It's the slightly bigger names that subscribe to this ill-conditioned interpretation that freak me out, e.g. DeWitt, Zurek, perhaps partly Wheeler (he did talk about the wavefunction of the Universe) and I have a hunch that Sidney Coleman is something more of a fan of many worlds than you would like to think. The language that he uses in his In Your Face talk is kinda MWI-ish, e.g. at about 12:15 he says something like "we were in the branch that got spin up". Also he never says that he takes the wavefunction to be non-physical and his position seems to be "associated" with Everett, obviously, Yakir "Weak Measurement" Aharonov, David Albert, and Zurek. Now Zurek did a lot of great stuff on decoherence, but he subscribes to a modified many worlds interpretation of QM. Anyway, Sidney says a lot of smart things, but I'm really worried about these MWI-style statements. This might not be as serious as I take it to be, though it's hard to know, and even if it is of course none of Sidney's opinions reduce the amount of inconsistency of MWI. But it points the way to a curious psychological phenomenon, or problem, if you will. And that is that otherwise smart people, very smart even, who can extract wonders from the mathematics underlying our physical theories, reduce to complete morons when it comes to interpretational issues, the debating of which usually consists of very simple and irrelevant mathematics obscured to an arbitrary degree by metaphysical/science-philosophical vocabulary that they probably aren't really qualified to use. Take the recent work of 't Hooft and Weinberg for instance. I find that very mystifying. I find Zurek to be the most curious of these figures. In his paper at he advocates a variant of the MWI, of course called “relative-state” to make it more bland. Amongst other things he claims to derive the Born rule non-circularly (funny how after Deutsch et al’s failures it sounds like this is a specific type of its derivation :)) with the aid of envariance, a theoretical aid much in the spirit of the decoherence program. Unfortunately he truly doesn’t sound like he’s talking complete crap. He might be reaching for the deepest interpretational layers of quantum mechanics that can be reached without denying the objective existence of the wavefunction. Of course that might be worse than not reaching for them at all, since the consistency of the Copenhagen interpretation makes it completely unnecessary and it’s probably a ton of bullcrap anyway, only neatly worded and convoluted to the point when it looks convincing. But it does look convincing. Do you have an opinion on Zurek’s derivation? Especially, do you think you can identify a point at which possible inconsistencies arise? I tried debunking it myself but haven’t spotted the obvious problem yet. The only paper negatively addressing Zurek’s interpretation that I could find was by Ulrich Mohrhoff, a curious guy who I think is doing a good job in patiently explaining to the anti-quantum zealots why the probabilistic interpretation is the thing. But while ok, he seems to be doing stuff a bit differently from, say, consistent historians, so I am very much interested in your opinion. reader Luboš Motl said... Dear notallama, I would endorse a big part of what you write. Just a few comments. The "relative state interpretation" isn't Zurek's renamed stuff. It's the title, perhaps with formulation instead of interpretation, of Everett's 1957 thesis. So Zurek seems to be analyzing the same thing. Still, his 2007 paper starts by assuming the probabilistic interpretation, as far as I can read it. I also agree that Sidney Coleman himself used some MWI-like-sounding language. But as far as I can say, it's always just the language. He may have used the word "branch", perhaps because he was inspired to use it, but I don't see any indication that he would actually interpret the squared amplitudes as anything else than the probabilities or that he would try to look for a model where the wave function is "more real". Many of us have adopted certain phrases, especially because the strongest "proselytizers" when it comes to quantum mechanics are those who don't understand QM properly. (I remember a Czech physicist at my Alma Mater in Prague, Bedrich Velicky who knew very many famous world physicists, who always complained how universities don't teach the "real deal", but when it came to his "real deal", it was some naive "realist model", not remember which one.) So each of us picks a tolerable one among them, Coleman probably picked the Everett language as the most tolerable one but I don't think that it has influenced his thinking. I agree that those people are smart and brain-powerful when it comes to some technically more demanding questions but they just become complete idiots when the topic switches to interpretation. And I exactly agree with your observation that their technological capabilities suddenly evaporate and the most difficult maths is on the level of "squared amplitudes", and in most cases, they don't even square it right or they don't care whether it should be squared, and so on. It's completely weird. They probably see other "otherwise very smart" people who are doing the same thing so they feel justified to be equally breathtaking idiots. It's an infection of a sort. The 2007 Zurek paper is full of lots of redundant gibberish but of course I think that it's among the saner papers on the interpretational issues. It explains that MWI can't do almost anything right, as I read it, but one may supplement it with his insights - which are described by 50 different names or metaphors, decoherence, quantum Darwinism, einselection, envariance, and so on, and so on, but the essence is always the same mechanism - to get a sensible "interpretation". Also, if I guess right, the \|c\|^2 probabilities are extracted by looking at many states of an entangled complicated system including the environment chosen so large that each micro-outcome corresponds to a large number of microstates of the whole big system with the same absolute value of the amplitudes, by symmetries, the probabilities of each are probably claimed to be the same, which allows to "derive" \|c\|^2 in general by summing over many terms with the same absolute value. I think it's silly to think that this is more fundamental than the general rule for the situation in which the amplitudes have different absolute values - because they almost certainly have different absolute values, so it's contrived to assume that they should have the same absolute value. But it's a part of the hatred against everything that is quantum, including the simple Born rule. Some people just don't want it to be fundamental - well, one of the postulates or derived statements that are so closed to postulates by derivations that it makes no sense not to call them fundamental - and Zurek "partially" joins this idiotic movement in the paper. reader gbush said... You're basically just saying that it is impossible to observe an electron that is simultaneously spin up and spin down, and that every observation will confirm that charge is conserved. I doubt any MWI proponent would disagree with either statement or feel that it contradicts their interpretation. I think there's a valid philosophical objection to an interpretation that talks about the reality of alternate possibilities that can never be observed (if you can't possibly observe them, in what sense do they deserve to be called real?). But I don't see how you'd derive a mathematical contradiction to it, since at its heart it's just a very literal interpretation of the mathematics of the wave function. Personally, I don't see MWI as being much different than people talking about virtual particles in QFT. You'll never observe them, so are they real, or is it just a convenient way to visualize the math behind your theory? Probably the latter, but I'm not going to go to war over it and call people stupid monkeys if they talk about virtual particles. reader Luboš Motl said... MWI proponents may "feel" ;-) various things but science is not about feelings and the contradiction is there. It's not true at all that this multi-world fantasy is a "literal interpretation" of the wave function. It's a wrong interpretation designed as crutches for the stupid people but it has nothing to do with the right probabilistic interpretation and indeed, it contradicts it. As always, the key point of MWI that makes it incompatible with the real world - and with quantum mechanics - is the idea that there objectively exists some classical information that is independent of the observers and observations. This ain't the case. Sensible people call about virtual particles but they understand that they are not real physical particles. They're mathematical constructs contributing to probability amplitudes for processes involving real particles. But the point of the many worlds is different. It's the very point of MWI that those worlds are "real" in the classical sense, and this assumption may be shown and has been shown to contradict observations. If you're not getting it, you are a stupid ape. reader kmut said... You write: "But the statement "an electron exists in the region of slit A and an electron exists in the region of slit B" is just wrong (...) It's a point-like particle, there is only one electron (by charge conservation etc.), and it can't be in both slits at the same moment." But what about deBroigle's waves of matter? An electron can be described as a wave, and waves are certainly not just single points of space. All kinds of waves occuppy a region of space, so a wave can be in both slit A and B at the same time (just like Russia lays in both Europe and Asia, because it's not a point but an area) and there's nothing wrong with it. It's obvious in the case of double slit experiments performed with waves of water, for many people it's obvoius in the case of light, and I think there is no reason to think different in the case of electrons. reader Luboš Motl said... Dear Kmut, one could say that it was the very main purpose of this statement of mine to emphasize that the electron is not a classical wave. Prince de Broglie misunderstood those things much like you do, even after 1924 when he proposed his wave, which is why throwing his name around can't turn your invalid statements into valid ones. A classical water wave goes through both slits - one may detect "something" by an appropriate detector in both of them. But when an electron goes through the pair of slits, there is nothing that could ever be detected in both slits simultaneously. If you use a detector of any kind, call it a detector of waves, particles, disturbances, spirits, whatever, and if these detectors only operate in the regions around the two respective slits, they will never beep simultaneously. Also, the electron, unlike a classical wave, will always create a single point at the photographic plate. It's just not true that "there is nothing wrong with electron's being a classical wave". There's a lot of wrong things. A whopping 50% of statements one can make about waves are plain untrue about the electron. Just to be sure, many laymen don't get it: one wrong thing would invalidate your claim. But there are lots of waves to invalidate it, it's just wrong. A classical wave may be a method to think about the behavior of an electron or a quantum particle in some respects but it's surely not a valid model for all of its behavior. An electron is not a classical wave and the wave function isn't a classical wave, either. reader kmut said... Dear Lubos I really didnt want to make any ad hominem arguments. It certainly wasn't my purspose. I'm not a physicist but a person who would like to become one in the future, so my knowledge in this area is basic, especially when compared to yours. But you must know that it inst an unusual way for the people who teach QM to look on this problem from a differend side than you do. You say that electrons aren't classical waves because they cannot be measured in a classical way. But there are people who say that electrons are like classical waves because the Dirac equation, and more basic Schroedinger one describe a classical i.e. deterministic and unique time evolution so the real difference is in the act od measurement. In the quantum world the measurement is more "drastic" than in classical one. To observe waves on water we only need some light with energy too small to change the pattern in statistically significant way. But in quantum world energy of a wave we need to measure position of the electron is enough big to interfere with it. I've read recently some words by Wojciech Zurek, and I had an impression that he understads QM in in similar way stating that macroscopic objects are all quantum and the reason for which they dont behave like waves and for which they have unique lociaction, is that they're not enough separated from the environment, which is responsible for all this huge amount of interactions that forbid the macroscopic objects to behave like waves. Isnt this view just dual to yours? And how can one interprete interfference patternw in double slit experiment with electrons? If electron is point-like poarticle then what forbids it to behave like classical point-like particle and forces it to change its momentum? Is there any book particular where I could find all the answers to theese questions? Thank you in advance reader Luboš Motl said... "But there are people who say that electrons are like classical waves..." There are many people who say many dumb things and indeed, it's the main purpose of all these blog entries of mine to correct the widespread misconceptions. It's disappointing if you don't appreciate it and it's surprising that you seem to read this blog anyway even though the correction of stupidities said by people, especially if it is many people, is self-evidently the defining driver behind this blog. The wave functions evolve according to analogous "deterministic" equations as classical fields and waves but their physical interpretation is completely different so the "determinism" of Schrödinger's equation – or the Dirac equation promoted to a quantum equation for an actual system – does not translate to classical determinism of the real world which simply doesn't hold.
837b7199eb5698f7	Computational chemistry From Wikipedia, the free encyclopedia Jump to: navigation, search Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids. Its necessity arises from the well-known fact that apart from relatively recent results concerning the hydrogen molecular ion (see references therein for more details), the quantum many-body problem cannot be solved analytically, much less in closed form. While its results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials. Examples of such properties are structure (i.e. the expected positions of the constituent atoms), absolute and relative (interaction) energies, electronic charge distributions, dipoles and higher multipole moments, vibrational frequencies, reactivity or other spectroscopic quantities, and cross sections for collision with other particles. The methods employed cover both static and dynamic situations. In all cases the computer time and other resources (such as memory and disk space) increase rapidly with the size of the system being studied. That system can be a single molecule, a group of molecules, or a solid. Computational chemistry methods range from highly accurate to very approximate; highly accurate methods are typically feasible only for small systems. Ab initio methods are based entirely on theory from first principles. Other (typically less accurate) methods are called empirical or semi-empirical because they employ experimental results, often from acceptable models of atoms or related molecules, to approximate some elements of the underlying theory. Both ab initio and semi-empirical approaches involve approximations. These range from simplified forms of the first-principles equations that are easier or faster to solve, to approximations limiting the size of the system (for example, periodic boundary conditions), to fundamental approximations to the underlying equations that are required to achieve any solution to them at all. For example, most ab initio calculations make the Born–Oppenheimer approximation, which greatly simplifies the underlying Schrödinger equation by assuming that the nuclei remain in place during the calculation. In principle, ab initio methods eventually converge to the exact solution of the underlying equations as the number of approximations is reduced. In practice, however, it is impossible to eliminate all approximations, and residual error inevitably remains. The goal of computational chemistry is to minimize this residual error while keeping the calculations tractable. In some cases, the details of electronic structure are less important than the long-time phase space behavior of molecules. This is the case in conformational studies of proteins and protein-ligand binding thermodynamics. Classical approximations to the potential energy surface are employed, as they are computationally less intensive than electronic calculations, to enable longer simulations of molecular dynamics. Furthermore, cheminformatics uses even more empirical (and computationally cheaper) methods like machine learning based on physicochemical properties. One typical problem in cheminformatics is to predict the binding affinity of drug molecules to a given target. History [edit] Building on the founding discoveries and theories in the history of quantum mechanics, the first theoretical calculations in chemistry were those of Walter Heitler and Fritz London in 1927. The books that were influential in the early development of computational quantum chemistry include Linus Pauling and E. Bright Wilson's 1935 Introduction to Quantum Mechanics – with Applications to Chemistry, Eyring, Walter and Kimball's 1944 Quantum Chemistry, Heitler's 1945 Elementary Wave Mechanics – with Applications to Quantum Chemistry, and later Coulson's 1952 textbook Valence, each of which served as primary references for chemists in the decades to follow. With the development of efficient computer technology in the 1940s, the solutions of elaborate wave equations for complex atomic systems began to be a realizable objective. In the early 1950s, the first semi-empirical atomic orbital calculations were carried out. Theoretical chemists became extensive users of the early digital computers. A very detailed account of such use in the United Kingdom is given by Smith and Sutcliffe.[1] The first ab initio Hartree–Fock calculations on diatomic molecules were carried out in 1956 at MIT, using a basis set of Slater orbitals. For diatomic molecules, a systematic study using a minimum basis set and the first calculation with a larger basis set were published by Ransil and Nesbet respectively in 1960.[2] The first polyatomic calculations using Gaussian orbitals were carried out in the late 1950s. The first configuration interaction calculations were carried out in Cambridge on the EDSAC computer in the 1950s using Gaussian orbitals by Boys and coworkers.[3] By 1971, when a bibliography of ab initio calculations was published,[4] the largest molecules included were naphthalene and azulene.[5][6] Abstracts of many earlier developments in ab initio theory have been published by Schaefer.[7] In 1964, Hückel method calculations (using a simple linear combination of atomic orbitals (LCAO) method for the determination of electron energies of molecular orbitals of π electrons in conjugated hydrocarbon systems) of molecules ranging in complexity from butadiene and benzene to ovalene, were generated on computers at Berkeley and Oxford.[8] These empirical methods were replaced in the 1960s by semi-empirical methods such as CNDO.[9] In the early 1970s, efficient ab initio computer programs such as ATMOL, Gaussian, IBMOL, and POLYAYTOM, began to be used to speed up ab initio calculations of molecular orbitals. Of these four programs, only GAUSSIAN, now massively expanded, is still in use, but many other programs are now in use. At the same time, the methods of molecular mechanics, such as MM2, were developed, primarily by Norman Allinger.[10] One of the first mentions of the term "computational chemistry" can be found in the 1970 book Computers and Their Role in the Physical Sciences by Sidney Fernbach and Abraham Haskell Taub, where they state "It seems, therefore, that 'computational chemistry' can finally be more and more of a reality."[11] During the 1970s, widely different methods began to be seen as part of a new emerging discipline of computational chemistry.[12] The Journal of Computational Chemistry was first published in 1980. Fields of application [edit] The term theoretical chemistry may be defined as a mathematical description of chemistry, whereas computational chemistry is usually used when a mathematical method is sufficiently well developed that it can be automated for implementation on a computer. In theoretical chemistry, chemists, physicists and mathematicians develop algorithms and computer programs to predict atomic and molecular properties and reaction paths for chemical reactions. Computational chemists, in contrast, may simply apply existing computer programs and methodologies to specific chemical questions. There are two different aspects to computational chemistry: • Computational studies can be carried out to find a starting point for a laboratory synthesis, or to assist in understanding experimental data, such as the position and source of spectroscopic peaks. • Computational studies can be used to predict the possibility of so far entirely unknown molecules or to explore reaction mechanisms that are not readily studied by experimental means. Thus, computational chemistry can assist the experimental chemist or it can challenge the experimental chemist to find entirely new chemical objects. Several major areas may be distinguished within computational chemistry: • The prediction of the molecular structure of molecules by the use of the simulation of forces, or more accurate quantum chemical methods, to find stationary points on the energy surface as the position of the nuclei is varied. • Storing and searching for data on chemical entities (see chemical databases). • Identifying correlations between chemical structures and properties (see QSPR and QSAR). • Computational approaches to help in the efficient synthesis of compounds. • Computational approaches to design molecules that interact in specific ways with other molecules (e.g. drug design and catalysis). Accuracy [edit] The words exact and perfect do not appear here, as very few aspects of chemistry can be computed exactly. However, almost every aspect of chemistry can be described in a qualitative or approximate quantitative computational scheme. Molecules consist of nuclei and electrons, so the methods of quantum mechanics apply. Computational chemists often attempt to solve the non-relativistic Schrödinger equation, with relativistic corrections added, although some progress has been made in solving the fully relativistic Dirac equation. In principle, it is possible to solve the Schrödinger equation in either its time-dependent or time-independent form, as appropriate for the problem in hand; in practice, this is not possible except for very small systems. Therefore, a great number of approximate methods strive to achieve the best trade-off between accuracy and computational cost. Accuracy can always be improved with greater computational cost. Significant errors can present themselves in ab initio models comprising many electrons, due to the computational expense of full relativistic-inclusive methods. This complicates the study of molecules interacting with high atomic mass unit atoms, such as transitional metals and their catalytic properties. Present algorithms in computational chemistry can routinely calculate the properties of molecules that contain up to about 40 electrons with sufficient accuracy. Errors for energies can be less than a few kJ/mol. For geometries, bond lengths can be predicted within a few picometres and bond angles within 0.5 degrees. The treatment of larger molecules that contain a few dozen electrons is computationally tractable by approximate methods such as density functional theory (DFT). There is some dispute within the field whether or not the latter methods are sufficient to describe complex chemical reactions, such as those in biochemistry. Large molecules can be studied by semi-empirical approximate methods. Even larger molecules are treated by classical mechanics methods that employ what are called molecular mechanics. In QM/MM methods, small portions of large complexes are treated quantum mechanically (QM), and the remainder is treated approximately (MM). Methods [edit] A single molecular formula can represent a number of molecular isomers. Each isomer is a local minimum on the energy surface (called the potential energy surface) created from the total energy (i.e., the electronic energy, plus the repulsion energy between the nuclei) as a function of the coordinates of all the nuclei. A stationary point is a geometry such that the derivative of the energy with respect to all displacements of the nuclei is zero. A local (energy) minimum is a stationary point where all such displacements lead to an increase in energy. The local minimum that is lowest is called the global minimum and corresponds to the most stable isomer. If there is one particular coordinate change that leads to a decrease in the total energy in both directions, the stationary point is a transition structure and the coordinate is the reaction coordinate. This process of determining stationary points is called geometry optimization. The determination of molecular structure by geometry optimization became routine only after efficient methods for calculating the first derivatives of the energy with respect to all atomic coordinates became available. Evaluation of the related second derivatives allows the prediction of vibrational frequencies if harmonic motion is estimated. More importantly, it allows for the characterization of stationary points. The frequencies are related to the eigenvalues of the Hessian matrix, which contains second derivatives. If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum. If one eigenvalue is negative (i.e., an imaginary frequency), then the stationary point is a transition structure. If more than one eigenvalue is negative, then the stationary point is a more complex one, and is usually of little interest. When one of these is found, it is necessary to move the search away from it if the experimenter is looking solely for local minima and transition structures. The total energy is determined by approximate solutions of the time-dependent Schrödinger equation, usually with no relativistic terms included, and by making use of the Born–Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation. This leads to the evaluation of the total energy as a sum of the electronic energy at fixed nuclei positions and the repulsion energy of the nuclei. A notable exception are certain approaches called direct quantum chemistry, which treat electrons and nuclei on a common footing. Density functional methods and semi-empirical methods are variants on the major theme. For very large systems, the relative total energies can be compared using molecular mechanics. The ways of determining the total energy to predict molecular structures are: Ab initio methods [edit] The programs used in computational chemistry are based on many different quantum-chemical methods that solve the molecular Schrödinger equation associated with the molecular Hamiltonian. Methods that do not include any empirical or semi-empirical parameters in their equations – being derived directly from theoretical principles, with no inclusion of experimental data – are called ab initio methods. This does not imply that the solution is an exact one; they are all approximate quantum mechanical calculations. It means that a particular approximation is rigorously defined on first principles (quantum theory) and then solved within an error margin that is qualitatively known beforehand. If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained (the best that is possible with a finite word length on the computer, and within the mathematical and/or physical approximations made). Diagram illustrating various ab initio electronic structure methods in terms of energy. Spacings are not to scale. The simplest type of ab initio electronic structure calculation is the Hartree–Fock (HF) scheme, an extension of molecular orbital theory, in which the correlated electron–electron repulsion is not specifically taken into account; only its average effect is included in the calculation. As the basis set size is increased, the energy and wave function tend towards a limit called the Hartree–Fock limit. Many types of calculations (known as post-Hartree–Fock methods) begin with a Hartree–Fock calculation and subsequently correct for electron–electron repulsion, referred to also as electronic correlation. As these methods are pushed to the limit, they approach the exact solution of the non-relativistic Schrödinger equation. In order to obtain exact agreement with experiment, it is necessary to include relativistic and spin orbit terms, both of which are only really important for heavy atoms. In all of these approaches, in addition to the choice of method, it is necessary to choose a basis set. This is a set of functions, usually centered on the different atoms in the molecule, which are used to expand the molecular orbitals with the LCAO ansatz. Ab initio methods need to define a level of theory (the method) and a basis set. The Hartree–Fock wave function is a single configuration or determinant. In some cases, particularly for bond breaking processes, this is quite inadequate, and several configurations need to be used. Here, the coefficients of the configurations and the coefficients of the basis functions are optimized together. The total molecular energy can be evaluated as a function of the molecular geometry; in other words, the potential energy surface. Such a surface can be used for reaction dynamics. The stationary points of the surface lead to predictions of different isomers and the transition structures for conversion between isomers, but these can be determined without a full knowledge of the complete surface. A particularly important objective, called computational thermochemistry, is to calculate thermochemical quantities such as the enthalpy of formation to chemical accuracy. Chemical accuracy is the accuracy required to make realistic chemical predictions and is generally considered to be 1 kcal/mol or 4 kJ/mol. To reach that accuracy in an economic way it is necessary to use a series of post-Hartree–Fock methods and combine the results. These methods are called quantum chemistry composite methods. Density functional methods [edit] Density functional theory (DFT) methods are often considered to be ab initio methods for determining the molecular electronic structure, even though many of the most common functionals use parameters derived from empirical data, or from more complex calculations. In DFT, the total energy is expressed in terms of the total one-electron density rather than the wave function. In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density. DFT methods can be very accurate for little computational cost. Some methods combine the density functional exchange functional with the Hartree–Fock exchange term and are known as hybrid functional methods. Semi-empirical and empirical methods [edit] Semi-empirical quantum chemistry methods are based on the Hartree–Fock formalism, but make many approximations and obtain some parameters from empirical data. They are very important in computational chemistry for treating large molecules where the full Hartree–Fock method without the approximations is too expensive. The use of empirical parameters appears to allow some inclusion of correlation effects into the methods. Semi-empirical methods follow what are often called empirical methods, where the two-electron part of the Hamiltonian is not explicitly included. For π-electron systems, this was the Hückel method proposed by Erich Hückel, and for all valence electron systems, the extended Hückel method proposed by Roald Hoffmann. Molecular mechanics [edit] These methods can be applied to proteins and other large biological molecules, and allow studies of the approach and interaction (docking) of potential drug molecules (e.g. [1] and [2]). Methods for solids [edit] Computational chemical methods can be applied to solid state physics problems. The electronic structure of a crystal is in general described by a band structure, which defines the energies of electron orbitals for each point in the Brillouin zone. Ab initio and semi-empirical calculations yield orbital energies; therefore, they can be applied to band structure calculations. Since it is time-consuming to calculate the energy for a molecule, it is even more time-consuming to calculate them for the entire list of points in the Brillouin zone. Chemical dynamics [edit] Once the electronic and nuclear variables are separated (within the Born–Oppenheimer representation), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator (physics) associated to the time-dependent Schrödinger equation (for the full molecular Hamiltonian). In the complementary energy-dependent approach, the time-independent Schrödinger equation is solved using the scattering theory formalism. The potential representing the interatomic interaction is given by the potential energy surfaces. In general, the potential energy surfaces are coupled via the vibronic coupling terms. The most popular methods for propagating the wave packet associated to the molecular geometry are: Molecular dynamics [edit] Molecular dynamics (MD) use either quantum mechanics, Newton's laws of motion or a mixed model to examine the time-dependent behavior of systems, including vibrations or Brownian motion and reactions. MD combined with density functional theory leads to hybrid models. Interpreting molecular wave functions [edit] The Atoms in molecules or QTAIM model of Richard Bader was developed in order to effectively link the quantum mechanical picture of a molecule, as an electronic wavefunction, to chemically useful concepts such as atoms in molecules, functional groups, bonding, the theory of Lewis pairs and the valence bond model. Bader has demonstrated that these empirically useful chemistry concepts can be related to the topology of the observable charge density distribution, whether measured or calculated from a quantum mechanical wavefunction. QTAIM analysis of molecular wavefunctions is implemented, for example, in the AIMAll software package. Software packages [edit] There are many self-sufficient software packages used by computational chemists. Some include many methods covering a wide range, while others concentrating on a very specific range or even a single method. Details of most of them can be found in: See also [edit] Cited references [edit] 1. ^ Smith, S. J.; Sutcliffe B. T., (1997). "The development of Computational Chemistry in the United Kingdom". Reviews in Computational Chemistry 70: 271–316. 2. ^ Schaefer, Henry F. III (1972). The electronic structure of atoms and molecules. Reading, Massachusetts: Addison-Wesley Publishing Co. p. 146. 3. ^ Boys, S. F.; Cook G. B., Reeves C. M., Shavitt, I. (1956). "Automatic fundamental calculations of molecular structure". Nature 178 (2): 1207. Bibcode:1956Natur.178.1207B. doi:10.1038/1781207a0. 4. ^ Richards, W. G.; Walker T. E. H and Hinkley R. K. (1971). A bibliography of ab initio molecular wave functions. Oxford: Clarendon Press. 5. ^ Preuss, H. (1968). International Journal of Quantum Chemistry 2: 651. Bibcode:1968IJQC....2..651P. doi:10.1002/qua.560020506. 6. ^ Buenker, R. J.; Peyerimhoff S. D. (1969). "Ab initio SCF calculations for azulene and naphthalene". Chemical Physics Letters 3: 37. Bibcode:1969CPL.....3...37B. doi:10.1016/0009-2614(69)80014-X. 7. ^ Schaefer, Henry F. III (1984). Quantum Chemistry. Oxford: Clarendon Press. 8. ^ Streitwieser, A.; Brauman J. I. and Coulson C. A. (1965). Supplementary Tables of Molecular Orbital Calculations. Oxford: Pergamon Press. 9. ^ Pople, John A.; David L. Beveridge (1970). Approximate Molecular Orbital Theory. New York: McGraw Hill. 10. ^ Allinger, Norman (1977). "Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms". Journal of the American Chemical Society 99 (25): 8127–8134. doi:10.1021/ja00467a001. 11. ^ Fernbach, Sidney; Taub, Abraham Haskell (1970). Computers and Their Role in the Physical Sciences. Routledge. ISBN 0-677-14030-4. 12. ^ "vol 1, preface". Reviews in Computational Chemistry. doi:10.1002/9780470125786. Other references [edit] Specialized journals on computational chemistry [edit] External links [edit]
07e08cdabb84d913	Tell me more × Are there any examples of fermionic particles or quasiparticles for which the interaction potential is a globally smooth function? i.e. no singularities or branch points. As an example, in Flügge's Practical Quantum Mechanics, problem 148 has two repulsive particles on a circle. This is supposed to model the two helium electrons in the ground state. The equation he gives is $$ -\frac{\hbar^2}{2mr^2}\left(\frac{\partial^2 \psi}{\partial x_1^2}+\frac{\partial^2 \psi}{\partial x_2^2}\right)+V_0\cos(x_1-x_2)\psi=E\psi$$ I don't quite follow why this potential does not have a singularity when $x_2\rightarrow x_1$. Are there other such examples? share\|improve this question To clarify, you want a physical system for which there is a hamiltonian $H$ which is a very good approximation over some energy-range and whose term fourth order in fermions, $\rho(r_1)V(r_1-r_2)\rho(r_2)$, has $V(r)$ a $C^{\infty}$ function over all of space? – BebopButUnsteady Jun 27 '11 at 14:40 Yes. Ideally, $V(r_1-r_2)$ is $C^\infty$. At the very least is there any such model for fermionic particles where $V(r_1-r_2)$ is continuous at $r_1=r_2$? – Greg von Winckel Jun 27 '11 at 15:03 I'm not sure I completely understand the question, but if the two electrons have their spin degrees of freedom in a singlet, then the spatial wavefunction is symmetric under exchange of 1 & 2. There are no nodes in the ground-state wavefunction, so an effective potential doesn't have to introduce any singularities. – wsc Jun 27 '11 at 16:14 3 Answers Well there's no particular reason for a textbook problem to actually model a physical system... But one can certainly write something like this as a completely valid approximation. Take Flugge's example, with He3 so its fermionic[fn.2]. Say the size of the atoms is very small, much smaller than the scales on which the ground state wavefunction varies, which is reasonable enough. Now there should really be a term $V_{repulse}(x_1-x_2)\psi$ where $V_{repulse}$ gets really big when $\|x_1-x_2\|\rightarrow 0$ to capture the fact that you can't put the two atoms on top of each other [fn.2]. But this is going to be really short ranged, almost zero if $\|x_1-x_2\|$ is significantly bigger then the size of the atom. On the other hand we know that $\psi(x_1,x_2)\rightarrow 0$ when $x_1 \rightarrow x_2$. So in precisely the region where $V_{repulse}$ would matter, $\psi$ is basically zero. So we can basically ignore $V_{repulse}\psi$. More exactly, the term is proportional to (size of atom)/(size of circle) squared, which could be very small. So you don't always have to include a repulsive term. It can actually be quite negligible, even though it seems like a fact you can't ignore. [fn. 1] There are also times when you can ignore the repulsive interactions of bosons, although its not suppressed like the fermions. [fn. 2] Its not really true that it should diverge as $x_1\rightarrow x_2$. If you really got the two atoms on top of each they would stop behaving like pointlike atoms, so your model would stop being applicable, rather than anything going to infinity. share\|improve this answer On reading again it seems maybe you want the potential to go to infinity to enforce Pauli exclusion. If that's the case, then the answer is that the Pauli exclusion is a purely geometrical fact, and has nothing to do with potentials. – BebopButUnsteady Jun 27 '11 at 20:58 On further reading I'm not sure what the question is, so you should clarify it. – BebopButUnsteady Jun 27 '11 at 22:17 I have written a spectral code for computing eigenstates of 1D fermion systems with arbitrary confinement and interaction potentials. I am looking for model problems to test the code on and have already tried solving the (no spin) $n$-particle problem $$\left\{-\frac{\hbar^2}{2m}\nabla^2 + \sum\limits_{j=1}^n V_{ext}(x_j) + \sum\limits_{k=j+1}^n V_{int}(x_j-x_k)\right\}\psi(\mathbf{x})=E\psi(\mathbf{x})$$. When I use Coulomb interaction for $V_{int}$, the method converges quadratically. I am looking for problems with smooth $V_{int}$ to see if the convergence improves. – Greg von Winckel Jun 28 '11 at 6:23 Why not just put in some smooth interaction, say $\frac{1}{(x^2 +1)^2}$ and see if it converges? Or if you're looking for an analytically solved model to compare against, there's a fairly extensive set of 1D fermionic systems that are analytically tractable. The confining potential will probably make things difficult, but if you make the scale of the interaction much smaller than the confinement you can probably get something to work. – BebopButUnsteady Jun 28 '11 at 14:34 I have indeed tried a Lorenzian potential and observed spectral convergence. My hope, however, was not to try an arbitrary smooth potential, but one that is used in practice to model something. Some analytically solvable 1D fermionic systems would be of interest anyway. Do you have a reference for any of those? – Greg von Winckel Jun 29 '11 at 8:00 Apparently the Gaussian Effective Potential is used a fair amount. This would be something like $$V(x_1,x_2)=V_0 \exp(-\alpha(x_1-x_2)))$$. Thanks for the responses. share\|improve this answer In atomic physics effective 1D soft-core Coulomb potentials are routinely used for the interaction between particles, particularly when external fields are present. For example, for the interaction between an electron and a (space-fixed) proton at the center of coordinates: $$V(x) = - \frac{1}{\sqrt{x^2 + \epsilon^2}}$$ where atomic units are used and $\epsilon$ is usual fitted or taken as one. It is much simpler to integrate the time-dependent Schrödinger equation for this potential. share\|improve this answer Your Answer
03a4e2e6c9b88c50	From Wikipedia, the free encyclopedia - View original article Jump to: navigation, search On the other hand, what some physicists refer to as "apparent" or "effective" FTL[2][3][4][5] depends on the hypothesis that unusually distorted regions of spacetime might permit matter to reach distant locations in less time than light could in normal or undistorted spacetime. Although according to current theories matter is still required to travel subluminally with respect to the locally distorted spacetime region, apparent FTL is not excluded by general relativity. Examples of FTL proposals are the Alcubierre drive and the traversable wormhole, although their physical plausibility is uncertain. FTL travel of non-information[edit] In the context of this article, FTL is the transmission of information or matter faster than c, a constant equal to the speed of light in a vacuum, which is 299,792,458 metres per second (by definition) or about 186,282.4 miles per second. This is not quite the same as traveling faster than light, since: Neither of these phenomena violates special relativity or creates problems with causality, and thus neither qualifies as FTL as described here. In the following examples, certain influences may appear to travel faster than light, but they do not convey energy or information faster than light, so they do not violate special relativity. Daily sky motion[edit] For an earthbound observer, objects in the sky complete one revolution around the Earth in 1 day. Proxima Centauri, which is the nearest star outside the solar system, is about 4 light-years away.[6] On a geostationary view Proxima Centauri has a speed many times greater than c as the rim speed of an object moving in a circle is a product of the radius and angular speed.[6] It is also possible on a geostatic view for objects such as comets to vary their speed from subluminal to superluminal and vice versa simply because the distance from the Earth varies. Comets may have orbits which take them out to more than 1000 AU.[7] The circumference of a circle with a radius of 1000 AU is greater than one light day. In other words, a comet at such a distance is superluminal in a geostatic, and therefore non-inertial, frame. Light spots and shadows[edit] If a laser is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c.[8] Similarly, a shadow projected onto a distant object can be made to move across the object faster than c.[8] In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light.[8][9][10] Apparent FTL propagation of static field effects[edit] Main article: Static field Since there is no "retardation" (or aberration) of the apparent position of the source of a gravitational or electric static field when the source moves with constant velocity, the static field "effect" may seem at first glance to be "transmitted" faster than the speed of light. However, uniform motion of the static source may be removed with a change in reference frame, causing the direction of the static field to change immediately, at all distances. This is not a change of position which "propagates", and thus this change cannot be used to transmit information from the source. No information or matter can be FTL-transmitted or propagated from source to receiver/observer by an electromagnetic field. Closing speeds[edit] The rate at which two objects in motion in a single frame of reference get closer together is called the mutual or closing speed. This may approach twice the speed of light, as in the case of two particles travelling at close to the speed of light in opposite directions with respect to the reference frame. Imagine two fast-moving particles approaching each other from opposite sides of a particle accelerator of the collider type. The closing speed would be the rate at which the distance between the two particles is decreasing. From the point of view of an observer standing at rest relative to the accelerator, this rate will be slightly less than twice the speed of light. Special relativity does not prohibit this. It tells us that it is wrong to use Galilean relativity to compute the velocity of one of the particles, as would be measured by an observer traveling alongside the other particle. That is, special relativity gives the right formula for computing such relative velocity. It is instructive to compute the relative velocity of particles moving at v and -v in accelerator frame, which corresponds to the closing speed of 2v > c. Expressing the speeds in units of c, β = v/c: \beta_{rel} = { \beta + \beta \over 1 + \beta ^2 } = { 2\beta \over 1 + \beta^2 } \leq 1. Proper speeds[edit] If a spaceship travels to a planet one light-year (as measured in the Earth's rest frame) away from Earth at high speed, the time taken to reach that planet could be less than one year as measured by the traveller's clock (although it will always be more than one year as measured by a clock on Earth). The value obtained by dividing the distance traveled, as determined in the Earth's frame, by the time taken, measured by the traveller's clock, is known as a proper speed or a proper velocity. There is no limit on the value of a proper speed as a proper speed does not represent a speed measured in a single inertial frame. A light signal that left the Earth at the same time as the traveller would always get to the destination before the traveller. How far can one travel from the Earth?[edit] [original research?] Since one might not travel faster than light, one might conclude that a human can never travel further from the earth than 40 light-years if the traveler is active between the age of 20 and 60. A traveler would then never be able to reach more than the very few star systems which exist within the limit of 20-40 light-years from the Earth. This is a mistaken conclusion: because of time dilation, the traveler can travel thousands of light-years during their 40 active years. If the spaceship accelerates at a constant 1 g (in its own changing frame of reference), it will, after 354 days, reach speeds a little under the speed of light (for an observer on Earth), and time dilation will increase their lifespan to thousands of Earth years, seen from the reference system of the Solar System, but the traveler's subjective lifespan will not thereby change. If the traveler returns to the Earth, they will land thousands of years into the future. Their speed will not be seen as higher than the speed of light by observers on Earth, and the traveler will not measure their speed as being higher than the speed of light, but will see a length contraction of the universe in their direction of travel. And as the traveler turns around to return, the Earth will seem to experience much more time than the traveler does. So, although their (ordinary) speed cannot exceed c, the four-velocity (distance as seen by Earth divided by their proper, i.e. subjective, time) can be much greater than c. This is seen in statistical studies of muons traveling much further than c times their half-life (at rest), if traveling close to c.[11] Phase velocities above c[edit] The phase velocity of an electromagnetic wave, when traveling through a medium, can routinely exceed c, the vacuum velocity of light. For example, this occurs in most glasses at X-ray frequencies.[12] However, the phase velocity of a wave corresponds to the propagation speed of a theoretical single-frequency (purely monochromatic) component of the wave at that frequency. Such a wave component must be infinite in extent and of constant amplitude (otherwise it is not truly monochromatic), and so cannot convey any information.[13] Thus a phase velocity above c does not imply the propagation of signals with a velocity above c.[14] Group velocities above c[edit] The group velocity of a wave (e.g., a light beam) may also exceed c in some circumstances. In such cases, which typically at the same time involve rapid attenuation of the intensity, the maximum of the envelope of a pulse may travel with a velocity above c. However, even this situation does not imply the propagation of signals with a velocity above c,[15] even though one may be tempted to associate pulse maxima with signals. The latter association has been shown to be misleading, basically because the information on the arrival of a pulse can be obtained before the pulse maximum arrives. For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind (distortion), the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than c without this effect.[16] Universal expansion[edit] History of the universe - gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang (17 March 2014).[17][18][19] The expansion of the universe causes distant galaxies to recede from us faster than the speed of light, if proper distance and cosmological time are used to calculate the speeds of these galaxies. However, in general relativity, velocity is a local notion, so velocity calculated using comoving coordinates does not have any simple relation to velocity calculated locally.[20] (See comoving distance for a discussion of different notions of 'velocity' in cosmology.) Rules that apply to relative velocities in special relativity, such as the rule that relative velocities cannot increase past the speed of light, do not apply to relative velocities in comoving coordinates, which are often described in terms of the "expansion of space" between galaxies. This expansion rate is thought to have been at its peak during the inflationary epoch thought to have occurred in a tiny fraction of the second after the Big Bang (models suggest the period would have been from around 10−36 seconds after the Big Bang to around 10−33 seconds), when the universe may have rapidly expanded by a factor of around 1020 to 1030.[21] There are many galaxies visible in telescopes with red shift numbers of 1.4 or higher. All of these are currently traveling away from us at speeds greater than the speed of light. Because the Hubble parameter is decreasing with time, there can actually be cases where a galaxy that is receding from us faster than light does manage to emit a signal which reaches us eventually.[22][23] However, because the expansion of the universe is accelerating, it is projected that most galaxies will eventually cross a type of cosmological event horizon where any light they emit past that point will never be able to reach us at any time in the infinite future,[24] because the light never reaches a point where its "peculiar velocity" towards us exceeds the expansion velocity away from us (these two notions of velocity are also discussed in Comoving distance#Uses of the proper distance). The current distance to this cosmological event horizon is about 16 billion light-years, meaning that a signal from an event happening at present would eventually be able to reach us in the future if the event was less than 16 billion light-years away, but the signal would never reach us if the event was more than 16 billion light-years away.[23] Astronomical observations[edit] Apparent superluminal motion is observed in many radio galaxies, blazars, quasars and recently also in microquasars. The effect was predicted before it was observed by Martin Rees[clarification needed] and can be explained as an optical illusion caused by the object partly moving in the direction of the observer,[25] when the speed calculations assume it does not. The phenomenon does not contradict the theory of special relativity. Interestingly, corrected calculations show these objects have velocities close to the speed of light (relative to our reference frame). They are the first examples of large amounts of mass moving at close to the speed of light.[26] Earth-bound laboratories have only been able to accelerate small numbers of elementary particles to such speeds. Quantum mechanics[edit] Certain phenomena in quantum mechanics, such as quantum entanglement, might give the superficial impression of allowing communication of information faster than light. According to the no-communication theorem these phenomena do not allow true communication; they only let two observers in different locations see the same system simultaneously, without any way of controlling what either sees. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality or allow FTL it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. The uncertainty principle implies that individual photons may travel for short distances at speeds somewhat faster (or slower) than c, even in a vacuum; this possibility must be taken into account when enumerating Feynman diagrams for a particle interaction.[27] However, it was shown in 2011 that a single photon may not travel faster than c.[28] In quantum mechanics, virtual particles may travel faster than light, and this phenomenon is related to the fact that static field effects (which are mediated by virtual particles in quantum terms) may travel faster than light (see section on static fields above). However, macroscopically these fluctuations average out, so that photons do travel in straight lines over long (i.e., non-quantum) distances, and they do travel at the speed of light on average. Therefore, this does not imply the possibility of superluminal information transmission. There have been various reports in the popular press of experiments on faster-than-light transmission in optics—most often in the context of a kind of quantum tunnelling phenomenon. Usually, such reports deal with a phase velocity or group velocity faster than the vacuum velocity of light.[citation needed] However, as stated above, a superluminal phase velocity cannot be used for faster-than-light transmission of information. There has sometimes been confusion concerning the latter point. Additionally a channel that permits such propagation cannot be laid out faster than the speed of light.[citation needed] Quantum teleportation transmits quantum information at whatever speed is used to transmit the same amount of classical information, likely the speed of light. This quantum information may theoretically be used in ways that classical information can not, such as in quantum computations involving quantum information only available to the recipient. Hartman effect[edit] Main article: Hartman effect The Hartman effect is the tunnelling effect through a barrier where the tunnelling time tends to a constant for large barriers.[29] This was first described by Thomas Hartman in 1962.[30] This could, for instance, be the gap between two prisms. When the prisms are in contact, the light passes straight through, but when there is a gap, the light is refracted. There is a nonzero probability that the photon will tunnel across the gap rather than follow the refracted path. For large gaps between the prisms the tunnelling time approaches a constant and thus the photons appear to have crossed with a superluminal speed.[31] However, an analysis by Herbert G. Winful from the University of Michigan suggests that the Hartman effect cannot actually be used to violate relativity by transmitting signals faster than c, because the tunnelling time "should not be linked to a velocity since evanescent waves do not propagate".[32] The evanescent waves in the Hartman effect are due to virtual particles and a non-propagating static field, as mentioned in the sections above for gravity and electromagnetism. Casimir effect[edit] Main article: Casimir effect In physics, the Casimir effect or Casimir-Polder force is a physical force exerted between separate objects due to resonance of vacuum energy in the intervening space between the objects. This is sometimes described in terms of virtual particles interacting with the objects, owing to the mathematical form of one possible way of calculating the strength of the effect. Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small. Because the effect is due to virtual particles mediating a static field effect, it is subject to the comments about static fields discussed above. EPR Paradox[edit] Main article: EPR paradox The EPR paradox refers to a famous thought experiment of Einstein, Podolski and Rosen that was realized experimentally for the first time by Alain Aspect in 1981 and 1982 in the Aspect experiment. In this experiment, the measurement of the state of one of the quantum systems of an entangled pair apparently instantaneously forces the other system (which may be distant) to be measured in the complementary state. However, no information can be transmitted this way; the answer to whether or not the measurement actually affects the other quantum system comes down to which interpretation of quantum mechanics one subscribes to. An experiment performed in 1997 by Nicolas Gisin at the University of Geneva has demonstrated non-local quantum correlations between particles separated by over 10 kilometers.[33] But as noted earlier, the non-local correlations seen in entanglement cannot actually be used to transmit classical information faster than light, so that relativistic causality is preserved; see no-communication theorem for further information. A 2008 quantum physics experiment also performed by Nicolas Gisin and his colleagues in Geneva, Switzerland has determined that in any hypothetical non-local hidden-variables theory the speed of the quantum non-local connection (what Einstein called "spooky action at a distance") is at least 10,000 times the speed of light.[34] Delayed choice quantum eraser[edit] Delayed choice quantum eraser (an experiment of Marlan Scully) is a version of the EPR paradox in which the observation or not of interference after the passage of a photon through a double slit experiment depends on the conditions of observation of a second photon entangled with the first. The characteristic of this experiment is that the observation of the second photon can take place at a later time than the observation of the first photon,[35] which may give the impression that the measurement of the later photons "retroactively" determines whether the earlier photons show interference or not, although the interference pattern can only be seen by correlating the measurements of both members of every pair and so it can't be observed until both photons have been measured, ensuring that an experimenter watching only the photons going through the slit does not obtain information about the other photons in an FTL or backwards-in-time manner.[36][37] FTL communication possibility[edit] Faster light (Casimir vacuum and quantum tunnelling)[edit] Raymond Y. Chiao was first to measure the quantum tunnelling time, which was found to be between 1.5 to 1.7 times the speed of light. Einstein's equations of special relativity postulate that the speed of light in a vacuum is invariant in inertial frames. That is, it will be the same from any frame of reference moving at a constant speed. The equations do not specify any particular value for the speed of the light, which is an experimentally determined quantity for a fixed unit of length. Since 1983, the SI unit of length (the meter) has been defined using the speed of light. The experimental determination has been made in vacuum. However, the vacuum we know is not the only possible vacuum which can exist. The vacuum has energy associated with it, unsurprisingly called the vacuum energy. This vacuum energy can perhaps be changed in certain cases.[43] When vacuum energy is lowered, light itself has been predicted to go faster than the standard value c. This is known as the Scharnhorst effect. Such a vacuum can be produced by bringing two perfectly smooth metal plates together at near atomic diameter spacing. It is called a Casimir vacuum. Calculations imply that light will go faster in such a vacuum by a minuscule amount: a photon traveling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 1036.[44] Accordingly there has as yet been no experimental verification of the prediction. A recent analysis[45] argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signalling. However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze. Other authors argue that Scharnhorst's original analysis which seemed to show the possibility of faster-than-c signals involved approximations which may be incorrect, so that it is not clear whether this effect could actually increase signal speed at all.[46] The physicists Günter Nimtz and Alfons Stahlhofen, of the University of Cologne, claim to have violated relativity experimentally by transmitting photons faster than the speed of light.[31] They say they have conducted an experiment in which microwave photons—relatively low energy packets of light—travelled "instantaneously" between a pair of prisms that had been moved up to 3 ft (1 m) apart. Their experiment involved an optical phenomenon known as "evanescent modes", and they claim that since evanescent modes have an imaginary wave number, they represent a "mathematical analogy" to quantum tunnelling.[31] Nimtz has also claimed that "evanescent modes are not fully describable by the Maxwell equations and quantum mechanics have to be taken into consideration."[47] Other scientists such as Herbert G. Winful and Robert Helling have argued that in fact there is nothing quantum-mechanical about Nimtz's experiments, and that the results can be fully predicted by the equations of classical electromagnetism (Maxwell's equations).[48][49] Nimtz told New Scientist magazine: "For the time being, this is the only violation of special relativity that I know of." However, other physicists say that this phenomenon does not allow information to be transmitted faster than light. Aephraim Steinberg, a quantum optics expert at the University of Toronto, Canada, uses the analogy of a train traveling from Chicago to New York, but dropping off train cars at each station along the way, so that the center of the ever shrinking main train moves forward at each stop; in this way, the speed of the center of the train exceeds the speed of any of the individual cars.[50] Herbert G. Winful argues that the train analogy is a variant of the "reshaping argument" for superluminal tunneling velocities, but he goes on to say that this argument is not actually supported by experiment or simulations, which actually show that the transmitted pulse has the same length and shape as the incident pulse.[48] Instead, Winful argues that the group delay in tunneling is not actually the transit time for the pulse (whose spatial length must be greater than the barrier length in order for its spectrum to be narrow enough to allow tunneling), but is instead the lifetime of the energy stored in a standing wave which forms inside the barrier. Since the stored energy in the barrier is less than the energy stored in a barrier-free region of the same length due to destructive interference, the group delay for the energy to escape the barrier region is shorter than it would be in free space, which according to Winful is the explanation for apparently superluminal tunneling.[51][52] A number of authors have published papers disputing Nimtz's claim that Einstein causality is violated by his experiments, and there are many other papers in the literature discussing why quantum tunneling is not thought to violate causality.[53] It was later claimed by the Keller group in Switzerland that particle tunneling does indeed occur in zero real time. Their tests involved tunneling electrons, where the group argued a relativistic prediction for tunneling time should be 500-600 attoseconds (an attosecond is one quintillionth (10−18) of a second). All that could be measured was 24 attoseconds, which is the limit of the test accuracy.[54] Again, though, other physicists believe that tunneling experiments in which particles appear to spend anomalously short times inside the barrier are in fact fully compatible with relativity, although there is disagreement about whether the explanation involves reshaping of the wave packet or other effects.[51][52][55] Give up (absolute) relativity[edit] Because of the strong empirical support for special relativity, any modifications to it must necessarily be quite subtle and difficult to measure. The best-known attempt is doubly special relativity, which posits that the Planck length is also the same in all reference frames, and is associated with the work of Giovanni Amelino-Camelia and João Magueijo. One consequence of this theory is a variable speed of light, where photon speed would vary with energy, and some zero-mass particles might possibly travel faster than c.[citation needed] However, even if this theory is accurate, it is still very unclear whether it would allow information to be communicated, and appears not in any case to allow massive particles to exceed c. There are speculative theories that claim inertia is produced by the combined mass of the universe (e.g., Mach's principle), which implies that the rest frame of the universe might be preferred by conventional measurements of natural law. If confirmed, this would imply special relativity is an approximation to a more general theory, but since the relevant comparison would (by definition) be outside the observable universe, it is difficult to imagine (much less construct) experiments to test this hypothesis. Space-time distortion[edit] Although the theory of special relativity forbids objects to have a relative velocity greater than light speed, and general relativity reduces to special relativity in a local sense (in small regions of spacetime where curvature is negligible), general relativity does allow the space between distant objects to expand in such a way that they have a "recession velocity" which exceeds the speed of light, and it is thought that galaxies which are at a distance of more than about 14 billion light-years from us today have a recession velocity which is faster than light.[56] Miguel Alcubierre theorized that it would be possible to create an Alcubierre drive, in which a ship would be enclosed in a "warp bubble" where the space at the front of the bubble is rapidly contracting and the space at the back is rapidly expanding, with the result that the bubble can reach a distant destination much faster than a light beam moving outside the bubble, but without objects inside the bubble locally traveling faster than light. However, several objections raised against the Alcubierre drive appear to rule out the possibility of actually using it in any practical fashion. Another possibility predicted by general relativity is the traversable wormhole, which could create a shortcut between arbitrarily distant points in space. As with the Alcubierre drive, travelers moving through the wormhole would not locally move faster than light which travels through the wormhole alongside them, but they would be able to reach their destination (and return to their starting location) faster than light traveling outside the wormhole. Dr. Gerald Cleaver, associate professor of physics at Baylor University, and Richard Obousy, a Baylor graduate student, theorize that by manipulating the extra spatial dimensions of string theory around a spaceship with an extremely large amount of energy, it would create a "bubble" that could cause the ship to travel faster than the speed of light. To create this bubble, the physicists believe manipulating the 10th spatial dimension would alter the dark energy in three large spatial dimensions: height, width and length. Cleaver said positive dark energy is currently responsible for speeding up the expansion rate of our universe as time moves on.[57] Heim theory[edit] In 1977, a paper on Heim theory theorized that it may be possible to travel faster than light by using magnetic fields to enter a higher-dimensional space.[58] MiHsC/Quantised inertia[edit] A new theory has been proposed that Modifies inertia by assuming it is due to Unruh radiation subject to a Hubble scale Casimir effect (MiHsC, or quantised inertia). MiHsC predicts a minimum possible acceleration[59] even at light speed, implying that this speed can be exceeded. Lorentz symmetry violation[edit] The possibility that Lorentz symmetry may be violated has been seriously considered in the last two decades, particularly after the development of a realistic effective field theory that describes this possible violation, the so-called Standard-Model Extension.[60][61][62] This general framework has allowed experimental searches by ultra-high energy cosmic-ray experiments[63] and a wide variety of experiments in gravity, electrons, protons, neutrons, neutrinos, mesons, and photons.[64] The breaking of rotation and boost invariance causes direction dependence in the theory as well as unconventional energy dependence that introduces novel effects, including Lorentz-violating neutrino oscillations and modifications to the dispersion relations of different particle species, which naturally could make particles move faster than light. In some models of broken Lorentz symmetry, it is postulated that the symmetry is still built into the most fundamental laws of physics, but that spontaneous symmetry breaking of Lorentz invariance[65] shortly after the Big Bang could have left a "relic field" throughout the universe which causes particles to behave differently depending on their velocity relative to the field;[66] however, there are also some models where Lorentz symmetry is broken in a more fundamental way. If Lorentz symmetry can cease to be a fundamental symmetry at Planck scale or at some other fundamental scale, it is conceivable that particles with a critical speed different from the speed of light be the ultimate constituents of matter. In current models of Lorentz symmetry violation, the phenomenological parameters are expected to be energy-dependent. Therefore, as widely recognized,[67][68] existing low-energy bounds cannot be applied to high-energy phenomena; however, many searches for Lorentz violation at high energies have been carried out using the Standard-Model Extension.[64] Lorentz symmetry violation is expected to become stronger as one gets closer to the fundamental scale. Another recent theory (see EPR paradox above) resulting from the analysis of an EPR communication set up, has the simple device based on removing the effective retarded time terms in the Lorentz transform to yield a preferred absolute reference frame.[69][70] This frame cannot be used to do physics (i.e., compute the influence of light-speed limited signals) but it provides an objective, absolute frame all could agree upon, if superluminal communication is possible. If this sounds indulgent, it allows simultaneity, absolute space and time and a deterministic universe (along with decoherence theory) whilst the status-quo permits time travel/causality paradoxes, subjectivity in the measurement process and multiple universes. Superfluid theories of physical vacuum[edit] Main article: Superfluid vacuum In this approach the physical vacuum is viewed as the quantum superfluid which is essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of nature but rather the approximate description valid only for the small fluctuations of the superfluid background.[71] Within the framework of the approach a theory was proposed in which the physical vacuum is conjectured to be the quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger equation. It was shown that the relativistic gravitational interaction arises as the small-amplitude collective excitation mode[72] whereas relativistic elementary particles can be described by the particle-like modes in the limit of low momenta.[73] The important fact is that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one - they can reach the speed of light limit at finite energy; also the faster-than-light propagation is possible without requiring moving objects to have imaginary mass.[74][75] Time of flight of neutrinos[edit] MINOS experiment[edit] Main article: MINOS In 2007 MINOS collaboration reported results measuring the flight-time of 3 GeV neutrinos yielding a speed exceeding that of light by 1.8-sigma significance.[76] However, those measurements were considered to be statistically consistent with neutrinos traveling at the speed of light.[77] After the detectors for the project were upgraded in 2012, MINOS corrected their initial result and found agreement with the speed of light. Further measurements are going to be conducted.[78] OPERA neutrino anomaly[edit] On September 22, 2011, a paper[79] from the OPERA Collaboration indicated detection of 17 and 28 GeV muon neutrinos, sent 730 kilometers (454 miles) from CERN near Geneva, Switzerland to the Gran Sasso National Laboratory in Italy, traveling faster than light by a factor of 2.48×10−5 (approximately 1 in 40,000), a statistic with 6.0-sigma significance.[80] On 18 November 2011, a second follow-up experiment by OPERA scientists confirmed their initial results.[81][82] However, scientists were skeptical about the results of these experiments, the significance of which was disputed.[83] In March 2012, the ICARUS collaboration failed to reproduce the OPERA results with their equipment, detecting neutrino travel time from CERN to the Gran Sasso National Laboratory indistinguishable from the speed of light.[84] Later the OPERA team reported two flaws in their equipment set-up that had caused errors far outside their original confidence interval: a fiber optic cable attached improperly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast.[85] Main article: Tachyon In special relativity, it is impossible to accelerate an object to the speed of light, or for a massive object to move at the speed of light. However, it might be possible for an object to exist which always moves faster than light. The hypothetical elementary particles with this property are called tachyonic particles. Attempts to quantize them failed to produce faster-than-light particles, and instead illustrated that their presence leads to an instability.[86][87] Various theorists have suggested that the neutrino might have a tachyonic nature,[88][89][90][91][92] while others have disputed the possibility.[93] General relativity[edit] General relativity was developed after special relativity to include concepts like gravity. It maintains the principle that no object can accelerate to the speed of light in the reference frame of any coincident observer.[citation needed][clarification needed] However, it permits distortions in spacetime that allow an object to move faster than light from the point of view of a distant observer.[citation needed][clarification needed] One such distortion is the Alcubierre drive, which can be thought of as producing a ripple in spacetime that carries an object along with it. Another possible system is the wormhole, which connects two distant locations as though by a shortcut. Both distortions would need to create a very strong curvature in a highly localized region of space-time and their gravity fields would be immense. To counteract the unstable nature, and prevent the distortions from collapsing under their own 'weight', one would need to introduce hypothetical exotic matter or negative energy. General relativity also recognizes that any means of faster-than-light travel could also be used for time travel. This raises problems with causality. Many physicists believe that the above phenomena are impossible and that future theories of gravity will prohibit them. One theory states that stable wormholes are possible, but that any attempt to use a network of wormholes to violate causality would result in their decay.[citation needed] In string theory, Eric G. Gimon and Petr Hořava have argued[94] that in a supersymmetric five-dimensional Gödel universe, quantum corrections to general relativity effectively cut off regions of spacetime with causality-violating closed timelike curves. In particular, in the quantum theory a smeared supertube is present that cuts the spacetime in such a way that, although in the full spacetime a closed timelike curve passed through every point, no complete curves exist on the interior region bounded by the tube. Variable speed of light[edit] In conventional physics, the speed of light in a vacuum is assumed to be a constant. However, theories exist which postulate that the speed of light is not a constant. The interpretation of this statement is as follows. The speed of light is a dimensional quantity and so, as has been emphasized in this context by João Magueijo, it cannot be measured.[95] Measurable quantities in physics are, without exception, dimensionless, although they are often constructed as ratios of dimensional quantities. For example, when the height of a mountain is measured, what is really measured is the ratio of its height to the length of a meter stick. The conventional SI system of units is based on seven basic dimensional quantities, namely distance, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.[96] These units are defined to be independent and so cannot be described in terms of each other. As an alternative to using a particular system of units, one can reduce all measurements to dimensionless quantities expressed in terms of ratios between the quantities being measured and various fundamental constants such as Newton's constant, the speed of light and Planck's constant; physicists can define at least 26 dimensionless constants which can be expressed in terms of these sorts of ratios and which are currently thought to be independent of one another.[97] By manipulating the basic dimensional constants one can also construct the Planck time, Planck length and Planck energy which make a good system of units for expressing dimensional measurements, known as Planck units. Magueijo's proposal used a different set of units, a choice which he justifies with the claim that some equations will be simpler in these new units. In the new units he fixes the fine structure constant, a quantity which some people, using units in which the speed of light is fixed, have claimed is time-dependent. Thus in the system of units in which the fine structure constant is fixed, the observational claim is that the speed of light is time-dependent. While it may be mathematically possible to construct such a system, it is not clear what additional explanatory power or physical insight such a system would provide, assuming that it does indeed accord with existing empirical data. See also[edit] Science fiction 1. ^ 2. ^ Gonzalez-Diaz, P. F. (2000). "Warp drive space-time". Physical Review D 62 (4): 044005. arXiv:gr-qc/9907026. Bibcode:2000PhRvD..62d4005G. doi:10.1103/PhysRevD.62.044005. 3. ^ Loup, F.; Waite, D.; Halerewicz, E. Jr. (2001). "Reduced total energy requirements for a modified Alcubierre warp drive spacetime". arXiv:0107097 [gr-qc]. 4. ^ Visser, M.; Bassett, B.; Liberati, S. (2000). "Superluminal censorship". Nuclear Physics B: Proceedings Supplement 88: 267–270. arXiv:gr-qc/9810026. Bibcode:2000NuPhS..88..267V. doi:10.1016/S0920-5632(00)00782-9. 5. ^ Visser, M.; Bassett, B.; Liberati, S. (1999). "Perturbative superluminal censorship and the null energy condition". AIP Conference Proceedings 493: 301–305. arXiv:gr-qc/9908023. doi:10.1063/1.1301601. ISBN 1-56396-905-X. 6. ^ a b See Salters Horners Advanced Physics A2 Student Book, Oxford etc. (Heinemann) 2001, pp. 302 and 303 7. ^ see 8. ^ a b c Gibbs, Philip (1997). Is Faster-Than-Light Travel or Communication Possible?. University of California, Riverside. Retrieved 20 August 2008. 9. ^ Salmon, Wesley C. (2006). Four Decades of Scientific Explanation. University of Pittsburgh Pre. p. 107. ISBN 0-8229-5926-7. , Extract of page 107 10. ^ Steane, Andrew (2012). The Wonderful World of Relativity: A Precise Guide for the General Reader. Oxford University Press. p. 180. 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Richard (2002). Time Travel in Einstein's Universe. pp. pp. 82–83. 41. ^ Petkov, Vesselin; Relativity and the Nature of Spacetime, p. 219 42. ^ Raine, Derek J.; Thomas, Edwin George; and Thomas, E. G.; An Introduction to the Science of Cosmology, p. 94 43. ^ "What is the 'zero-point energy' (or 'vacuum energy') in quantum physics? Is it really possible that we could harness this energy?". Scientific American. 1997-08-18. Retrieved 2009-05-27. 44. ^ Scharnhorst, Klaus (1990-05-12). "Secret of the vacuum: Speedier light". Retrieved 2009-05-27. 45. ^ Visser, Matt; Liberati, Stefano; Sonego, Sebastiano (2001-07-27). "Faster-than-c signals, special relativity, and causality". Annals of Physics 298: 167–185. arXiv:gr-qc/0107091. Bibcode:2002AnPhy.298..167L. doi:10.1006/aphy.2002.6233. 46. ^ Fearn, Heidi (2007). "Can Light Signals Travel Faster than c in Nontrivial Vacuua in Flat space-time? Relativistic Causality II". LaserPhys. 17 (5): 695–699. arXiv:0706.0553. Bibcode:2007LaPhy..17..695F. doi:10.1134/S1054660X07050155. 47. ^ Nimtz, Günter; Superluminal Tunneling Devices, 2001 48. ^ a b Winful, Herbert G. (2007-09-18). "Comment on "Macroscopic violation of special relativity" by Nimtz and Stahlhofen". arXiv:0709.2736 [quant-ph]. 49. ^ Helling, Robert C.; "Faster than light or not" (blog) 50. ^ Anderson, Mark (18–24 August 2007). "Light seems to defy its own speed limit". New Scientist 195 (2617). p. 10. 52. ^ a b For a summary of Herbert G. Winful's explanation for apparently superluminal tunneling time which does not involve reshaping, see 53. ^ A number of papers are listed at Literature on Faster-than-light tunneling experiments 54. ^ Eckle, P.; et al., "Attosecond Ionization and Tunneling Delay Time Measurements in Helium", Science, 322 (2008) 1525 55. ^ Sokolovski, D. (8 February 2004). "Why does relativity allow quantum tunneling to 'take no time'?". Proceedings of the Royal Society A 460 (2042): 499–506. 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Alan; The Neutrino As A Tachyon, Physics Letters B 150, 431 (1985) 89. ^ Chodos, Alan; Kostelecký, V. Alan; IUHET 280 (1994). "Nuclear Null Tests for Spacelike Neutrinos". Physics Letters B 336 (3–4): 295–302. arXiv:hep-ph/9409404. Bibcode:1994PhLB..336..295C. doi:10.1016/0370-2693(94)90535-5. 90. ^ Chodos, Alan; Kostelecký, V. Alan; Potting, R.; and Gates, E.; Null experiments for neutrino masses, Modern Physics Letters A7, 467 (1992) 91. ^ List of articles on the tachyonic neutrino idea (may be incomplete). InSPIRE database. Parity Violation and Neutrino Mass Tsao Chang 92. ^ Chang, Taso; Parity Violation and Neutrino Mass, Nuclear Science and Techniques, Vol. 13, No. 3 (2002) 129 93. ^ Hughes, R. J.; and Stephenson, G. J., Jr.; Against tachyonic neutrinos, Physics Letters B 244, 95-100 (1990) 94. ^ Gimon, Eric G.; Hořava, Petr (2004). "Over-rotating black holes, Gödel holography and the hypertube". arXiv:hep-th/0405019 [hep-th]. 95. ^ Magueijo, João; Albrecht, Andreas (1999). 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26bd3326c610f485	Take the 2-minute tour × At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are contour integrals and Laplace transform, but of course this should be a "real" complex analysis course which I could later refer to in honours courses. I am now confident (after this discussion, especially after Gauss complaints given in Keith's comment) that the name "complex" is quite discouraging to average students. Why do we need to study numbers which do not belong to the real world? Of course, we all know that the thesis is wrong and I have in mind some examples where the use of complex variable functions simplify solving considerably (I give two below). The drawback of all them is assuming already some knowledge from students. So I would be really happy to learn elementary examples which may convince students in usefulness of complex numbers and functions in complex variable. As this question runs in the community wiki mode, I would be glad to see one example per answer. Thank you in advance! Here comes the two promised example. The 2nd one was reminded by several answers and comments about relations with trigonometric functions (but also by notification "The bounty on your question Trigonometry related to Rogers--Ramanujan identities expires within three days"; it seems to be harder than I expect). Example 1. Find the Fourier expansion of the (unbounded) periodic function $$ f(x)=\ln\Bigl\|\sin\frac x2\Bigr\|. $$ Solution. The function $f(x)$ is periodic with period $2\pi$ and has poles at the points $2\pi k$, $k\in\mathbb Z$. Consider the function on the interval $x\in[\varepsilon,2\pi-\varepsilon]$. The series $$ \sum_{n=1}^\infty\frac{z^n}n, \qquad z=e^{ix}, $$ converges for all values $x$ from the interval. Since $$ \Bigl\|\sin\frac x2\Bigr\|=\sqrt{\frac{1-\cos x}2} $$ and $\operatorname{Re}\ln w=\ln\|w\|$, where we choose $w=\frac12(1-z)$, we deduce that $$ \operatorname{Re}\Bigl(\ln\frac{1-z}2\Bigr)=\ln\sqrt{\frac{1-\cos x}2} =\ln\Bigl\|\sin\frac x2\Bigr\|. $$ Thus, $$ \ln\Bigl\|\sin\frac x2\Bigr\| =-\ln2-\operatorname{Re}\sum_{n=1}^\infty\frac{z^n}n =-\ln2-\sum_{n=1}^\infty\frac{\cos nx}n. $$ As $\varepsilon>0$ can be taken arbitrarily small, the result remains valid for all $x\ne2\pi k$. Example 2. Let $p$ be an odd prime number. For an integer $a$ relatively prime to $p$, the Legendre symbol $\bigl(\frac ap\bigr)$ is $+1$ or $-1$ depending on whether the congruence $x^2\equiv a\pmod{p}$ is solvable or not. One of elementary consequences of (elementary) Fermat's little theorem is $$ \biggl(\frac ap\biggr)\equiv a^{(p-1)/2}\pmod p. \qquad\qquad\qquad {()} $$ Show that $$ \biggl(\frac2p\biggr)=(-1)^{(p^2-1)/8}. $$ Solution. In the ring $\mathbb Z+\mathbb Zi=\Bbb Z[i]$, the binomial formula implies $$ (1+i)^p\equiv1+i^p\pmod p. $$ On the other hand, $$ (1+i)^p =\bigl(\sqrt2e^{\pi i/4}\bigr)^p =2^{p/2}\biggl(\cos\frac{\pi p}4+i\sin\frac{\pi p}4\biggr) $$ and $$ 1+i^p =1+(e^{\pi i/2})^p =1+\cos\frac{\pi p}2+i\sin\frac{\pi p}2 =1+i\sin\frac{\pi p}2. $$ Comparing the real parts implies that $$ 2^{p/2}\cos\frac{\pi p}4\equiv1\pmod p, $$ hence from $\sqrt2\cos(\pi p/4)\in\{\pm1\}$ we conclude that $$ 2^{(p-1)/2}\equiv\sqrt2\cos\frac{\pi p}4\pmod p. $$ It remains to apply ($$): $$ \biggl(\frac2p\biggr) \equiv2^{(p-1)/2} \equiv\sqrt2\cos\frac{\pi p}4 =\begin{cases} 1 & \text{if } p\equiv\pm1\pmod8, \cr -1 & \text{if } p\equiv\pm3\pmod8, \end{cases} $$ which is exactly the required formula. share\|improve this question Maybe an option is to have them understand that real numbers also do not belong to the real world, that all sort of numbers are simply abstractions. – Mariano Suárez-Alvarez Jul 1 '10 at 14:50 Probably your electrical engineering students understand better than you do that complex numbers (in polar form) are used to represent amplitude and frequency in their area of study. – Gerald Edgar Jul 1 '10 at 15:36 Not an answer, but some suggestions: try reading the beginning of Needham's Visual Complex Analysis (usf.usfca.edu/vca/) and the end of Levi's The Mathematical Mechanic (amazon.com/Mathematical-Mechanic-Physical-Reasoning-Problems/dp/…). – Qiaochu Yuan Jul 1 '10 at 17:05 Your example has a hidden assumption that a student actually admits the importance of calculating F.S. of $\ln\left\|\sin{x\over 2}\right\|$, which I find dubious. The examples with an oscillator's ODE is more convincing, IMO. – Paul Yuryev Jul 2 '10 at 3:02 @Mariano, Gerald and Qiaochu: Thanks for the ideas! Visual Complex Analysis sounds indeed great, and I'll follow Levi's book as soon as I reach the uni library. @Paul: I give the example (which I personally like) and explain that I do not consider it elementary enough for the students. It's a matter of taste! I've never used Fourier series in my own research but it doesn't imply that I doubt of their importance. We all (including students) have different criteria for measuring such things. – Wadim Zudilin Jul 2 '10 at 5:06 32 Answers 32 The nicest elementary illustration I know of the relevance of complex numbers to calculus is its link to radius of convergence, which student learn how to compute by various tests, but more mechanically than conceptually. The series for $1/(1-x)$, $\log(1+x)$, and $\sqrt{1+x}$ have radius of convergence 1 and we can see why: there's a problem at one of the endpoints of the interval of convergence (the function blows up or it's not differentiable). However, the function $1/(1+x^2)$ is nice and smooth on the whole real line with no apparent problems, but its radius of convergence at the origin is 1. From the viewpoint of real analysis this is strange: why does the series stop converging? Well, if you look at distance 1 in the complex plane... More generally, you can tell them that for any rational function $p(x)/q(x)$, in reduced form, the radius of convergence of this function at a number $a$ (on the real line) is precisely the distance from $a$ to the nearest zero of the denominator, even if that nearest zero is not real. In other words, to really understand the radius of convergence in a general sense you have to work over the complex numbers. (Yes, there are subtle distinctions between smoothness and analyticity which are relevant here, but you don't have to discuss that to get across the idea.) Similarly, the function $x/(e^x-1)$ is smooth but has a finite radius of convergence $2\pi$ (not sure if you can make this numerically apparent). Again, on the real line the reason for this is not visible, but in the complex plane there is a good explanation. share\|improve this answer Thanks, Keith! That's a nice point which I always mention for real analysis students as well. The structure of singularities of a linear differential equation (under some mild conditions) fully determines the convergence of the series solving the DE. The generating series for Bernoulli numbers does not produce sufficiently good approximations to $2\pi$, but it's just beautiful by itself. – Wadim Zudilin Jul 2 '10 at 5:14 You can solve the differential equation y''+y=0 using complex numbers. Just write $$(\partial^2 + 1) y = (\partial +i)(\partial -i) y$$ and you are now dealing with two order one differential equations that are easily solved $$(\partial +i) z =0,\qquad (\partial -i)y =z$$ The multivariate case is a bit harder and uses quaternions or Clifford algebras. This was done by Dirac for the Schrodinger equation ($-\Delta \psi = i\partial_t \psi$), and that led him to the prediction of the existence of antiparticles (and to the Nobel prize). share\|improve this answer Students usually find the connection of trigonometric identities like $\sin(a+b)=\sin a\cos b+\cos a\sin b$ to multiplication of complex numbers striking. share\|improve this answer Not sure about the students, but I do. :-) – Wadim Zudilin Jul 1 '10 at 12:21 This is an excellent suggestion. I can never remember these identities off the top of my head. Whenever I need one of them, the simplest way (faster than googling) is to read them off from $(a+ib)(c+id)=(ac-bd) + i(ad+bc)$. – alex Jul 1 '10 at 20:35 When I first started teaching calculus in the US, I was surprised that many students didn't remember addition formulas for trig functions. As the years went by, it's gotten worse: now the whole idea of using an identity like that to solve a problem is alien to them, e.g. even if they may look it up doing the homework, they "get stuck" on the problem and "don't get it". What is there to blame: calculators? standard tests that neglect it? teachers who never understood it themselves? Anyway, it's a very bad omen. – Victor Protsak Jul 2 '10 at 1:43 @Victor: It can be worse... When I taught Calc I at U of Toronto to engineering students, I was approached by some students who claimed they had heard words "sine" and "cosine" but were not quite sure what they meant. – Yuri Bakhtin Jul 2 '10 at 8:51 From "Birds and Frogs" by Freeman Dyson [Notices of Amer. Math. Soc. 56 (2009) 212--223]: One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926. Schrödinger was a bird who started from the idea of unifying mechanics with optics. A hundred years earlier, Hamilton had unified classical mechanics with ray optics, using the same mathematics to describe optical rays and classical particle trajectories. Schrödinger’s idea was to extend this unification to wave optics and wave mechanics. Wave optics already existed, but wave mechanics did not. Schrödinger had to invent wave mechanics to complete the unification. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrödinger’s idea seemed to be going nowhere. But then came the surprise. Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrödinger as well as to everybody else. According to Schrödinger, his fourteen-year-old girl friend Itha Junger said to him at the time, "Hey, you never even thought when you began that so much sensible stuff would come out of it." All through the nineteenth century, mathematicians from Abel to Riemann and Weierstrass had been creating a magnificent theory of functions of complex variables. They had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first. share\|improve this answer Here are two simple uses of complex numbers that I use to try to convince students that complex numbers are "cool" and worth learning. 1. (Number Theory) Use complex numbers to derive Brahmagupta's identity expressing $(a^2+b^2)(c^2+d^2)$ as the sum of two squares, for integers $a,b,c,d$. 2. (Euclidean geometry) Use complex numbers to explain Ptolemy's Theorem. For a cyclic quadrilateral with vertices $A,B,C,D$ we have $$\overline{AC}\cdot \overline{BD}=\overline{AB}\cdot \overline{CD} +\overline{BC}\cdot \overline{AD}$$ share\|improve this answer And even more amazingly, one can completely solve the diophantine equation $x^2+y^2=z^n$ for any $n$ as follows: $$x+yi=(a+bi)^n, \ z=a^2+b^2.$$ I learned this from a popular math book while in elementary school, many years before studying calculus. – Victor Protsak Jul 2 '10 at 1:21 If the students have had a first course in differential equations, tell them to solve the system $$x'(t) = -y(t)$$ $$y'(t) = x(t).$$ This is the equation of motion for a particle whose velocity vector is always perpendicular to its displacement. Explain why this is the same thing as $$(x(t) + iy(t))' = i(x(t) + iy(t))$$ hence that, with the right initial conditions, the solution is $$x(t) + iy(t) = e^{it}.$$ On the other hand, a particle whose velocity vector is always perpendicular to its displacement travels in a circle. Hence, again with the right initial conditions, $x(t) = \cos t, y(t) = \sin t$. (At this point you might reiterate that complex numbers are real $2 \times 2$ matrices, assuming they have seen this method for solving systems of differential equations.) share\|improve this answer One of my favourite elementary applications of complex analysis is the evaluation of infinite sums of the form $$\sum_{n\geq 0} \frac{p(n)}{q(n)}$$ where $p,q$ are polynomials and $\deg q > 1 + \deg p$, by using residues. share\|improve this answer One cannot over-emphasize that passing to complex numbers often permits a great simplification by linearizing what would otherwise be more complex nonlinear phenomena. One example familiar to any calculus student is the fact that integration of rational functions is much simpler over $\mathbb C$ (vs. $\mathbb R$) since partial fraction decompositions involve at most linear (vs quadratic) polynomials in the denominator. Similarly one reduces higher-order constant coefficient differential and difference equations to linear (first-order) equations by factoring the linear operators over $\mathbb C$. More generally one might argue that such simplification by linearization was at the heart of the development of abstract algebra. Namely, Dedekind, by abstracting out the essential linear structures (ideals and modules) in number theory, greatly simplified the prior nonlinear theory based on quadratic forms. This enabled him to exploit to the hilt the power of linear algebra. Examples abound of the revolutionary power that this brought to number theory and algebra - e.g. for one little-known gem see my recent post explaining how Dedekind's notion of conductor ideal beautifully encapsulates the essence of elementary irrationality proofs of n'th roots. share\|improve this answer • If they have a suitable background in linear algebra, I would not omit the interpretation of complex numbers in terms of conformal matrices of order 2 (with nonnegative determinant), translating all operations on complex numbers (sum, product, conjugate, modulus, inverse) in the context of matrices: with special emphasis on their multiplicative action on the plane (in particular, "real" gives "homotety" and "modulus 1" gives "rotation"). • The complex exponential, defined initially as limit of $(1+z/n)^n$, should be a good application of the above geometrical ideas. In particular, for $z=it$, one can give a nice interpretation of the (too often covered with mystery) equation $e^{i\pi}=-1$ in terms of the length of the curve $e^{it}$ (defined as classical total variation). • A brief discussion on (scalar) linear ordinary differential equations of order 2, with constant coefficients, also provides a good motivation (and with some historical truth). • Related to the preceding point, and especially because they are from engineering, it should be worth recalling all the useful complex formalism used in Electricity. • Not on the side of "real world" interpretation, but rather on the side of "useful abstraction" a brief account of the history of the third degree algebraic equation, with the embarrassing "casus impossibilis" (three real solutions, and the solution formula gives none, if taken in terms of "real" radicals!) should be very instructive. Here is also the source of such terms as "imaginary". share\|improve this answer @Wadim: The $(1+z/n)^n$ definition of the exponential is exactly what you get by applying Euler's method to the defining diff Eq of the exponential function, if you travel along the straight line from 0 to z in the domain, and use n equal partitions. – Steven Gubkin Aug 27 '12 at 13:24 If you really want to "demystify" complex numbers, I'd suggest teaching what complex multiplication looks like with the following picture, as opposed to a matrix representation: If you want to visualize the product "z w", start with '0' and 'w' in the complex plane, then make a new complex plane where '0' sits above '0' and '1' sits above 'w'. If you look for 'z' up above, you see that 'z' sits above something you name 'z w'. You could teach this picture for just the real numbers or integers first -- the idea of using the rest of the points of the plane to do the same thing is a natural extension. You can use this picture to visually "demystify" a lot of things: • Why is a negative times a negative a positive? --- I know some people who lost hope in understanding math as soon as they were told this fact • i^2 = -1 • (zw)t = z(wt) --- I think this is a better explanation than a matrix representation as to why the product is associative • \|zw\| = \|z\| \|w\| • (z + w)v = zv + wv • The Pythagorean Theorem: draw (1-it)(1+it) = 1 + t^2 etc. One thing that's not so easy to see this way is the commutativity (for good reasons). After everyone has a grasp on how complex multiplication looks, you can get into the differential equation: $\frac{dz}{dt} = i z , z(0) = 1$ which Qiaochu noted travels counterclockwise in a unit circle at unit speed. You can use it to give a good definition for sine and cosine -- in particular, you get to define $\pi$ as the smallest positive solution to $e^{i \pi} = -1$. It's then physically obvious (as long as you understand the multiplication) that $e^{i(x+y)} = e^{ix} e^{iy}$, and your students get to actually understand all those hard/impossible to remember facts about trig functions (like angle addition and derivatives) that they were forced to memorize earlier in their lives. It may also be fun to discuss how the picture for $(1 + \frac{z}{n})^n$ turns into a picture of that differential equation in the "compound interest" limit as $n \to \infty$; doing so provides a bridge to power series, and gives an opportunity to understand the basic properties of the real exponential function more intuitively as well. But this stuff is less demystifying complex numbers and more... demystifying other stuff using complex numbers. Here's a link to some Feynman lectures on Quantum Electrodynamics (somehow prepared for a general audience) if you really need some flat out real-world complex numbers share\|improve this answer They're useful just for doing ordinary geometry when programming. A common pattern I have seen in a great many computer programs is to start with a bunch of numbers that are really ratios of distances. Theses numbers get converted to angles with inverse trig functions. Then some simple functions are applied to the angles and the trig functions are used on the results. Trig and inverse trig functions are expensive to compute on a computer. In high performance code you want to eliminate them if possible. Quite often, for the above case, you can eliminate the trig functions. For example $\cos(2\cos^{-1} x) = 2x^2-1$ (for $x$ in a suitable range) but the version on the right runs much faster. The catch is remembering all those trig formulae. It'd be nice to make the compiler do all the work. A solution is to use complex numbers. Instead of storing $\theta$ we store $(\cos\theta,\sin\theta)$. We can add angles by using complex multiplication, multiply angles by integers and rational numbers using powers and roots and so on. As long as you don't actually need the numerical value of the angle in radians you need never use trig functions. Obviously there comes a point where the work of doing operations on complex numbers may outweigh the saving of avoiding trig. But often in real code the complex number route is faster. (Of course it's analogous to using quaternions for rotations in 3D. I guess it's somewhat in the spirit of rational trigonometry except I think it's easier to work with complex numbers.) share\|improve this answer How about how the holomorphicity of a function $f(z)=x+yi$ relates to, e.g., the curl of the vector $(x,y)\in\mathbb{R}^2$? This relates nicely to why we can solve problems in two dimensional electromagnetism (or 3d with the right symmetries) very nicely using "conformal methods." It would be very easy to start a course with something like this to motivate complex analytic methods. share\|improve this answer Thanks, Jeremy! I'll definitely do the search, the magic word "the method of conformal mapping" is really important here. – Wadim Zudilin Jul 1 '10 at 11:45 I think most older Russian textbooks on complex analysis (e.g. Lavrentiev and Shabat or Markushevich) had examples from 2D hydrodynamics (Euler-D'Alambert equations $\iff$ Cauchy-Riemann equations). Also, of course, the Zhukovsky function and airwing profile. They serve more as applications of theory than motivations, since nontrivial mathematical work is required to get there. – Victor Protsak Jul 2 '10 at 2:04 Several motivating physical applications are listed on wikipedia You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q). Here is a nice quote by Yuri Manin picked from here On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics. (Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 ) share\|improve this answer I never took a precalculus class because every identity I've ever needed involving sines and cosines I could derive by evaluating a complex exponential in two different ways. Perhaps you could tell them that if they ever forget a trig identity, they can rederive it using this method? share\|improve this answer In answer to "Why do we need to study numbers which do not belong to the real world?" you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe. I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check. (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes. After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.) share\|improve this answer Perhaps you are referring to Feynman's book QED? – S. Carnahan Jul 2 '10 at 4:41 Tristan Needham's book Visual Complex Analysis is full of these sorts of gems. One of my favorites is the proof using complex numbers that if you put squares on the sides of a quadralateral, the lines connecting opposite centers will be perpendicular and of the same length. After proving this with complex numbers, he outlines a proof without them that is much longer. The relevant pages are on Google books: http://books.google.com/books?id=ogz5FjmiqlQC&lpg=PP1&dq=visual%20complex%20analysis&pg=PA16#v=onepage&q&f=false share\|improve this answer This is not exactly an answer to the question, but it is the simplest thing I know to help students appreciate complex numbers. (I got the idea somewhere else, but I forgot exactly where.) It's something even much younger students can appreciate. Recall that on the real number line, multiplying a number by -1 "flips" it, that is, it rotates the point 180 degrees about the origin. Introduce the imaginary number line (perpendicular to the real number line) then introduce multiplication by i as a rotation by 90 degrees. I think most students would appreciate operations on complex numbers if they visualize them as movements of points on the complex plane. share\|improve this answer Try this: compare the problems of finding the points equidistant in the plane from (-1, 0) and (1, 0), which is easy, with finding the points at twice the distance from (-1, 0) that they are from (1, 0). The idea that "real" concepts are the only ones of use in the "real world" is of course a fallacy. I suppose it is more than a century since electrical engineers admitted that complex numbers are useful. share\|improve this answer I do see an undeniable benefit. If you are later asked about it in $\mathbb{R}^3$ then you use vectors and dot product. The historical way would have been to use quaternions; indeed, this is how the notion of dot product crystallized in the work of Gibbs, and more relevantly for your EE students, Oliver Heaviside. – Victor Protsak Jul 2 '10 at 1:26 Maybe artificial, but a nice example (I think) demonstrating analytic continuation (NOT just the usual $\mathrm{Re}(e^{i \theta})$ method!) I don't know any reasonable way of doing this by real methods. As a fun exercise, calculate $$ I(\omega) = \int_0^\infty e^{-x} \cos (\omega x) \frac{dx}{\sqrt{x}}, \qquad \omega \in \mathbb{R} $$ from the real part of $F(1+i \omega)$, where $$ F(k) = \int_0^\infty e^{-kx} \frac{dx}{\sqrt{x}}, \qquad \mathrm{Re}(k)>0 $$ (which is easily obtained for $k>0$ by a real substitution) and using analytic continuation to justify the same formula with $k=1+i \omega$. You need care with square roots, branch cuts, etc.; but this can be avoided by considering $F(k)^2$, $I(\omega)^2$. Of course all the standard integrals provide endless fun examples! (But the books don't have many requiring genuine analytic continuation like this!) share\|improve this answer I rather suspect analytic continuation is a conceptual step above what the class in question coud cope with... – Yemon Choi Jul 8 '10 at 1:22 Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression formula, you can expand f(x)=1-x^2+... . This series converges for \|x\|<1 but diverges for all other x. Why this is so? The function looks nice and smooth everywhere on the real line. This example is taken from the Introduction of the textbook by B. V. Shabat. share\|improve this answer From the perspective of complex analysis, the theory of Fourier series has a very natural explanation. I take it that the students had seen Fourier series first, of course. I had mentioned this elsewhere too. I hope the students also know about Taylor theorem and Taylor series. Then one could talk also of the Laurent series in concrete terms, and argue that the Fourier series is studied most naturally in this setting. First, instead of cos and sin, define the Fourier series using complex exponential. Then, let $f(z)$ be a complex analytic function in the complex plane, with period $1$. Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i \infty$ corresponds to $q = 0$. The Fourier expansion of $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$. Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis. I am am electrical engineer; I have an idea what they all study; so I can safely override any objections that this won't be accessible to electrical engineers. Moreover, the above will reduce their surprise later in their studies when they study signal processing and wavelet analysis. share\|improve this answer I always like to use complex dynamics to illustrate that complex numbers are "real" (i.e., they are not just a useful abstract concept, but in fact something that very much exist, and closing our eyes to them would leave us not only devoid of useful tools, but also of a deeper understanding of phenomena involving real numbers.) Of course I am a complex dynamicist so I am particularly partial to this approach! Start with the study of the logistic map $x\mapsto \lambda x(1-x)$ as a dynamical system (easy to motivate e.g. as a simple model of population dynamics). Do some experiments that illustrate some of the behaviour in this family (using e.g. web diagrams and the Feigenbaum diagram), such as: • The period-doubling bifurcation • The appearance of periodic points of various periods • The occurrence of "period windows" everywhere in the Feigenbaum diagram. Then let x and lambda be complex, and investigate the structure both in the dynamical and parameter plane, observing • The occurence of beautiful and very "natural"-looking objects in the form of Julia sets and the (double) Mandelbrot set; • The explanation of period-doubling as the collision of a real fixed point with a complex point of period 2, and the transition points occuring as points of tangency between interior components of the Mandelbrot set; • Period windows corresponding to little copies of the Mandelbrot set. Finally, mention that density of period windows in the Feigenbaum diagram - a purely real result, established only in the mid-1990s - could never have been achieved without complex methods. There are two downsides to this approach: * It requires a certain investment of time; even if done on a superficial level (as I sometimes do in popular maths lectures for an interested general audience) it requires the better part of a lecture * It is likely to appeal more to those that are mathematically minded than engineers who could be more impressed by useful tools for calculations such as those mentioned elsewhere on this thread. However, I personally think there are few demonstrations of the "reality" of the complex numbers that are more striking. In fact, I have sometimes toyed with the idea of writing an introductory text on complex numbers which uses this as a primary motivation. share\|improve this answer Having been through the relevant mathematical mill, I subsequently engaged with Geometric Algebra (a Clifford Algebra interpreted strictly geometrically). Once I understood that the square of a unit bivector is -1 and then how rotors worked, all my (conceptual) difficulties evaporated. I have never had a reason to use (pure) complex numbers since and I suspect that most engineering/physics/computing types would avoid them if they were able. Likely you have the above group mixed together with pure mathematicians that feel quite at home with the non-physical aspects of complex numbers and wouldn't dream of asking such an impertinent question:-) share\|improve this answer This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. Most are probably already familiar with its content, but it might be useful to state it again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late. If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$ It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers. If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for. Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept. Reference: Dummit & Foote: Abstract Algebra, 13.1 share\|improve this answer I don't think you can answer this in a single class. The best answer I can come up with is to show how complicated calculus problems can be solved easily using complex analysis. As an example, I bet most of your students hated solving the problem $\int e^{-x}cos(x) dx.$ Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $\int e^{-x}cos(x) dx=\Re \int e^{-x(1-i)}dx.$ They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world. share\|improve this answer An interesting example of usage of complex numbers can be found in http://arxiv.org/abs/math/0001097 (Michael Eastwood, Roger Penrose, Drawing with Complex Numbers). share\|improve this answer Is it too abstract to motivate complex numbers in terms of the equations we can solve depending on whether we choose to work in ${\mathbb N, \mathbb Z, \mathbb Q, \mathbb R, \mathbb C}$? The famous "John and Betty" (http://mathforum.org/johnandbetty/) takes such an approach. share\|improve this answer As an example to demonstrate the usefulness of complex analysis in mechanics (which may seem counterintuitive to engineering students, since mechanics is introduced on the reals), one may consider the simple problem of the one dimensional harmonic oscillator, whose Hamiltonian equations of motion are diagonalized in the complex representation, equivalently one needs to integrate a single (holomorphic) first order ODE instead of a single second order or two first order ODEs. share\|improve this answer Motivating complex analysis The physics aspect of motivation should be the strongest for engineering students. No complex numbers, no quantum mechanics, no solid state physics, no lasers, no electrical or electronic engineering (starting with impedance), no radio, TV, acoustics, no good simple way of understanding of the mechanical analogues of RLC circuits, resonance, etc., etc. Then the "mystery" of it all. Complex numbers as the consequence of roots, square, cubic, etc., unfolding until one gets the complex plane, radii of convergence, poles of stability, all everyday engineering. Then the romance of it all, the "self secret knowledge", discovered over hundreds of years, a new language which even helps our thinking in general. Then the wider view of say Smale/Hirsch on higher dimensional differential equations, chaos etc. They should see the point pretty quickly. This is a narrow door, almost accidentally discovered, through which we see and understand entire new realms, which have become our best current, albeit imperfect,descriptions of how to understand and manipulate a kind of "inner essence of what is" for practical human ends, i.e. engineering. (True, a little over the top, but then pedagogical and motivational). For them to say that they just want to learn a few computational tricks is a little like a student saying, "don't teach me about fire, just about lighting matches". It's up to them I suppose, but they will always be limited. There might be some computer software engineer who needs a little more, but then I suppose there is also modern combinatorics. :-) share\|improve this answer Here's a visual thing I handed out to students in a much more elementary class than the one the question mentions: share\|improve this answer Your Answer
74dad8d499c5713f	Ashtekar variables From Scholarpedia Abhay Ashtekar (2015), Scholarpedia, 10(6):32900. doi:10.4249/scholarpedia.32900 revision #150550 [link to/cite this article] Jump to: navigation, search In the spirit of Scholarpedia, this invited article is addressed to students and younger researchers. It provides the motivation and background material, a summary of the main physical ideas, mathematical structures and results, and an outline of applications of the connection variables for general relativity. These variables underlie both the canonical/Hamiltonian and the spinfoam/path integral approaches in loop quantum gravity. This article describes a new formulation of general relativity, introduced in the mid 1980s (Ashtekar A, 1986; Ashtekar A, 1987; Ashtekar A, 1991). The main motivation was to launch a non-perturbative, background independent approach to unify principles of general relativity with those of quantum physics. The approach has since come to be known as Loop Quantum Gravity (LQG) and is being pursued using both Hamiltonian methods (Canonical LQG) and path integral methods (Spinfoams). Details of this program can be found, e.g., in an introductory text addressed to undergraduates (Gambini R, Pullin J, 2012), a more advanced review article (Ashtekar A, Lewandowski J, 2004), and monographs (Rovelli C, 2004; Thiemann T, 2007; Rovelli C, Vidotto F, 2014). This reformulation of general relativity has also had some applications to black hole physics (Ashtekar A, Krishnan B, 2004; Ashtekar A, Krasnov K, 1999; Ashtekar A, Reuter M, Rovelli C, 2015), numerical relativity (Yoneda G, Shinkai H-A, 1999; Yoneda G, Shinkai H-A, 1999; Yoneda G, Shinkai H-A, 2000), and cosmology (Ashtekar A, Singh P, 2011; Bethke L, Magueijo J, 2011; Ashtekar A, Reuter M, Rovelli C, 2015). It is closely related to developments in Twistor theory, particularly Penrose's non-linear graviton (Penrose R, 1976) and more recent advances in calculations of scattering amplitudes using twistorial techniques (Adamo T, Casali E, Skinner D, 2014; Arkani-Hamed N, Trnka J, 2013).1 The article is organized as follows. Section 2 discusses the motivation behind this reformulation and explains some of the key ideas in general terms. This non-technical introduction is addressed to readers interested only in the underlying ideas and the global context. Section 3 presents a brief but self-contained summary of the reformulation. The level of this discussion is significantly more technical; in particular it assumes familiarity with basic differential geometry and mathematical underpinnings of general relativity. Section 4 provides a short summary and two directions for future research. There is a fascinating but largely unknown historical episode involving Albert Einstein and Erwin Schrödinger that is closely related to this reformulation. It is described in Appendix A. Motivation and Key ideas Smooth Lorentzian metrics $g_{ab}$ (with signature $-,+,+,+$) constitute the basic mathematical variables of general relativity. $g_{ab}$ endows the space-time manifold with a pseudo-Riemannian geometry and the gravitational field is encoded in its curvature. Key predictions of general relativity which have revolutionized our understanding of the universe can be traced back to the fact that the metric $g_{ab}$ ---and hence space-time geometry--- is a dynamical entity in this theory. For example, it is this feature that makes it possible for the universe to expand. It is this feature that lets ripples of curvature propagate, providing us with gravitational waves. In the strong curvature regions, it is this feature that warps the space-time geometry so much that even light is trapped, leading to the formation of black holes. Let me use an analogy with particle mechanics to further explain the fundamental role played by metrics in general relativity. Just as position serves as the configuration variable for a particle, the positive, definite three dimensional metric of space, $q_{ab}$, can be taken to be the configuration variable in general relativity. Given the initial position and velocity of a particle, Newton's laws provide us with its trajectory in the position space. Similarly, given a three dimensional metric $q_{ab}$ and its time derivative $K_{ab}$ at an initial instant, Einstein's equations provide us with a four dimensional space-time, or, a trajectory in the infinite dimensional space of 3-geometries.2 This superspace of 3-metrics is thus the arena for describing dynamics in general relativity. Thus, general relativity can be regarded as the dynamical theory of 3-metrics $q_{ab}$ and was therefore labeled geometrodynamics by John Wheeler (Wheeler J A, 1964). However, this central role played by the metric also sets general relativity apart from all other fundamental forces of Nature. For, theories of electro-weak and strong interactions are also geometric, but the basic dynamical variable in these theories is a (matrix-valued) vector potential, or a connection. Thus, these theories describe connection-dynamics (in the fixed space-time geometry of Minkowski space). The connections enable one to parallel-transport geometrical/physical entities along curves. In electrodynamics, the entity is a charged particle such as an electron; in chromodynamics, it is a particle with internal color, such as a quark. Generally, if we move the object around a closed loop in an external gauge field, its state does not return to its initial value but is rotated by a non-trivial unitary matrix. The unitary matrix is a measure of the flux of the field strength ---i.e. the curvature of the connection--- across a surface bounded by the loop. In the case of electrodynamics, the connection $A_{a}$ is a 1-form on 4-dimensional Minkowski space-time that takes values in the Lie algebra of ${\rm U(1)}$, and the curvature is the field strength $F_{ab}$. In chromodynamics, the connection $A_{a}^{i}$ is a 1-form that takes values in the Lie algebra of the non-Abelian group ${\rm SU(3)}$ and its curvature $F_{ab}^{i}$ encapsulates the strong force. The Lagrangian governing dynamics of all these connections dynamics is just quadratic in curvature. But the space-time metric of general relativity also determines a specific connection named after Levi-Civita. It enables one to parallel transport vectors in the tangent space of a point to those at another point in presence of a gravitational field encoded in the space-time metric. Furthermore, the Einstein-Hilbert Lagrangian also features the curvature of this connection. Therefore, it is natural to ask if one can recast general relativity as connection-dynamics thereby bringing it closer to the theories of other basic forces of Nature. Early attempts along these lines were motivated by considerations of unifying general relativity with electrodynamics, dating back long before the advent of Yang Mills theories (that govern the electro-weak and strong interactions). The earliest of these investigations were by Einstein, Eddington and Schrödinger, where only the Levi-Civita type connection was used as the basic dynamical variable (see, e.g., (Schrödinger E, 1947; Schrödinger E, 1948)). But the equations became complicated and physics remained opaque. The fascinating episode involving Einstein and Schrödinger I referred to in section 1, occurred in this phase. It is surprising that this episode is not widely known today but it appears to have cast a long shadow on the subject, dampening for quite some time subsequent attempts to construct theories of relativistic gravity based connection-dynamics. The attempts were revived decades later, after the advent of geometrical formulations of Yang Mills theories, along two directions. In the first, by mimicking Yang Mills theory closely, one was led to new theories of gravity (see, e.g., (Yang C N, 1974; Mielke E W, Maggiolo A A R, 2004)), which, it was hoped would be better suited for incorporating quantum aspects of gravity. However, these ideas have lost their original appeal because, on the one hand, general relativity has continued to meet with impressive observational successes, and, on the other hand, the original hopes that the new theories would readily lead to satisfactory quantum gravity completions of general relativity were not realized. The second direction involved recasting general relativity itself as a theory of connections. Here, the motivation came from ‘Palatini type’ actions for general relativity in which both the metric and the connection are regarded as independent dynamical variables. The relation between them which says that the metric serves as a ‘potential’ for the connection now emerges as a field equation. This framework was extended using in part the strategies employed in the early attempts described above. It was shown that general relativity coupled with electrodynamics could be treated as a gauge theory in the modern sense of the term, with only the electromagnetic and gravitational connections as basic variables (see, e.g., (Ferraris M, Kijowski J, 1981; Ferraris M, Kijowski J, 1982)). The space-time metric did not appear in the action but emerged as a ‘derived’ quantity. However, when one carries out the Legendre transform to pass to the Hamiltonian theory, one essentially recovers the same phase space as geometrodynamics. Einstein equations again provide dynamical trajectories on the superspace of metrics. Consequently one cannot import techniques from gauge theories in the passage to quantum theory. This status quo changed in the mid-1980s (Ashtekar A, 1986; Ashtekar A, 1987). The new twist was motivated by entirely different considerations, rooted in the appreciation of the role played by chirality ---i.e. self-duality (or anti self-duality)--- in simplifying Einstein dynamics. The simplification was made manifest by apparently diverse results: Penrose's twistorial construction of the ‘non-linear graviton’ (Penrose R, 1976; Penrose R, Rindler W, 1987); Newman's construction of ‘H-spaces’ (Ko M, Ludvigsen M, Newman E T, Tod K P, 1981); and results on ‘asymptotic quantization’ (Ashtekar A, 1981; Ashtekar A, 1981; Ashtekar A, 1981; Ashtekar A, 1987) based on the structure of the radiative degrees of freedom of the gravitational field in exact general relativity. The first two of these constructions provided a precise sense in which the chiral sectors of general relativity are ‘exactly integrable’, while the third showed that chirality considerably simplifies the description of the asymptotic quantum states of the gravitational field in the exact, fully non-linear general relativity, bringing them closer to the Fock spaces of gravitons used in perturbative treatments of quantum gravity. In the new formulation (Ashtekar A, 1986; Ashtekar A, 1987) the connections of interest parallel transport chiral spinors ---rather than vectors--- in a gravitational field. The use of the spin-connection was motivated by certain simplifications in the Hamitlonian formulation of general relativity (Sen A, 1982; Ashtekar A, Horowitz G T, 1982). These mathematical objects have direct physical content because the left handed fermions of the standard model of particle physics are represented by chiral spinors. The new framework is a generalization of Palatini's in that it uses orthonormal tetrads ---i.e., ‘square roots of metrics’--- in place of metrics, and spin connections in place of the Levi Civita connections. Then, in the Hamiltonian framework, the phase space is the same as in the ${\rm SU(2)}$ Yang-Mills theory. Thus, the dynamics of general relativity is now represented by trajectories in the infinite dimensional superspace, but now of connections rather than of metrics. But, as discussed in detail in section 3, the equations that govern the dynamics of the gravitational field on this common phase space are very different from those governing the Yang-Mills dynamics. In the Yang-Mills case, the space-time geometry is fixed once and for all and equations make use of the background Minkowski metric. On the phase space of general relativity, by contrast, there is no background geometry and so all equations have to be written entirely in terms of (spin) connections and their canonically conjugate momenta. However, the dynamical trajectories can be projected down to the configuration space, which is now the infinite dimensional superspace of ${\rm SU(2)}$ connections, exactly as in the Yang-Mills theory. In this sense, the kinematical arena for all interactions ---including gravitational--- is now a connection superspace. Furthermore, in the case of general relativity, the projected trajectories can be interpreted as geodesics on the connection superspace, with respect to the ‘super-metric’ that features in the so called Hamiltonian constraint. In this sense, the new setting brings out the underlying simplicity of the dynamics of general relativity.3 This simplicity can also be seen directly in the form of equations. In contrast to geometrodynamics, where the equations involve rather complicated non-polynomial functions of the spatial metric, in the new connection-dynamics all equations are low order polynomials in the basic phase space variables. Finally, because the phase space is the same as in Yang-Mills theory, now it is possible to import into gravity certain powerful mathematical techniques from Yang-Mills theories. All these features play an important role in the emergence of quantum geometry. In LQG, this specific quantum geometry is crucial both in the Hamiltonian theory and spinfoams, and in applications of these theories to black holes and cosmology. This concludes the broad-brush overview of the underlying ideas. General relativity as a dynamical theory of spin-connections Let us begin with a quote from Einstein's 1946 biographical notes on how to formulate relativistic theories of gravity (Einstein A, 1973): The major question for anyone doing research in this field is: Of which mathematical type are the variables… which permit the expression of physical properties of space… Only after that, which equations are satisfied by these variables? Recall that before the advent of Yang-Mills theories, Einstein, Schrödinger, Palatini and others focused on the Levi-Civita connections that enable one to parallel transport space-time vectors along curves. However, in Yang-Mills theories, the parallel transported objects have ‘internal indices’. As explained in section 2, in the gravitational context, the natural objects ‘with internal indices’ are spinors. Thus, in the approach described in this article, the answer to the first major question of Einstein's is: spin-connections. As for the second question, in approaches motivated directly by non-Abelian gauge theories (Yang C N, 1974; Mielke E W, Maggiolo A A R, 2004), one mimics the Yang-Mills strategy and introduces actions that are quadratic in the curvature of the connections. Then the field equations turn out to be of order four (or higher) in the metric. In the present approach, by contrast, one retains general relativity. Thus, the equations satisfied by the spin-connections and their conjugate momenta will be equivalent to Einstein's, which contain only second derivatives of the metric. Preliminaries: 2+1 dimensional General Relativity It is instructive to begin with 2+1 dimensions, first because many of the essential conceptual aspects of connection dynamics are more transparent there, and second, because the discussion brings out the key difficulties that arise in 3+1 dimensions. Indeed, even today, new ideas for background independent, non-perturbative quantum gravity are often first tried in 2+1 dimensions. Now, the 2+1 dimensional Lorentz group ${\rm SO(1,2)}$ ---or its double cover, $SU(1,1)$--- serves as the internal gauge group. It acts on orthonormal co-frames $e_{a}^{I}$, where the index $a$ refers to the 1-form nature of the co-frame and the index $I$ takes values in the Lie algebra ${\rm so(1,2)}$ of ${\rm SO(1,2)}$. The space-time metric is given by $g_{ab} = e_{a}^{I}e_{b}^{J} \eta_{IJ}$, where $\eta_{IJ}$, the Cartan-Killing metric on ${\rm so(1,2)}$, has signature $-,+,+$. Note that $\eta_{IJ}$ constitutes the kinematic structure of the theory, fixed once and for all, in the ‘internal space’. It enables one to freely raise and lower the internal indices. The co-frames $e_{a}^{I}$, on the other hand are dynamical variables that directly determine the space-time metric $g_{ab}$ which is guaranteed to have signature $-,+,+$ simply from the signature of $\eta_{IJ}$. The transition from the metric $g_{ab}$ to its ‘square-root’ $e_{a}^{I}$ becomes essential for incorporation of spinors in the theory. Physically, of course, spinors serve as fundamental matter fields. However, interestingly, even if one restricts oneself to the purely bosonic sector, spinors provide a more transparent way to establish a number of mathematical results, the most notable being the positive energy theorem in 2+1 dimensions (Ashtekar A, Varadarajan M, 1994) (that follows the key ideas introduced by Sen, Nester, Sparling and Witten in 3+1 dimensions (Witten E, 1981)). In the Palatini formulation, the fundamental dynamical variables are the co-frames $e_{a}^{I}$ and a connection 1-form $A_{a}^{IJ}$ that takes values in the Lie algebra of ${\rm so(1,2)}$. For pure gravity, the action is given by \[ \tag{1} S_{P}^{(2+1)} (e, A) = \frac{1}{8\pi G_{3}}\int_{M} {\rm d}^{3}x\, \epsilon_{IJK} \tilde{\eta}^{abc}e^{I}_{a} F^{JK}_{bc} \] where $\tilde{\eta}^{abc}$ is the (metric independent) space-time Levi-Civita tensor density of weight 1 and $F_{ab}^{IJ} = 2 \partial_{[a}A_{b]}^{IJ} + A_{a}{}^{IK}A_{bKJ}$ is the curvature 2-form of $A_{a}^{IJ}$. When one carries out a Legendre transform of this action, one is led to the Hamiltonian framework. The pull-back of $A_{a}^{IJ}\epsilon_{IJ}{}^{K}$ to a spatial 2-manifold, which for simplicity we denote by $A_{a}{}^{K}$, turns out to be the configuration variable. The conjugate momentum is the ‘electric field’ $\tilde{E}^{a}_{I} = \tilde{\eta}^{ab}e_{b\,I}$, the dual of the pull-back of the co-triad to the spatial 2-manifold. Thus, now the configuration space is simply the space of connections and the (spatial) metric is ‘derived’ from the canonically conjugate momentum $\tilde{E}^{a}_{I}$.4 Recall that in general relativity, some of Einstein's equations constrain initial data and the remaining equations describe dynamics. Furthermore, because there are no background fields, in the phase space framework, the Hamiltonian generating dynamics is a linear combination of constraints. In the metric variables, constraints are rather complicated, non-polynomial functions of the 2-metric and its canonically conjugate momentum. In connection-dynamics under considerations, by contrast, the constraints are very simple: \[ \tag{2} \mathcal{D}_{a} \tilde{E}^{a}_{I} =0, \qquad \text{and} \qquad F^{IJ}_{ab} = 0, \] where $\mathcal{D}$ is the gauge covariant derivative defined by the connection $A_{a}^{IJ}$ and $F_{ab}^{IJ}$ its curvature and $\tilde{E}^{a}_{I} = \tilde{\eta}^{ab}e_{b\,I}$, the dual of the co-triad, can be regarded as the ‘electric field’ in the Yang-Mills terminology. (Recall that the lower case indices now refer to the 2-dimensional tangent space of the spatial manifold, where all fields in the Hamiltonian theory live.) The first constraint is just the familiar Gauss law while the second says that the field strength of the pulled-back connection vanishes. Since the Hamiltonian generating dynamics is just a linear combination of these constraints, the dynamical equations are also low order polynomials in the connection variables although they are equivalent to the standard Einstein's equations which are non-polynomial in the metric variables. (For further details, see, e.g., (Ashtekar A, 1991; Ashtekar A, 1995).) 3+1 dimensional General Relativity: The Lagrangian framework For simplicity, I will continue to restrict myself to source-free general relativity. The connection-dynamics framework is, however, quite robust: all its basic features ---including the fact that all equations are low order polynomials in the basic canonical variables--- remain unaltered by the inclusion of a cosmological constant and coupling of gravity to Klein-Gordon fields, (classical or Grassmann-valued) Dirac fields and Yang-Mills fields with any internal gauge group (Ashtekar A, Romano J D, Tate R S, 1989; Ashtekar A, 1991). This is true both for the Lagranagian framework discussed in this sub-section and the Hamiltonian framework discussed in the next sub-section. Let us begin by extending the underlying ideas from 2+1 to to 3+1 dimensions. Now the ${\rm SO(2,1)}$ connection is replaced by the ${\rm SO(3,1)}$ Lorentz connection ${}^{4}\omega_{a}^{IJ}$ and the co-triad $e_{a}^{I}$ by a co-tetrad $e_{a}^{I}$, where $I,J,...$ now denote the internal ${\rm so(3,1)}$ indices labeling the co-tetrads, and $a,b,..$ denote 4-dimensional space-time indices. However, because the underlying manifold $M$ is now 4-dimensional, the Palatini action contains two co-tetrads, rather than just one: \[ \tag{3} S_P(e, {}^4\omega) := {1\over 16\pi G}\int_{M} {\rm d}^4 x\, \epsilon_{IJKL}\tilde\eta^{abcd}e_{aI} e_{bJ}({}^4\!R_{cd}{}^{KL}), \] where $G$ is Newton's constant, $\tilde\eta^{abcd}$ is the metric independent Levi-Civita density on space-time and ${}^4R_{ab}{}^{IJ}$ is the curvature tensor of the ${\rm SO(3,1)}$ connection ${}^4\omega_a^{IJ}$. (Note that the internal indices can again be raised and lowered freely using the fixed, kinematical metric $\eta_{IJ}$ on the internal space.) Hence, when one performs the Legendre transform, the momentum $\tilde{\Pi}^a_{IJ}$ conjugate to the connection $A_a^{IJ}$ is the dual $\tilde{\eta}^{abc} \epsilon^{IJ}{}_{KL}e_b^K e_c^L$ of a product of two co-triads rather than of a single co-triad as in the 2+1 dimensional case. The theory then has an additional constraint --saying that the momentum is “decomposable” in this manner-- which spoils the first class nature of the constraint algebra. Following the Dirac procedure, one can solve for the second class constraint and obtain new canonical variables. It is in this elimination that one loses the connection 1-form altogether and is led to geometrodynamics. (For details, see (Ashtekar A, Balachandran A P, Jo S G, 1989) and chapters 3 and 4 in (Ashtekar A, 1991).) However, these complications disappear if one requires the connection to take values only in the self dual (or, alternatively anti-self dual) part of ${\rm SO(3,1)}$. Furthermore, the resulting connection dynamics is technically significantly simpler than geometrodynamics. It is this simplicity that leads to LQG. Thus, in connection-dynamics, the answer to the first question posed by Einstein in his autobiographical notes is that (the configuration) variables of the theory should be chiral connections. I will now elaborate on this observation. Let me first explain what I mean by self duality here. If one begins with a Lorentz connection ${}^4\omega_a^{IJ}$, the self dual connection ${}^4\!A_a^{IJ}$ is given by dualizing over the internal indices: \[ \tag{4} {}^4\!A_a^{IJ} = \frac{1}{2G}({}^4\omega_a^{IJ} - \tfrac{i}{2} \epsilon^{IJ}{}_{KL}{}^4\omega_a^{KL}), \] where $G$ is Newton's constant. (This factor has been introduced for later convenience and plays no role in this discussion of the mathematical meaning of self duality.) However, one regards the self dual connections themselves as fundamental; they are subject just to the following algebraic condition on internal indices: \[ \tag{5} \tfrac{1}{2} \epsilon^{IJ}{}_{KL} {}^4\!A_a^{KL} = i\;{}^4\!A_a^{IJ}. \] Let me emphasize that, unlike in the analysis of self dual Yang-Mills fields on a given space-time, the notion of self duality here refers to the internal rather than space-time indices: to define the duality operation, we use the kinematical internal metric $\eta_{IJ}$ (and its alternating tensor $\epsilon^{IJ}{}_{KL}$) rather than the dynamical space-time metric (to be introduced later). The new action is obtained simply by substituting the real ${\rm SO(3,1)}$ connection ${}^4\omega_a^{IJ}$ by the self dual connection $A_a^{IJ}$ in the Palatini action (modulo overall constants): \[ \tag{6} S(e, {}^4\!A) := \frac{1}{16\pi} \int_{\rm M} {\rm d}^4 x\, \epsilon_{IJKL}\tilde\eta^{abcd} e_{aI} e_{bJ}({}^4\!F_{ab}{}^{KL}), \] where, \[ \tag{7} {}^4F_{abI}{}^J := 2 \partial_{[a} {}^4\!A_{b]I}{}^J + G{}^4A_{aI} {}^M {}^4A_{bM}{}^J - G{}^4A_{bI} {}^M{}^4A_{aM}{}^J \] is the field strength of the connection ${}^4\!A_{aI}{}^J$. Thus, $G$ plays the role of the coupling constant. Note incidentally that because of the factors of $G$, ${}^4A_{aI}{}^J$ and ${}^4F_{abI}{}^J$ do not have the usual dimensions of connections and field strength. By setting the variation of the action with respect to ${}^4\!A_{a}{}^{IJ}$ to zero we obtain the result that ${}^4\!A_{a}{}^{IJ}$ is the self dual part of the (torsion-free) connection ${}^4\Gamma_a{}^{IJ}$ compatible with the tetrad $e_a^I$. Thus, ${}^4\!A_{a}{}^{IJ}$ is completely determined by $e_a^I$. Setting the variation with respect to $e_a^I$ to zero and substituting for the connection from the first equation of motion, we obtain the result that the space-time metric $g^{ab} =e^a{}_I e^b{}_J\eta^{IJ}$ satisfies the vacuum Einstein's equation. Thus, as far as the classical equations of motion are concerned, the self dual action (6) is completely equivalent to the Palatini action (3). This result seems surprising at first. Indeed, since ${}^4\!A_{a}{}^{IJ}$ is the self dual part of ${}^4\omega_a^{IJ}$, it follows that the curvature ${}^4\!F_{ab} {}^{IJ}$ is the self dual part of the curvature ${}^4\!R_{ab}{}^{IJ}$. Thus, the self dual action is obtained simply by adding to the Palatini action an extra (imaginary) term. This term is not a pure divergence. How can it then happen that the equations of motion remain unaltered? This comes about as follows. First, the compatibility of the connections and the tetrads forces the “internal” self duality to be the same as the space-time self duality, whence the curvature ${}^4\!F_{abI}{}^J$ can be identified with the self dual part, on space-time indices, of the Riemann tensor of the space-time metric. Hence, the imaginary part of the field equation says that the trace of the dual of the Riemann tensor must vanish. This, however, is precisely the (first) Bianchi identity! Thus, the imaginary part of the field equation just reproduces an equation which holds in any case; classically, the two theories are equivalent. However, the extra term does change the definition of the canonical momenta in the Legendre transform --i.e., gives rise to a canonical transform on the Palatini phase space-- and this change, in turn, enables one to regard general relativity as a theory governing the dynamics of 3-connections rather than of 3-geometries. (For details, see (Ashtekar A, 1986; Ashtekar A, 1987; Ashtekar A, 1991; Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007; Ashtekar A, Romano J D, Tate, R S, 1989)). Hamiltonian framework Since in the Lorentzian signature self dual fields are necessarily complex, it is convenient to begin with complex general relativity --i.e. by considering complex, Ricci-flat metrics $g_{ab}$ on a real 4-manifold $M$-- and take the “real section” of the resulting phase-space at the end. Let $e_a^I$ then be a complex co-tetrad on $M$ and ${}^4\!A_{a}{}^{IJ}$ a self dual ${\rm SO(3,1)}$ connection, and let the action be given by (6). Let us assume that the space-time manifold $M$ has the topology $\Sigma\times\mathbb{R}$ and carry out the Legendre transform. This procedure is remarkably straightforward especially when compared to geometrodynamics. The resulting canonical variables are then complex fields on a (“spatial”) 3-manifold $\Sigma$. To begin with, the configuration variable turns out to be a 1-form $A_a^{IJ}$ on $\Sigma$ which takes values in the self dual part of the (complexified) ${\rm SO(3,1)}$ Lie-algebra and its canonical momentum $\tilde{E}_a^{IJ}$ is a self dual vector density which takes values also in the self dual part of the ${\rm SO(3,1)}$ Lie algebra. (Thus, in the Hamiltonian framework, the lower case latin indices refer to the spatial 3-manifold $\Sigma$.) The key improvement over the Palatini framework is that there are no additional constraints on the algebraic form of the momentum (Ashtekar A, 1991; Ashtekar A, 1989). Hence, all constraints are now first class and the analysis retains its simplicity. For technical convenience, one can set up, once and for all, an isomorphism between the self dual sub-algebra of the Lie algebra of ${\rm SO(3,1)}$ and the Lie algebra of ${\rm SO(3)}$. When this is done, we can take our configuration variable to be a complex, ${\rm SO(3)}$-valued connection $A_a^i$ and its canonical momentum, a complex spatial triad $\tilde{E}_i^a$ with density weight one, where ‘$a$’ is the manifold index and ‘$i$’ is the triad or the ${\rm SO(3)}$ internal index. The (only non-vanishing) fundamental Poisson brackets are: \[ \tag{8} \{\tilde{E}^a{}_i(x),\,A_b{}^j(y)\}=-i\delta^a{}_b \delta_i{}^j\delta^3(x,y). \] The geometrical interpretation of these canonical variables is as follows. As we saw above, in any solution to the field equations, ${}^4\!A_{a}{}^{IJ}$ turns out to be the self dual apart of the spin-connection defined by the tetrad, whence $A_a^i$ has the interpretation of being a potential for the self dual part of the Weyl curvature. $\tilde{E}_i^a$ can be thought of as a “square-root” of the 3-metric (times its determinant) on $\Sigma$. More precisely, the relation of these variables to the familiar geometrodynamical variables, the 3-metric $q_{ab}$ and the extrinsic curvature $K_{ab}$ on $\Sigma$, is as follows: \[ \tag{9} GA_a{}^i = \Gamma_a{}^i - i K_a{}^i \quad {\rm and} \quad \tilde{E}^a{}_i \tilde{E}^{bi} = (q) q^{ab} \] where, as before, $G$ is Newton's constant, $\Gamma_a{}^i$ is the spin-connection determined by the triad, $K_a{}^i$ is obtained by transforming the space index ‘$b$’ of the extrinsic curvature $K_{ab}$ into an internal index by the triad $E^a_i := (1/\sqrt{q})\tilde{E}^a_i$, and $q$ is the determinant of $q_{ab}$. Note, however, that, as far as the mathematical structure is concerned, we can also think of $A_a^i$ as a (complex) ${\rm so(3)}$-Yang-Mills connection and $\tilde{E}_i^a$ as its conjugate electric field. Thus, the phase space has a dual interpretation. It is this fact that enables one to import into general relativity and quantum gravity ideas from Yang-Mills theory and quantum chromodynamics and may, ultimately, lead to a unified mathematical framework underlying the quantum description of all fundamental interactions. In what follows, we shall alternate between the interpretation of $\tilde{E}_i^a$ as a triad and as the electric field canonically conjugate to the connection $A_i^a$. Since the configuration variable $A_a^i$ has nine components per space point and since the gravitational field has only two degrees of freedom, we expect seven first class constraints. This expectation is indeed correct. The constraints are given by: \[ \tag{10} \begin{split} {\cal G}_i(A,\tilde{E}) &:= \mathcal{D}_a \tilde E^a{}_i=0 \\ {\cal V}_a (A,\tilde{E}) &:= \tilde{E}^b{}_i\, F_{ab}{}^i\equiv \mathrm{tr}\, E\times B =0 \\ {\cal S} (A,\tilde{E}) &:= \epsilon^{ijk}\tilde E^a{}_i\,\tilde E^b{}_j\,F_{abk} \equiv \mathrm{tr}\, E\times E\cdot B =0, \end{split} \] where $F_{ab}{}^i:=2\partial_{[a} A_{b]}{}^i + G\epsilon^{ijk} A_{a{}j}A_{b{}k}$ is the field strength constructed from $A_a^i$, $B$ stands for the magnetic field $\tilde\eta^{abc}F_{bc}^i$, constructed from $F_{ab}^i$, and $\mathrm{tr}$ refers to the standard trace operation in the fundamental representation of ${\rm SO(3)}$. Note that all these equations are simple polynomials in the basic variables; the worst term occurs in the last constraint and is only quadratic in each of $\tilde{E}_i^a$ and $A_a^i$ . The three equations are called, respectively, the Gauss constraint, the vector constraint and the scalar constraint. The first, Gauss law, arises because we are now dealing with triads rather than metrics. It simply tells us that the internal ${\rm SO(3)}$ triad rotations are “pure gauge”. Modulo these internal rotations, the vector constraint generates spatial diffeomorphisms on $\Sigma$ while the scalar constraint is responsible for diffeomorphisms in the “time-like directions”. Thus, the overall situation is the same as in triad geometrodynamics. From geometrical considerations we know that the “kinematical gauge group” of the theory is the semi-direct product of the group of local triad rotations with that of spatial diffeomorphisms on $\Sigma$. This group has a natural action on the canonical variables $A_a^i$ and $\tilde{E}_i^a$ and thus admits a natural lift to the phase-space. This is precisely the group formed by the canonical transformations generated by the Gauss and the vector constraints. Thus, six of the seven constraints admit a simple geometrical interpretation. What about the scalar constraint? Note that, being quadratic in momenta, it is of the form $G^{\alpha\beta} p_\alpha p_\beta=0$ on a generic phase space, where, the connection supermetric $\epsilon^{ijk}F_{ab k}$ plays the role of $G^{\alpha\beta}$ and the momenta $\tilde{E}^{a}{}_{i}$ of $P_{\alpha}$. Consequently, the motions generated by the scalar constraint in the phase space correspond precisely to the null geodesics of the “connection supermetric”. As in geometrodynamics, the space-time interpretation of these canonical transformations is that they correspond to “multi-fingered” time-evolution. Thus, we now have an attractive representation of the Einstein evolution as a null geodesic motion in the (connection) configuration space.5 If $\Sigma$ is spatially compact, the Hamiltonian is given just by a linear combination of constraints. In the asymptotically flat situation, on the other hand, constraints generate only those diffeomorphisms which are asymptotically identity. To obtain the generators of space and time translations, one has to add suitable boundary terms. In a 3+1 framework, these translations are coded in a lapse-shift pair. The lapse --which tends to a constant value at infinity-- tells us how much of a time translation we are making while the shift --which approaches a constant vector field at infinity-- tells us the amount of space-translation being made. Given a lapse6 $\underset{\sim}{N}$ and a shift $N^a$, the Hamiltonian is given by: \[ \tag{11} \begin{split} H(A,\tilde E) &= i \int_{\Sigma} {\rm d}^3 x \, (N^a F_{ab}{}^i\tilde E^b{}_i -\tfrac{i}{2}\underset{\sim}{N} \epsilon^{ijk}F_{ab k} \tilde E^a{}_i \tilde E^b{}_j) \\ & \qquad - \oint_{\partial\Sigma} {\rm d}^2S_a\,(\underset{\sim}{N} \epsilon^{ijk} A_{bk} \tilde {E}^a{}_i \tilde{E}^b{}_j + 2 i N^{[a} \tilde E^{b]}{}_i A_b{}^i). \end{split} \] The dynamical equations are easily obtained since the Hamiltonian is also a low order polynomial in the canonical variables. We have \[ \tag{12} \begin{split} \dot{A}_a^i &= -i\epsilon^{ijk}\underset{\sim}{N}\tilde{E}^b_jF_{ab}{}_{k} - N^bF^i_{ab} \\ \dot{E}^a_i &= i\epsilon_i^{jk}\mathcal{D}_b(\underset{\sim}{N}\tilde{E}^a_j \tilde{E}^b_k) -2\mathcal{D}_b(N^{[a}\tilde{E}^{b]i}) \end{split} \] Again, relative to their analogs in geometrodynamics, these equations are significantly simpler. So far, we have discussed complex general relativity. To recover the Lorentzian theory, we must now impose reality conditions, i.e., restrict ourselves to the real, Lorentzian section of the phase-space. Let me explain this point by means of an example. Consider a simple harmonic oscillator. One may, if one so wishes, begin by considering a complex phase-space spanned by two complex co-ordinates $q$ and $p$ and introduce a new complex co-ordinate $z= q - ip$. ($q$ and $p$ are analogous to the triad $\tilde{E}_i^a$ and the extrinsic curvature $K_a{}^i$, while $z$ is analogous to $A_a^i$.) One can use $q$ and $z$ as the canonically conjugate pair, express the Hamiltonian in terms of them and discuss dynamics. Finally, the real phase-space of the simple harmonic oscillator may be recovered by restricting attention to those points at which $q$ is real and $ip = q-z$ is pure imaginary (or, alternatively, $\dot{q}$ is also real.) In the present phase-space formulation of general relativity, the situation is analogous. In terms of the familiar geometrodynamic variables, the reality conditions are simply that the 3-metric be real and the extrinsic curvature --the time derivative of the 3-metric-- be real. If these conditions are satisfied initially, they continue to hold under time-evolution. In terms of the present canonical variables, these become: i) the 3-metric $\tilde E^a{}_i \tilde E^{bi}$ (with density weight 2 be real, and, ii) its Poisson bracket with the Hamiltonian $H$ be real, i.e., \[ \tag{13} \begin{split} (\tilde{E}^a{}_i \tilde{E}^{bi})^\star &= \tilde{E}^a{}_i \tilde{E}^{bi} \\ \big(\epsilon^{ijk}\tilde{E}^{(a}{}_i \mathcal{D}_c(\tilde{E}^{b)}{}_k\tilde{E}^c{}_j ) \big)^\star &= - \epsilon^{ijk}\tilde E^{(a}{}_i \mathcal{D}_c(\tilde{E}^{b)}{}_k\tilde{E}^c{}_j), \end{split} \] where $\star$ denotes complex-conjugation. (Note, incidentally, that in Euclidean relativity, these conditions can be further simplified since self dual connections are now real: The reality conditions require only that we restrict ourselves to real triads and real connections.) As far as the classical theory is concerned, we could have restricted to the “real slice” of the phase-space right from the beginning. In quantum theory, on the other hand, it may be simpler to first consider the complex theory, solve the constraint equations and then impose the reality conditions as suitable Hermitian-adjointness relations. Thus, the quantum reality conditions would be restrictions on the choice of the inner-product on physical states. Could we have arrived at the phase-space description of real general relativity in terms of ($A_a^i$, $\tilde{E}_i^a$) without having to first complexify the theory? The answer is in the affirmative. This is in fact how the new canonical variables were first introduced (Ashtekar A, 1986; Ashtekar A, 1987). The idea is to begin with the standard Palatini action for real tetrads and real Lorentz-connections, perform the Legendre transform and obtain the phase-space of real relativity à la Arnowitt, Deser and Misner. The basic canonical variables in this description can be taken to be the density weighted triads $\tilde{E}_i^a$ and their canonical conjugate momenta $\pi_a^i$. The interpretation of $\pi_a^i$ is as follows: In any solution to the field equations, i.e., “on shell,” $K_{ab}:= \pi_{(a}^i E_{b)i}$ turns out to be the extrinsic curvature. Up to this point, all fields in question are real. On this real phase space, one can make a (complex) canonical transformation to pass to the new variables: $(\tilde{E}^a_i, \pi_a^i)\to (\tilde{E}^a_i , GA_a^i := \Gamma_a^i - i \pi_a^i \equiv (\delta F/\delta\tilde{E}^a_i) - i \pi_a^i)$, where the generating function $F(\tilde{E})$ is given by: $F(\tilde{E}) = \int_{\Sigma} {\rm d}^3x \tilde{E}^a_i \Gamma_a^i$, and where $\Gamma_a^i$ are the spin-coefficients determined by the triad $\tilde{E}_i^a$. Thus, $A_a^i$ is now just a complex coordinate on the traditional, real phase space. This procedure is completely analogous to the one which lets us pass from the canonical coordinates $(q,p)$ on the phase space of the harmonic oscillator to another set of canonical coordinates $(q, z = dF/dq - ip)$, with $F(q) = \frac{1}{2}q^2$, and makes the analogy mentioned above transparent. Finally, the second of the reality conditions, (4.2.9), can now be re-expressed as the requirement that $GA_a^i - \Gamma_a^i$ be purely imaginary, which follows immediately from the expression of $A_a^i$ in terms of the real canonical variables $(\tilde{E}^a_i, K_a^i)$. I will conclude this sub-section with a few remarks. • In broad terms the Hamiltonian framework developed above is quite similar to that for the 2+1 theory: in both cases, the configuration variable is a connection, the momentum can be regarded as a “square root” of the metric, and the constraints, the Hamiltonian and the equations of motion are all low order polynomials in these basic variables. However, there are a number of key differences. The first is that the connection in the 3+1 theory is the self dual ${\rm SO(3,1)}$ spin-connection while that in the 2+1 theory is the real ${\rm SO(2,1)}$ spin-connection. More importantly, while in the 2+1-theory the connection is constrained to be flat a simple counting argument shows that there is no such implication in the 3+1 theory. This is the reason behind the main difference between the two theories: Unlike in the 2+1 case, the 3+1 theory has local degrees of freedom and hence gravitons. • A key feature of this framework is that all equations of the theory --the constraints, the Hamiltonian and hence the evolution equations and the reality conditions-- are simple polynomials in the basic variables $\tilde{E}_i^a$ and $A_a^i$. This is in striking contrast to the ADM framework where the constraints and the evolution equations are non-polynomial in the basic canonical variables. An interesting --and potentially, powerful-- consequence of this simplicity is the availability of a nice algorithm to obtain the “generic” solution to the vector and the scalar constraint (Capovilla R, Dell J, Jacobson T, 1989). Choose any connection $A_a^i$ such that its magnetic field $\tilde B^a{}_i := \tilde\eta^{abc} F_{bci}$, regarded as a matrix, is non-degenerate. A “generic” connection $A_a^i$ will satisfy this condition; it is not too restrictive an assumption. Now, we can expand out $\tilde{E}_i^a$ as $\tilde{E}_i^a = M_i{}^j \tilde B^a{}_j$ for some matrix $M_i{}^j$. The pair ($A_a^i$ , $\tilde{E}_i^a$ ) then satisfies the vector and the scalar constraints if and only if $M_i{}^j$ is of the form $M_i{}^j = [\phi^2 -\frac{1}{2} \mathrm{tr} \phi^2 ]_i{}^j$, where $\phi_i{}^j$ is an arbitrary trace-free, symmetric field on $\Sigma$. Thus, as far as these four constraints are concerned, the “free data” consists of $A_a^i$ and $\phi_i{}^j$. • The phase-space of general relativity is now identical to that of complex-valued Yang-Mills fields (with internal group ${\rm SO(3)}$). Furthermore, one of the constraint equations is precisely the Gauss law that one encounters on the Yang-Mills phase-space. Thus, we have a natural embedding of the constraint surface of Einstein's theory into that of Yang-Mills theory: Every initial datum ($A_a^i$ , $\tilde{E}_i^a$) for Einstein's theory is also an initial datum for Yang-Mills theory which happens to satisfy, in addition to the Gauss law, a scalar and a vector constraint. From the standpoint of Yang-Mills theory, the additional constraints are the simplest diffeomorphism and gauge invariant expressions one can write down in absence of a background structure such as a metric. Note that the degrees of freedom match: the Yang-Mills field has $2$(helicity) $\times 3$ (internal) $= 6$ degrees and the imposition of four additional first-class constraints leaves us with $6-4 =2$ degrees of freedom of Einstein's theory. I want to emphasize, however, that in spite of this close relation of the two initial value problems, the Hamiltonians (and thus the dynamics) of the two theories are very different. Nonetheless, the similarity that does exist can be exploited to obtain interesting results relating the two theories. For example, there are interesting and surprising relations between instantons in the two theories. (See, e.g. (Samuel J, 2000)). • Since all equations are polynomial in $A_a^i$ and $\tilde{E}_i^a$ they continue to be meaningful even when the triad (i.e. the “electric field”) $\tilde{E}_i^a$ becomes degenerate or even vanishes. As in the 2+1 theory, this feature enables one to use in quantum theory a representation in which states are functionals of $A_a^i$ and $\hat{E}^a_i$ is represented simply by a functional derivative with respect to $A_a^i$, thereby shifting the emphasis from triads to connections. In fact, Capovilla, Dell and Jacobson (Capovilla R, Dell J, Jacobson T, 1989; Capovilla R, Dell J, Jacobson T, Mason L, 1991) have introduced a Lagrangian framework which reproduces the Hamiltonian description discussed above but which never even introduces a space-time metric or tetrads! This formulation of “general relativity without metric” lends strong support to the viewpoint that the traditional emphasis on metric-dynamics, however convenient in classical physics, is not indispensable. This completes the discussion of the Hamiltonian description of general relativity which casts it as a theory of self dual connections. We have transformed triads from configuration to momentum variables and found that self dual connections serve as especially convenient configuration variables. In effect, relative to the Arnowitt-Deser-Misner geometrodynamical description, we are looking at the theory “upside down” or “inside out”. And this unconventional way of looking reveals that the theory has a number of unexpected and, potentially, profound features: it is much closer to gauge theories (particularly the topological ones) than was previously imagined; its constraints are the simplest background independent expressions one can write down on the phase space of a gauge theory; its dynamics has a simple geometrical interpretation on the space of connections; etc. It opens new doors particularly for the task of quantizing the theory. We are led to shift emphasis from metrics and distances to connections and holonomies and this, in turn suggests fresh approaches to unifying the mathematical framework underlying the four basic interactions (see, e.g., (Peldán P, 1993)). Real connection-variables Because of the topic assigned to me by the Editors, so far I have focused on chiral connection-variables. As we saw, in the classical theory, they provide a viable reformulation of general relativity and, by bringing the theory closer to the successful gauge theories describing other interactions, they provide new tools for the passage to quantum theory. However, since a chiral connection is complex-valued in the Lorentzian signature, the holonomies it defines take values in a non-compact subgroup of ${\rm SL(2, C)}$ generated by the self-dual sub-space of its Lie-algrbra. This creates a major obstacle in developing a well-defined integration theory ---that respects gauge and diffeomorphism invariance--- on the infinite dimensional space of these connections. Without this integration theory, we cannot construct the Hilbert space of quantum states, introduce physically interesting operators thereon, and analyze properties of these operators. Since the difficulty stems from the non-compact nature of the subgroup ${\rm SL(2, C)_{sd}}$ of ${\rm SL(2, C)}$ in which holonomies take values, a natural strategy is to perform a ‘Wick transform’ in the internal space that sends ${\rm SL(2, C)_{sd}}$ to an ${\rm SU(2)}$ subgroup of ${\rm SL(2, C)}$ (which is compact). This strategy has been adopted in most of the mainstream work in LQG since mid-1990s. Concretely, the desired Wick transform is performed by sending self dual connections $A_{a}^{i}$ to connections ${}^{\gamma}\!A_a{}^i$, simply by replacing $i$ in (9) with a real parameter $\gamma$, called the Barbero-Immirzi parameter (Immirzi G, 1997; Barbero F, 1995) (which is assumed to be positive without loss of generality): \[ \tag{14} G\, {}^{\gamma}\!A_a^i := \Gamma_a{}^i - \gamma K_a{}^i. \] While the subgroup of ${\rm SL(2, C)}$ one thus obtains depends on the choice of $\gamma$, it is always an ${\rm SU(2)}$ subgroup, whence the integration theory is insensitive to the specific choice of $\gamma$. Note that this ‘Wick transform’ is performed on the internal space, where it is well-defined also in curved space-times; it is distinct from the standard space-time Wick transform performed in Minkowskian quantum field theories which does not have a well-defined extension to general curved space-times. Nonetheless the basic motivation is the same as in Minkowskian quantum field theories: one can regard it as a method of regularizing the functional integrals that are ill-defined in the Lorentzian sector. After this passage, it was possible to develop a rigorous integration theory as well as introduce notions from geometry on the (infinite dimensional) configuration space of connections ${}^{\gamma}\!A_{a}^{i}$ and systematically develop a specific quantum theory of Riemannian geometry (Ashtekar A, Lewandowski L, 1994; Ashtekar A, Lewandowski L, 1995; Ashtekar A, Lewandowski L, 1997; Ashtekar A, Lewandowski L, 1997). The construction of the Hilbert space of quantum states and the definition of the elementary (holonomy and flux) operators is insensitive to the choice of $\gamma$. However, now $\gamma$ enters in the relation between the momentum ${}^{\gamma}\!\Pi^{a}_{i}$ conjugate to ${}^{\gamma}\!A_{a}^{i}$ and the orthonormal triad $\tilde{E}^{a}_{i}$. As a result it also enters the expressions of various geometric operators. However, the qualitative features of quantum geometry do not depend on the specific choice of $\gamma$. Just as the flux of the magnetic field is quantized in a type II superconductor, the flux of the ‘electric field’ $\tilde{E}^{a}_{i}$ is quantized in the quantum Riemannian geometry. Since the electric field also serves as a (density weighted triad) in the classical theory, determining the spatial metric $q_{ab}$, the quantum Riemannian geometry now acquires an interesting and very non-trivial discreteness. More precisely, the eigenvalues of geometric operators such as areas of 2-surfaces or volumes of 3-dimensional regions are discrete. This discreteness has non-trivial consequences on quantum dynamics. In particular, in cosmological models, quantum geometry creates a brand new repulsive force which is negligible under normal circumstances but rises quickly in the Planck regime and overwhelms the classical attraction. In intuitive terms, under normal circumstances, general relativity provides an excellent approximation to quantum dynamics. But if this dynamics drives a curvature scalar to the Planck regime, quantum geometry effects become prominent and ‘dilute’ the curvature scalar, preventing the formation of a strong curvature singularity. (For details, see (Ashtekar A, Singh P, 2011) and references therein.) Similarly, in the path integral approach, one is now naturally led to sum over the specific, discrete quantum geometries provided by the detailed LQG framework. As a result, one can express the transition amplitudes as a sum, each term in which is ultraviolet finite. (For details, see (Ashtekar A, Reuter M, Rovelli C, 2015) and references therein). Thus, the connection-dynamics formulation of general relativity leads one along new paths that combine techniques from gauge theory and the underlying diffeomorphism invariance. This combination has led to unforeseen results representing concrete advances through LQG. However, the ‘internal Wick transform’ strategy has two limitations. First, the form of the constraints (and evolution equations) is now considerably more complicated already in the classical theory. These complications seemed so formidable that while the idea of moving away from chiral connections was considered, it was not pursued initially. Almost a decade after the introduction of chiral connections, the strategy was worked out in complete detail by Barbero (Barbero F, 1995) at the classical level. Soon thereafter, Thiemann (Thiemann T, 1996; Thiemann T, 1998; Thiemann T, 1998; Thiemann T, 1998; Thiemann T, 2007) introduced several astute techniques to handle the complications in the canonical approach within LQG. It is only then that the strategy became mainstream. A second limitation of the strategy is that while the connection ${}^{\gamma}\!A_a^i$ is well-defined on the spatial 3-manifold and continues to have a simple relation (14) to the ADM variables, it does not have a natural 4-dimensional geometrical interpretation even in solutions to the field equations (Samuel J, 2000). As I illustrated above, significant advances have occurred in spite of these limitations. Still the situation could be improved significantly by seriously pursuing the viewpoint that the passage to ${}^{\gamma}\!A_a^i$ is only a mathematical construct and all questions should be phrased and all answers be given using the Lorentzian chiral connection. This idea is acquiring momentum over the last few years (see, e.g., (Geiller M, Noui K, 2014; Wieland W M, 2012; Wieland W M, 2015) but we are still far from a complete picture. The connection-dynamics formulation of general relativity provide a fresh perspective on the deep underlying simplicity of Einstein's equations. As I emphasized in the beginning of this article, the central feature of general relativity is that gravity is encoded in the very geometry of space-time. Therefore, a quantum theory of gravity should have at its core a quantum theory of geometry. The connection-dynamics formulation opens new vistas to construct this theory. Specifically, it provides brand new tools for this purpose: holonomies defined by the gravitational connection (i.e., Wilson loops), and quantization of the flux of the conjugate electric field across 2-surfaces. These tools have led to a rich Riemannian quantum geometry (Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007). The novel aspects associated with the fundamental discreteness of this geometry have already led to some unforeseen consequences. These include the resolution of strong curvature singularities in a variety of cosmological models (Ashtekar A, Singh P, 2011; Singh P, Singh P, 2009); new insights into the microstructure of the geometry of quantum horizons (Ashtekar A, Krishnan B, 2004; Ashtekar A, Krasnov K, 1999; Ashtekar A, Reuter M, Rovelli C, 2015); and, a derivation of the graviton propagator in the background independent, non-perturbative setting of spinfoams (Ashtekar A, Reuter M, Rovelli C, 2015; Bianchi E, Magliaro E, Perini C, 2009; Bianchi E, Magliaro E, Perini C, 2012). The formulation also provides a small generalization of Einstein's theory: Since the equations are polynomial in the canonical variables, they do not break down if the (density weighted) triad $\tilde{E}^{a}_{i}$ were to become degenerate or even vanish. Consequently, unlike the Arnowitt-Deser-Misner formalism, evolution remains well-defined even when the 3-metric becomes degenerate or even vanishes at some points during evolution. This extension of general relativity was studied in some detail (Bengtsson I, Jacobson T, 1997; Bengtsson I, Jacobson T, 1998), and in particular, the causal structure of these generalized solutions has been analyzed (Matschull H J, 1996). These investigations may well be useful in the future analysis of various phases of quantum gravity. Another tantalizing aspect of the connection formulation is that is also leads to a Hamiltonian formulation of general relativity in terms of fields (with density weight 1) on the spatial 3-manifold which have only internal indices (Ashtekar A, Henderson A, Sloan D, 2009). This formulation is particularly well-suited to analyze the behavior of the gravitational field as one approaches space-like singularities. Indeed, there is now a specific formulation of the Belinskii-Khalatnikov-Lifshits (BKL) conjecture in terms of these variables provided by the (self-dual) connection dynamics and numerical simulations have been performed using these variables, exhibiting the conjectured BKL behavior (Ashtekar A, Henderson A, Sloan D, 2009; Ashtekar A, Henderson A, Sloan D, 2011). While other formulations of the BKL conjecture are motivated primarily by considerations involving differential equations, this formulation comes with a Hamiltonian framework and is therefore well suited for the analysis of the fate of generic space-like singularities in LQG. The extension to include degenerate metrics and the Hamiltonian framework involving only fields with internal indices have a strong potential to shed new light on the fate of generic, space-like singularities of general relativity but have remained largely unexplored so far. Finally, the connection dynamics formulation of general relativity, discussed here, has been generalized in several ways: generalizations of Einstein dynamics in four dimensions through interesting deformations of the constraint algebra (Krasnov K, 2007); general relativity in higher space-time dimensions (Bodendorfer N, Thiemann T, Thurn A, 2013) ; and, inclusion of supersymmetries (Bodendorfer N, Thiemann T, Thurn A, 2012). There is considerable ongoing research in these interesting directions. However, these developments fall beyond the charge assigned to me by the Editors. Appendix A: Connection-dynamics: A historical anecdote I will now describe the early attempt by Einstein and Schrödinger, mentioned in section 1, to formulate theories of relativistic gravity in terms of connections rather than space-time metrics.7 The episode is of interest not only for its scientific content but also for sociology of science. In the 1940s, both men were working on unified field theories. They were intellectually very close. Indeed, Einstein wrote to Schrödinger saying that he was perhaps the only one who was not ‘wearing blinkers’ in regard to fundamental questions in science and Schrödinger credited Einstein for inspiration behind his own work that led to the Schrödinger equation. Einstein was at the Institute of advanced Study in Princeton and Schrödinger at the Institute for Advanced Study in Dublin. During the years 1946-47, they frequently exchanged ideas on unified field theory and, in particular, on the issue of whether connections should be regarded as fundamental in place of space-time metrics. In fact the dates on their letters often show that the correspondence was going back and forth with astonishing speed. It reveals how quickly they understood the technical material the other hand sent, how they sometimes hesitated and how they teased each other. Here are a few quotes: The whole thing is going through my head like a millwheel: To take $\Gamma$ [the connection] alone as the primitive variable or the $g$'s [metrics] and $\Gamma$'s ? ... ---Schrödinger, May 1st, 1946. How well I understand your hesitating attitude! I must confess to you that inwardly I am not so certain ... We have squandered a lot of time on this thing, and the results look like a gift from devil's grandmother. ---Einstein, May 20th, 1946 Einstein was expressing doubts about using the Levi-Civita connection alone as the starting point which he had advocated at one time. Schrödinger wrote back that he laughed very hard at the phrase ‘devil's grandmother’. In another letter, Einstein called Schrödinger ‘a clever rascal’. Schrödinger was delighted: ‘No letter of nobility from emperor or king ... could to me greater honor...’. This continued all through 1946. Then, in the beginning of 1947, Schrödinger thought he had made a breakthrough.8 He wrote to Einstein: Today, I can report on a real advance. May be you will grumble frightfully for you have explained recently why you don't approve of my method. But very soon, you will agree with me... ---Schrödinger, January 26th, 1947 Schrödinger sincerely believed that this advance was revolutionary. In his view the ‘breakthrough’ was to drop the requirement that the (Levi-Civita) connection be symmetric, i.e., to allow for torsion. This does not seem to be that profound an idea. But at the time Schrödinger believed it was. In discussions at Dublin, he referred to similar previous attempts by Einstein and Eddington ---in which this symmetry was assumed--- and suggested that they did not work simply because of this strong restriction: I will give you a good simile. A man wants a steed to take a hurdle. He looks at it and says ‘Poor thing, it has four legs, it will be very difficult for him to control all four of them. ... I will teach him in successive steps. I will bind his hind legs together. He will learn to jump on his fore legs alone. That will be much simpler. Later on, perhaps, he will learn it with all four. This describes the situation perfectly. The poor thing, $\Gamma^i_{kl}$, got its hind legs bound together by the symmetry condition, $\Gamma^i_{kl} = \Gamma^i_{lk}$, taking away 24 of its 64 degrees of freedom. The effect was, it could not jump and it was put away as good for nothing. The paper was presented to the Royal Irish academy on January 27th, the day after he wrote to Einstein. The Irish prime minister (the Taoiseach) Eamon de Valera, a mathematician, and a number of newspaper reporters were present. Privately, Schrödinger spoke of a second Nobel prize. The next day, the following headlines appeared: Twenty persons heard and saw history being made in the world of physics. ... The Taoiseach was in the group of professors and students. .. [To a question from the reporter] Professor Schrödinger replied “This is the generalization. Now the Einstein theory becomes simply a special case ...” ---Irish Press, January 28th, 1947 Not surprisingly, the headlines were picked up by New York Times which obtained photocopies of Schrödinger's paper and sent them to prominent physicists --including of course Einstein-- for comments. As Walter Moore, Schrödinger's biographer puts it, Einstein could hardly believe that such grandiose claims had been made based on a what was at best a small advance in an area of work that they both had been pursuing for some time along parallel lines. He prepared a carefully worded response to the request from New York Times: ... It seems undesirable to me to present such preliminary attempts to the public in any form. It is even worse when an impression is created that one is dealing with definite discoveries concerning physical reality. Such communiqués given in sensational terms give the lay public misleading ideas about the character of research. The reader gets the impression that every five minutes there is a revolution in Science, somewhat like a coup d’état in some of the smaller unstable republics. ... Einstein's comments were also carried by the international press. On seeing them, Schrödinger felt deeply chastened and wrote a letter of apology to Einstein. Unfortunately, as an excuse for his excessive claims, he said he had to ‘indulge in a little hot air in my present somewhat precarious [financial] situation’. It seems likely that this explanation only worsened the situation. Einstein never replied. He also stopped scientific communication with Schrödinger for three years. The episode must have been shocking to those few who were exploring general relativity and unified field theories at the time. Could it be that this episode effectively buried the desire to follow-up on connection formulations of general relativity until an entirely new generation of physicists who were blissfully unaware of this episode came on the scene? Acknowledgments In the work summarized here, I profited a great deal from discussions with a large number of colleagues. I would especially like to thank Amitabha Sen, Gary Horowitz, Joe Romano, Ranjeet Tate, Jerzy Lewandowski, Joseph Samuel, Carlo Rovelli and Lee Smolin. This work was supported in part by the NSF grant PHY-1205388 and the Eberly research funds of Penn state. 1 At this point, I should emphasize that the bibliography provided in this article is very far from being exhaustive; these references only provide a point of entry for various topics. For more exhaustive lists, the reader can use the one complied in (Beetle C, Corichi A, 1997) for papers written in the first decade of LQG, and references given in (Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007; Rovelli C, Vidotto F, 2014; Ashtekar A, Reuter M, Rovelli C, 2015) for later developments. 2 Actually, only six of the ten Einstein's equations provide the evolution equations. The other four do not involve time-derivatives at all and are thus constraints on the initial values of $q_{ab}$ and its time derivative $K_{ab}$, the extrinsic curvature. However, if the constraint equations are satisfied initially, they continue to be satisfied at all times. 3 This simplicity is not manifest in geometrodynamics where the dynamical trajectories on the superspace of metrics are not geodesics but rather represent a motion in a potential that is, moreover, non-polynomial in the basic configuration variable. 4 Note that the conjugate momentum is a vector density, and can thus be thought of as the ‘electric field’ in the Yang-Mills terminology. We could have worked with the 1-forms $e_{a}^{I}$ without dualization through $\tilde{\eta}^{ab}$. Here, and also in the 3+1 theory, the dualization is carried out only to bring out the similarity with the traditional Hamiltonian formulations of Yang-Mills theories. 5 At first sight, it may appear that this interpretation requires $G^{\alpha\beta}$ to be non-degenerate since it is only in this case that one can compute the connection compatible with $G^{\alpha\beta}$ unambiguously and speak of null geodesics. However, in the degenerate case, there exists a natural generalization of this notion of null geodesics which is routinely used in Hamiltonian mechanics. 6 In this framework, the lapse naturally arises as a scalar density $\underset{\sim}{N}$ of weight $-1$. It is $\underset{\sim}{N}$ that is the basic, metric independent field. The “geometric” lapse function $N$ is metric dependent and given by $N=\sqrt{q}\underset{\sim}{N}$. Note also that, unlike in geometrodynamics, Newton's constant never appears explicitly in the expressions of constraints, Hamiltonians, or equations of motion; it features only through the expression for $F_{ab}{}^i$ in terms of the connection. 7 For further details and more complete quotes, see e.g. Chapter 11 in (Moore W, 1999). 8 He used only a non-symmetric Levi-Civita connection $\Gamma^i_{kl}$ on space-time as the basic variable and regarded the square-root of the Ricci tensor $R_{ik}$ of the connection as the Lagrangian density. Space-time metric did not even feature in the main equations. The theory was to naturally unify gravitation with electromagnetism. • Ashtekar, A. (1986). New variables for Classical and quantum gravity. Phys. Rev. Lett. 57: 2244-2247. • Ashtekar, A. (1987). A new Hamiltonian formulation of general relativity. Phys. Rev. D 36: 1587-1603. • Ashtekar, A. (1991). Lectures on Non-perturbative Canonical Gravity, Notes prepared in collaboration with R.S. Tate. 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0b8afbcfe19ee7c0	The attempt to reduce the Scientific Process to Computation "For centuries, scientists have attempted to identify and document analytical laws that underlie physical phenomena in nature..." and so goes the rather bombastic opening salvo from a recent Science article titled Distilling Free-Form Natural Laws from Experimental Data by Schmidt and Lipson. It's the kind of work that follows the well-trodden path of the logical positivists who tried to subvert science into a branch of logic. Although the logical positivist took a near fatal beating from Gödel's theorem, there are some who want to keep the dream alive. Or in this case, it is the attempt to reduce the scientific process to a computable process. According to my machine-learning life-line (Dr. Mark Reid), this article represents a huge advance. The article describes an algorithm that deduces analytical equations from the analysis of observations made on several mechanical systems. These guys were able to identify the subtle tweaks needed to let the system find invariants in a reasonable amount of time, a major breakthrough in machine-learning. Yet the paper suggests that these methods can be applied to "all physical laws", which rhetorically suggests the method can be widened to many different branches of science. This, I think is a massive overstatement. Let me explain. Many physics undergraduates cut their teeth on Goldstein's Classical Mechanics, an exhaustive encylopedia of mechanical systems that has served as the standard text of classical mechanics, that slowly builds the formal machinery of mechanics from Newton's equation to the abstract formalism of Lagrangians and Hamiltonians. Goldstein is not a pleasant read. Later on, if sufficiently motivated, they might crack open Feynman Lectures on Physics (a rarity in the science literature in that the book is genuinely fun) where much of the messy guts of mechanics is exposed. But only a few physics undergrads will ever venture onto Lev Landau's slim volume Course of Theoretical Physics: Mechanics where the formal properties of mechanics are properly explained in some 80 terse pages, so terse that I've had to read the book several times. It's the kind of mechanics book where Newton's three laws of motion are not even mentioned. I bring up Landau's "Mechanics" because not many people have studied it and it is there that Landau points out that if you have any system that is deterministic with respect to coordinates and velocity, you will end up with a conserved Langragian, from which you can derive a conserved Hamiltonian. If you look at the systems studied in the Science paper, they are all mechanical systems, and the observed data are coordinates and velocities. If you assume the system is deterministic, then you can be sure that there must be a conserved Lagrangian or Hamiltonian based on the coordinates and velocities. The algorithms identified by Schmidt and Lipson will only work on deterministic mechanical system, which would be obvious to anyone who's studied Landau. Unfortunately, there are not many systems that have such beautiful analytical properties, so it is hard to see how this system can be applied to other systems. For instance, could it work for the Schrödinger equation, the workhorse equation in everything from chemistry to semiconductor physics? The Schrödinger equation is deterministic in a very loose sense, and it is the wave function that is conserved, not the observable probabilities! In biology, we have an incredible amount of data on genomes, on genes, and interaction maps. Unfortuantely, we do not have any equivalent Lagrangians for them. In some ways, this article illustrates one of the points made by the great Canadian philosopher of science, Ian Hacking, that physics was the first science to be developed was no accident. It was because the data for theorizing about planets are the easiest to measure in the natural world. This data came in the form of careful measurements of the motion of the stars and planets, made not originally for science, but for commerical purposes in the need for accurate navigational charts. These precious measurements of planetary motions allowed Kepler and Brahe and Newton to theorize about planetary orbitals and interplanetary forces. Fortunately for them, the forces that dictate planetary orbitals, at least from a non-relativistic approximation, are beautiful determistic systems that, as Landau could well appreciate, could be derived from the coordinates and velocities only.
c5264cfa675cf00f	This Quantum World/Implications and applications/Why energy is quantized From Wikibooks, open books for an open world < This Quantum World‎ \| Implications and applications Jump to: navigation, search Why energy is quantized[edit] Limiting ourselves again to one spatial dimension, we write the time independent Schrödinger equation in this form: {d^2\psi(x)\over dx^2}=A(x)\,\psi(x),\qquad A(x)={2m\over\hbar^2}\Big[V(x)-E\Big]. Since this equation contains no complex numbers except possibly \psi itself, it has real solutions, and these are the ones in which we are interested. You will notice that if V>E, then A is positive and \psi(x) has the same sign as its second derivative. This means that the graph of \psi(x) curves upward above the x axis and downward below it. Thus it cannot cross the axis. On the other hand, if V<E, then A is negative and \psi(x) and its second derivative have opposite signs. In this case the graph of \psi(x) curves downward above the x axis and upward below it. As a result, the graph of \psi(x) keeps crossing the axis — it is a wave. Moreover, the larger the difference E-V, the larger the curvature of the graph; and the larger the curvature, the smaller the wavelength. In particle terms, the higher the kinetic energy, the higher the momentum. Let us now find the solutions that describe a particle "trapped" in a potential well — a bound state. Consider this potential: Potential energy well.svg Observe, to begin with, that at x_1 and x_2, where E=V, the slope of \psi(x) does not change since d^2\psi(x)/dx^2=0 at these points. This tells us that the probability of finding the particle cannot suddenly drop to zero at these points. It will therefore be possible to find the particle to the left of x_1 or to the right of x_2, where classically it could not be. (A classical particle would oscillates back and forth between these points.) Next, take into account that the probability distributions defined by \psi(x) must be normalizable. For the graph of \psi(x) this means that it must approach the x axis asymptotically as x\rightarrow\pm\infty. Suppose that we have a normalized solution for a particular value E. If we increase or decrease the value of E, the curvature of the graph of \psi(x) between x_1 and x_2 increases or decreases. A small increase or decrease won't give us another solution: \psi(x) won't vanish asymptotically for both positive and negative x. To obtain another solution, we must increase E by just the right amount to increase or decrease by one the number of wave nodes between the "classical" turning points x_1 and x_2 and to make \psi(x) again vanish asymptotically in both directions. The bottom line is that the energy of a bound particle — a particle "trapped" in a potential well — is quantized: only certain values E_k yield solutions \psi_k(x) of the time-independent Schrödinger equation:
bbb0063209cc7191	Creating Reality Stephen Gaskin on Energy and Attention • “Within each one of us is a spark of God. Some people call it inborn intelligence: a capacity to look out and see something. That capacity is so strong that if you look at someone and you see something in them that you like, you don’t have to say anything, or give them a bouquet or write them a poem or send them a card. If you just see something in them that you like, that thing will become stronger and it will come out at you; and they will do it more for you.” • “Everybody needs attention — it’s a human requirement, just like oxygen and water. The need for it begins as soon as we’re born, and if we don’t get it in a fair way, we’ll learn outlaw habits of getting it. People will do outrageous things to get attention, because it is life force and energy. The reason to be discriminating about what you give your attention to, is to give real help to a person. That’s how we all be each other’s teachers: what we dig in each other, we reinforce.” • “Paying attention to what we choose to pay it to is probably the greatest freedom we have.” • “Attention is energy. What you put your attention on, you get more of. Each one of us is a fountain of energy, a valve through which universal life energy is metered into the world, and we can each point our self at whatever we want to. We add life force to our surroundings — to everything we pay attention to. If you put your attention on the best, highest, finest, most beautiful thing that you can, that will be amplified.” • “We all control what happens in the future by what we pay attention to in the present. If you perceive it to be improving and a groove, it improves and is a groove.” • “If you see that something should be a way, assume it’s going to be that way.” • “If you but know it, in your highest and your finest and your most honest places in your own heart, God is speaking to you. Even now. All the time, in your highest and finest places.” • “…Rather than figuring it out, and saying, “Is this right?” or “Where would this be in the light of contemporary philosophy?” — that first flash is your best bet. I try to trust myself and trust myself until I can just move on that first flash.” • “If we all moved together in our interaction on that first flash, we would be incredibly fast and smart. If every time you asked a question, the next thing that came back was the answer instead of “Huh?” or if they just said, “I don’t know,” and let you clear the circuit to do the next thing — if we just all answered honestly and correctly the first time, it would be so easy, so incredibly fast and smart — we would just be fabulous.” • “You have to learn to trust your mind — don’t try to force it and push it in various ways. The more you trust it and the more you let it run on its automatic pilot, the faster and smarter and heavier it gets. It lets you out when you trust it. It’s a good one — trust it. • “Any time something is hard for you to do, bring yourself to bear; pay attention to it. Concentrate yourself. Come on to it with all your energy focused. That’s all karate and breaking bricks is — is having all your attention focused when you hit. You can break bricks if your attention is focused. If your attention is not focused and the swing is the same, you might break your hand.” • “One of the reasons for the spiritual practice of non-attachment—trying not to be personally attached about your thing, or pain or whatever happens to you — is so that you school yourself so that nothing can happen to you from the outside that can make you lose your energy, because as long as you have your energy on, you can do it.” • “There isn’t really supposed to be an intermediary between you and God; although some religions teach the necessity of an intermediary. Some religions think of Jesus as a gateman to Heaven — who you have to get straight with before you can go in — instead of being the spiritual vibration itself, which if you are in contact with, you automatically become in contact with Heaven — and if you’re in good enough shape to touch it, it will touch you back.” • “You have to be sure you’re not pretending to don’t be confident so that nobody will think you’re on a trip. Some people go around pretending that they don’t know where it’s at so that nobody will think they’re on a trip, when they do sometimes really know where it’s at. But they don’t really know where it’s at because they pretend not to. If you’re doing a good thing, swing on; get heavy.” • “God is not separate from the Universe. God is only One. The Universe itself is God’s mind; and the flow of everything is God’s thoughts. And praying to us really means to try to be an intelligent synapse in God’s mind, a synapse that is not going to trigger for violence, no matter what. Love, connect. And we affect the mind of God by being free will synapses.” Morphic Resonance and Morphic Fields • The concept of morphic resonance has much in common with the Akashic Record; or quantum physicist David Bohm’s implicate order; or, as Joseph Campbell once suggested, the Hindu concept of maya — the field of space-time that gives birth to the forms of the world. • Excerpt from Morphic Resonance & Morphic Fields: Collective Memory & the Habits of Nature. Rupert Sheldrake writes: • “The fields responsible for the development and maintenance of bodily form in plants and animals are called morphogenetic fields. Morphic Fields and the Implicate Order • Excerpt from Morphic Fields and the Implicate Order: A Dialogue with David Bohm and Rupert Sheldrake. • David Bohm was an eminent quantum physicist. As a young man he worked closely with Albert Einstein at Princeton University. With Yakir Aharonov he discovered the Aharonov-Bohm effect. He was later Professor of Theoretical Physics at Birkbeck College, London University, and was the author of several books, including Causality and Chance in Modern Physics and Wholeness and the Implicate Order. • Bohm: But from the point of view of the implicate order, I think you would have to say that this formative field is a whole set of potentialities, and that in each moment there’s a selection of which potential is going to be realized, depending to some extent on the past history, and to some extent on creativity. • Sheldrake: But this set of potentialities is a limited set, because things do tend towards a particular endpoint. I mean cat embryos grow into cats, not dogs. So there may be variation about the exact course they can follow, but there is an overall goal or endpoint. • Bohm: But there would be all sorts of contingencies that determine the actual cat. • Sheldrake: Exactly. Contingencies of all kinds, environmental influences, possibly genuinely chance fluctuations. But nevertheless the endpoint of the chreode would define the general area in which it’s going to end up. • Bohm: In terms of the totality beyond time, the totality in which all is implicate, what unfolds or comes into being in any present moment is simply a projection of the whole. That is, some aspect of the whole is unfolded into that moment and that moment is just that aspect. Likewise, the next moment is simply another aspect of the whole. And the interesting point is that each moment resembles its predecessors but also differs from them. I explain this using the technical terms ‘injection’ and ‘projection’. Each moment is a projection of the whole, as we said. But that moment is then injected or introjected back into the whole. The next moment would then involve, in part, a re-projection of that injection, and so on in-definitely. Each moment will therefore contain a projection of the re-injection of the previous moments, which is a kind of memory; so that would result in a general replication of past forms, which seems similar to what you’re talking about. • Sheldrake: So this re-injection into the whole from the past would mean there is a causal relation ship between what happens in one moment and what subsequently happens? • Bohm: Yes, that is the causal relation. When abstracted from the implicate order, there seems to be at least a tendency, not necessarily an exact causal relationship, for a certain content in the past to be followed by a related content in the future. • Sheldrake: Yes. So if something happens in one place at one time what happens there is then re-injected into the whole. • Bohm: But it has been somewhat changed; it is not re-injected exactly, because it was previously projected. • Sheldrake: Yes, it is somewhat changed, but it is fed back into the whole. That can have an influence which, since it is mediated by the whole, can be felt somewhere else. It doesn’t have to be local. • Bohm: Right, it could be anywhere. • Sheldrake: Well that does sound very similar to the concept of morphic resonance, where things that happen in the past, even if they’re separated from each other in space and time, can influence similar things in the present, over, through, or across — however one cares to put it — space and time. There’s this non-local connection. This seems to me to be very important because it would mean that these fields have causal (but non-local) connections with things that have happened before. They wouldn’t be somehow inexplicable manifestations of an eternal, timeless set of archetypes. Morphogenetic fields, which give repetitions of habitual forms and patterns, would be derived from previous fields (what you call ‘cosmic memory’). The more often a particular form or field happened, the more likely it would be to happen again, which is what I am trying to express with this idea of morphic resonance and automatic averaging of previous forms. • Bohm: If we extended quantum mechanics through the implicate order, we would bring in just that question of how past moments have an effect on the present (i.e., via injection and re-projection). At present, physics says the next moment is entirely independent, but with some probability of being such and such. There’s no room in it for the sort of thing you’re talking about, of having a certain accumulated effect of the past; but the implicate order extension of quantum mechanics would have that possibility. And further, suppose somehow I were to combine the implicate order extension of quantum mechanics [which would account for the accumulated effects of the past] with this quantum potential [which would account for these effects being non-local in nature], then I think I would get things very like what you are talking about. • Sheldrake: Yes, that would be very exciting! Of all the ways I’ve come across I think that’s the most promising way of being able to mesh together these sort of ideas. I haven’t come across any other way which seems to show such possible connections. • Bohm: If we can bring in time, and say that each moment has a certain field of potentials (represented by the Schrödinger equation) and also an actuality, which is more restricted (represented by the particle itself); and then say that the next moment has its potential and its actuality, and we must have some connection between the actually of the previous moments and the potentials of the next — that would be introjection, not of the wave function of the past, but of the actuality of the past into that field from which the present is going to be projected. That would do exactly the sort of thing you’re talking about. Because then you could build up a series of actualities introjected which would narrow down the field potential more and more, and these would form the basis of subsequent projections. That would account for the influence of the past on the present. • Sheldrake: Yes, yes. In the Presence of the Past • Excerpts from In the Presence of the Past: Interview with Rupert Sheldrake. • INTERVIEWER RMN: Could you give a specific example of, and describe the morphogenetic process in terms of, the development of a well-established species, like a potato, for example? • RS: Well, the idea is that each species, each member of a species draws on the collective memory of the species, and tunes in to past members of the species, and in turn contributes to the further development of the species. So in the case of a potato, you’d have a whole background resonance from past species of potatoes, most of which grow wild in the Andes. And then in that particular case, because it’s a cultivated plant, there’s been a development of a whole lot of varieties of potatoes, which are cultivated, and as it so happens potatoes are propagated vegetatively, so they’re clones. • So each clone of potatoes, each variety, each member of the clone will resonate with all previous members of the clone, and that resonance is against a background of resonance with other members of the potato species, and then that’s related to related potato species, wild ones that still grow in the Andes. So, there’s a whole kind of background resonance, but what’s most important is the resonance from the most similar ones, which is the past members of that variety. And this is what makes the potatoes of that variety develop the way they do, following the habits of their kind. • Usually these things are ascribed to genes. Most people assume that inheritance depends on chemical genes and DNA, and say there’s no problem, it’s all just programmed in the DNA. What I’m saying is that that view of biological development is inadequate. The DNA is the same in all the cells of the potato, in the shoots, in the roots, in the leaves, and the flowers. The DNA is exactly the same, yet these organs develop differently. So something more than DNA must be giving rise to the form of the potato, and that is what I call the morphic field, the organizing field. • An example of how you’d test the theory would depend on looking at some change in the species that hadn’t happened before, a new phenomenon, and seeing how it spreads through the species. So, for example, if you train rats to learn a new trick in one place, then rats of that breed should learn it more quickly everywhere in the world, just because the first ones have learned it. The more that learn it, the easier it should get. • INTERVIEWER DJB: What are morphic fields made of, and how is it that they can exist everywhere all at once? Do they work on a principle similar to Bell’s Theorem? • RS: Well, you could ask the question, what are any fields made of? You know, what is the electromagnetic field made of, or what is the gravitational field made of? Nobody knows, even in the case of the known fields of physics. It was thought in the nineteenth century that they were made of ether. But then Einstein showed that the concept of the ether was superfluous; he said the electromagnetic field isn’t made out of ether, it’s made out of itself. It just is. The magnetic field around a magnet, for example, is not made of air, and it’s not made of matter. When you scatter iron fillings, you can reveal this field, but it’s not made of anything except the field. And then if you say, well maybe all fields have some common substance, or common property, then that’s the quest for a unified field theory. • Then if you say, “Well, what is it that all fields are made of?” the only answer that can be given is space-time, or space and time. The substance of fields is space; fields are modifications of space or of the vacuum. And according to Einstein’s general theory of relativity, the gravitational field, the structure of space-time in the whole universe, is not in space and time; it is space-time. There’s no space and time other than the structure of fields. So fields are patterns of space-time. And so the morphic field, like other fields, will be structures in space and time. They have their own kind of ontological status, the same kind of status as electromagnetic and gravitational fields. • INTERVIEWER DJB: Wait. But those are localized aren’t they? I mean, you sprinkle iron fillings about a magnet, and you can see the field around it. How is it that a morphic field can exist everywhere all at once? • RS: It doesn’t. The morphic fields are localized. They’re in and around the system they organize. So the morphic field of you is in and around your body. The morphic field around a tomato plant is in and around that plant. What I’m suggesting is that morphic fields in different tomato plants resonate with each other across space and time. I’m not suggesting that the field itself is delocalized over the whole of space and time. It’s suggesting that one field influences another field through space and time. Now, the medium of transmission is obscure. I call it morphic resonance, this process of resonating. What this is replacing in conventional physics is the so-called “laws of nature,” which are believed to be present in all places, and at all times. • INTERVIEWER RMN: That leads on to the next question I have about how to use the concept of attractors, as expressed in the current research of dynamical systems, in the theory of formative causation. • RS: Well, the idea of attractors, which is developed in modern mathematical dynamics, is a way of modeling the way systems develop, by modeling the end states toward which they tend. This is an attempt to understand systems by understanding where they’re headed to in the future, rather than just where they’ve been pushed from in the past. So, the attractor, as the name implies, pulls the system towards itself. A very simple, easy-to-understand, example is throwing marbles, or round balls into a pudding basin. The balls will roll round and round, and they’ll finally come to rest at the bottom of the basin. The bottom of the basin is the attractor, in what mathematicians call the basin of attraction. • The basin is, in fact, their principal metaphor. So the ball rolls down to the bottom. It doesn’t matter where you throw it in, or at what speed you throw it in, or by what route it takes–what this model does is tell you where it’s going to end up. This kind of mathematical modeling is extremely appropriate, I think, to the understanding of biological morphogenesis, or the formation of crystals or molecules, or the formation of galaxies, or the formation of ideas, or human behavior, or the behavior of entire societies. Because all of them seem to have this kind of tendency to move towards attractors, which we think of consciously as goals and purposes. But, throughout the natural world these attractors exist, I think, largely unconsciously. The oak tree is the attractor of the acorn. So the growing oak seedling is drawn towards its formal attractor, its morphic attractor, which is the mature oak tree. • INTERVIEWER RMN: So, it is like the future in some sense. • RS: It’s like the future pulling, but it’s not the future. It’s a hard concept to grasp, because what we think of as the future pulling is not necessary what will happen in the future. You can cut the acorn down before it ever reaches the oak tree. So, it’s not as if its future as oak tree is pulling it. It’s some kind of potentiality to reach an end state, which is inherent in its nature. The attractor in traditional language is the entelechy, in Aristotle’s language, and in the language of the medieval scholastics. Entelechy is the aspect of the soul, which is the end which draws everything towards it. • So all people would have their own entelechy, which would be like their own destiny or purpose. Each organism, like an acorn, would have the entelechy of an oak tree, which means this end state — entelechy means the end which is within it — it has its own end, purpose, or goal. And that’s what draws it. But that end, purpose, or goal is somehow not necessarily in the future. It is in a sense in the future. In another sense it’s not the actual future of that system, although it becomes so. • INTERVIEWER RMN: Perhaps the most compelling implication of your hypothesis is that nature is not governed by eternally fixed laws but more by habits that are able to evolve as conditions change. In what ways do you think the human experience of reality could be affected as a result of this awareness? • RS: Well, I think first of all the idea of habits developing along with nature gives us a much more evolutionary sense of nature herself. I think that nature – the entire cosmos, the natural world we live in – is in some sense alive, and that it’s more like a developing organism, with developing habits, than like a fixed machine governed by fixed laws, which is the old image of the cosmos, the old world view. • Second, I think the notion of natural habits enables us to see how there’s a kind of presence of the past in the world around us. The past isn’t just something that happens and is gone. It’s something which is continually influencing the present, and is in some sense present in the present. • Thirdly, it [the notion of natural habits] gives us a completely different understanding of ourselves, our own memories, our own collective memories, and the influence of our ancestors, and the past of our society. And it also gives an important new insight into the importance of rituals, and forms through which we connect ourselves with the past, forms in which past members of our society become present through ritual activity. I think it also enables us to understand how new patterns of activity can spread far more quickly than would be possible under standard mechanistic theories, or even under standard psychological theories. Because if many people start doing, thinking, or practicing something, it’ll make it easier for others to do the same thing. • INTERVIEWER RMN: And the way different discoveries are found simultaneously. • RS: Yes. I mean, that’s another aspect. It will also mean things that some people do will resonate with others, as in independent discoveries, parallel cultural development, etc. Vibrations — Links When You Have the Right Vibe, It’s Not a Coincidence: Synchonicities, Energy Healing, and Other Strangeness in the Field • Excerpted from Active Consciousness: Awakening the Power Within. “One piece of evidence for the holographic nature of nonstandard fields that have been proposed in recent years — the zero-point field (a candidate for the unified field), the psi field of psychic phenomena, Ervin Laszlo’s Akashic field, and the morphic field proposed by Rupert Sheldrake — is that they all share a common feature: sensitivity to similarity in vibration. • “Biologist Rupert Sheldrake’s theory of morphic resonance also depends upon similarity in vibration. Members of the same species, being ‘on the same wavelength’, are able to tap into information that pertains uniquely to them. And while members of an entire species might be able to tune into a fairly broad spectrum of frequencies (think of Carl Jung’s notion of the collective unconscious that humans supposedly tap into), smaller, more tightly connected groups — such as members of the same family or loving couples — resonate in more focused zones of vibration; they have access to their own ‘private frequency.’ In fact, Sheldrake goes even further and suggests that morphic fields can explain how human memory operates. Instead of being stored in our brains, he suggests that memories are stored in the morphic field. Our brains then pick them up via resonance, like radios tuning to their own private stations. • “The reality we experience each day may be flooded with fields of meaning. One field might embody the horror and violence of 9/11. Another field might be associated with a hope for rebirth. Each field of meaning has a particular vibration to it, and objects, individuals, emotions, dreams, and events with similar vibrations will tend to resonate with one another and then co-occur. This is what creates synchronicities.” In Resonance Excerpts from In Resonance: Interview with Rupert Sheldrake. • Roozbeh Gazdar: In his paper, Morphic Fields and Morphic Resonance – An Introduction, Sheldrake has explained, “The morphic fields of mental activity are not confined to the insides of our heads. They extend far beyond our brain through intention and attention. We are already familiar with the idea of fields extending beyond the material objects in which they are rooted… Likewise the fields of our minds extend far beyond our brains.” • INTERVIEWER RG: Would you see your work as being a “scientific” validation of Indian beliefs such as reincarnation, existence of a universal soul, and so on? RUPERT SHELDRAKE: I don’t think my work in itself provides a “scientific” validation of reincarnation. It leads to a theory of collective memory, and leaves open the possibility that sometimes individual memories from one person could be transferred to another in a more specific way. But it raises a new question for the hypothesis for the doctrine of reincarnation. According to my view, memories can be transferred by morphic resonance, but it does not prove that the person who has these memories is the same person as the previous personality whose memories they have access to. Memory transfer does indeed seem to occur, as in the cases studied by Professor Ian Stevenson of children who remember previous lives. But this does not necessarily prove that these cases are ones of reincarnation. They simply show that there has been a transfer of memory. My work would not automatically imply a universal soul. The idea of mophic fields would imply that the entire universe has a field, which could perhaps be taken to correspond to the universal soul. But it would not necessarily imply that the field of the universe was conscious. Most aspects of morphic fields are unconscious, since they organise habits. Most of our own habits take place unconsciously and much of our mind is unconscious. From Cellular Aging to the Physics of Angels Excerpts from Quest Magazine: From Cellular Aging to the Physics of Angels: A Conversation with Rupert Sheldrake Interviewed by John David Ebert • JE: For Rupert Sheldrake, the “laws” of the universe may not in fact be laws at all, but rather deeply ingrained habits of action which have been built up over the many eons in which the universe has spun itself out. Like the ancient riverbeds on the surface of Mars left behind by the pressures of flowing water over billions of years, so too, the “laws” of the universe may be thought of as runnels engraved in the texture of space-time by endless, unchanging repetition. And the longer particular patterns persist, the greater their tendency to resist change. Sheldrake terms this habitual tendency of nature “morphic resonance,” whereby present forms are shaped through the influence of past forms. Morphic resonance is transmitted by means of “morphogenetic fields,” which are analogous to electromagnetic fields in that they transmit information, but differ in that they do so without using energy, and are therefore not diminished by transmission through time or space. Sheldrake illustrates his idea with the analogy of a television set. Though we can alter the images on our screens by adjusting components or distorting them — just as we can alter or distort phenotypical characteristics through genetic engineering — it by no means follows that the images are coming from inside the television set. They are in fact encoded as information coming from electromagnetic frequencies which the skillful arrangement of the transistors and circuits within the television set enables us to pick up and render visible. Likewise, it is not at all necessary for us to assume that the physical characteristics of organisms are contained inside the genes, which may in fact be analogous to transistors tuned in to the proper frequencies for translating invisible information into visible form. Thus, morphogenetic fields are located invisibly in and around organisms, and may account for such hitherto unexplainable phenomena as the regeneration of severed limbs by worms and salamanders, phantom limbs, the holographic properties of memory, telepathy, and the increasing ease with which new skills are learned as greater quantities of a population acquire them. • JE: Joseph Campbell once suggested that the idea of morphogenetic fields reminded him of the Hindu concept of maya — the field of space-time that gives birth to the forms of the world…. You see evolutionary history as a tension between the two forces of habit — -or morphic resonance — and creativity, which involves the appearance of new morphic fields. But in the case of mass extinctions you suggested once that the ghosts of dead species would still be haunting the world, that the fields of the dinosaurs would still be potentially present if you could tune into them. Would you mind commenting on how it might be possible for extinct species to reappear? RS: Well, I haven’t in mind some kind of Jurassic Park scenario. What I was thinking of was that the fields would remain present, but the conditions for tuning into them are no longer there if the species is extinct, so they’re not expressed. However, it’s a well known fact in evolutionary studies that some of the features of extinct species can reappear again and again. Sometimes this happens in occasional mutations, sometimes it turns up in the fossil record. And when these features of extinct species reappear, they’re usually given the name, “atavism,” which implies a kind of throwback to an ancestral form. Atavisms were well known to Darwin, and he was very interested in them for the same reasons I am, that they seem to imply a kind of memory of what went before. • JE: Do you think that morphic fields could account for the existence of ghosts in any way? RS: Well, the fields represent a kind of memory. If places have memories, then I suppose it’s possible for ghostly-type phenomena to be built into their fields. This is a very hazy area of speculation and not one I’ve thought through rigorously. And I’ve had no incentive to think it through rigorously because it’s so hard to think of repeatable experiments with ghosts. But ghosts do seem to be a kind of memory thing, and morphic fields have to do with memory, so there may well be a connection. • JE: Karl Pribram suggests that memories are spread throughout the brain like waves, or holograms, and you go further in suggesting that memories may not be stored in the brain at all, but rather that the brain acts as a tuning device and picks up memories analogously to the way a television tunes in to certain frequencies. Furthermore, you’ve suggested that if memories aren’t stored in the brain at all, this leaves the door open for the possibility of the existence of the soul. Can you explain how your ideas on the existence of the soul fit into this paradigm? RS: Well, we should clarify the terms here. The traditional view in Europe was that all animals and plants have souls — not just people — and that these souls were what organized their bodies and their instincts. In some ways, therefore, the traditional idea of soul is very similar to what I mean by morphic fields. The traditional view of the soul in Aristotle and in St. Thomas Aquinas was not the idea of some immortal spiritual principle. It was that the soul is a part of nature, a part of physics, in the general sense. It’s that which organizes living bodies. In that sense, all morphic fields of plants and animals are like souls. However, in the case of human beings, the additional question arises as to whether it’s possible for the soul to persist after bodily death. Now, normally souls are associated with bodies. And the theory I’m putting forward is one that would see the soul normally associated with the body and memories coming about by morphic resonance. If it’s possible for the soul to survive the death of the body, then you could have a persistence of memory and of consciousness. From the point of view of the theory I’m putting forward, there’s nothing in the theory that says the soul has to survive the death of the body, and there’s nothing that says that it can’t. So this is simply an open question. But it’s not one that can be decided a priori. • JE: In your book The Presence of the Past, you have an interesting theory of reincarnation. You suggest that people who have memories of past lives may actually be tuning in to the memories of other people in the morphogenetic field, and that they may not actually represent reincarnated people at all. Would you care to comment on that? RS: Yes. I’m suggesting that through morphic resonance we can all tune in to a kind of collective memory, memories from many people in the past. It’s theoretically possible that we could tune into the memories of specific people. That might be explained subjectively as a memory of a past life. But this way of thinking about it doesn’t necessarily mean this has to be reincarnation. The fact that you can tune into somebody else’s memories doesn’t prove that you are that person. Again, I would leave the question open. But, you see, this provides a middle way of thinking about the evidence for memories of past lives, for example, that collected by Ian Stevenson and others. Usually the debate is polarized between people who say this is all nonsense because reincarnation is impossible — the standard scientific, skeptical view (I should say, the standard skeptical view; it’s not particularly scientific) — and the other people who say this evidence proves what we’ve always believed, namely, the reality of reincarnation. I’m suggesting that it’s possible to accept the evidence and accept the phenomenon, but without jumping to the conclusion that it has to be reincarnation. • JE: So your theory that information can be transmitted by these nonmaterial morphic fields makes theoretically plausible a paradigm in which phenomena such as telepathy or ESP can be understood. Can you explain how your paradigm makes sense out of this type of phenomena? RS: Well, if people can tune in to what other people have done in the past, then telepathy is a kind of logical extension of that. If you think of somebody tuning in to somebody else’s thought a fraction of a second ago, then it becomes almost instantaneous and approaches the case of telepathy. So telepathy doesn’t seem to be particularly difficult in principle to explain, if there’s a world in which morphic resonance takes place. I think that some of the other phenomena of parapsychology are hard to explain from the point of view of morphic fields and morphic resonance. For example, anything to do with precognition or premonition doesn’t fit in to an idea of influences just coming in from the past. So, I don’t think this is going to give a blanket explanation of all parapsychological phenomena, but I think it’s going to make some of it at least, seem normal, rather than paranormal. A Thinking Person’s Guide to Discovering “God” Excerpts from An Interview with Dr. Rupert Sheldrake: A Thinking Person’s Guide to Discovering “God”. Rupert Sheldrake quotes are in double quotation marks. • “I don’t think many people arrive at a discovery of God through reason alone, but through various kinds of mystical experience, including a sense of divine presence, the experience of transcendent beauty through art, music or nature; visions; psychedelic experiences; meditation; through love and the experience of being loved; through religious rituals and liturgies; spontaneous mystical experiences, and so on. After one or more of these kinds of experiences people may enquiry further and at this stage religious stories, doctrines, liturgies and theology can be a big help.” • “For me it’s an important point about Jesus’ teachings about the kingdom of heaven that one of the primary metaphors is of a wedding feast, a party in which people of all ages are included, and at which people are happy. His first miracle, the turning of water into wine at the wedding feast at Cana in Galilee reinforces this image. The extra wine no doubt made it a better party.” • “The school of theology that makes most sense to me is panentheism, the idea that God is in nature, and nature is in God. The being of God on which all nature depends, and on which our own being depends, is not like that of an emperor or overlord but rather that of something that sustains all things.” • “In other words, I think nature is sustained from moment to moment by the being of God not just made by God in the beginning and then functioning automatically as a mechanistic universe or even as an autonomous living universe. A physical analogy might be the electromagnetic field. This is the ground of all electromagnetic being, including light. The electromagnetic field does not relate to light in the manner of a Roman Emperor or overlord of vassals, but rather as the basis of its very being and activity.” • “If God is light, then God is also the electromagnetic field that is the basis of light, and all the things that we can see through that light. God’s nature or image in the Christian tradition is that of the Holy Trinity: the Father, or the ground of all being; the Son or logos, the source of all form pattern and order, as well as words; and the Spirit the principle of movement, energy, and activity.” • “Light, it seems to me, is one of the main manifestations of the Holy Spirit, along with wind, movement, breath, fire and other energetic processes. So if God is light, God is also that through which we can see the light and interpret it. This is expressed particularly clearly in the Kena Upanishad: ‘What cannot be seen with the eye, but that whereby the eye can see; know that alone to be Brahman, the spirit; and not what people here adore.'” • “Yes, we can block out the light of God or ignore it, and there are many ways in which we do this, perhaps the commonest through a preoccupation with all the things that keep us so busy physically, emotionally and mentally. Even though modern people have more leisure than most people in the past, much of it is filled up with ceaseless activity including entertainment and social media, as well as excessive work.” • “God by definition lies beyond our powers of conception, as the ground of being and the source of all consciousness and activity. We have a variety of available models, and as a Christian the one I found most helpful is the Holy Trinity. There are parallels in other traditions like Satchitananda, as I just mentioned. We cannot explain the diversity and creativity of the world in terms of a single undifferentiated unity, but through a God who already includes a differentiation of being and function.” Consciousness — Links Memory, Morphic Resonance, and the Collective Unconscious • 1 hour-20-minute audio. “When we’re thinking about the nature of consciousness, I’m rather influenced by the Tibetan theory that we have ‘turiya’, the deep sleep state. It’s this state of not blankness but of infinite conscious possibility. People who meditate a lot become conscious within sleep, and that is the kind of state of ultimate, nonbounded consciousness. But normally we’re unconscious of it, but it’s potentially accesible thru consciousness. • “Our waking life is limited to the bodily conditions we’re in, etc. And dreams are somewhere between those two realms: the realm of infinite possibility and the realm of much more limited actuality in our waking life. Dreams have this much greater openness to possibility so they’re closer to the deep sleep state than our waking state and so they have this intermediate quality to them which makes them so interesting and intriguing.” Why NDEs Are Not Hallucinations • Stanislav Grof: “I had my training as a psychiatrist, a physician and then as a Freudian analyst. When I became interested in non-ordinary states and started serving powerful mystical experiences, also having some myself, my first idea was that it (consciousness) has to be hard-wired in the brain. I spent quite a bit of time trying to figure out how something like that is possible. • “Today, I came to the conclusion that it is not coming from the brain. In that sense, it supports what Aldous Huxley believed after he had some powerful psychedelic experiences and was trying to link them to the brain. He came to the conclusion that maybe the brain acts as a kind of reducing valve that actually protects us from too much cosmic input. So, I don’t see, for example, that experiences of archetypal realms, heavens, paradises, experiences of archetypal beings, such as deities, demons from different cultures, that people typically have in these states that they can be somehow explained as something that comes from the brain. I don’t think you can locate the source of consciousness. I am quite sure it is not in the brain not inside of the skull. It actually, according to my experience, would lie beyond time and space, so it is not localizable. You actually come to the source of consciousness when you dissolve any categories that imply separation: individuality, time, space and so on. You just experience it as a presence. • “People who have these experiences can either perceive that source or they can actually become the source, completely dissolved and experience that source.” Is the Sun Conscious? • 36-minute audio clip. “Rupert Sheldrake explores the possibility that the sun and other stars are conscious, as opposed to the usual assumption that they are unconscious and inanimate. This talk was at the Royal Geological Society in London in December 2015, to the Gaia Network.” Collective Unconscious • “In ‘The Significance of Constitution and Heredity in Psychology’ (November 1929), Jung wrote: • ‘And the essential thing, psychologically, is that in dreams, fantasies, and other exceptional states of mind the most far-fetched mythological motifs and symbols can appear autochthonously at any time, often, apparently, as the result of particular influences, traditions, and excitations working on the individual, but more often without any sign of them. These ‘primordial images’ or ‘archetypes,’ as I have called them, belong to the basic stock of the unconscious psyche and cannot be explained as personal acquisitions. Together they make up that psychic stratum which has been called the collective unconscious. • “Jung linked the collective unconscious to ‘what Freud called “archaic remnants” — mental forms whose presence cannot be explained by anything in the individual’s own life and which seem to be aboriginal, innate, and inherited shapes of the human mind’. He credited Freud for developing his ‘primal horde’ theory in Totem and Taboo and continued further with the idea of an archaic ancestor maintaining its influence in the minds of present-day humans. Every human being, he wrote, ‘however high his conscious development, is still an archaic man at the deeper levels of his psyche. • “As modern humans go through their process of individuation, moving out of the collective unconscious into mature selves, they establish a persona — which can be understood simply as that small portion of the collective psyche which they embody, perform, and identify with. • “The collective unconscious exerts overwhelming influence on the minds of individuals. These effects of course vary widely, since they involve virtually every emotion and situation. At times, the collective unconscious can terrify, but it can also heal.” Science Proves that Human Consciousness and our Material World Are Intertwined: See For Yourself • Arjun Walia: “Everything we call real is made of things that cannot be regarded as real.” – Niels Bohr. “The revelation that the universe is not an assembly of physical parts, but instead comes from an entanglement of immaterial energy waves stems from the work of Albert Einstein, Max Planck and Werner Heisenberg, amongst others.”
b898b126881e1540	Skip to Content The Division of Information Technology Research Cyberinfrastructure Scientific Applications A++/P++ A++ and P++ are both C++ array class libraries, providing the user with array objects to simplify the development of serial and parallel numerical codes. A5pipeline A5 is a pipeline for assembling DNA sequence data generated on the Illumina sequencing platform. This README will take you through the steps necessary for running A5. ACML 4.4.0, 5.0.0 ACML provides a free set of thoroughly optimized and threaded math routines for HPC, scientific, engineering and related compute-intensive applications. ACML is ideal for weather modeling, computational fluid dynamics, financial analysis, oil and gas applications and more. ADINA 8.7 The ADINA System offers a one-system program for comprehensive finite element analyses of structures, fluids, heat transfer, electromagnetics and multiphysics. Amber 11 "Amber" refers to two things: a set of molecular mechanical force fields for the simulation of biomolecules (which are in the public domain, and are used in a variety of simulation programs); and a package of molecular simulation programs which includes source code and demos. ampliconnoise-1.25 AmpliconNoise is a collection of programs for the removal of noise from 454 sequenced PCR amplicons. It involves two steps: the removal of noise from the sequencing itself and the removal of PCR point errors. ANTs Advanced Normalization Tools (ANTs) extracts information from complex datasets that include imaging. AUTO-07p AUTO is a software for continuation and bifurcation problems in ordinary differential equations autostem calculate high resolution (atomic or near atomic) conventional and scanning transmission electron microscope (CTEM and STEM) images of thin specimens from first principles using the multislice method for electrons (with simplifying assumptions for the interactive versions) in the energy range of approximately 100 keV to 1OOO keV. BEAGLE an application programming interface (API) and library for high-performance statistical phylogenetic inference - See more at: BEAST 1.6.1 BEAST is a cross-platform program for Bayesian MCMC analysis of molecular sequences. It is entirely orientated towards rooted, time-measured phylogenies inferred using strict or relaxed molecular clock models biom-format-0.9.3 The BIOM format is designed for general use in broad areas of comparative -omics. For example, in marker-gene surveys, the primary use of this format is to represent OTU tables: the observations in this case are OTUs and the matrix contains counts corresponding to the number of times each OTU is observed in each sample. Biopython Biopython is a set of freely available tools for biological computation written in Python by an international team of developers. It is a distributed collaborative effort to develop Python libraries and applications which address the needs of current and future work in bioinformatics. blast 2.2.25/.26/.28 Basic Local Alignment Search Tool is a sequence comparison algorithm optimized for speed used to search sequence databases for optimal local alignments to a query. boost 1.53 Boost provides free peer-reviewed portable C++ source libraries. BWA 0.7.0 BWA is a program for aligning sequencing reads against a large reference genome (e.g. human genome). It has two major components, one for read shorter than 150bp and the other for longer reads. cdbtools-10.11.2010-release CDB (Constant DataBase) indexing and retrieval tools for multi-FASTA files cdhit-3.1-release CD-HIT is a very widely used program for clustering and comparing protein or nucleotide sequences.CD-HIT is very fast and can handle extremely large databases. CD-HIT helps to significantly reduce the computational and manual efforts in many sequence analysis tasks and aids in understanding the data structure and correct the bias within a dataset. chimeraslayer-4.29.2010-release Chimera Slayer involves the following series of steps that operate to flag chimeric 16S rRNA sequences: (A) the ends of a query sequence are searched against an included database of reference chimera-free 16S sequences to identify potential parents of a chimera; (B) candidate parents of a chimera are selected as those that form a branched best scoring alignment to the NAST-formatted query sequence; the NAST alignment of the query sequence is improved in a ‘chimera-aware’ profile-based NAST realignment to the selected reference parent sequences; and (D) an evolutionary framework is used to flag query sequences found to exhibit greater sequence homology to an in silico chimera formed between any two of the selected reference parent sequences. clearcut-1.0.9-release Clearcut is a stand-alone reference implementation of relaxed neighbor joining (RNJ) Clustal W 2.1 Clustal W is a general purpose multiple alignment program for DNA or proteins. Cogent Cogent is a toolkit for statistical analysis of biological sequences. COMSOL 4.3 COMSOL Multiphysics is a finite element analysis, solver and Simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. COMSOL Multiphysics also offers an extensive interface to MATLAB and its toolboxes for a large variety of programming, preprocessing and postprocessing possibilities. CUDA 4.2, 5 CUDA (aka Compute Unified Device Architecture) is a parallel computing platform and programming model created by NVIDIA and implemented by the graphics processing units (GPUs) that they produce CULA R11, R14, sparse CULA™ is an implementation of the Linear Algebra PACK-age (LAPACK) interface for CUDA™-enabled NVIDIA® graphics processing units (GPUs). cytoscape-2.7.0-release Cytoscape is an open source software platform for visualizing complex networks and integrating these with any type of attribute data. delft3d 4.0, 5.0 Delft3D is a world leading 3D modeling suite to investigate hydrodynamics, sediment transport and morphology and water quality for fluvial, estuarine and coastal environments. drisee-1.2-release DRISEE is a tool that utilizes artifactual duplicate reads (ADRs) to provide a platform independent assessment of sequencing error in metagenomic (or genomic) sequencing data. DRISEE is designed to consider shotgun data. exonerate 2.2 exonerate is a generic tool for pairwise sequence comparison.It allows you to align sequences using a many alignment models, using either exhaustive dynamic programming, or a variety of heuristics. FastTree FastTree is open-source software that infers approximately-maximum-likelihood phylogenetic trees from alignments of nucleotide or protein sequences. FastTree can handle alignments with up to a million of sequences in a reasonable amount of time and memory. For large alignments, FastTree is 100-1,000 times faster than PhyML 3.0 or RAxML 7. FSL FSL is a comprehensive library of analysis tools for FMRI, MRI and DTI brain imaging data. gamess GAMESS is a program for ab initio molecular quantum chemistry. A variety of molecular properties, ranging from simple dipole moments to frequency dependent hyperpolarizabilities may be computed. Many basis sets are stored internally, together with effective core potentials or model core potentials, so that essentially the entire periodic table can be considered. gaussian 09 Starting from the fundamental laws of quantum mechanics, Gaussian 09 predicts the energies, molecular structures, vibrational frequencies and molecular properties of molecules and reactions in a wide variety of chemical environments. GCC 4.5.3, 4.6.2, 4.7.0, 4.7.1 The GNU Compiler Collection includes front ends for C, C++, Objective-C, Fortran, Java, Ada, and Go, as well as libraries for these languages (libstdc++, libgcj,...). gg_otus-4feb2011-release GreenGenes OTU picker module for Qiime Globus 5.2.3 The open source Globus® Toolkit is a fundamental enabling technology for the "Grid," letting people share computing power, databases, and other tools securely online across corporate, institutional, and geographic boundaries without sacrificing local autonomy. GMAC 1.1.1 GMAC is a user-level library that implements an Asymmetric Distributed Shared Memory model to be used by CUDA programs. An ADSM model allows CPU code to access data hosted in accelerator (GPU) memory. GPUmat4 GPUmat allows standard MATLAB code to run on GPUs. The engine is written in C/C++ and based on NVIDIA CUDA. hdf5 1.8.4, 1.8.6 HDF5 is a data model, library, and file format for storing and managing data. It supports an unlimited variety of datatypes, and is designed for flexible and efficient I/O and for high volume and complex data. IMa2 8.27.12 an implementation of the MCMC method for the analysis of genetic data under the Isolation with Migration model of population divergence. IMa2 applies this model to genetic data drawn from a pair of closely related populations or species. The results are estimates of the marginal posterior probability densities for each of the model parameters. infernal-1.0.2-release Infernal ("INFERence of RNA ALignment") is for searching DNA sequence databases for RNA structure and sequence similarities. It is an implementation of a special case of profile stochastic context-free grammars called covariance models (CMs). Intel compiler 11.1-059, 11.1-072, 12.1.5, 13.0.1, 13.1.0 The Intel® Composer XE suites are available in several configurations that combine industry leading C, C++ and Fortran compilers, programming models including Intel® Cilk™ Plus and OpenMP*, performance libraries including Intel® Math Kernel Library (Intel® MKL), Intel® Integrated Performance Primitives (Intel® IPP) and Intel® Threading Building Blocks (Intel® TBB) for leadership application performance on systems using Intel® Core™ and Xeon® processors, Intel® Xeon Phi™ coprocessors and compatible processors. Intel Memory Configuration Tool This application assists in finding optimal memory configurations for Intel ® Xeon ® Processor series platforms. Java developer kit 1.6, 1.7 For Java Developers. Includes a complete JRE plus tools for developing, debugging, and monitoring Java applications. jinja2 Jinja2 is a template engine written in pure Python. It provides a Django inspired non-XML syntax but supports inline expressions and an optional sandboxed environment. LAMMPS LAMMPS is a classical molecular dynamics code, and an acronym for Large-scale Atomic/Molecular Massively Parallel Simulator. LAMMPS has potentials for soft materials (biomolecules, polymers) and solid-state materials (metals, semiconductors) and coarse-grained or mesoscopic systems. It can be used to model atoms or, more generically, as a parallel particle simulator at the atomic, meso, or continuum scale. libjacket 1.1 a broad and fast C/C++ library for GPU computing. With over 500 C/C++ functions, LIBJACKET represents the largest GPU computing library in the world. This CUDA-based library integrates seamlessly in any application enabling optimized utilization of NVIDIA CUDA-capable GPUs, including powerful Tesla compute devices. libsvm 3.0 LIBSVM is an integrated software for support vector classification, (C-SVC, nu-SVC), regression (epsilon-SVR, nu-SVR) and distribution estimation (one-class SVM). It supports multi-class classification. magma 1.0.0 MAGMA provides implementations for CUDA, OpenCL, and Intel Xeon Phi. MaSuRCA 2.0.0 MaSuRCA is whole genome assembly software. It combines the efficiency of the de Bruijn graph and Overlap-Layout-Consensus (OLC) approaches. MaSuRCA can assemble data sets containing only short reads from Illumina sequencing or a mixture of short reads and long reads matlab R2008b, R2010b, R1011a, R2012a MATLAB® is a high-level language and interactive environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. metis 4.0 METIS is a set of serial programs for partitioning graphs, partitioning finite element meshes, and producing fill reducing orderings for sparse matrices. mothur 1.29 This project seeks to develop a single piece of open-source, expandable software to fill the bioinformatics needs of the microbial ecology community. It has incorporated the functionality of dotur, sons, treeclimber, s-libshuff, unifrac, and much more. 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MUSCLE can align hundreds of sequences in seconds. mvapich2 MVAPICH2 (MPI-3 over InfiniBand) is an MPI-3 implementation based on MPICH ADI3 layer. NAMD 2.83, 2.9 a parallel molecular dynamics code designed for high-performance simulation of large biomolecular systems. Based on Charm++ parallel objects, NAMD scales to hundreds of processors on high-end parallel platforms and tens of processors on commodity clusters using gigabit ethernet. NAMD is file-compatible with AMBER, CHARMM, and X-PLOR netcdf 3.6.3, 4.1.1 NetCDF is a set of software libraries and self-describing, machine-independent data formats that support the creation, access, and sharing of array-oriented scientific data. NetLogo NetLogo is a programmable modeling environment for simulating natural and social phenomena. Numpy NumPy is the fundamental package for scientific computing with Python. It contains among other things: a powerful N-dimensional array object, sophisticated (broadcasting) functions, tools for integrating C/C++ and Fortran code, and useful linear algebra, Fourier transform, and random number capabilities. Besides its obvious scientific uses, NumPy can also be used as an efficient multi-dimensional container of generic data. Arbitrary data-types can be defined. This allows NumPy to seamlessly and speedily integrate with a wide variety of databases. open64 Open64 has been well-recognized as an industrial-strength production compiler. It is the final result of research contributions from a number of compiler groups around the world. Formerly known as Pro64, Open64 was initially created by SGI from SGI's MIPSPro compiler, and licensed under the GNU Public License (GPL v2). opencurrent 1.1 OpenCurrent is an open source C++ library for solving Partial Differential Equations (PDEs) over regular grids using the CUDA platform from NVIDIA. openFOAM 2.1.1 OpenFOAM is first and foremost a C++ library, used primarily to create executables, known as applications. The applications fall into two categories: solvers, that are each designed to solve a specific problem in continuum mechanics; and utilities, that are designed to perform tasks that involve data manipulation. openMM 3.1.1 OpenMM is a toolkit for molecular simulation. It can be used either as a stand-alone application for running simulations, or as a library you call from your own code. Itprovides a combination of extreme flexibility (through custom forces and integrators), openness, and high performance (especially on recent GPUs) that make it truly unique among simulation codes. openmpi 1.6.1 The Open MPI Project is an open source MPI-2 implementation that is developed and maintained by a consortium of academic, research, and industry partners. Open MPI is therefore able to combine the expertise, technologies, and resources from all across the High Performance Computing community in order to build the best MPI library available. Open MPI offers advantages for system and software vendors, application developers and computer science researchers. Pandaseq PANDAseq assembles paired-end reads rapidly and with the correction of most errors. Uncertain error corrections come from reads with many low-quality bases identified by upstream processing. paraview 3.12 ParaView is an open-source, multi-platform data analysis and visualization application. ParaView users can quickly build visualizations to analyze their data using qualitative and quantitative techniques. The data ParaView was developed to analyze extremely large datasets using distributed memory computing resources. 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ProtTest makes this selection by finding the model in the candidate list with the smallest Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) score or Decision Theory Criterion (DT). At the same time, ProtTest obtains model-averaged estimates of different parameters (including a model-averaged phylogenetic tree) and calculates their importance(Posada and Buckley 2004). ProtTest differs from its nucleotide analog jModeltest (Posada 2008) in that it does not include likelihood ratio tests, as not all models included in ProtTest are nested. Pynast PyNAST is a reimplementation of NAST, introducing new features that increase its portability and flexibility. Its availability as an open source application with three convenient interfaces will allow the application of the NAST algorithm on a wider basis, to larger datasets, and in novel domains. Python 2.7.2, 2.7.3, 2.7.5, 3.2.1, 3.2.3 Python is a remarkably powerful dynamic programming language that is used in a wide variety of application domains. Python is often compared to Tcl, Perl, Ruby, Scheme or Java. Pytz PyTZ is a Python library that allows cross-platform and accurate timezone calculations. Pyzmq Pyzmq provides python bindings for ØMQ and allows you to leverage ØMQ in python applications. qchem 4.0, 4.0.1 Q-Chem is a comprehensive ab initio quantum chemistry package for accurate predictions of molecular structures, reactivities, and vibrational, electronic and NMR spectra. The new release of Q-Chem 4 represents the state-of-the-art of methodology from the highest performance DFT/HF calculations to high level post-HF correlation methods: qhull Qhull computes the convex hull, Delaunay triangulation, Voronoi diagram, halfspace intersection about a point, furthest-site Delaunay triangulation, and furthest-site Voronoi diagram. The source code runs in 2-d, 3-d, 4-d, and higher dimensions. Qhull implements the Quickhull algorithm for computing the convex hull. It handles roundoff errors from floating point arithmetic. It computes volumes, surface areas, and approximations to the convex hull. Qiime QIIME is an open source software package for comparison and analysis of microbial communities, primarily based on high-throughput amplicon sequencing data (such as SSU rRNA) generated on a variety of platforms, but also supporting analysis of other types of data (such as shotgun metagenomic data). qrupdate Library for updating of QR and Cholesky decompositions quantum espresso 4.3.2, 5.0.2 An integrated suite of Open-Source computer codes for electronic-structure calculations and materials modeling at the nanoscale. It is based on density-functional theory, plane waves, and pseudopotentials. QUAST 2.2 QUAST performs fast and convenient quality evaluation and comparison of genome assemblies. r-2.151, 3.0.1 R is a system for statistical computation and graphics. It consists of a language plus a run-time environment with graphics, a debugger, access to certain system functions, and the ability to run programs stored in script files. a raxml-7.3.0-release RAxML is a fast implementation of maximum-likelihood (ML) phylogeny estimation that operates on both nucleotide and protein sequence alignments. rdpclassifier-2.2-release The RDP Classifier is a naive Bayesian classifier that can rapidly and accurately provides taxonomic assignments from domain to genus, with confidence estimates for each assignment Roche 454 (2 versions) Analysis software for 454 Sequencing ROMS 3.6 ROMS illustrates various computational pathways: standalone or coupled to atmospheric and/or wave models. rtax-0.981-release RTAX is specifically designed for assigning taxonomy to paired-end reads, but additionally works on single-end reads SPAdes 2.5.0 SPAdes – St. Petersburg genome assembler – is intended for both standard isolates and single-cell MDA bacteria assemblies. sphinx-1.0.4-release Sphinx is a tool that makes it easy to create documentation for Python projects suitesparse Mathematic packages for Matlab and Metis sunstudio 12.1 Sun Studio software includes compilers, libraries, and tools for application development in C, C++, and Fortran, on Solaris, OpenSolaris, and Linux systems, on SPARC and x86 platforms. terachem TeraChem is general purpose quantum chemistry software designed to run on NVIDIA GPU architectures under a 64-bit Linux operating system. Tophat TopHat is a fast splice junction mapper for RNA-Seq reads. It aligns RNA-Seq reads to mammalian-sized genomes using the ultra high-throughput short read aligner Bowtie, and then analyzes the mapping results to identify splice junctions between exons. Tornado Tornado is a Python web framework and asynchronous networking library trilinos 11.0 The Trilinos Project is an effort to develop and implement robust algorithms and enabling technologies using modern object-oriented software design, while still leveraging the value of established libraries such as PETSc, Metis/ParMetis, SuperLU, Aztec, the BLAS and LAPACK. 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Trinity partitions the sequence data into many individual de Bruijn graphs, each representing the transcriptional complexity at at a given gene or locus, and then processes each graph independently to extract full-length splicing isoforms and to tease apart transcripts derived from paralogous genes. turbomole TURBOMOLE has been designed for robust and fast quantum chemical applications. uclust-1.2.22-release Extreme High-speed Sequence Clustering, Aligment and Database Search vienna-1.8.4-release A package for RNA secondary structure prediction and comparison visit 2.4.0, 2.5.2, 2.6.1 VisIt is a free interactive parallel visualization and graphical analysis tool for viewing scientific data on Unix and PC platforms. Users can quickly generate visualizations from their data, animate them through time, manipulate them, and save the resulting images for presentations. VisIt contains a rich set of visualization features so that you can view your data in a variety of ways. It can be used to visualize scalar and vector fields defined on two- and three-dimensional (2D and 3D) structured and unstructured meshes. VisIt was designed to handle very large data set sizes in the terascale range and yet can also handle small data sets in the kilobyte range. wolfram mathematica 8 Mathematica 8 introduces free-form linguistic input—a whole new way to compute. Enter plain English; get immediate results—no syntax required. It's a new entry point into the complete Mathematica workflow, now upgraded with 500 additional functions and 7 application areas—including the world's most advanced statistics capability and state-of-the-art image processing. zephyr 2.0.3 The current release of OpenMM Zephyr enables acceleration of molecular dynamics on specific NVIDIA and ATI GPU cards and operating systems.
1b19c9957081938f	Keywords \| • Physics Quantum entanglement Quantum entanglement is a fundamental phenomenon in quantum mechanics discovered by Einstein and Schrödinger in the 1930s. Two physical systems, such as two particles, are found to be in a quantum state in which they form a single system in a certain subtle way. Any measurements on one of the systems will affect the other irrespective of the distance between them. Before entanglement, two non-interacting physical systems are in independent quantum states, but after entanglement these two states are in a way "entangled" and it is no longer possible to describe them independently. This is why, as indicated above, non-local properties appear and a measurement on one system instantly influences the other system, even at a distance of light-years. The entanglement phenomenon is one of the most disturbing in quantum mechanics and is the basis of its Copenhagen interpretation Quantum entanglement is at the heart of the famous experiments known as the EPR paradox and Schrödinger's cat or Wigner's friend. The entanglement phenomenon is based on the mathematical and physical principles of quantum mechanics. That is to say the notions of state vectors and tensor products of these state vectors on the one hand, and the principles of superposition of states and reduction of the state vector on the other hand. Remember that in quantum mechanics, which is the extension of Heisenberg's matrix mechanics and Schrödinger's wave mechanics, there is a complete reworking of the kinematics and dynamics of the physical and mathematical quantities associated with observable phenomena and physical systems. Quantum mechanics, even though it deals with wave-particle duality, is not a theory that can be reduced to particle wave mechanics. The dual nature of matter and light shown in the case of charged particle theory and electromagnetic radiation theory is only a consequence of the reworking of the differential and integral laws associated with physical phenomena and a physical system. The introduction of the concept of wave function for a particle is then only a very special case of the introduction of the concept of state vector for a physical system with dynamic variables giving rise to a measurable phenomenon, whatever this system and these variables, as long as a notion of energy and interaction between this system and a classical measuring instrument exists. It is because the differential and integral laws describing the change in space and time of an observable quantity in classical physics naturally have the form of the kinematic laws of a discrete or continuous set of material points, that correspondences are found between the general quantum formulation of these laws and the quantum laws of electrons and photons. It is important to remember that in classical physics already a phenomenon is measured and defined from the modification in the kinematic and dynamic state of a particle of the material being tested. An electromagnetic field is defined by its effect on a charged test particle of matter at a point in space and therefore, in particular, a field of light waves. Temperature can also be defined by the dilation of a material body at one point, and here too, an observable quantity is, in the last analysis, defined by the kinematics of a material point and the sum total of the energy and momentum exchanges. The solution to the wave-particle duality problem therefore lies in the two central ideas in the Copenhagen interpretation and quantum mechanics in the form given by Dirac, Von Neumann and Weyl from the work of Bohr, Heisenberg and Born. -in nature there is fundamentally neither wave nor corpuscle in the classical sense. These concepts are only useful and are still used in the theory because they must necessarily establish a correspondence between the form of the quantum laws and the form of classical laws that must emerge from the former. Just as a test particle serves to define an electromagnetic field, a classical measuring instrument serves to define a quantum system by the way in which this quantum system will affect the measuring instrument. Inevitably, the kinematic and dynamic description of this instrument will involve the classical wave and particle concepts. Quantum formalism must therefore express both all of this and the fundamental non-existence of the classical particle and wave, just as relativity is based on the non-existence of absolute space and time. This property of formalism is largely satisfied by the Heisenberg inequalities. -the wave-corpuscle duality is not derived from any subtle association of particles and waves, i.e. there are no special laws restricted to the laws of motion and the structure of particles of matter and to the waves of interaction fields (electromagnetic, nuclear etc.), but there are laws of change in time and space of any physical quantity which are modified, and in particular the general form of a differential law and an integral law. It is because this framework is quantified that it necessarily applies to any physical system at all. It is very important to remember in this connection that the existence of an energy is an essential property in all the laws of physics. The universality of energy and the fact that any definition of the measurement of a phenomenon is based, in the final analysis, on an interaction with energy automatically ensures that the laws of quantum mechanics apply when describing the change in any arbitrary system. This is why wave mechanics, which finally is based largely on the existence noted by de Broglie of a strong analogy between Maupertuis' principle for the motion of a particle of matter and Fermat's principle for a light beam, is merely a very special case of quantum mechanics since the latter does not finally apply to the laws governing the motion of particles in space and time but to the change in all directly or indirectly measurable physical quantities. In particular, the laws of quantum mechanics naturally contain the possibility of creating or destroying a particle and of its transformation into another particle, which is not a phenomenon that can be described using the Fermat or Maupertuis principles. The construction and form of quantum theory are thus based on the ideas that: -the laws of physics do not fundamentally apply to something in space and time. -particles and waves are not fundamental structures but approximations of the form of the laws and objects of the physical world. -energy is at the heart of the quantification process and ensures/explains the universal character of quantification (the quantification of certain classical dynamic variables, probability amplitudes for observing these values). However, the laws of quantum mechanics emerged historically and can be introduced for teaching purposes as a first approximation with the wave and matrix mechanics of particles in classical space and time. But it is central to understand as quickly as possible that these mechanics are not the true structure of quantum mechanics. The way we proceed is reminiscent of thermodynamics which functions independently of whether or not the physical system has any atomic structure. The total energy of the system, called an equation of state of the system, is considered and there is a set of fundamental variables called variables of state related by the energy function and other equations of state of the thermodynamic system. The system is defined as a black box (what is inside is not important) and only the sum totals of input and output energy and the values of the variables of state are measured. Nevertheless, quantum mechanics does achieve a synthesis of the wave and corpuscular structure for the change of physical values. In particular, this means that the physics and mathematics of waves and fields must appear in the form of these laws such that, when they are applied to particular systems such as classical electrons, protons and electromagnetic fields, we find the wave mechanics of these systems. Thus the principle of the superposition of fields in electrodynamics and optics must reappear to describe the state of a quantum system. The entire structure of Fourier analysis must especially be present. Similarly, the structure of analytical mechanics with the Hamiltonian function of the energy of a classical mechanical system must be kept and play a central role. Bearing in mind the above considerations, the way in which quantum mechanics is constructed starts to become clear. The observable variables Ai and a total energy H called the Hamiltonian are associated with a physical system. In the case of a particle having momentum variables Pi and position variables Qi placed in a potential V(Qi), the function H of the particle is written: where T(Pi) is the kinetic energy of the particle. In its initial form, the Schrödinger equation for such a particle involved an object called an energy operator H, derived from the previous function, and giving rise to a differential equation for a function Ψ (Qi) called the wave function of which the square gives the probability of measuring the particle with the value Qi of its position. The formulation of quantum mechanics makes use of all this and generalises it. We still have an energy operator H but the wave function is merely a special case of the state vector (think of thermodynamics) of any physical system. To clearly show the departure from the concept of wave function this vector is denoted by Ι Ψ >. This is Dirac's vector notation for introducing Fourier analysis abstracted from Hilbert's functional analysis for linear partial differential equations. An observable dynamic variable A, transcribed in the form of a linear operator A, can then have a series of values an during a measurement. Experience shows that there is a probability IcnI 2 of observing each value an, and that the state vector of the system is written as a vector sum of the base vectors associated with each value an such that: Ι Ψ > = ∑ cn Ι an > ∑ ΙcnΙ 2=1 with n=1,2 .... as required for introducing probabilities. The base vectors Ι an > and the values an are called the eigenvectors and the eigenvalues of the linear operator A . It is in this sense that we speak of a superposition of states in quantum mechanics. The coefficients cn are complex numbers of which the square gives the probability of finding the system in the state cn Ι an > of its dynamic variable A. This can be the position, the speed and quantum state variable that can be associated to express the characteristics of the system. In the case of electrons, the phenomena of diffraction and interference which they display depend precisely on this principle of superposition of states applied to their states of position. Except that it is not a question of a series of discrete values xn for Q1=x=A1 but a continuous distribution. It is also for this reason that, generally speaking we refer to probability amplitudes for cn by analogy with light waves where the square of an amplitude gives the intensity of the light at a given point. Schrödinger's equation in its general form is then an equation of change written: (ih/2π) d Ι Ψ >/dt = H Ι Ψ > If we have correctly understood the long arguments developed above we should not be surprised that as soon as we can define an energy and physical variables for any system, Schrödinger's equation above will apply and is absolutely not confined to notions of the change in space and time of a particle in a potential. In particular, if the system were a quantum animal that could take the form either of a quantum whale or a quantum dolphin, in the sense where there would be two energy states for the same physical system, such as a quantum aquatic mammal, Schrödinger's equation would apply! And this is what happens in neutrino or K meson oscillation phenomena, and also in the multiplets of isospin such as quarks and leptons in the electroweak theory and in QCD. It is clear that this has nothing to do with notions of wave-corpuscle duality and wave mechanics. During a measurement the state vector makes a quantum jump to now consist only of I an >. By analogy with a superposition of plane waves in a wave packet, we speak of reduction of the wave packet for the wave function of the position of a particle and, generally speaking, of reduction of the state vector for a quantum system. With these fundamental notions in mind, we can study the phenomenon of entanglement in somewhat more detail. Consider a simple quantum system, a quantum coin in a game of quantum heads or tails. The base state vectors will be Ι f > and Ι p > for heads and tails. The coin can be in a state of quantum superposition such that its state vector is: Ι Ψ > =c1 Ι f >+c2 Ι p > where Ic2I2 will give the probability of observing the coin in the state of heads, for example. If we use two coins A and B; we will then have two state vectors: Ι ΨA > = c1a Ι fa >+c2a Ι pa > et Ι ΨB > =c1b Ι fb >+c2b Ι pb > The two coins are considered as initially having no interactions, which means that we will have two independent Hamiltonians Ha and Hb. Let H be the Hamiltonian of the system made up of these two coins and I psi > its state vector. Then H =Ha+Hb and the state vector of the complete system and the most general form of the solution to the Schrödinger equation is a rather special product called a tensor product (χ) of the state vectors of each coin. Ι Ψ > = ( c1a Ι fa >+c2a Ι pa > ) (χ) ( c1b Ι fb >+c2b Ι pb > ) = c1a c1b Ι fa > (χ) Ι fb >+c1a c2b Ι fa > (χ) Ι pb >+ c2a c1b Ι pa > (χ) Ι fb >+ c2a c2b Ι pa > (χ) Ι pb > This is just the abstract re-transcription of the technique of separation of variables in a partial derivative equation. If the Hamiltonian can no longer be broken down into a sum of Hamiltonians of coins with no interactions, during a brief instant when the coins might be electrically charged for example, the state vector of the system can no longer be described exactly as a tensor product of the state vectors of its parts with no interactions. And this is exactly what we call the entangled state! But this requires some important explanations. The state vector is always the sum of tensor products of the base states, heads or tails of a coin with no interaction, but the coefficients giving the amplitudes of the probabilities of finding the results of observations of the two coins can no longer be broken down into products of the amplitudes of the states of each coin before interaction, i.e. entanglement. If the two entangled coins are separated and transported to opposite antipodes, a measurement on one will instantly affect the quantum state of the other. This means that the results of measurements on the second coin will no longer be independent of measurements made on the first one. The EPR paradox and Bell's inequalities are essentially based on an analogous situation with physical systems formally giving rise to the same mathematical equations. Here we see the full power of the abstract formulation of quantum theory, and above all the nature of quantum theory itself, in the sense that general principles are at work in a large variety of different physical systems and they result in mathematical equations that are largely independent of the form and of the physical system and of the physical variables of the system. So that if we wish to analyse any given quantum phenomenon, the principles of quantum mechanics can be tested with the physical system and the type of dynamic variable that are the easiest to produce experimentally. And indeed, the EPR paradox was initially formulated with variables of position and momentum for a pair of particles. But it keeps its essential meaning if we take the spin variables of a pair of particles, whether they be electrons or photons, for example. This is the reason why David Bohm proposed to test the paradox in this form, and this is what Alain Aspect did in 1982 with a pair of entangled polarised photons. Fill out my online form.
5040eb84e5d7a4f9	$^{95}_{41}$Nb $\beta^-$-decays to the first excited state of $^{95}_{42}$Mo. This state has an excitation energy of 768 keV and the de-excitation to the ground state goes via photon emission or internal conversion. The spin and parity of $^{95}_{41}$Nb and $^{95}_{42}$Mo can be determined from the odd proton/neutron to be $9/2^+$ and $5/2^+$ respectively (I used the energy levels from Shell model). How can I determine spin and parity of the excited state? I also want to calculate the energy for the internal conversion electron. I suppose that it is not simply the excitation energy. Computing the excitation spectrum for a nucleus — that is, its energy levels and their quantum numbers — is hard. Consider that the Schrödinger equation has no known exact solutions for atoms other than hydrogen: a helium atom, with three charges instead of two, is too complex to be treated except in approximation. The nuclear many-body problem is much thornier. Not only are there more participants in the system (ninety-five, for $^{95}$Nb), but in addition to the electrical interaction among the protons you have the pion-mediated attractive strong force, the rho- and omega-mediated hard-core repulsion, three-body forces, etc. (I'm impressed that you got the correct spins and parities for the ground states from the shell model; nice work!) So when normal people want to know the spin and parity and energy of a nuclear state, we look it up. The best source is the National Nuclear Data Center hosted by Brookhaven National Lab, which maintains several different databases of nuclear data (each with its own steep learning curve). Searching the Evaluated Nuclear Structure Data File by decay for $^{95}$Nb brings up level schemes, with references and lots of ancillary data, for both niobium and molybdenum. These confirm that you've gotten the $J^P$ for the ground states correct. Two excited states are listed for $^{95}$Mo: one at $200\rm\, keV$ with $J^P = 3/2^+$, and the one you mention at $766\rm\,keV$ with $J^P=7/2^+$. You can follow the references to see the experimental arguments for assigning those spins and parities. You can make some general, hand-waving predictions about spins by thinking about angular momentum conservation in the transitions. The matrix element for a particular transition is generally proportional to the overlap between the initial wavefunction and the final wavefunction. In nuclear decays the initial state is the nucleus, which is tiny and more-or-less spherical with uniform density, while the final state includes the daughter nucleus and the wavefunctions for the decay products. If the decay products carry orbital angular momentum $\ell$, the radial part of the wavefunction goes like $r^\ell$ near the origin. Dimensional analysis then says that the overlap between the nucleus and the decay wavefunction is proportional to $(kR)^\ell$, where $R$ is the nuclear radius and $k = p/\hbar = 2\pi/\lambda$ is the wavenumber of the decay product. (Note that nuclear decay products typically have $\lambda \gg R$, so you can treat the decay product wavefunction as roughly uniform averaged over the nucleus.) If the probability of a decay is proportional to $(kR)^\ell$, that means that • decays where the product's momentum $p=\hbar k$ is large are preferred over decays where the product's momentum is small • decays where the orbital angular momentum $\ell$ is small are preferred over decays where $\ell$ is large For your $\rm Nb\to Mo$ transition, the decay to the excited state is preferred over the $\frac92\to\frac52$ ground-state-to-ground-state transition, which suggests that the excited state probably has spin $\frac72, \frac92, \frac{11}2$. The most probable of these is $\frac72$, since angular momentum tends to relax during decay processes — and indeed, that's the spin of the $766\rm\,keV$ excited state. Your Answer
d5ab206dd4f34e38	The following comment by Wildcat made me think about whether density functional theory (DFT) can be considered an ab initio method. @Martin-マーチン, this is sort of nitpicking, but DFT (where the last "T" comes from "Theory") can be considered as an ab-initio method since the theory itself is built from the first principles. The problem with the theory is that the exact functional is unknown, and as a result, in practice we do DFA calculations ("A" from "Approximation") with some approximate functional. It it DFA which is not an ab-initio method then, not DFT. :) I always thought that ab initio refers to wave function based methods only. In principle the wave function is not necessary for the basis of DFT, but it was later introduced by Kohn and Sham for practical reasons. The IUPAC goldbook offers a definition of ab initio quantum mechanical methods: ab initio quantum mechanical methods Synonym: non-empirical quantum mechanical methods Methods of quantum mechanical calculations independent of any experiment other than the determination of fundamental constants. The methods are based on the use of the full Schroedinger equation to treat all the electrons of a chemical system. In practice, approximations are necessary to restrict the complexity of the electronic wavefunction and to make its calculation possible. According to this, most density functional approximations (DFA) cannot be termed ab initio since almost all involve some empirical parameters and/or fitting. DFT on the other hand is independent of any of this. What I have my problems with is the second sentence. It states, that treatment of all electrons is necessary. This is technically not the case for DFT, because here only the electron density is treated. All electrons and the wavefunction are implicitly treated. An earlier definition of ab initio can be found in Leland C. Allen and Arnold M. Karo, Rev. Mod. Phys., 1960, 32, 275. By ab initio we imply: First, consideration of all the electrons simultaneously. Second, use of the exact nonrelativistic Hamiltonian (with fixed nuclei), $$\mathcal{H} = -\frac12\sum_i{\nabla_i}^2 - \sum_{i,a}\frac{Z_a}{\mathbf{r}_{ia}} + \sum_{i>j}\frac{1}{\mathbf{r}_{ij}} + \sum_{a,b}\frac{Z_aZ_b}{\mathbf{r}_{ab}}$$ the indices $i$, $j$ and $a$, $b$ refer, respectively, to the electrons and to the nuclei with nuclear charges $Z_a$, and $Z_b$. Third, an effort should have been made to evaluate all integrals rigorously. Thus, calculations are omitted in which the Mulliken integral approximations or electrostatic models have been used exclusively. These approximate schemes are valuable for many purposes, but present experience indicates that they are not sufficiently accurate to give consistent results in ab initio work. This definition obviously does not include DFT, but this is probably due to the fact it was published before the Hohenberg-Kohn theorems. But in general this definition is still largely the same as in the goldbook. Another point which confuses me are titles like: "Potential Energy Surfaces of the Gas-Phase SN2 Reactions $\ce{X- + CH3X ~$=$~ XCH3 + X-}$ $\ce{(X ~$=$~ F, Cl, Br, I)}$: A Comparative Study by Density Functional Theory and ab Initio Methods" Liqun Deng , Vicenc Branchadell , Tom Ziegler, J. Am. Chem. Soc., 1994, 116 (23), 10645–10656. And then again we have titles like: "Ab Initio Density Functional Theory Study of the Structure and Vibrational Spectra of Cyclohexanone and its Isotopomers" F. J. Devlin and P. J. Stephens, J. Phys. Chem. A, 1999, 103 (4), 527–538. Unfortunately Koch and Holthausen, who wrote the probably most concise book on DFT, A Chemist's Guide to Density Functional Theory, never really refer to DFT as ab initio or clearly draw the line. The closest they come is on page 18: In the context of traditional wave function based ab initio quantum chemistry a large variety of computational schemes to deal with the electron correlation problem has been devised during the years. Since we will meet some of these techniques in our forthcoming discussion on the applicability of density functional theory as compared to these conventional techniques, we now briefly mention (but do not explain) the most popular ones. But that does not really answer my question. Throughout the book they use the term only in the form of conventional ab initio theory or in combination of explicitly stating wave function and variations thereof. In my quite extensive research about DFT selection criteria I never came about the term 'ab initio DFT'. So the question remains: Is density functional theory an ab initio method? First note that the acronym DFA I used in my comment originates from Axel D. Becke paper on 50 year anniversary of DFT in chemistry: Let us introduce the acronym DFA at this point for “density-functional approximation.” If you attend DFT meetings, you will know that Mel Levy often needs to remind us that DFT is exact. The failures we report at meetings and in papers are not failures of DFT, but failures of DFAs. Axel D. Becke, J. Chem. Phys., 2014, 140, 18A301. So, there are in fact two questions which must be addressed: "Is DFT ab initio?" and "Is DFA ab initio?" And in both cases the answer depend on the actual way ab initio is defined. • If by ab initio one means a wave function based method that do not make any further approximations than HF and do not use any empirically fitted parameters, then clearly neither DFT nor DFA are ab initio methods since there is no wave function out there. • But if by ab initio one means a method developed "from first principles", i.e. on the basis of a physical theory only without any additional input, then • DFT is ab initio; • DFA might or might not be ab initio (depending on the actual functional used). Note that the usual scientific meaning of ab initio is in fact the second one; it just happened historically that in quantum chemistry the term ab initio was originally attached exclusively to Hartree–Fock based (i.e. wave function based) methods and then stuck with them. But the main point was to distinguish methods that are based solely on theory (termed "ab initio") and those that uses some empirically fitted parameters to simplify the treatment (termed "semi-empirical"). But this distinction was done before DFT even appeared. So, the demarcation line between ab initio and not ab initio was drawn before DFT entered the scene, so that non-wave-function-based methods were not even considered. Consequently, there is no sense to question "Is DFT/DFA ab initio?" with this definition of ab initio historically limited to wave-function-based methods only. Today I think it is better to use the term ab initio in quantum chemistry in its more usual and more general scientific sense rather then continue to give it some special meaning which it happens to have just for historical reasons. And if we stick to the second definition of ab initio then, as I already said, DFT is ab initio since nothing is used to formulate it except for the same physical theory used to formulate HF and post-HF methods (quantum mechanics). DFT is developed from the quantum mechanical description without any additional input: basically, DFT just reformulates the conventional quantum mechanical wave function description of a many-electron system in terms of the electron density. But the situation with DFA is indeed a bit more involved. From the same viewpoint a DFA method with a functional which uses some experimental data in its construction is not ab initio. So, yes, DFA with B3LYP would not qualify as ab initio, since its parameters were fitted to a set of some experimentally measure quantities. However, a DFA method with a functional which does not involve any experimental data (except the values of fundamental constants) can be considered as ab initio method. Say, a DFA using some LDA functional constructed from a homogeneous electron gas model, is ab initio. It is by no means an exact method since it is based on a physically very crude approximation, but so does HF from the family of the wave function based methods. And if the later is considered to be ab initio despite the crudeness of the underlying approximation, why can't the former be also considered ab initio? • $\begingroup$ I read and referred to that paper, too. I love that quote. I agree with you. (I still leave the accepting part for next week, to encourage more people to vote.) Would you say that the IUPAC definition should be updated, i.e. include the electron density explicitly in the last sentence? $\endgroup$ – Martin - マーチン Jul 9 '15 at 12:13 • $\begingroup$ @Martin-マーチン, it depends on how do you interpret this second sentence. In fact, I do not see any problem here with respect to DFT being ab initio method in accordance with this definition. Do we use the Schrödinger equation to treat the electrons? Yes, we do; rather indirectly, but we use it. We don't solve the Schrödinger equation, but we use it a starting point in the development of DFT: the key constituent, electron density, is defined in terms of the solution of the Schrödinger equation. $\endgroup$ – Wildcat Jul 9 '15 at 12:26 • 1 $\begingroup$ @Martin-マーチン, now, for the second part of this sentence that insists on treating "all the electrons", I again see no problem with DFT. We indeed treat all the electrons. Yes, we do so in a tricky way: throughout all the process we use only one-electron density without constructing any many-electron entity that describes our many-electron system as a whole (not like in HF where we construct the many-electron wave function out of one-electron function). But we do so because we already proved that many-electron systems can be treated in such a way: one-electron density is enough. $\endgroup$ – Wildcat Jul 9 '15 at 12:30 • $\begingroup$ Yes, I had no problem with the second one, since HK clarifies that quite nicely. I was referring to the last sentence with limitations, considering that for example B3LYP is not ab initio. $\endgroup$ – Martin - マーチン Jul 9 '15 at 12:37 • $\begingroup$ @Martin-マーチン, got it. So, let us say that DFT is ab initio. Now the situation with DFA is indeed a bit more involved. From the same viewpoint a DFA method with a functional which uses some experimental data in its construction is not ab initio. So, yes, DFA with B3LYP would not qualify as ab initio, since its parameters were fitted to a set of some experimentally measure quantities. $\endgroup$ – Wildcat Jul 9 '15 at 13:00 The convention used by many is that ab initio refers solely to wave-function based methods of various sorts and that first principles refers to either wave-function or DFT methods with little approximation. I can't find a citation at the moment, but I know this convention is fairly widely used in, e.g., J. Phys. Chem. journals. The IUPAC gold book doesn't have "first principles," but Google Scholar gives over 224,000 hits for "first principles DFT". • 3 $\begingroup$ In physics, "from first principles" is a synonym for "ab initio"; just an English equivalent of the Latin phrase. In quantum chemistry we habitually avoid using "ab initio" term with DFT since it still can potentially cause some needless terminological battles due to historical strict meaning of "ab initio" term. $\endgroup$ – Wildcat Jul 9 '15 at 15:41 • 1 $\begingroup$ But you're perfectly right, of course. In QC the convention is to avoid calling DFT methods "ab initio" and exclusively use "from first principles" term for them, while both terms can be used for wf-based methods. $\endgroup$ – Wildcat Jul 9 '15 at 15:59 Your Answer
aff44113b91222cb	The objective of this project is to design and develop a cost-effective commercial fire detector system that monitors multiple characteristics of fires/flames in residential environments for a zero error early fire detection and identification system. The detector integrates existing current state of the art smoke detectors with an intelligent wireless solar-blind dual-band photodetector system for advanced early fire detection/recognition, controlled by FPGA type portable circuitry, with neural network-based identification capability. The three main characteristics of a fire are an increase in temperature, generation of smoke particles and optical emission. Fires initiated by exposure of combustible material to temperatures around or slightly above the flashpoint of the material, typically tend to generate smoke before they ignite into flame. However, in some situations where the ignition temperature is greater than the flashpoint of the object, complete combustion occurs with very little smoke. Other situations, such for example one where a draft exists prior to the fire, may also hinder the capability of the smoke alarm system, , removing most if not all smoke particles, causing tragic delay in fire alarm. In such cases early fire detection is practically not possible with current state of the art ionization smoke alarms or optical smoke alarms. In such scenarios integration of existing smoke detectors with optical emission sensing and identification technology is a paramount for a false free and nuisance-alarm free fire detector. Fires produce emissions ranging from ultraviolet to IR. Such emissions can only be detected over the wide-range of ambient light background, by fast multi-range optical detectors allowing time- and spectrally-resolved measurements in particular optical regions. As a result, not only the spectral range, but also the detector speed, spatial resolution and alignment become critical for fast fire detection as well as for avoiding costly false alarms. Currently used photo-multiplier tubes (PMTs) have high sensitivity. However, they are bulky, require high voltage operation, have low mechanical and temperature strength, and cannot be easily integrated into current fire detectors. The recently developed dual-band detectors that are composed of discrete UV and IR solid-state components are bulky, not capable of detecting the multi-band optical signal with high spatial resolution, and are not suitable for networking. Employment of a miniature, chip-based dual-color high-temperature visible- or even solar-blind optical sensor system would allow for fast and false alarm-free fire detection and recognition, thus providing a fast and reliable response in separated UV and IR bands with high spatial resolution, and “smart”, artificial neural networks (ANN)-based signal analysis. Moreover, development of such sensors promotes fabrication of multi-pixel dual-band UV/IR focal plane arrays with a visible- or solar-blind imaging capability. There are two primary approaches of integrating the optical sensor system with existing state of the art smoke detectors. The first method is to remotely locate high sensitivity dual-band UV/IR focal plane arrays and smoke detectors in areas that are prone to possible fires, such as kitchens and bedrooms. These devices will then communicate with one central control system that analyses the nature and type of flame and sound an alarm accordingly. The second method is to integrate the smoke and the high sensitivity dual-band UV/IR focal plane array detector into a unit controlled by one system, and then place them in a close proximity of possible fire sources. Group III-nitride materials are superior for advanced UV detector fabrication, due to their wide direct band gap along with high thermal, chemical, mechanical, and radiation tolerance. Research and development performed by several groups, indicate that effective optical emission and detection can be achieved in a wide spectral band ranging from 200 to 1770nm, which includes also the near IR range. The Radio Frequency Molecular Beam Epitaxy (RF MBE) method used in our laboratory for nitride material growth allows fabrication of multilayer structures that incorporate binary, ternary, or even quaternary nitride compounds, with a precise control over the layer, thickness, chemical composition, crystalline quality, and doping during a single-process growth on commercial sapphire or silicon substrates. Our preliminary data from GaN, AlGaN, and InGaN based photodiode structures grown on Si and sapphire, indicate that sensitivity in both the UV and IR ranges can be achieved from a single structure. Measurements performed in our laboratory on GaN/InGaN-based heterostructure chips, show that they can be operated at temperatures over 300°C without internal or external cooling. The challenge is to take advantage of all the advanced nitride material growth and processing capabilities, as well as the unique optical, chemical, and thermal properties of the nitrides, in order to develop wireless, miniature, inexpensive, and reliable integrated multi-band solar-blind fire detectors. We own two US patents (US 7,381,966 and US 7,566,875) on this technology. Other preliminary results are from the development of chip integrated optoelectronic multi-band chemical sensors. In this project, an integrated device structure based on wavelength-selective LED and photodetector chips (Figure 4) is controlled by a FDMA-based circuitry with an Artificial Neural Network (ANN)-based signal acquisition and analysis. Variable signal patterns are generated by combined effects of fluorescence, absorption, and scattering resulting from interaction of the multi-wavelength optical emission with the analytes. ANN was employed for the categorization of different analytes of various concentrations using Stuttgart Neural Network Simulator (SNNS) tool (Figure 4a). For 8 different analytes at 4 or 5 different concentrations, totaling 35 different samples, after 2000 cycles of training the network the results were: 96% accuracy for the testing set and 100% accuracy for the training set (Figure 4b). The current efforts on this project are directed towards the development of an intelligent portable multifunctional bio-chemical sensor system with time-resolved capability in a ps time resolution range. The schematic of a single-pixel dual-band UV/IR photodetector is shown in Figure 5. Double side-polished n-type <111> Si wafers are used as substrates. AlN and GaN buffer layers are grown by RF MBE on Si to compensate for the lattice mismatch and reduce the effect of the substrate material on the active AlGaN film grown on the top of the structure. The content of Al in this film determines the UV cut-off wavelengths of the device and can be varied between 30 and 60%. Spectroscopic ellipsometry is used to determine the band gap of the AlGaN layer. Reflection High Energy Electron Diffraction (RHEED) is used to monitor the crystalline quality of the layers during growth. Post-growth characterization includes photoluminescence (PL), optical transmission, spectroscopic ellipsometry, and Hall effect measurements. In order to form the IR sensitive part of the photodetector, a Pt /Au layer is deposited by e-beam evaporation on the backside of Si and patterned to form Pt Schottky barrier contacts to n-type Si. Then Ti / Au dots are deposited through a stencil mask in between the Pt/Au dots in order to form the ohmic contacts to n-type Si. The UV sensitive part of the photodetector is processed as follows. A silicon dioxide (SiO2) layer is first deposited on top of the AlGaN layer by a PECVD method in order to provide insulation for the Schottky barrier contacts. Rows of round openings aligned with the metal dots on the bottom of the Si wafer are formed then in the SiO2 layer using photolithography. Conductive transparent tin oxide (SnO2) layer is deposited then by spray pyrolysis on top of the patterned SiO2 layer. This layer is also patterned by photolithography in order to produce round contacts aligned with the Pt/Au dots on the bottom of the wafer, and with the corresponding windows in the SiO2 layer in every second row. Finally Ti /Au contacts are deposited by e-beam evaporation on the top of the wafer. Figure 6a shows a diced 4 pixel array before mounting on the AlN chip (view insert in Fig. 2), indicating the UV pixels, ohmic contacts and bulk contacts. Figure 6b shows the array of UV diodes and ohmic contacts fabricated on a silicon wafer. A standard TO-8 housing with a 5 mm opening in the cap is used for packaging. The Pt contacts on the backside of the silicon chip are bonded using thermo-compressive bonding to the Au pads deposited and patterned on top of a thermally-conductive electrically insulating AlN ceramic carrier plate. The Au pads on the ceramic plate are then micro-bonded with a 30 µ thick Au wire with two of the TO-8 housing legs, while the Ti/Au contacts on top of the chip are micro-bonded with other two legs of the housing. We should optimize Si substrate properties, III nitride growth parameters, and device processing in order to achieve higher efficiency of the dual-band UV/IR photodetectors. The optimization of the substrate parameters is done first by using theoretical simulations that will be directed towards reduction of the leakage between the UV and the IR structures currently resulting in some background sensitivity to the visible (or solar) light. For this purpose Si substrates with guarding p-n-junctions can be applied. The boron implantation parameters are modeled using SILVACOTM device simulation software. The thickness, doping type and concentration of the Si substrate and the nitride layers will be optimized. The Poisson and Schrödinger equations representing the device structures should be simultaneously solved to determine the electric field distribution and calculate the responsivity of each diode. The nitride growth parameters are experimentally optimized mainly for two purposes: a) to accomplish more effective and reproducible doping of the nitride layers, b) reduce the defect density that greatly affects the device efficiency. The device processing optimization focuses on providing higher mechanical and thermal stability as well as more efficient utilization of the advanced electrical, optical, and semiconductor properties of all materials used in the diode fabrication process. In particular, loss resulting from the reflection of the light passing multiple interfaces can be reduced by employment of anti-reflection coatings. The metal combinations used for contact fabrication should be selected to sustain elevated temperatures and harsh environments. We also consider various approaches for fabrication of micro-miniature surface-mount dual band UV/IR photodetectors that can be simply integrated into large networks and mounted into hard-to-reach areas. In addition, we investigate the possibility for integration of such detectors with micro-miniature Si-based wireless transmitter chips. Two main issues targeted in this project are early fire detection and elimination of nuisance alarms. The earliest alarm time reported for ionization type smoke detectors is 37s, which in turn affects the egress time. However, with early fire detection technology using photodetectors as discussed in this proposal, alarm time can be as low as 1s. This allows for possible extinction of fire before tragic loss of lives or property damage can be done. Secondly by monitoring UV and IR emissions from the source of alarm nuisance alarms can be eliminated. The only major drawback of a photodetector based alarm system is that the source of alarm has to be in the line of sight of the detector. Thus, emphasizing the need for an integrated photodetection and smoke detection system. For effective fire detection the signals generated by the photodetector in response to the flame, need to be captured continuously or with a very high sampling rate. Our approach is based on two different measurements: steady state (SS) and time-resolved (TR) measurements. The system’s output (fire-alarm) is based on data from both measurements that can be classified by ANN in real-time. In steady-state photocurrent measurements, the common method of capturing and digitizing a DC signal is to use a switched integrator in combination with an analog to digital converter (ADC). The principle is based on collecting the signal to an integration capacitance for an integration period selected by the user (Figure 7) followed by digitizing. Current ADCs are very accurate in digitizing low-level currents. In our photo-detection set up, we envision currents in the range from picoamperes for very low-emission levels to a few hundred nanoamperes for photodetectors placed in close proximity to fires and bright sparks. To measure such a wide dynamic range of current-inputs, an ADC with a high dynamic range will be necessary. In this part, we will design, fabricate and test a suitable ADC, which can perform the necessary capturing of low intensity level flames and sparks. This can be performed separately from the TR measurement, which will be combined with the SS measurement at a later stage. The measurements also require necessary ADC control signals from a portable setup. There are many different ways of controlling signals and acquiring data by using a portable design based on employment of microcontrollers, FPGAs, PLDs and other programmable devices. Utilization of time-resolved measurements for flame identification requires employment of FPGA based control capable of providing fast response-time and stable operation. A suitable FPGA should be selected depending on the requirements for time resolution in the TR measurements, and to have a sufficient capacity for performing ANN and other related tasks. Based on our preliminary flame dynamics studies, a TR system with a resolution in the microsecond range is required for recognizing flame dynamics patterns used for early fire detection. Employment of fast Schottky barrier structures provide an additional benefit for the TR measurements. Amplification of the photodetector signals (in pA or nA) plays an important role in these measurements. We have already demonstrated a method of amplifying such signals using a bootstrapped-cascoded technique[1]. This technique was successfully employed for amplification of current pulses with 5-10ns fall/rise time. Reduction of noise background is also a key criterion, which is investigated in detail during this task. Optimization of photodiode amplifier circuit parameters should be performed for the amplification of low-level current from low level fires with the elimination of background noise at sampling high-speeds. Another key is to combine the TR and SS measurement controls in a single FPGA chip, overcoming noise from different sources. A sample set-up proposed for combining the steady-state and time-resolved measurements is shown in Figure 8[2]. Here an EEPROM is required to start the system in a portable mode. The SS and TR measurements are made separately and stored in FPGA, which performs the ANN algorithm for fire detection. [1] C. Joseph, M. Boukadoum, J. Charlson, D. Starikov, and A. Bensaoula, “High-speed front end for LED-Photodiode based fluorescence lifetime measurement system”, Proc. IEEE International Symposium on Circuits and Systems, May 2007 (ISCAS 2007), pp. 3578-358. http://ieeexplore.ieee.org/document/4253454/ [2] Data sheet for DDC101 from Texas Instruments. http://www.ti.com/lit/ds/sbas029/sbas029.pdf We use several approaches for early error-free fire detection and identification. One of the approaches is employment of solar/visible blind photodetectors insensitive to common background light sources. The second approach is miniaturization of these photodetectors in order to integrate them into large networks parts of which will be placed in close proximity from potential fire sources and at places that cannot be monitored by conventional fire protection equipment The third approach is based on independent simultaneous detection of the optical flame emission in two separate UV and IR spectral bands. This would remove most false alarms related to optical emissions from lightning, welding, and electrical arcs, as well as from various heat sources and lighting. Finally these devices will be integrated into existing commercial smoke detectors to ensure an early false free residential alarm system. Figure 9 shows an example a design approach would that can be used for such integration. We strongly believe that in addition to the above approaches, the dual band photodetector capability combined with intelligent analysis can be used: a) to distinguish a flame from other light/heat sources; b) to identify the fire source. Such features can be enabled after investigation of various typical optical emission scenarios taking place during an interval that includes time preceding the ignition and the time right after the ignition. For example, if an electrical spark is the cause of the ignition, a short, high-intensity, broad-band light pulse followed by the flame emission light will be detected by the sensor. If the fire is caused by excessive heat, a strong, slow, monotonically increasing IR signal followed by a mixed UV/IR signal- after flame ignition, will be detected. In addition, flames from sources of different chemical composition will have distinct optical signatures (time and energy). Depending on which gas radicals are excited in the burning process, the intensities of the UV and IR peaks will change during the initial time period- after ignition, before normal flickering takes effect. The proposed time resolution acquisition system should easily discriminate between signal patterns, characteristic to various potential fire sources, such as fuels, plastics, wood, electrical isolation, etc Understanding and classifying these signal patterns is an important goal for this project. Flame flickering at a specific frequency (typically from few to hundreds of Hz) will be used, in addition to the initial signal pattern recognition, to confirm presence of a flame. To the best of our knowledge, none or very little research has been performed on optical flame dynamics from various ignition sources. All existing relevant data attributed to combustion processes, that are quite different from flames, can be used only for identification of various gas radicals that are excited during combustion (Table 1[1],[2]). In our preliminary studies, a repeatable delay between the IR and the UV signal components from a matchstick flame was measured first with a high-speed memory oscilloscope (Figure 10). In order to further investigate different flame dynamics we used an analog to digital data acquisition card (DAC USB 1208LS). The instrument was interfaced with LABVIEW™ and data was first collected and stored on the card FIFO buffer before being read. This allowed for the card to rapidly monitor the flame using commercial UV and IR photodiodes with peak wavelengths of 375, and 850 nm, respectively. In order to detect very low light intensities we designed and implemented a photodiode amplifier circuit using low noise high gain OPA177GP Op Amps. Table 2 indicates that delays between the IR and UV signal components measured with the setup from flames of aviation JP-8 fuel and gasoline are positive and negative, respectively, and also have differing average values. [1] Measurement and analysis of OH emission spectra following laser-induced optical breakdown in air Parigger, Christian G. (Center for Laser Applications, University of Tennessee, Space Institute); Guan, Guoming; Hornkohl, James O. Source: Applied Optics, v 42, n 30, Oct 20, 2003, p 5986-5991. [2] Photographic Observation and Emission Spectral Analysis of HCCI Combustion, Combustion Science and Technology. v 177, pg1699-1723, 2005 Further work should be undertaken to characterize flames from other materials relevant to residential environments e. g. ignition of fuel, oil, plastic, etc. For this purpose a higher speed data acquisition card with a resolution in a nanosecond range, currently used in our time-resolved fluorescence sensors, will be employed. Once we gain sufficient knowledge on flame dynamics, we will, in the Phase II project, design and fabricate a low cost fire flame simulator based on electronically controlled mixing of light from different LEDs with emission bands ranging from UV to IR. Such simulator will be used for certification and testing of various photodetectors for fire/flame detection without the hassle and safety concerns related to testing of real flames. For early-fire detection, a very important task is to integrate the data from both SS and TR measurements and classify the flame using the trained ANN, in real-time. The SS data contains the flame signals from UV and IR photodetectors. This alone is partially implemented in some current systems, however they are still subjected to false alarms due to effects from various ambient heat and light sources. ANN is very effective in recognizing patterns which are reproducible. As described above, the repeatable features of flames during their initial formation and development can be used for distinguishing flames from other light (heat) sources. Other features that depend on the chemical composition of the burning substance can be used for source identification. Data from different possible burning sources will be collected and used for training the ANN. This data will be employed for error-free identification of the flame source. The data contains information collected during the period from just before the ignition to complete development of the flame. The flow-chart in Figure 11 shows the different sources of data, with a view on how the problem could be approached. We will be also investigating different ways of implementing the fire-detection and fire-prevention methods. The ANN based decision would be evaluated as a stand-alone method or as a second-layer of fire-detection in addition to SS-method and frequency counting. The architecture required to integrate the data from different sources (SS, TR and Frequency counter) will be designed and developed, for an effective method of fire detection and prevention. In addition to architecture design for integration of data and analysis, the selection and optimization of ANN algorithms and architecture will be performed. The final algorithm with all the coefficients derived from the trained ANN will be implemented in an FPGA to allow for a complete portable device. Radical OH H2O CO, CO2 CH Wavelength (nm) 306-322 809.7 350-450 431.5 JP-8 0.23 0.32 0.12 0.33 0.34 0.27 0.31 0.33 0.2 0.22 Gasoline -0.35 -0.21 -0.37 -0.35 -0.28 0.36 -0.32 -0.28 -0.26 -0.3
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Jay Wallace Ted Warfield Roy Weatherford C.F. von Weizsäcker William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Michael Arbib Walter Baade Bernard Baars Leslie Ballentine Gregory Bateson John S. Bell Mara Beller Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson Gregory Chaitin Jean-Pierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Olivier Darrigol Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Philipp Frank Steven Frautschi Edward Fredkin Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Jacques Hadamard Mark Hadley Patrick Haggard Stuart Hameroff Augustin Hamon Sam Harris Hyman Hartman John-Dylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Art Hobson Jesper Hoffmeyer E. T. Jaynes William Stanley Jevons Roman Jakobson Pascual Jordan Ruth E. Kastner Stuart Kauffman Martin J. Klein William R. Klemm Simon Kochen Hans Kornhuber Stephen Kosslyn Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé Pierre-Simon Laplace David Layzer Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch Ulrich Mohrhoff Jacques Monod Emmy Noether Abraham Pais Howard Pattee Wolfgang Pauli Massimo Pauri Roger Penrose Steven Pinker Colin Pittendrigh Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Adolphe Quételet Jürgen Renn/a> Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang Herbert Simon Dean Keith Simonton B. F. Skinner Lee Smolin Ray Solomonoff Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark William Thomson (Kelvin) Giulio Tononi Peter Tse Vlatko Vedral Heinz von Foerster John von Neumann John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson Stephen Wolfram H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Free Will Mental Causation James Symposium Stern-Gerlach Experiment The first Stern-Gerlach experiment was in 1922, long before the discovery of electron spin with which it is now associated. It was an attempt to prove the existence of "space quantization," the limitation of the direction of angular momentum to a few space directions, as hypothesized by Niels Bohr and Arnold Sommerfeld. Even today, Stern-Gerlach is one of the experiments that most directly shows the quantization at the core of quantum mechanics. Understanding how it works sheds light on the problem of measurement. The Stern-Gerlach apparatus consists of an oven that heats a gas of neutral silver atoms. The rapidly moving atoms escaping from the oven are collimated (limited in the vertical dimension) and sent between two magnets, one of which has a sharp point that concentrates the magnetic field. If the field were homogeneous, there would be no effect of the atoms' trajectories. The inhomogeneous magnetic fields bends the trajectories proportional to the amount of spin. If the particles' spins had a range of classical values, the trajectories would be smeared out vertically. Because the spins are quantized, half the spins are deflected up, the other half deflected down, by a discrete amount. The quantization of spin is clearly visible as two distinct spots. The Stern-Gerlach experiment allows us to visualize the quantization, to see it directly, perhaps better than most quantum experiments. We can also study the superposition of probability amplitudes and their deterministic evolution according to the Schrödinger equation of motion as the components of the superposition are pulled apart into two different parts of space, then directly see the collapse of the wave-function when one component encounters a detector in its path. Designing a Quantum Measurement Apparatus The first step in quantum measurement is to build an apparatus that separates a quantum system physically into distinguishable paths or regions of space, where the different regions correspond to (are correlated with) the physical properties we want to measure. We do not actually distinguish the atoms as following one of the paths at this first step. That would cause the probability amplitude wave function to collapse. This first step is reversible, at least in principle. It is deterministic and an example of John von Neumann's process 2, evolution of the system according to the Schrödinger equation of motion. We need a beam of atoms (and the ability to reduce the intensity to one atom at a time). Spin-up atoms are deflected upward (shown in blue). Spin-down atoms go down (shown in red in a schematic diagram adapted from photons passing through birefringent filters as going straight). Any given atom has the possibility of being deflected up or down by the inhomogeneous magnetic field in the Stern-Gerlach apparatus. Quantum mechanics describes the single atom as being in a superposition of up and down states. Note that this first part of our apparatus accomplishes the separation of our two states into distinct physical regions. We have not actually measured yet, so a single atom passing through our measurement apparatus is described as in a linear combination (a superposition) of spin-up and spin-down states, \| ψ > = ( 1/√2) \| up > + ( 1/√2) \| down > (1) This does not mean that there are two atoms, one on each path. It is a statement about probabilities. There is an equal probability that the atom will be found (at random) with its spin up or its spin down. This is a superposition of probability amplitudes, which can interfere with one another, not a superposition of particles, which cannot. Whenever we measure, we do not find a fraction of a particle, but the whole particle. Nor does it become two particles, one spin-up and one spin-down, as in the popular but mistaken interpretation of the Schrödinger Cat as in a superposition of live and dead cats. An Information-Preserving, Reversible Example of Process 2 To show that Von Neumann's process 2 is reversible, we can add a second Stern-Gerlach apparatus, in line with the superposition of the physically separated states, Since we have not made a measurement and do not know the path of the photon, the phase information in the (generally complex) coefficients of equation (1) has been preserved, so when they combine in the second apparatus, they emerge in a state identical to that before entering the first apparatus (black arrow). An Information-Creating, Irreversible Example of Process 1 But now suppose we insert something between the two apparatuses that is capable of a measurement to produce observable information. We need a detector that locates the atom in one of the two paths. Let's consider an ideal photographic plate capable of precipitating visible silver grains upon the receipt of a single particle (and subsequent development). Today photography cannot detect single particles, but detectors using charge coupled devices (CCDs) are approaching this sensitivity. We could also use a simple Geiger counter. Note that we do not literally "see" a spin-up atom. All that we really see is a black spot on a photographic plate or an increment in the numeric display of a Geiger counter. We infer that what we see was caused by a spin-up atom, since our detector is located in the path such a particle would travel. We can write a quantum description of the plate as containing two sensitive collection areas, the part of the apparatus measuring spin-up atoms, \| Aup > (shown as the blue spot), and the part of the apparatus measuring spin-down atoms, \| Adown > (shown as the red spot) We treat the detection systems quantum mechanically, and say that each detector has two eigenstates, e.g., \| Aup0 >, corresponding to its initial state and correlated with no atoms, and the final state \| Aup1 >, in which it has detected a spin-up atom. When we actually detect the atom, say in a spin-up state with statistical probability 1/2, two "collapses" or "jumps" occur. The first is the jump of the probability amplitude wave function \| ψ > of the atom in equation (1) into the state \| up >. The second is the quantum jump of the spin-up detector from \| Aup0 > to \| Aup1 >. These two happen together, as the microscopic quantum states of individual atoms have become correlated with the states of the sensitive detectors in the macroscopic Stern-Gerlach apparatus. One can say that the atom has become entangled with the sensitive spin-up detector area, so that the wave function describing their interaction is a superposition of atom and apparatus states that cannot be observed independently. \| ψ > + \| Aup0 > => \| ψ, Aup0 > => \| up, Aup1 > These jumps destroy (unobservable) phase information (between the possible spin-up and spin-down states), raise the (Boltzmann) entropy of the apparatus, and increase information (Shannon entropy) in the form of the visible spot. The entropy increase takes the form of a large chemical energy release when a photographic spot is developed (or a cascade of electrons in a CCD or Geiger counter). We can animate these irreversible and reversible processes, here shown as polarized photons in a birefringent filter, but equally applicable to spin-up and spin-down atoms in the Stern-Gerlach apparatus. We see that our example agrees with Von Neumann. A measurement which finds the atom in a specific state spin-up is thermodynamically irreversible, whereas the deterministic evolution described by Schrödinger's equation up to the moment of detection is reversible. We thus establish a clear connection between a measurement, which increases the information by some number of bits (Shannon entropy), and the necessary compensating increase in the (Boltzmann) entropy of the macroscopic apparatus, and the cosmic creation process, where new particles form, reducing the entropy locally, and the energy of formation is radiated or conducted away as Boltzmann entropy. Note that the Boltzmann entropy can only be radiated away (ultimately into the night sky to the cosmic microwave background) because the expansion of the universe provides a sink for the entropy, as pointed out by David Layzer. Note also that this cosmic information-creating process requires no conscious observer. The universe is its own observer. All quantum measurements that become observations have a three-step character, which begins when the wave function describing a quantum system, evolving deterministically according to the Schrödinger equation, encounters (perhaps becomes entangled with) a measuring apparatus. 1. In standard quantum theory, the first required element is the collapse of the wave-function. This is the Dirac projection postulate, which John von Neumann called Process 1 in any measurement. Note that the collapse might not leave a determinate record. If nothing in the environment is macroscopically affected so as to leave an indelible record of the collapse, we can say that no information about the collapse is created. The overwhelming fraction of collapses are of this kind. Moreover, information might actually be destroyed. For example, collisions between atoms or molecules in a gas that erase past information about their paths. 2. If the collapse occurs when the quantum system is entangled with a macroscopic measurement apparatus, a well-designed apparatus will also "collapse" into a correlated "pointer" state that can be seen by an observer as new information. This is the second required element - a determinate record of the event. Note this is impossible without an irreversible thermodynamic process that involves: a) the creation of at least one bit of new information (negative entropy) and b) the transfer away from the measuring apparatus of an amount of positive entropy (generally much, much) greater than the information created. Notice that no conscious observer need be involved. We can generalize this second step to an event in the physical world that was not designed as a measurement apparatus by a physical scientist, but nevertheless leaves an indelible record of the collapse of a quantum state. This might be a highly specific single event, or the macroscopic consequence of billions of atomic-molecular level of events. 3. Finally, the third required element is that the indelible determinate record is looked at by an observer, presumably conscious, although the consciousness itself has nothing to do with the measurement (despite von Neumann's puzzling about some kind of "psycho-physical parallelism"). When we have all three of these essential elements, we have what we normally mean by a measurement and an observation, both involving a human being. When we have only the first two, we can say metaphorically that the "universe is measuring itself," creating an information record of quantum collapse events. For example, every hydrogen atom formed in the early recombination era is a record of the time period when macroscopic bodies could begin to form. A certain pattern of photons records the explosion of a supernova billions of light years away. When detected by the CCD in a telescope, it becomes a potential observation. Craters on the back side of the moon recorded collisions with solar system debris that could become observations only when the first NASA mission circled the moon. Normal \| Teacher \| Scholar
c1d025e4f69afa07	The effect of random wind forcing in the nonlinear Schrödinger equation Leo Dostal The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is...
a26a359cbdbcef1c	Floquet quasienergy spectrum, continuous or discrete? \| PhysicsOverflow • Register Please help promote PhysicsOverflow ads elsewhere if you like it. New printer friendly PO pages! Migration to Bielefeld University was successful! Please vote for this year's PhysicsOverflow ads! ... see more Tools for paper authors Submit paper Claim Paper Authorship Tools for SE users Search User Reclaim SE Account Request Account Merger Nativise imported posts Claim post (deleted users) Import SE post Public $\beta$ tools Report a bug with a feature Request a new functionality 404 page design Send feedback (propose a free ad) Site Statistics 174 submissions , 137 unreviewed 4,308 questions , 1,640 unanswered 5,092 answers , 21,632 comments 1,470 users with positive rep 641 active unimported users More ... Floquet quasienergy spectrum, continuous or discrete? + 3 like - 0 dislike I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The Floquet theorem says the solution to the Schrödinger equation will have the form $$\psi(r,t)=e^{-i\varepsilon t}u(r,t)\ ,$$ where u(r,t) is a function periodic in time. We can rewrite the Schrödinger equation as $$\mathscr{H}u(r,t)=[H(t)-\mathrm{i}\hbar\frac{\partial}{\partial t}]u(r,t)=\varepsilon u(r,t)\ ,$$ the $\mathscr{H}$ can be thought as a Hermitian operator in the Hilbert space $\mathcal{R}+\mathcal{T}$, where $\mathcal{T}$ is a Hilbert space with all square integrable periodic functions with periodicity $\tau$. Then the above equation can be thought analogy to the stationary Schrödinger equation, with the real eigenvalue $\varepsilon$ defined as Floquet quasienergy. My question is, since in stationary Schrödinger equation, we have continuous and discrete spectrum. How about floquet quasienergy? Another thing is, is this a measurable quantity? If it is, in what sense it is measurable? (I mean, in the stationary case, the eigenenergy difference is a gauge invariant quantity, what about quasienergy?) This post imported from StackExchange Physics at 2014-10-07 10:52 (UTC), posted by SE-user luming asked Oct 5, 2014 in Theoretical Physics by BaBQ (95 points) [ revision history ] edited Oct 7, 2014 by Dilaton Most voted comments show all comments What is $\mathcal{R}$? And what does "$+$" in $\mathcal{R}+\mathcal{T}$ mean? $\mathcal{R}$ is a linear space consists of square-integrable functions of $\vec{r}$ . "+" is direct sum Evolution equations with time-dependent generators are difficult to treat in a rigorous way. One standard source is the book by Pazy. A reference that seems more tailored on your question is this book. There exists nothing like the "space of square integrable periodic functions with periodicity $\tau$". Thus the question does not make sense as it stands. You should be more precise on that. Moreover the only "object" could make sense in place of that $+$ is the tensor product $\otimes$. @ValterMoretti I mean, $\mathcal{T}$ is consists of all the functions $a(t)$ with periodicity $\tau$, and the inner product is defined as $(a,b)=\int_0^\tau a^*(t)b(t)\mathrm{d}t/\tau$. Does this make sense? Most recent comments show all comments @ValterMoretti So do you have further comment on my question "discrete or continous"? If you still think this question is ill defined, please help me improve it. Well, if the functions $u$ you consider are elements of ${\cal R}\otimes L^2(0,\tau)$ (and this can be checked case by case), the numbers $\epsilon$ certainly belong to the point spectrum of ${\cal H}$ by definition. What is physically disputable is whether or not these $u$ could have the meaning of "stationary" states in any sense. I do not think so...(What is the relevant time evolution with respect to they are stationary? Time evolution is already embodied in the definition of the Hilbert space...) 1 Answer + 0 like - 0 dislike You can think of a Floquet energy in a similar way to a Bloch state. In the latter case, because space is periodic, the momentum states are repeated at every reciprocal lattice vector, $\textbf{G}$. For a Floquet state, because time is periodic, energy states are repeated every $n\hbar \omega$ where $n$ is an integer and $\hbar \omega$ depends on the time, $\tau$ (where $\tau$ in the experiment is the time between laser pulses). Here is an image from the attached paper in case you cannot view it, but I highly recommend reading the below paper if you are interested in Floquet states. You can see (barely) in the image below that the Dirac Cone (which was chosen to be the system studied here for no particular reason), is repeated at several values of $\hbar \omega$ above and below the "actual" Dirac Cone at $n=0$. You can see the $n=1$, $n=2$, and $n=-1$ states pretty clearly. enter image description here See the paper here: answered Oct 5, 2014 by Xcheckr (0 points) [ no revision ] I happened to see this paper before. So your answer seem to say quasi-energy is a gauge invariant in the same sense as Bloch energy. Is that what you mean? Also, what about my first question? @luming Yes, the quasi-energy must be gauge-invariant or it wouldn't be measurable! Also, perhaps I don't understand in what sense you mean discrete vs. continuous? It seems to me that the quasi-energies are not very different than Bloch states with different bands, except separated by constant energy steps. Your answer Live preview (may slow down editor) Preview Your name to display (optional): Anti-spam verification: user contributions licensed under cc by-sa 3.0 with attribution required Your rights
125a70540fdd1a84	Open main menu In physics, fractional quantum mechanics is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. This concept was discovered by Nick Laskin who coined the term fractional quantum mechanics.[1] Standard quantum mechanics can be approached in three different ways: the matrix mechanics, the Schrödinger equation and the Feynman path integral. The Feynman path integral[2] is the path integral over Brownian-like quantum-mechanical paths. Fractional quantum mechanics has been discovered by Nick Laskin (1999) as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. A path integral over the Lévy-like quantum-mechanical paths results in a generalization of quantum mechanics.[3] If the Feynman path integral leads to the well known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.[4] The Lévy process is characterized by the Lévy index α, 0 < α ≤ 2. At the special case when α = 2 the Lévy process becomes the process of Brownian motion. The fractional Schrödinger equation includes a space derivative of fractional order α instead of the second order (α = 2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.[5] This is the key point to launch the term fractional Schrödinger equation and more general term fractional quantum mechanics. As mentioned above, at α = 2 the Lévy motion becomes Brownian motion. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at α = 2. The quantum-mechanical path integral over the Lévy paths at α = 2 becomes the well-known Feynman path integral and the fractional Schrödinger equation becomes the well-known Schrödinger equation. Fractional Schrödinger equationEdit The fractional Schrödinger equation discovered by Nick Laskin has the following form (see, Refs.[1,3,4]) using the standard definitions: • Dα is a scale constant with physical dimension [Dα] = [energy]1 − α·[length]α[time]α, at α = 2, D2 =1/2m, where m is a particle mass, • the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 4]); Here, the wave functions in the position and momentum spaces; and are related each other by the 3-dimensional Fourier transforms: Fractional quantum mechanics in solid state systemsEdit The effective mass of states in solid state systems can depend on the wave vector k, i.e. formally one considers m=m(k). Polariton Bose-Einstein condensate modes are examples of states in solid state systems with mass sensitive to variations and locally in k fractional quantum mechanics is experimentally feasible. See alsoEdit 1. ^ Laskin, Nikolai (2000). "Fractional quantum mechanics and Lévy path integrals". Physics Letters A. 268 (4–6): 298–305. arXiv:hep-ph/9910419. doi:10.1016/S0375-9601(00)00201-2. 2. ^ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965 3. ^ Laskin, Nick (1 August 2000). "Fractional quantum mechanics". Physical Review E. American Physical Society (APS). 62 (3): 3135–3145. arXiv:0811.1769. doi:10.1103/physreve.62.3135. ISSN 1063-651X. 4. ^ Laskin, Nick (18 November 2002). "Fractional Schrödinger equation". Physical Review E. American Physical Society (APS). 66 (5): 056108. arXiv:quant-ph/0206098. doi:10.1103/physreve.66.056108. ISSN 1063-651X. 5. ^ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993 • Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 978-2-88124-864-1. • Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-51832-3. • Pinsker, F.; Bao, W.; Zhang, Y.; Ohadi, H.; Dreismann, A.; Baumberg, J. J. (25 November 2015). "Fractional quantum mechanics in polariton condensates with velocity-dependent mass". Physical Review B. American Physical Society (APS). 92 (19): 195310. arXiv:1508.03621. doi:10.1103/physrevb.92.195310. ISSN 1098-0121. Further readingEdit
f72b0442d7f38081	All Issues Volume 12, 2018 Volume 11, 2017 Volume 10, 2016 Volume 9, 2015 Volume 8, 2014 Volume 7, 2013 Volume 6, 2012 Volume 5, 2011 Volume 4, 2010 Volume 3, 2009 Volume 2, 2008 Volume 1, 2007 Inverse Problems & Imaging June 2017 , Volume 11 , Issue 3 Select all articles A direct D-bar method for partial boundary data electrical impedance tomography with a priori information Melody Alsaker, Sarah Jane Hamilton and Andreas Hauptmann 2017, 11(3): 427-454 doi: 10.3934/ipi.2017020 +[Abstract](570) +[HTML](3) +[PDF](1419.3KB) Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios. Reconstruction in the partial data Calderón problem on admissible manifolds Yernat M. Assylbekov 2017, 11(3): 455-476 doi: 10.3934/ipi.2017021 +[Abstract](206) +[HTML](2) +[PDF](452.4KB) We consider the problem of developing a method to reconstruct a potential \begin{document}$q$\end{document} from the partial data Dirichlet-to-Neumann map for the Schrödinger equation \begin{document}$(-Δ_g+q)u=0$\end{document} on a fixed admissible manifold \begin{document}$(M,g)$\end{document}. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential \begin{document}$q$\end{document}. This allows us to reconstruct \begin{document}$q$\end{document} locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct \begin{document}$q$\end{document} globally, if \begin{document}$M$\end{document} satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo [7]. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street [14] in the Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following [14] and improving their arguments, we use a certain family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry. Ambient noise correlation-based imaging with moving sensors Mathias Fink and Josselin Garnier 2017, 11(3): 477-500 doi: 10.3934/ipi.2017022 +[Abstract](161) +[HTML](9) +[PDF](451.9KB) Waves can be used to probe and image an unknown medium. Passive imaging uses ambient noise sources to illuminate the medium. This paper considers passive imaging with moving sensors. The motivation is to generate large synthetic apertures, which should result in enhanced resolution. However Doppler effects and lack of reciprocity significantly affect the imaging process. This paper discusses the consequences in terms of resolution and it shows how to design appropriate imaging functions depending on the sensor trajectory and velocity. Time-invariant radon transform by generalized Fourier slice theorem Ali Gholami and Mauricio D. Sacchi 2017, 11(3): 501-519 doi: 10.3934/ipi.2017023 +[Abstract](483) +[HTML](9) +[PDF](4112.8KB) Time-invariant Radon transforms play an important role in many fields of imaging sciences, whereby a function is transformed linearly by integrating it along specific paths, e.g. straight lines, parabolas, etc. In the case of linear Radon transform, the Fourier slice theorem establishes a simple analytic relationship between the 2-D Fourier representation of the function and the 1-D Fourier representation of its Radon transform. However, the theorem can not be utilized for computing the Radon integral along paths other than straight lines. We generalize the Fourier slice theorem to make it applicable to general time-invariant Radon transforms. Specifically, we derive an analytic expression that connects the 1-D Fourier coefficients of the function to the 2-D Fourier coefficients of its general Radon transform. For discrete data, the model coefficients are defined over the data coefficients on non-Cartesian points. It is shown numerically that a simple linear interpolation provide satisfactory results and in this case implementations of both the inverse operator and its adjoint are fast in the sense that they run in \begin{document}$O(N \;\text{log}\; N)$\end{document} flops, where \begin{document}$N$\end{document} is the maximum number of samples in the data space or model space. These two canonical operators are utilized for efficient implementation of the sparse Radon transform via the split Bregman iterative method. We provide numerical examples showing high-performance of this method for noise attenuation and wavefield separation in seismic data. Recovering the boundary corrosion from electrical potential distribution using partial boundary data Jijun Liu and Gen Nakamura 2017, 11(3): 521-538 doi: 10.3934/ipi.2017024 +[Abstract](190) +[HTML](3) +[PDF](378.7KB) We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specified on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Hölder type based on some a priori assumptions on the unknown impedance and the electrical current input specified in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data. Subspace clustering by (k,k)-sparse matrix factorization Haixia Liu, Jian-Feng Cai and Yang Wang 2017, 11(3): 539-551 doi: 10.3934/ipi.2017025 +[Abstract](337) +[HTML](4) +[PDF](498.9KB) High-dimensional data often lie in low-dimensional subspaces instead of the whole space. Subspace clustering is a problem to analyze data that are from multiple low-dimensional subspaces and cluster them into the corresponding subspaces. In this work, we propose a \begin{document}$(k,k)$\end{document}-sparse matrix factorization method for subspace clustering. In this method, data itself is considered as the "dictionary", and each data point is represented as a linear combination of the basis of its cluster in the dictionary. Thus, the coefficient matrix is low-rank and sparse. With an appropriate permutation, it is also blockwise with each block corresponding to a cluster. With an assumption that each block is no more than \begin{document}$k$\end{document}-by-\begin{document}$k$\end{document} in matrix recovery, we seek a low-rank and \begin{document}$(k,k)$\end{document}-sparse coefficient matrix, which will be used for the construction of affinity matrix in spectral clustering. The advantage of our proposed method is that we recover a coefficient matrix with \begin{document}$(k,k)$\end{document}-sparse and low-rank simultaneously, which is better fit for subspace clustering. Numerical results illustrate the effectiveness that it is better than SSC and LRR in real-world classification problems such as face clustering and motion segmentation. Probabilistic interpretation of the Calderón problem Petteri Piiroinen and Martin Simon 2017, 11(3): 553-575 doi: 10.3934/ipi.2017026 +[Abstract](241) +[HTML](3) +[PDF](443.5KB) In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes. This probabilistic interpretation comes in three equivalent formulations which open up novel perspectives on the classical question of unique determinability of conductivities from boundary data. We aim to make this work accessible to both readers with a background in stochastic process theory as well as researchers working on deterministic methods in inverse problems. Image segmentation with dynamic artifacts detection and bias correction Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss and Andrea L. Bertozzi 2017, 11(3): 577-600 doi: 10.3934/ipi.2017027 +[Abstract](518) +[HTML](4) +[PDF](7525.7KB) Region-based image segmentation is well-addressed by the Chan-Vese (CV) model. However, this approach fails when images are affected by artifacts (outliers) and illumination bias that outweigh the actual image contrast. Here, we introduce a model for segmenting such images. In a single energy functional, we introduce 1) a dynamic artifact class preventing intensity outliers from skewing the segmentation, and 2), in Retinex-fashion, we decompose the image into a piecewise-constant structural part and a smooth bias part. The CV-segmentation terms then only act on the structure, and only in regions not identified as artifacts. The segmentation is parameterized using a phase-field, and efficiently minimized using threshold dynamics. We demonstrate the proposed model on a series of sample images from diverse modalities exhibiting artifacts and/or bias. Our algorithm typically converges within 10-50 iterations and takes fractions of a second on standard equipment to produce meaningful results. We expect our method to be useful for damaged images, and anticipate use in applications where artifacts and bias are actual features of interest, such as lesion detection and bias field correction in medical imaging, e.g., in magnetic resonance imaging (MRI). 2016 Impact Factor: 1.094 Email Alert [Back to Top]
63928bae58b9a7de	Open Access Article This Open Access Article is licensed under a Creative Commons Attribution 3.0 Unported Licence Toward fully quantum modelling of ultrafast photodissociation imaging experiments. Treating tunnelling in the ab initio multiple cloning approach Dmitry V. Makhov a, Todd J. Martinez b and Dmitrii V. Shalashilin a aSchool of Chemistry, University of Leeds, Leeds, LS2 9JT, UK. E-mail: D.Makhov@leeds.ac.uk; D.Shalashilin@leeds.ac.uk bDepartment of Chemistry, Stanford University, Stanford, CA 94305, USA Received 11th April 2016 , Accepted 1st May 2016 First published on 2nd May 2016 We present an account of our recent effort to improve simulation of the photodissociation of small heteroaromatic molecules using the Ab Initio Multiple Cloning (AIMC) algorithm. The ultimate goal is to create a quantitative and converged technique for fully quantum simulations which treats both electrons and nuclei on a fully quantum level. We calculate and analyse the total kinetic energy release (TKER) spectra and Velocity Map Images (VMI), and compare the results directly with experimental measurements. In this work, we perform new extensive calculations using an improved AIMC algorithm that now takes into account the tunnelling of hydrogen atoms. This can play an extremely important role in photodissociation dynamics. I. Introduction Quantum non-adiabatic molecular dynamics is a powerful tool for understanding the details of the mechanisms of important photo-induced processes, such as the photodissociation of pyrrole and other heteroaromatic molecules. In these processes, quantum effects such as electronically non-adiabatic transitions and tunnelling are important, and an approach that goes beyond surface hopping, such as multiconfigurational time dependent Hartree (MCTDH),1 for example, is often required. MCTDH can be very accurate, and was recently used to simulate the dissociation of pyrrole.2 However it needs a parameterized potential energy surface as a starting point, which significantly restricts its practicality. A good alternative is represented by a variety of methods3–11 based on trajectory-guided Gaussian basis functions (TBF). Despite the fact that such approaches use classical trajectories, they are still fully quantum mechanical because these trajectories are employed only for propagating the basis, while the evolution of their amplitudes and, thus, of the total nuclear wave-function is determined by the time-dependent Schrödinger equation. An important advantage of trajectory-guided quantum dynamics methods is that they are fully compatible with direct or ab initio molecular dynamics where excited state energies, gradients, and non-adiabatic coupling terms are evaluated on the fly simultaneously with the nuclear propagation. The disadvantage is that trajectory based direct dynamics is very expensive due to the high cost of electronic structure calculations and typically can afford only a limited number of trajectories, which can be an obstacle to full convergence. Recently, we introduced the ab initio multiple cloning (AIMC)10 method, where TBFs are moving along Ehrenfest trajectories, as in the multiconfigurational Ehrenfest (MCE)8,9 approach, with bifurcation of the wave-functions taken into account via basis function cloning. While leading to the growth of the number of trajectories, the use of cloning helps to adopt the basis set to quantum dynamics significantly better than in the classical MCE approach. AIMC also uses a number of tricks to efficiently sample the trajectory basis and to use the information obtained on the fly: (1) similar to the previously developed trajectory based methods AIMC relies on importance sampling of initial conditions. (2) AIMC uses the so called time displaced or train basis sets,10,12,13 which increase the basis set size almost without any additional extra cost by reusing the ab initio data which has already been obtained. (3) The method calculates quantum amplitudes in a “post-processing technique” after the trajectories of the basis set functions have been found. As a result, the trajectories can be calculated one by one in parallel and good statistics can be accumulated. In this work, we present a new implementation of the AIMC approach that is improved to take into account the tunnelling of hydrogen atoms by identifying possible tunnelling points and placing additional TBFs of the other side of the barrier. We use this new implementation to simulate the dynamics of the photodissociation of pyrrole, a process where tunnelling can play a very important role. We calculate the TKER spectrum and velocity map image (VMI), and directly compare the results of our calculations with experimental observations.14 The paper is organized as follows. In Section II we describe the proposed implementation of the AIMC approach. Section III contains the computational details of our simulations. In Section IV, we present and discuss the results. Conclusions are given in Section V. II. Theory II.1 Working equations The AIMC method10 is based on the same ansatz as the multiconfigurational Ehrenfest (MCE) approach,8,9 in which the total wave-function \|Ψ(t)〉 is represented in a trajectory-guided basis \|ψn(t)〉: image file: c6fd00073h-t1.tif(1) The basis functions \|ψn(t)〉 are composed of nuclear and electronic parts: image file: c6fd00073h-t2.tif(2) The nuclear part \|χn(t)〉 is a Gaussian coherent state moving along an Ehrenfest trajectory: image file: c6fd00073h-t3.tif(3) where [R with combining macron]n(t) and [P with combining macron]n(t) are the phase space coordinate and momentum vectors of the basis function centre, γn(t) is a phase, and the parameter α determines the width of the Gaussians. The electronic part of basis functions \|ψn(t)〉 is represented as a superposition of several adiabatic eigenstates \|ϕI〉 with quantum amplitudes a(n)I. The time dependence of the Ehrenfest amplitudes a(n)I is given by the equations image file: c6fd00073h-t4.tif(4) where the matrix elements of electronic Hamiltonian Hel(n)IJ are expressed as: image file: c6fd00073h-t5.tif(5) here VI([R with combining macron]n) is the Ith potential energy surface and dIJ([R with combining macron]n) = 〈ϕI\|R\|ϕJ〉 is the non-adiabatic coupling matrix element (NACME). The motion of the centres of the Gaussians follows the standard Newton's equations: image file: c6fd00073h-t6.tif(6) where the force [F with combining macron]n is an Ehrenfest force that includes both the usual gradient term and the additional term related to the change of quantum amplitudes as a result of non-adiabatic coupling: image file: c6fd00073h-t7.tif(7) Finally, the phase γn evolves as: image file: c6fd00073h-t8.tif(8) Eqn (3)–(8) form a complete set, determining the basis and its time evolution. The evolution of the total wave-function \|Ψ(t)〉 (eqn (1)) is defined by both the evolution of the basis functions \|ψn(t)〉 and the evolution of the relevant amplitudes cn(t). The time dependence of the amplitudes cn(t) is given by the equation image file: c6fd00073h-t9.tif(9) which can be easily obtained by substituting (1) into the time dependent Schrödinger equation. The Hamiltonian matrix elements Hmn can be written as: image file: c6fd00073h-t10.tif(10) Assuming that the second derivative of the electronic wave-function \|ϕI〉 with respect to R can be disregarded, we get: image file: c6fd00073h-t11.tif(11) The matrix elements of the kinetic energy operator image file: c6fd00073h-t12.tif can be calculated analytically. For potential energy and non-adiabatic coupling matrix elements, we use a simple approximation:10 image file: c6fd00073h-t13.tif(12) image file: c6fd00073h-t14.tif(13) The approximation (12) represents a linear interpolation of the potential energy between the two points and can be improved further at the cost of calculating higher derivatives of the potential energy along the trajectories. It has been tested previously,10 and no visible change of the results was found when this approximation was applied compared to the saddle point approximation which expands around a distinct centroid for each pair of TBFs.4 The term image file: c6fd00073h-t15.tif in eqn (9), which originates from the time dependence of the basis, can be expressed as: image file: c6fd00073h-t16.tif(14) image file: c6fd00073h-t17.tif(15) Notice that in the AIMC approach, all off-diagonal matrix elements entering eqn (9) are calculated from the electronic structure data at the TBF centres, which is needed for the propagation of the basis. Thus, quantum coupling between the configurations comes at almost no extra cost. Moreover, eqn (9) can be solved after the trajectories have been calculated, provided the appropriate electronic structure information has been saved. The detailed derivation of MCE equations together with the expressions for relevant matrix elements can be found in our previous works.10,11 II.2 Basis set sampling and cloning The Ehrenfest basis set is guided by an average potential, which can be advantageous when quantum transitions are frequent. However, it becomes unphysical in regions of low non-adiabatic coupling when two or more electronic states have significant amplitudes: in this case, the difference of the shapes of potential energy surfaces for different electronic states should lead to branching of the wavepacket. In order to reproduce the bifurcation of the wave-function after leaving the non-adiabatic coupling region, AIMC methods adopt the cloning procedure,10 where the appropriate basis function is replaced by two basis functions, each guided (mostly) by a single potential energy surface. After the cloning event, an Ehrenfest configuration image file: c6fd00073h-t18.tif yields two configurations: image file: c6fd00073h-t19.tif(16) image file: c6fd00073h-t20.tif(17) The first clone configuration has non-zero amplitudes for only one electronic state, and the second clone contains contributions of all other electronic states. The amplitudes of the two new configurations become: image file: c6fd00073h-t21.tif(18) so that the contribution of the two clones \|ψn〉 and \|ψn〉 to the whole wave-function (1) remains the same as the contribution of original function: image file: c6fd00073h-t22.tif(19) We apply the cloning procedure shortly after a trajectory passes near a conical intersection, when the non-adiabatic coupling is lower than a threshold, and, at the same time, the so-called breaking force image file: c6fd00073h-t23.tif(20) which is the force pulling the Ith state away from the remaining states, is sufficiently strong. The cloning procedure is very much in spirit of the spawning, used in the Ab Initio Multiple Spawning approach (AIMS). Cloning does not require any back-propagation of spawned/cloned basis functions, unlike many4 (but not all15,16) implementations of spawning. As has been described in our previous work,7 we rely on importance sampling when generating the initial conditions. Using the linearity of the Schrödinger equation, we first represent the initial wave-function as a superposition of Gaussians and then propagate each of them independently, “bit-by-bit”.7 We use a time-displaced basis set (coherent state trains), where several Gaussian basis functions are moving along the same trajectory but with a time-shift Δt, allowing us to reuse the same electronic structure data for each of the basis functions in the “train.” Fig. 1 shows a time displaced basis guided by a trajectory and its bifurcation via cloning. The best possible result with AIMC can be achieved when a swarm of trains is used to propagate each “bit” of the initial wave-function. image file: c6fd00073h-f1.tif Fig. 1 A sketch of the AIMC propagation scheme. The wave-function is represented as a superposition of Gaussian coherent states, which form a train moving along the trajectory. After passing the intersection, the train branches in the process of cloning. The figure shows a single train with cloning. In the most detailed AIMC calculation, a basis of several cloning trains interacting with each other is used. II.3 Tunnelling The tunnelling of hydrogen atoms can play an important role in photodissociation processes. As mentioned above, MCE, AIMC and AIMS are fully quantum methods because classical trajectories are used only to propagate the basis, while the amplitudes cn(t) are found by solving the time dependent Schrödinger equation. When Gaussian basis functions are present on two sides of the potential barrier, the interaction between them can provide quantum tunnelling through the barrier. However, in the case of direct ab initio dynamics, the basis is usually very small, far from being complete. As a result, no basis functions normally would be present on the other side, and they must be placed there by hand in order to take tunnelling into account. In this paper we adopt the ideas17,18 previously used in the AIMS method to describe tunnelling for use with the AIMC technique. Fig. 2 illustrates the algorithm that we apply. First, we calculate the usual AIMC trajectories and find turning points, where the distance between the hydrogen atom and the radical reaches a local maximum. Then, for each of these turning points, we calculate the shape of the potential barrier: we increase manually the length of the N–H bond keeping all other degrees of freedom frozen, calculate potential energies, and find the point on the other side of the barrier with the same energy as in the turning point. If this point lies further than a set threshold from the turning point, we assume that tunnelling is not possible here, as the potential barrier is too wide. Otherwise, we use it as a starting point for an additional AIMC trajectory. The new trajectory is calculated both forward and backward in time, and the initial momenta are taken as the same as in the turning point, ensuring that new trajectories have the same total classical energies as their parent trajectories. This is exactly the procedure used in the multiple spawning approach, thus our method combines cloning for non-adiabatic events and spawning for tunnelling events. The forward propagation of new trajectories often involves branching as a result of cloning; backward propagation is performed without cloning and for a sufficiently short time, until new and parent trajectories separate in phase space. image file: c6fd00073h-f2.tif Fig. 2 Illustration of the algorithm used to treat tunnelling in our approach. (A) Identify turning point; (B) find a point with the same potential energy on the opposite side of the barrier; (C) run an additional trajectory through this point; (D) solve time-dependent Schrödinger equation in the basis of a coherent state trains10 moving along the trajectories on both sides of the barrier. When all the trajectories are calculated, we solve eqn (9) for quantum amplitudes cn(t) in a time-displaced basis set (coherent state trains). This is similar to our previous approach10,11 but with the difference that now the basis is better adapted to treat tunnelling. The train basis on the new trajectory is placed in such way that it reaches the tunnelling point at the same time as the train basis on the parent trajectory. Because the new trajectory differs from its parent by only one coordinate at a tunnelling point, namely by the length of the N–H bond, there is a significant overlap between Gaussian basis functions belonging to these two trajectories. This interaction is retained for a significant time while the coherent state trains are passing the tunnelling point, ensuring the transfer of quantum amplitude across the barrier. III. Computational details Using our AIMC approach, we have simulated the dynamics of pyrrole following excitation to the first excited state. Trajectories were calculated using the AIMS-MOLPRO19 computational package, which has been modified to incorporate Ehrenfest dynamics. Electronic structure calculations were performed with the complete active space self-consistent field (CASSCF) method using the cc-pVDZ basis set. As in our previous works,9,11 we used an active space of eight electrons in seven orbitals (three ring π orbitals and two corresponding π* orbitals, one σ orbital and a corresponding σ* orbital). State averaging was performed over four singlet states using equal weights, i.e. the electronic wave-function is SA4-CAS(8,7)/cc-pVDZ. The width of Gaussian functions α was taken as 4.7 bohr−2 for hydrogen, 22.7 bohr−2 for carbon, and 19.0 bohr−2 for nitrogen atoms, as suggested in ref. 20. Three electronic states were taken into consideration during the dynamics – the ground state and the two lowest singlet excited states. The initial positions and momenta were randomly sampled from the ground state vibrational Wigner distribution in the harmonic approximation using vibrational frequencies and normal modes were calculated at the same CASSCF level of theory. We approximate the photoexcitation by simply lifting the ground state wavepacket to the excited state, as would be appropriate for an instantaneous excitation pulse within the Condon approximation. Of course, the fine details of the initial photoexcited wavepacket are lost in this approximation, however, we do not expect these details to have much effect on the observables shown in this paper. We have run 900 initial Ehrenfest trajectories, each propagated with a time-step of ∼0.06 fs (2.5 a.u.) for 200 fs or until the dissociation occurred, defined as an N–H distance exceeding 4.0 Å. For a small number of trajectories, simulations exhibiting N–H dissociation were carried out to the full 200 fs in order to investigate the dynamics of the radical. Cloning was applied to TBFs when the breaking acceleration of eqn (20) exceeded a threshold of 5 × 10−6 a.u. and the norm of the non-adiabatic coupling vector was simultaneously less than 2 × 10−3 a.u. For all initial trajectories, as well as for their branches resulting from cloning, we identified turning points for the N–H bond length and calculated the width of the potential barrier. Additional trajectories on the other side of the barrier were placed if the width of the barrier did not exceed 0.5 bohr, which corresponds to an overlap of ∼0.3 between Gaussian basis functions. The new trajectories were propagated backward for 20 fs to accommodate the train basis set, and forward until dissociation or until the trajectory time exceeds 200 fs. For each initial trajectory with all its branches and tunnelling sub-trajectories, we solved eqn (9) using a train basis set of N = 21 Gaussians per branch, separated by 10 time steps, which corresponds to an average overlap of ∼0.6 between the nearest Gaussians in the train. The total size of the basis is constantly changing because of the inclusion of new branches. The final amplitudes cn give statistical weights for each of the branches, which are used in the analysis that follows. IV. Results As a result of cloning, 900 initial configurations give rise to 1131 trajectory branches. This corresponds to an average of ∼0.25 cloning events per initial trajectory. For these branches, we have found 7702 local maxima of N–H bond length, of which 2376 have been identified as possible tunnelling points. For all these points, we run sub-trajectories, which finally gives 3203 additional branches, 4334 branches in total. The majority of these branches undergo N–H dissociation within our computational time of 200 fs: the total statistical weight of dissociative trajectories is 92%, of which 53% is the contribution of tunnelling sub-trajectories. The kinetic energy distribution of the ejected hydrogen atom is presented in Fig. 3 together with the experimental TKER spectrum.14 Both distributions clearly exhibit two contributions: a large peak at higher energies, and a small contribution at lower energies. It is important to note that adopting the basis set to tunnelling shifts the high-energy peak of TKER spectrum toward the lower energies by about ∼1000 cm−1 and makes the low-energy peak slightly more pronounced. While the calculated energies are still on average about 1.5 times higher than experimental values, this difference can be ascribed to the lack of dynamic electron correlation in the CASSCF potential energy surfaces. We previously showed11 that a more accurate MS-CASPT2 PES would lead to a shift in the kinetic energy peak of approximately ∼1800–1900 cm−1 towards lower energies, significantly improving the agreement with experiment. image file: c6fd00073h-f3.tif Fig. 3 Total kinetic energy release (TKER) spectrum of hydrogen atoms after dissociation calculated with (solid) and without (dash) taking tunnelling into account. Both spectra are averaged over the same ensemble of initial configuration. The curves are smoothed by replacing delta-functions with Gaussian functions (σ = 200 cm−1). The inset shows the experimentally measured spectrum.14 Analysis of the electronic state amplitudes in the Ehrenfest configurations (eqn (2)) shows that the bifurcation of the wave-function while passing through a conical intersection plays an important role in the formation of a two-peak spectrum: the high kinetic energy product is predominantly in the ground state, while the low energy peak is formed by mostly low-weight branches with substantial contribution from excited electronic states. Fig. 4 presents an example of such a bifurcating trajectory. At about 55 fs after photoexcitation, this trajectory reaches an intersection for the first time. After passing the intersection, the ground and first excited states of the original TBF are approximately equally populated, so the cloning procedure is applied creating instead two TBF, one in the ground state and one in the excited state. At this point, the potential energy surfaces for the ground and excited states have opposite gradients. This leads to the acceleration of the hydrogen atom for the TBF associated with the ground state and, at the same time, slows it down for the exited state TBF. As a result, although both branches are leading to dissociation, the kinetic energies of the ejected atoms are significantly different: the ground state branch contributes to the high energy peak of the distribution in Fig. 3, while the excited state branch contributes to the low energy peak. For the ground state branch, the remaining vibrational energy of the radical is low, so it remains in the ground state for the rest of the run and does not reach the intersection again. For the excited state branch, the energy taken away by the hydrogen atom is lower, leaving the pyrrolyl radical with sufficient energy to pass through numerous intersections with population transfer between the ground and both excited states. Naturally, quenching to the ground state will happen eventually for this branch but the time scale of this process is much longer than that for the dissociation, while the TKER spectrum is only affected by the radical dynamics until the H atom is lost. image file: c6fd00073h-f4.tif Fig. 4 An example of trajectory bifurcation on conical intersection. Electronic state populations (a), the kinetic energy of the H atom (b) and the N–H distance (c) as a function of time. Fast and slow branches are referred as (1) and (2) respectively. The black vertical line indicates the moment when cloning was applied. In order to calculate the velocity map image with respect to the laser pulse polarization, we must average the velocity distribution of hydrogen atoms relative to the axes of the molecule, given by calculations, over all possible orientations of the molecule: image file: c6fd00073h-t24.tif(21) where α, β and γ are Euler angles, θ is the angle between the atom velocity vector v and the transition dipole of the molecule, ξ(α,β,γ) is the angle between the transition dipole and light polarization vectors, and ϕ(θ,α,β,γ) is the angle between the light polarization vector and atom velocity. Here we take into account that the probability of excitation is proportional to cos2(ξ). Integrating over Euler angles and replacing, as usual, the δ-function for \|v\| with a narrow Gaussian function, we obtain image file: c6fd00073h-t25.tif(22) Fig. 5 shows the simulated velocity map with respect to the laser pulse polarization assuming that the transition dipole is normal to the molecular plane. The simulations reproduce well the main feature of the velocity map image, which is the anisotropy of the intense high energy part. Our results are also consistent with experiment14 in the low energy region showing an isotropic distribution, although admittedly the statistics of both experiment and simulation are poorer in the region of low energy. image file: c6fd00073h-f5.tif Fig. 5 Simulated velocity map image with respect to the laser pulse polarization assuming that the transition dipole moment is normal to the molecule plane. The experimental VMI14 is shown in the inset. V. Conclusion We simulated the photodissociation dynamics of pyrrole excited to the lowest singlet excited state (1A11A2) using a new implementation of the AIMC approach, which now is modified to take into account the tunnelling of hydrogen atoms more accurately. AIMC is a fully quantum technique but its computational cost in our implementation is compatible with classical “on the fly” molecular dynamics, which allows the accumulation of sufficient statistics to clarify the details of photo-induced processes in pyrrole. The treatment of tunnelling in our implementation provides a promising starting point for the further development of fully quantum methods for non-adiabatic dynamics and tunnelling with the ultimate goal of reaching well converged quantitative results. The current version of AIMC is already accurate enough to reproduce features of the experimentally observed TKER spectrum and velocity map images. DM and DS acknowledge the support from EPSRC through grants EP/J001481/1 and EP/N007549/1. 1. G. A. Worth, H.-D. Meyer, H. Köppel, L. S. Cederbaum and I. Burghardt, Using the MCTDH wavepacket propagation method to describe multimode non-diabatic dynamics, Int. Rev. Phys. Chem., 2008, 27, 569–606 CrossRef CAS. 2. G. Wu, S. P. 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Shalashilin, Nonadiabatic dynamics with the help of multiconfigurational Ehrenfest method: improved theory and fully quantum 24D simulation of pyrazine, J. Chem. Phys., 2010, 132(24), 244111 CrossRef PubMed. 8. D. V. Shalashilin, Multiconfigurational Ehrenfest approach to quantum coherent dynamics in large molecular systems, Faraday Disc., 2011, 153, 105 RSC. 9. K. Saita and D. V. Shalashilin, On-the-fly ab initio molecular dynamics with multiconfigurational Ehrenfest method, J. Chem. Phys., 2012, 137, 8 CrossRef PubMed. 10. D. V. Makhov, W. J. Glover, T. J. Martinez and D. V. Shalashilin, Ab initio multiple cloning algorithm for quantum nonadiabatic molecular dynamics, J. Chem. Phys., 2014, 141(5), 054110 CrossRef PubMed. 11. D. V. Makhov, K. Saita, T. J. Martinez and D. V. Shalashilin, Ab initio multiple cloning simulations of pyrrole photodissociation: TKER spectra and velocity map imaging, Phys. Chem. Chem. Phys., 2015, 17, 3316 RSC. 12. D. V. Shalashilin and M. S. Child, Basis set sampling in the method of coupled coherent states: coherent state swarms, trains and pancakes, J. Chem. Phys., 2008, 128, 054102 CrossRef PubMed. 13. M. Ben-Nun and T. J. Martinez, Exploiting Temporal Non-Locality to Remove Scaling Bottlenecks in Nonadiabatic Quantum Dynamics, J. Chem. Phys., 1999, 110, 4134–4140 CrossRef CAS. 14. G. M. Roberts, C. A. Williams, H. Yu, A. S. Chatterley, J. D. Young, S. Ullrich and V. G. Stavros, Probing ultrafast dynamics in photoexcited pyrrole: timescales for (1)pi sigma* mediated H-atom elimination, Faraday Discuss., 2013, 163, 95–116 RSC. 15. M. Ben-Nun and T. J. Martínez, A Continuous Spawning Method for Nonadiabatic Dynamics and Validation for the Zero-Temperature Spin-Boson Problem, Isr. J. Chem., 2007, 47, 75–88 CrossRef CAS. 16. S. Yang, J. D. Coe, B. Kaduk and T. J. Martínez, An “Optimal” Spawning Algorithm for Adaptive Basis Set Expansion in Nonadiabatic Dynamics, J. Chem. Phys., 2009, 130, 134113 CrossRef PubMed. 17. M. 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0edd5bdaafe9ae67	Quantum Gravity and String Theory 1302 Submissions [18] viXra:1302.0167 [pdf] submitted on 2013-02-27 02:15:35 Precursor to the Theory of Everything. This is Just a Brief Outline – not a Formal Theory with Specialized Terms and Maths Equations. Authors: Rodney Bartlett Comments: 7 Pages. also published at www.researchgate.net This article began as a question to an astronomy magazine about the Milky Way which I thought of when reading an article of theirs about astronomers looking billions of years into the past at dwarf galaxies. This led to my second question suggesting universal expansion can be reduced to an inflating cosmic fifth dimension. A 1919 paper by Einstein then combines with fractal geometry to equate it with multiple quantum fifth dimensions. It’s then suggested that the WMAP spacecraft shows dark energy is this fifth dimension and that it possesses mathematical form. This maths borrows ideas from string theory to conclude there are “currents” of binary digits flowing in two Mobius loops that are connected into an infinite number of figure-8 Klein bottles. In this way, in addition to fifth dimensional hyperspace being mathematical, the four dimensions of space-time are also shown to be composed of maths. This offers an explanation of wormholes and cosmic strings; and each figure-8 Klein bottle is said to be a subuniverse (our own being 13.7 billion years old) within an infinite, eternal universe often called the multiverse when subuniverses (which share one set of physics’ laws) are wrongly called parallel universes and claimed to have differing laws. Dark energy is considered to be not merely the fifth dimension but the radiation of binary digits from that hyperspace to create everything (in the form of mathematically-generated gravitational and electromagnetic waves). Since “everything” refers to mass and all the other properties of matter’s particles, they produce the non-Standard Model Higgs field. Binary digits are the cause of the effect known as gravitation and the idea of retrocausality – the time component of space’s quantum entanglement – can remove any separation of cause and effect to make gravitation equal to dark energy … as long as gravity is assumed to be a repelling force. This makes it compulsory to explain its obvious attraction. I’ve addressed various details concerning this in my articles but I’ll just address one point here – Earth’s orbit. This article ends by mentioning the warped nonlinearity of time and the possibility of future human involvement in creation; the existence of God (non-supernatural) via the inverse-square law’s infinite aspect coupled with eternal quantum entanglement; the Law of Conservation; and evolution’s continuing existence as adaptation. Category: Quantum Gravity and String Theory [17] viXra:1302.0152 [pdf] submitted on 2013-02-23 06:55:54 The Spherical Solution of the Quantum Gravity and the Revised Gravity Field Equation Authors: sangwha Yi Comments: 6 Pages. In the general relativity theory, using Einstein’s revised gravity field equation (add the cosmological term), discover the spherical solution of the quantum gravity. Category: Quantum Gravity and String Theory [16] viXra:1302.0151 [pdf] replaced on 2013-02-24 01:56:49 E=mQ² as a Basis for Integration of q Electrical-Mass into Q Quantum Potential as E=Q Authors: Andraz Pibernik Comments: 5 Pages. The most simple approach to a Quantum Mechanical Solution ever done using Al Zeeper's true Energy Equation E=qA²Z². This is a new approach to simulation where Q or Quantum Potential Energy is used to design Quantum simulated realities without the need to solve the Schrödinger equation. I present an unique integraion of q or Electrical-Mass aka cosmic-ocean that permeates all space and its pressure is pushing us down to earth wich is gravity. Volume x Pressure = Energy. In my model which is based on the most simple mathematics, I show that Electrical Energy = Quantum Potential Energy when relativistic mass m is replaced by the lighter and universal composite material called q or Electrical-Mass. This is the first step towards my newly invented notion of sapiedelic society that is based on self-betterment and betterment of all kind. An attempt like this has never been done before that is why I would like to dedicate this sicentific paper to my brother and future Univ. Dipl. Ing. of Electrotechnics for his 30th birthday. Category: Quantum Gravity and String Theory [15] viXra:1302.0150 [pdf] submitted on 2013-02-22 18:23:52 E=mQ² How Big Bang and Toe Are Overruled in Our Timeless Universe Authors: Andraz Pibernik Comments: 108 Pages. Please do not conduct experiments mentioned in the paper at home. If so, self-betterment and betterment of all kind are guaranteed for the price of one's image or even one's life. This work answers two major scientific questions: 1) Do the theory of the beginning and final theory exist, as time represents only a numerical order of material change and does not exist as such in the vast and infinite universe? 2) How does the scientific society evaluate this as also distance and constant velocity are referred to as dummy units created by the conscious observers? Furthermore a fact from a source among the desperate physicists is provided: ... correct, today we know that the hope into the Theory of Everything has been an overambitious project, although right now, there is no evidence for any “new physics” at higher energies to be on the horizon ... I would like to thank to all scientists who participate in this largest project on the human soil, still being conducted. Main thanks go to TOE Quest forum community and all test entities during the 14 year old simulation and my last 4 month implementation of the simulation results again on the TOE Quest forum test polygon ... I overruled the Theory of Everything, because such blunder does not exist, let alone that such answer would be provided by the physicists. To give such an answer one needs at least 50 years of multidisciplinary work and research ... 30 down 20 to go. I often had a feeling that the science had never been united as it is is with this project that will pave the way for future generations to understand the struggle a single scientist had to go through, before being admitted by the scientific society. The work should remind the reader of the necessity of sparking the scientific disputes in order to fertilize one’s work. Yet it should also show to the old generation and rigid professors who stick to the comic, I repeat, comic constants and theories, being theories for millennia, based on an infinite amount of constants that more conventional and artistic approach is needed. A proof to them that they should sometimes think like kids in order to find a proper solution to a problem that could be solved by any child at the age of six, when kids gain their consciousness. Tell your kind that we are being pushed to the celestial body by the pressure of q Electrical-Mass, not pulled by the mass of the body, a false definition of gravity. Herewith you have already done the first step towards infinity. There was no beginning, just an everlasting ongoing. Nothing is constant in this universe, even the expansion is either accelerated or decelerated. "Try not, do." Quote: Master Yoda. Finally I dedicate this work to my late father and his cousin and my true uncle from Brazil, whom I miss very much and plan to visit soon. I devote all my free time and available financial ressources to this research. Category: Quantum Gravity and String Theory [14] viXra:1302.0146 [pdf] replaced on 2013-02-22 00:36:15 E=mQ² Quantum Gravity and Quantum Sapiedelic Consciousness Equation as E=√Q Authors: Andraz Pibernik Comments: 3 Pages. For this experiment a new notion that can not be found by google was invented. Instead of humans being stoned apes or living in a psychedelic society, I now propose a sapiedelic Consciousness/Mind/Thought society where non-fiscal betterment takes place. First I show a model how simple Quantum Mathematics and Geometry give birth to a Quantum Gravity equation where Einstein's Cosmological constant is put to Pi/Planck Time squared or Planck Linear and Angular Frequency in order to obtain Quantum Golden Ratio Proportioning or Organic Growth that corresponds Planck Acceleration over Planck Area or √(c x h_bar x G). Than I develop a model of a Planck Sphere with Hamid-Planck Length that puts dice^roulette (author Hamid from Iran) and annihilates the Heissenberg uncertainty principle of Quantum Mechanics, declaring Planck Length for the one and only Physics constant in this universe at (1/6)^37 x 10^-6 meters. Later on and on the page 2, I develop a quantum brain and consciousness model where electrical-thought-mass is applied and 4 Planck values square rooted just in order to keep the mind within the idea of the Special Relativity. Once we remove the square root, electrical-mass that has no velocity ceiling and builds the cosmic ocean we all swim in, gets integrated into the velocity of thought as E=Q in the realm of c or the speed of light to the 4th power integer. (to be discussed in my next paper as this non relativistic concept is being tested at IJS) On the page 3, I post some pictures, where I present a Quantum Gravity Model that proves we do not think straight, but in circles or double helix spiral due to the universal motion or spin, latter represents the seat of consciousness. Enjoy the paper and share it with your friends. I dedicate this work to my little brother and future Univ. Dipl. Ing. Tadej. Category: Quantum Gravity and String Theory [13] viXra:1302.0129 [pdf] submitted on 2013-02-19 11:25:50 A New Basis for Cosmology Without Dark Energy Authors: Michael A. Ivanov Comments: 3 pages, 1 figure It is shown that small quantum effects of the model of low-energy quantum gravity by the author give a possibility of another interpretation of cosmological observations without an expansion of the Universe and dark energy. Category: Quantum Gravity and String Theory [12] viXra:1302.0127 [pdf] submitted on 2013-02-19 06:02:43 The Photon Consists of a Positive and a Negative Charge Authors: Hans W Giertz Comments: 6 Pages. In my paper “Gravity caused by TEM waves operating on dipoles in atoms”, viXra:1302.0066 [pdf], the nature of gravity is explained and that it is caused by TEM waves. The study displays that the photon consists of a positive charge and a negative charge. A gravitational singularity in the universe generates synchronized, extremely low frequency gravitational waves consisting of extremely low frequency photons. This photon was exposed to a magnetic field resulting in that the photon was split into a separate positive charge and a separate negative charge. These charges were exposed to electric fields, revealing their nature. It is proposed that the photon’s positive charge rotates clockwise and the negative charge rotates counter clockwise perpendicular to the photons radial direction forming a double helix. The photon’s radial speed is equal to the speed of light. The charge’s rotational speed is equal to the photon frequency ν. The study describes how the photon’s charges were measured. The study describes a method to measure how e.g. an incandescent light bulb and a radio transmitter antenna generate photons, revealing the true nature of photons. The study describes the relationship between photon, gravity and the atom’s elementary particles. Category: Quantum Gravity and String Theory [11] viXra:1302.0118 [pdf] submitted on 2013-02-17 19:20:04 Bounds Upon Graviton Mass – Using the Difference Between Graviton Propagation Speed and HFGW Transit Speed to Observe Post-Newtonian Corrections to Gravitational Potential Fields: Updated to Take Into Account Early Universe Cosmology and Penrose Cyclic co Authors: A.W.Beckwith Comments: 9 Pages. Has three appendix entries, justifying title change. Also, to be submitted to prespacetime journal later., One table included The author presents a post-Newtonian approximation based upon an earlier argument in a paper by Clifford Will as to Yukawa revisions of gravitational potentials, in part initiated by gravitons having explicit mass dependence in their Compton wave length. Prior work with Clifford Will’s idea was stymied by the application to binary stars and other such astrophysical objects, with non-useful frequencies topping off near 100 Hertz, thereby rendering Yukawa modifications of Gravity due to gravitons effectively an experimental curiosity which was not testable with any known physics equipment. This work improves on those results. Category: Quantum Gravity and String Theory [10] viXra:1302.0103 [pdf] replaced on 2013-04-20 20:42:51 The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/hubble Scales Authors: Carlos Castro Comments: 30 Pages. Submitted to Foundations of Physics. It is explained how to construct n-ary extensions of Born's reciprocal relativity based on n-ary algebras. We extend the construction of Born's Reciprocal Phase Space Relativity to the case of Clifford Spaces which involve the use of $polyvectors$ and a $lower/upper$ length scale. We present the generalized polyvector-valued velocity and acceleration/force boosts in Clifford Phase Spaces and find an $explicit$ Clifford algebraic realization of the velocity and acceleration/force boosts. Finally, we provide a Clifford Phase-Space Gravitational Theory based in gauging the generalization of the Quaplectic group and invoking Born's reciprocity principle between coordinates and momenta (maximal speed of light velocity and maximal force). The generalized gravitational vacuum field equations are explicitly displayed. We conclude with a brief discussion on the role of higher-order Finsler geometry in the construction of extended relativity theories with an upper and lower bound to the higher order accelerations (associated with the higher order tangent and cotangent spaces). We explain how to find the procedure that will allow us to find the $n$-ary analog of the Quaplectic group transformations which will now mix the $X, P, Q, .......$ coordinates of the higher order tangent (cotangent) spaces in this extended relativity theory based on Born's reciprocal gravity and $n$-ary algebraic structures. Category: Quantum Gravity and String Theory [9] viXra:1302.0101 [pdf] submitted on 2013-02-15 20:11:55 Basics of Atomic Cosmology Authors: U.V.S. Seshavatharam, S. Lakshminarayana, B.V.S.T. Sai Comments: 18 Pages. 100 years of quantum mechanics, nuclear physics and cosmology can be refined and unified. Part-2: Current cosmological changes may be reflected in any existing atom. Hubble length (c/H_t) can be considered as the gravitational or electromagnetic interaction range. In this report an attempt is made to verify the cosmic acceleration in a quantum mechanical approach. The four key assumptions are : 1) Reduced Planck’s constant increases with cosmic time. 2) Being a primordial evolving black hole and angular velocity being (H_t), universe is always rotating with light speed. 3) Atomic gravitational constant is squared Avogadro number times the classical gravitational constant and 4) Atomic gravitational constant shows discrete behavior. This may be the root cause of discrete nature of revolving electron’s total energy. With reference to the present atomic and nuclear physical constants, obtained (H_0) = 69.642 km/sec/Mpc and can be compared with the recent value (H_0)= (69.32 +/-0.80) km/sec/Mpc. Category: Quantum Gravity and String Theory [8] viXra:1302.0099 [pdf] replaced on 2013-02-17 21:37:13 Strings Are Binary Digits Whose Currents in Two 2-D Mobius Loops Produce a 4-D Figure-8 Klein Bottle that Composes Each of the Subuniverses in the One Universe Authors: Rodney Bartlett Comments: 36 Pages. Reason for replacement - added details about sunspots, black holes, Earth's orbit, and SOHO spacecraft supporting idea of matter's wave packets The strings of physics’ string theory are the binary digits of 1 and 0 used in computers and electronics. The digits are constantly switching between their representations of the “on” and “off” states. This switching is usually referred to as a flow or current. Currents in the two 2-dimensional programs called Mobius loops are connected into a four-dimensional figure-8 Klein bottle by the infinitely-long irrational and transcendental numbers. Such an infinite connection translates - via bosons being ultimately composed of 1’s and 0’s depicting pi, e, √2 etc.; and fermions being given mass by bosons interacting in matter particles’ “wave packets” – into an infinite number of 8-Kleins. Each Klein 1) is one of the universe’s subuniverses (our own is 13.7 billion years old), 2) is made flexible through its binary digits which seamlessly, or almost seamlessly, join it to surrounding subuniverses and eliminate its central hole, and 3) possesses warped time and space because its foundation is the programmed curves in its mathematical Mobius loops (along with the twists they generate [p.7]). The universe functions according to the rules of fractal geometry. So the Mobius does not exist only at the cosmic level. It also manifests at the quantum scale, giving us photons and protons etc. Space and time are no longer separate, but are an indivisible space-time. So if space and the universe are infinite, how can time not be eternal? The past and the future must both extend forever (the idea of time being finite arises from confusion of our subuniverse with the one infinite universe). BITS (BInary digiTS) only suggest existence of the divine if time is linear. Although a non-supernatural God is proposed via the inverse-square law coupled with eternal quantum entanglement, Einstein taught us that time is warped. Warped time is nonlinear, making it at least possible that the BITS composing space-time and all particles originate from the computer science of humans. I suspect many readers will be content with reading this abstract. While there are more details, and mathematics, in the content; my natural style of writing is to avoid jargon and maths. I also tend to get philosophical. While I personally feel that there’s a lot of precious information in the content, I realize it won’t all be to everyone’s liking. Other subjects dealt with in this article are - the “Pioneer anomaly”, refinement of gravitational physics, dark energy and dark matter, quantum phenomena like mass and electric charge and quantum spin, Kepler’s laws of planetary motion, deflection of starlight by the sun, tides, falling bodies, Earth’s orbit, ancient Greek philosophers, Newton, Kepler, Galileo, Aristotle, Parmenides, Zeno of Elea, time travel into the past as well as the future, the elimination of distances in space, humanity’s construction of this universe we live in, The Law of Conservation of Matter-Energy, and support for the science-fiction-like idea of the electronic binary digits of 1 and 0 being the building blocks of our universe. Category: Quantum Gravity and String Theory [7] viXra:1302.0086 [pdf] submitted on 2013-02-13 09:00:28 Ulianov String Theory. Uma Nova Representação Para Partículas Fundamentais Authors: Policarpo Yōshin Ulianov Comments: 15 Pages. This is a Portuguese version of the paper http://vixra.org/abs/1201.0101 This paper introduces a new model for fundamental particles representation, named Ulianov String Theory(UST). In the UST model, the space is composed of eight dimensions, being four of them “rolled up” dimensions while the other four are ”ordinary” dimensions. Moreover, in the UST, time dimension is modeled as a complex variable and can also be “rolled up”. This new string model also defines point-like particles named Ulianov Holes (uholes), which, from the imaginary time collapse, is transformed into strings. These strings allow generation of a series of structures, which can be associated to configurations of the observed matter and energy in our universe. Category: Quantum Gravity and String Theory [6] viXra:1302.0073 [pdf] submitted on 2013-02-12 05:16:03 Theory of New Space Authors: Edward William Johnson Comments: 25 Pages. Description of Gravity as an operator and mechanism Abstract Gravity as an identifiable force or mechanism is one of the greatest physical mysteries alluded physicists since Euclid. This paper reintroduces and reconfirms Sir Isaac Newton’s idea and belief that Space provides us with a ‘Background Absolute’, later discounted by Albert Einstein. It is explained here by the constant emergence of New Primary Space. This same mechanism determines how gravitational information is exchanged but also to, the ability of a photon to transmit and transit in spacetime and determines the local maximum speed of ‘C’. This is central and a key feature in this paper. Both Gravity & Light are subject to this fundamental ‘Absolute’ space mechanism which determines the abilities of both. It introduces the concept of ‘Primary Space’ with a constantly emergent population of 1 dimensional points of time which is the physical key to unlocking this long lasting mystery. If this philosophy is correct then it is necessary to change our current audit of spatial dimensions, and remove temporal time as being one of them. The details presented provide explanation of this system which is dependent upon the following layout of dimensions. Ut, x, y, z and temporal time is excluded from these four. [ Ut is the symbol for Universal Time Constant ] this being the Primary dimension providing Newton’s Absolute. Category: Quantum Gravity and String Theory [5] viXra:1302.0070 [pdf] submitted on 2013-02-11 20:30:15 Octonion in Superstring Theory Authors: B. C. Chanyal, P. S. Bisht, O. P. S. Negi Comments: 13 Pages. bcchanyal@gmail.com In this paper, we have made an attempt to discuss the role of octonions in superstring theory (i.e. a theory of everything to describe the unification of all four types of forces namely gravitational, electromagnetic, weak and strong) where, we have described the octonion representation of the superstring(SS) theory as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions. We have discussed the octonionic differential operator, octonionic valued potential wave equation, octonionic field equation and other various quantum equations of superstring theory in simpler, compact and consistent manner. Consequently, the generalized Dirac-Maxwell’s equations are studied with the preview of superstring theory by means of octonions. Category: Quantum Gravity and String Theory [4] viXra:1302.0066 [pdf] replaced on 2013-02-21 04:20:36 Gravity Caused by Tem Waves Operating on Dipoles in Atoms Authors: Hans W Giertz Comments: 9 Pages. The study displays the existence of a gravitational singularity in the universe generating synchronized and extremely low frequency plane TEM (transverse electromagnetic) waves. It is proposed that atomic intrinsic electromagnetic fields create resonance with these plane TEM waves, causing particles and atoms to receive and to re-emit synchronized plane TEM waves. The energy flow of synchronized plane TEM waves, travelling in opposite directions between e.g. two atoms, creates mutual force of attraction, i.e. gravity. Consequently, gravity is not an intrinsic atomic feature; however, the result of fully passive atoms exposed to electromagnetic energy. The study describes how plane TEM waves emitted by the gravitational singularity were measured. The study also displays how gravity was measured and how gravity was simulated using an electronic device. The present electromagnetic law of gravity is compared with Newtonian geometric law of gravity. Category: Quantum Gravity and String Theory [3] viXra:1302.0047 [pdf] submitted on 2013-02-08 12:34:42 Space-Time as an Expanded State of Matter with a Physical Structure Interpreted as Simultaneously Exhibiting 10, 11, and 26 Dimensions Authors: Gary Heen Comments: 10 Pages. The model of this paper presupposes that space-time is not a mathematical abstraction, but that space-time is an expanded state of matter. The fundamental quantum of matter is designated the B-string, (for Brane/String complex). The B-string quanta of particle matter and space-time differ from one another only in the volumetric state of the B-string. It is demonstrated how the B-string can be interpreted as 10-dimensional, 11-dimensional, and 26-dimensional. The relationship of the B-string quanta to Planck's natural constants is shown, and a mathematical argument is presented demonstrating the conversion of space-time into particle matter. Category: Quantum Gravity and String Theory [2] viXra:1302.0013 [pdf] submitted on 2013-02-02 10:54:53 Octonion Dark Matter Comments: 13 Pages. bcchanyal@gmail.com In this paper, we have made an attempt to discuss the role of octonions in gravity and dark matter where, we have described the octonion space as the combination of two quaternionic spaces namely gravitational G-space and electromagnetic EM-space. It is shown that octonionic hot dark matter contains the photon and graviton (i.e. massless particles) while the octonionic cold dark matter is associated with the W-;Z (massive) bosons. Category: Quantum Gravity and String Theory [1] viXra:1302.0004 [pdf] replaced on 2013-02-20 15:43:19 Understanding Confirmed Predictions in Quantum Gravity Authors: Nigel B. Cook Comments: 7 Pages. Reference hyperlinks corrected. Feynman’s relativistic path integral replaces the non-relativistic 1st quantization indeterminancy principle (required when using a classical Coulomb field in quantum mechanics) with a simple physical mechanism, multipath interference between small mechanical interactions. Each mechanical interaction is represented by a Feynman Moller scattering diagram (Fig. 1) for a gauge boson emitted by one charge to strike an effective interaction cross-section of the other charge, a cross-sectionl that is proportional to the square of the interaction strength or running coupling. Each additional pair of vertices in a Feynman diagram reduce its relative contribution to the path integral by a factor of the coupling, so for a force with a very small couplings like observable (low energy) quantum gravity, only the 2-vertex Feynman diagram has an appreciable contribution, allowing a very simple calculation to check the observable (low energy) quantum gravity interaction strength. Evidence is given that quantum gravity arises from a repulsive U(1) gauge symmetry which also causes the cosmological acceleration. Category: Quantum Gravity and String Theory
5e1910042e60519e	Every Qualia Computing Article Ever The three main goals of Qualia Computing are to: 1. Catalogue the entire state-space of consciousness 2. Identify the computational properties of experience, and 3. Reverse engineer valence (i.e. discover the function that maps formal descriptions of states of consciousness to values along the pleasure-pain axis) Core Philosophy (2016) The Banality of Evil (quote) Person-moment affecting view (quote) Qualia Formalism in the Water Supply: Reflections on The Science of Consciousness 2018 (long) Qualia Research Institute presentations at The Science of Consciousness 2018 (Tucson, AZ) Modern Accounts of Psychedelic Action (quote) From Point-of-View Fragmentation to Global Visual Coherence: Harmony, Symmetry, and Resonance on LSD (mostly quote/long) What If God Were a Closed Individualist Presentist Hedonistic Utilitarian With an Information-Theoretic Identity of Indiscernibles Ontology? (quote) Every Qualia Computing Article Ever Qualia Computing Attending The Science of Consciousness 2018 Everything in a Nutshell (quote) Would Maximally Efficient Work Be Fun? (quote) The Universal Plot: Part I – Consciousness vs. Pure Replicators (long) No-Self vs. True Self (quote) Qualia Manifesto (quote) Traps of the God Realm (quote) Avoid Runaway Signaling in Effective Altruism (transcript) Burning Man (long) Mental Health as an EA Cause: Key Questions 24 Predictions for the Year 3000 by David Pearce (quote) Why I think the Foundational Research Institute should rethink its approach (quote/long) Quantifying Bliss: Talk Summary (long) Connectome-Specific Harmonic Waves on LSD (transcript) ELI5 “The Hyperbolic Geometry of DMT Experiences” Qualia Computing at Consciousness Hacking (June 7th 2017) Principia Qualia: Part II – Valence(quote) The Penfield Mood Organ (quote) The Most Important Philosophical Question The Forces At Work (quote) Psychedelic Science 2017: Take-aways, impressions, and what’s next (long) How Every Fairy Tale Should End Political Peacocks (quote) OTC remedies for RLS (quote) Their Scientific Significance is Hard to Overstate (quote) Memetic Vaccine Against Interdimensional Aliens Infestation (quote) Raising the Table Stakes for Successful Theories of Consciousness Qualia Computing Attending the 2017 Psychedelic Science Conference GHB vs. MDMA (quote) The Binding Problem (quote) The Hyperbolic Geometry of DMT Experiences: Symmetries, Sheets, and Saddled Scenes (long) Thinking in Numbers (quote) Praise and Blame are Instrumental (quote) The Tyranny of the Intentional Object Schrödinger’s Neurons: David Pearce at the “2016 Science of Consciousness” conference in Tucson Beyond Turing: A Solution to the Problem of Other Minds Using Mindmelding and Phenomenal Puzzles Core Philosophy David Pearce on the “Schrodinger’s Neurons Conjecture” (quote) Samadhi (quote) Panpsychism and Compositionality: A solution to the hard problem (quote) LSD and Quantum Measurements: Can you see Schrödinger’s cat both dead and alive on acid? (long) Empathetic Super-Intelligence Wireheading Done Right: Stay Positive Without Going Insane (long) Just the fate of our forward light-cone Information-Sensitive Gradients of Bliss (quote) A Single 3N-Dimensional Universe: Splitting vs. Decoherence (quote) Algorithmic Reduction of Psychedelic States (long) So Why Can’t My Boyfriend Communicate? (quote) The Mating Mind Psychedelic alignment cascades (quote) 36 Textures of Confusion Work Religion (quote) Qualia Computing in Tucson: The Magic Analogy In Praise of Systematic Empathy David Pearce on “Making Sentience Great” (quote) Philosophy of Mind Diagrams Ontological Runaway Scenario Peaceful Qualia: The Manhattan Project of Consciousness (long) Qualia Computing So Far You are not a zombie (quote) What’s the matter? It’s Schrödinger, Heisenberg and Dirac’s (quote) The Biointelligence Explosion (quote) A (Very) Unexpected Argument Against General Relativity As A Complete Account Of The Cosmos Status Quo Bias The Super-Shulgin Academy: A Singularity I Can Believe In (long) The effect of background assumptions on psychedelic research An ethically disastrous cognitive dissonance… Some Definitions (quote) Who should know about suffering? Ontological Qualia: The Future of Personal Identity (long) Google Hedonics Solutions to World Problems Why does anything exist? (quote) State-Space of Background Assumptions Personal Identity Joke Getting closer to digital LSD Psychedelic Perception of Visual Textures 2: Going Meta On Triviality (quote) State-Space of Drug Effects: Results How to secretly communicate with people on LSD Generalized Wada Test and the Total Order of Consciousness State-space of drug effects Psychophysics for Psychedelic Research: Textures (long) I only vote for politicians who have used psychedelics. EOM. Why not computing qualia? David Pearce’s daily morning cocktail (2015) (quote) Psychedelic Perception of Visual Textures Should humans wipe out all carnivorous animals so the succeeding generations of herbivores can live in peace? (quote) A workable solution to the problem of other minds The fire that breathes reality into the equations of physics (quote) Phenomenal Binding is incompatible with the Computational Theory of Mind David Hamilton’s conversation with Alf Bruce about the nature of the mind (quote) Manifolds of Consciousness: The emerging geometries of iterated local binding The Real Tree of Life Phenomenal puzzles – CIELAB The psychedelic future of consciousness Not zero-sum Discussion of Fanaticism (quote) What does comparatively matter in 2015? Suffering: Not what your sober mind tells you (quote) Reconciling memetics and religion. The Reality of Basement Reality The future of love And that’s why we can and cannot have nice things Breaking the Thought Barrier: Ethics of Brain Computer Interfaces in the workplace How bad does it get? (quote) God in Buddhism Practical metaphysics Little known fun fact Crossing borders (quote) A simple mystical explanation Bolded titles mean that the linked article is foundational: it introduces new concepts, vocabulary, heuristics, research methods, frameworks, and/or thought experiments that are important for the overall project of consciousness research. These tend to be articles that also discuss concepts in much greater depth than other articles. The “long” tag means that the post has at least 4,000 words. Most of these long articles are in the 6,000 to 10,000 word range. The longest Qualia Computing article is the one about Burning Man which is about 13,500 words long (and also happens to be foundational as it introduces many new frameworks and concepts). Quotes and transcripts are usually about: evolutionary psychology, philosophy of mind, ethics, neuroscience, physics, meditation, and/or psychedelic phenomenology. By far, David Pearce is the most quoted person on Qualia Computing. Fast stats: • Total number of posts: 120 • Foundational articles: 27 • Articles over 4,000 words: 15 • Original content: 73 • Quotes and transcripts: 47 Everything in a Nutshell David Pearce at Quora in response to the question: “What are your philosophical positions in one paragraph?“: “Everyone takes the limits of his own vision for the limits of the world.” All that matters is the pleasure-pain axis. Pain and pleasure disclose the world’s inbuilt metric of (dis)value. Our overriding ethical obligation is to minimise suffering. After we have reprogrammed the biosphere to wipe out experience below “hedonic zero”, we should build a “triple S” civilisation based on gradients of superhuman bliss. The nature of ultimate reality baffles me. But intelligent moral agents will need to understand the multiverse if we are to grasp the nature and scope of our wider cosmological responsibilities. My working assumption is non-materialist physicalism. Formally, the world is completely described by the equation(s) of physics, presumably a relativistic analogue of the universal Schrödinger equation. Tentatively, I’m a wavefunction monist who believes we are patterns of qualia in a high-dimensional complex Hilbert space. Experience discloses the intrinsic nature of the physical: the “fire” in the equations. The solutions to the equations of QFT or its generalisation yield the values of qualia. What makes biological minds distinctive, in my view, isn’t subjective experience per se, but rather non-psychotic binding. Phenomenal binding is what consciousness is evolutionarily “for”. Without the superposition principle of QM, our minds wouldn’t be able to simulate fitness-relevant patterns in the local environment. When awake, we are quantum minds running subjectively classical world-simulations. I am an inferential realist about perception. Metaphysically, I explore a zero ontology: the total information content of reality must be zero on pain of a miraculous creation of information ex nihilo. Epistemologically, I incline to a radical scepticism that would be sterile to articulate. Alas, the history of philosophy twinned with the principle of mediocrity suggests I burble as much nonsense as everyone else. Avoid Runaway Signaling in Effective Altruism Above: “Virtue Signaling” by Geoffrey Miller. This presentation was given at EAGlobal 2016 at the Berkeley campus. For a good introduction to the EA movement, we suggest this amazing essay written by Scott Alexander from SlateStarCodex, which talks about his experience at EAGlobal 2017 in San Francisco (note: we were there too, and the essay briefly discusses our encounter with him). We have previously discussed why valence research is so important to EA. In brief, we argue that in order to minimize suffering we need to actually unpack what it means for an experience to have low valence (ie. to feel bad). Unfortunately, modern affective neuroscience does not have a full answer to this question, but we believe that the approach that we use- at the Qualia Research Institute- has the potential to actually uncover the underlying equation for valence. We deeply support the EA cause and we think that it can only benefit from foundational consciousness research. We’ve already covered some of the work by Geoffrey Miller (see this, this, and this). His sexual selection framework for understanding psychological traits is highly illuminating, and we believe that it will, ultimately, be a crucial piece of the puzzle of valence as well. We think that in this video Geoffrey is making some key points about how society may perceive EAs which are very important to keep in mind as the movement grows. Here is a partial transcript of the video that we think anyone interested in EA should read (it covers 11:19-20:03): So, I’m gonna run through the different traits that I think are the most relevant to EA issues. One is low intelligence versus high intelligence. This is a remarkably high intelligence crowd. And that’s good in lots of ways. Like you can analyze complex things better. A problem comes when you try to communicate findings to the people in the middle of the bell curve or even to the lower end. Those folks are the ones who are susceptible to buying books like “Homeopathic Care for Cats and Dogs” which is not evidence-based (your cat will die). Or giving to “Guide Dogs for the Blind”. And if you think “I’m going to explain my ethical system through Bayesian rationality” you might impress people, you might signal high IQ, but you might not convince them. I think there is a particular danger of “runaway IQ-signaling” in EA. I’m relatively new to EA, I’m totally on board with what this community is doing, I think it’s awesome, it’s terrific… I’m very concerned that it doesn’t go the same path I’ve seen many other fields go, which is: when you have bright people, they start competing for status on the basis of brightness, rather than on the basis of actual contributions to the field. So if you have elitist credentialism, like if your first question is “where did you go to school?”. Or “I take more Provigil than you, so I’m on a nootropics arms race”. Or you have exclusionary jargon that nobody can understand without Googling it. Or you’re skeptical about everything equally, because skepticism seems like a high IQ thing to do. Or you fetishize counter-intuitive arguments and results. These are problems. If your idea of a Trolley Problem involves twelve different tracks, then you’re probably IQ signaling. A key Big Five personality trait to worry about, or to think about consciously, is openness to experience. Low openness tends to be associated with drinking alcohol, voting Trump, giving to ineffective charities, standing for traditional family values, and being sexually inhibited. High openness to experience tends to be associated with, well, “I take psychedelics”, or “I’m libertarian”, or “I give to SCI”, or “I’m polyamorous”, or “casual sex is awesome”. Now, it’s weird that all these things come in a package (left), and that all these things come in a package (right), but that empirically seems to be the case. openness_2Now, one issue here is that high openness is great- I’m highly open, and most of you guys are too- but what we don’t want to do is, try to sell people all the package and say “you can’t be EA unless you are politically liberal”, or “unless you are a Globalist”, or “unless you support unlimited immigration”, or “unless you support BDSM”, or “transhumanism”, or whatever… right, you can get into runaway openness signaling like the Social Justice Warriors do, and that can be quite counter-productive in terms of how your field operates and how it appears to others. If you are using rhetoric that just reactively disses all of these things [low openness attributes], be aware that you will alienate a lot of people with low openness. And you will alienate a lot of conservative business folks who have a lot of money who could be helpful. Another trait is agreeableness. Kind of… kindness, and empathy, and sympathy. So low agreeableness- and this is the trait with the biggest sex difference on average, men are lower on agreeableness than women. Why? Because we did a bit more hunting, and stabbing each other, and eating meat. And high A tends to be more “cuddle parties”, and “voting for Clinton”, and “eating Tofu”, and “affirmative consent rather than Fifty Shades”. EA is a little bit weird because this community, from my observations, combines certain elements of high agreeableness- obviously, you guys care passionately about sentient welfare across enormous spans of time and space. But it also tends to come across, potentially, as low agreeableness, and that could be a problem. If you analyze ethical and welfare problems using just cold rationality, or you emphasize rationality- because you are mostly IQ signaling- it comes across to everyone outside EA as low agreeableness. As borderline sociopathic. Because traditional ethics and morality, and charity, is about warm heartedness, not about actually analyzing problems. So just be aware: this is a key personality trait that we have to be really careful about how we signal it. High agreeableness tends to be things like traditional charity, where you have a deontological perspective, sacred moral rules, sentimental anecdotes, “we’re helping people with this well on Africa that spins around, children push on it, awesome… whatever”. You focus on vulnerable cuteness, like charismatic megaphone if you are doing animal welfare. You focus on in-group loyalty, like “let’s help Americans before we help Africa”. That’s not very effective, but it’s highly compelling… emotionally… to most people, as a signal. And the stuff that EA tends to do, all of this: facing tough trade-offs, doing expected utility calculations, focusing on abstract sentience rather than cuteness… that can come across as quite cold-hearted. EA so far, in my view- I haven’t run personality questionnaires on all of you, but my impression is- it tends to attract a fairly narrow range of cognitive and personality types. Obviously high IQ, probably the upper 5% of the bell curve. Very high openness, I doubt there are many Trump supporters here. I don’t know. Probably not. [Audience member: “raise your hands”. Laughs. Someone raises hands]. Uh oh, a lynching on the Berkeley campus. And in a way there might be a little bit of low agreeableness, combined with abstract concern for sentient welfare. It takes a certain kind of lack of agreeableness to even think in complex rational ways about welfare. And of course there is a fairly high proportion of nerds and geeks- i.e. Asperger’s syndrome- me as much as anybody else out here, with a focus on what Simon Baron-Cohen calls “systematizing” over “empathizing”. So if you think systematically, and you like making lists, and doing rational expected value calculations, that tends to be a kind of Aspie way to approaching things. The result is, if you make systematizing arguments, you will come across as Aspie, and that can be good or bad depending on the social context. If you do a hard-headed, or cold-hearted analysis of suffering, that also tends to signal so-called dark triad traits-narcissism, Machiavellianism, and sociopathy- and I know this is a problem socially, and sexually, for some EAs that I know! That they come across to others as narcissistic, Machiavellian, or sociopathic, even though they are actually doing more good in the world than the high agreeableness folks. [Thus] I think virtue signaling helps explain why EA is prone to runaway signaling of intelligence and openness. So if you include a lot more math than you really strictly need to, or more intricate arguments, or more mind-bending counterfactuals, that might be more about signaling your own IQ than solving relevant problems. I think it can also explain, according to the last few slides, why EA concerns about tractability, globalism, and problem neglectedness can seem so weird, cold, and unappealing to many people. 24 Predictions for the Year 3000 by David Pearce In response to the Quora question Looking 1000 years into the future and assuming the human race is doing well, what will society be like?, David Pearce wrote: The history of futurology to date makes sobering reading. Prophecies tend to reveal more about the emotional and intellectual limitations of the author than the future. […] But here goes… Year 3000 1) Superhuman bliss. Mastery of our reward circuitry promises a future of superhuman bliss – gradients of genetically engineered well-being orders of magnitude richer than today’s “peak experiences”. 2) Eternal youth. More strictly, indefinitely extended youth and effectively unlimited lifespans. Transhumans, humans and their nonhuman animal companions don’t grow old and perish. Automated off-world backups allow restoration and “respawning” in case of catastrophic accidents. “Aging” exists only in the medical archives. SENS Research Foundation – Wikipedia 3) Full-spectrum superintelligences. A flourishing ecology of sentient nonbiological quantum computers, hyperintelligent digital zombies and full-spectrum transhuman “cyborgs” has radiated across the Solar System. Neurochipping makes superintelligence all-pervasive. The universe seems inherently friendly: ubiquitous AI underpins the illusion that reality conspires to help us. Superintelligence: Paths, Dangers, Strategies – Wikipedia Artificial Intelligence @ MIRI Kurzweil Accelerating Intelligence 4) Immersive VR. “Magic” rules. “Augmented reality” of earlier centuries has been largely superseded by hyperreal virtual worlds with laws, dimensions, avatars and narrative structures wildly different from ancestral consensus reality. Selection pressure in the basement makes complete escape into virtual paradises infeasible. For the most part, infrastructure maintenance in basement reality has been delegated to zombie AI. Augmented reality – Wikipedia Virtual reality – Wikipedia 5) Transhuman psychedelia / novel state spaces of consciousness. Analogues of cognition, volition and emotion as conceived by humans have been selectively retained, though with a richer phenomenology than our thin logico-linguistic thought. Other fundamental categories of mind have been discovered via genetic tinkering and pharmacological experiment. Such novel faculties are intelligently harnessed in the transhuman CNS. However, the ordinary waking consciousness of Darwinian life has been replaced by state-spaces of mind physiologically inconceivable to Homo sapiens. Gene-editing tools have opened up modes of consciousness that make the weirdest human DMT trip akin to watching paint dry. These disparate states-spaces of consciousness do share one property: they are generically blissful. “Bad trips” as undergone by human psychonauts are physically impossible because in the year 3000 the molecular signature of experience below “hedonic zero” is missing. Qualia Computing 6) Supersentience / ultra-high intensity experience. The intensity of everyday experience surpasses today’s human imagination. Size doesn’t matter to digital data-processing, but bigger brains with reprogrammed, net-enabled neurons and richer synaptic connectivity can exceed the maximum sentience of small, simple, solipsistic mind-brains shackled by the constraints of the human birth-canal. The theoretical upper limits to phenomenally bound mega-minds, and the ultimate intensity of experience, remain unclear. Intuitively, humans have a dimmer-switch model of consciousness – with e.g. ants and worms subsisting with minimal consciousness and humans at the pinnacle of the Great Chain of Being. Yet Darwinian humans may resemble sleepwalkers compared to our fourth-millennium successors. Today we say we’re “awake”, but mankind doesn’t understand what “posthuman intensity of experience” really means. What earthly animal comes closest to human levels of sentience? 7) Reversible mind-melding. Early in the twenty-first century, perhaps the only people who know what it’s like even partially to share a mind are the conjoined Hogan sisters. Tatiana and Krista Hogan share a thalamic bridge. Even mirror-touch synaesthetes can’t literally experience the pains and pleasures of other sentient beings. But in the year 3000, cross-species mind-melding technologies – for instance, sophisticated analogues of reversible thalamic bridges – and digital analogs of telepathy have led to a revolution in both ethics and decision-theoretic rationality. Could Conjoined Twins Share a Mind? Mirror-touch synesthesia – Wikipedia Ecstasy : Utopian Pharmacology 8) The Anti-Speciesist Revolution / worldwide veganism/invitrotarianism. Factory-farms, slaughterhouses and other Darwinian crimes against sentience have passed into the dustbin of history. Omnipresent AI cares for the vulnerable via “high-tech Jainism”. The Anti-Speciesist Revolution has made arbitrary prejudice against other sentient beings on grounds of species membership as perversely unthinkable as discrimination on grounds of ethnic group. Sentience is valued more than sapience, the prerogative of classical digital zombies (“robots”). What is High-tech Jainism? The Antispeciesist Revolution ‘Speciesism: Why It Is Wrong and the Implications of Rejecting It’ 9) Programmable biospheres. Sentient beings help rather than harm each other. The successors of today’s primitive CRISPR genome-editing and synthetic gene drive technologies have reworked the global ecosystem. Darwinian life was nasty, brutish and short. Extreme violence and useless suffering were endemic. In the year 3000, fertility regulation via cross-species immunocontraception has replaced predation, starvation and disease to regulate ecologically sustainable population sizes in utopian “wildlife parks”. The free-living descendants of “charismatic mega-fauna” graze happily with neo-dinosaurs, self-replicating nanobots, and newly minted exotica in surreal garden of edens. Every cubic metre of the biosphere is accessible to benign supervision – “nanny AI” for humble minds who haven’t been neurochipped for superintelligence. Other idyllic biospheres in the Solar System have been programmed from scratch. CRISPR – Wikipedia Genetically designing a happy biosphere Our Biotech Future 10) The formalism of the TOE is known. (details omitteddoes Quora support LaTeX?) Dirac recognised the superposition principle as the fundamental principle of quantum mechanics. Wavefunction monists believe the superposition principle holds the key to reality itself. However – barring the epoch-making discovery of a cosmic Rosetta stone – the implications of some of the more interesting solutions of the master equation for subjective experience are still unknown. Theory of everything – Wikipedia M-theory – Wikipedia Why does the universe exist? Why is there something rather than nothing? Amazon.com: The Wave Function: Essays on the Metaphysics of Quantum Mechanics (9780199790548): Alyssa Ney, David Z Albert: Books 11) The Hard Problem of consciousness is solved. The Hard Problem of consciousness was long reckoned insoluble. The Standard Model in physics from which (almost) all else springs was a bit of a mess but stunningly empirically successful at sub-Planckian energy regimes. How could physicalism and the ontological unity of science be reconciled with the existence, classically impossible binding, causal-functional efficacy and diverse palette of phenomenal experience? Mankind’s best theory of the world was inconsistent with one’s own existence, a significant shortcoming. However, all classical- and quantum-mind conjectures with predictive power had been empirically falsified by 3000 – with one exception. Physicalism – Wikipedia Quantum Darwinism – Wikipedia Consciousness (Stanford Encyclopedia of Philosophy) Hard problem of consciousness – Wikipedia Integrated information theory – Wikipedia Principia Qualia Dualism – Wikipedia New mysterianism – Wikipedia Quantum mind – Wikipedia [Which theory is most promising? As with the TOE, you’ll forgive me for skipping the details. In any case, my ideas are probably too idiosyncratic to be of wider interest, but for anyone curious: What is the Quantum Mind?] 12) The Meaning of Life resolved. Everyday life is charged with a profound sense of meaning and significance. Everyone feels valuable and valued. Contrast the way twenty-first century depressives typically found life empty, absurd or meaningless; and how even “healthy” normals were sometimes racked by existential angst. Or conversely, compare how people with bipolar disorder experienced megalomania and messianic delusions when uncontrollably manic. Hyperthymic civilization in the year 3000 records no such pathologies of mind or deficits in meaning. Genetically preprogrammed gradients of invincible bliss ensure that all sentient beings find life self-intimatingly valuable. Transhumans love themselves, love life, and love each other. 13) Beautiful new emotions. Nasty human emotions have been retired – with or without the recruitment of functional analogs to play their former computational role. Novel emotions have been biologically synthesised and their “raw feels” encephalised and integrated into the CNS. All emotion is beautiful. The pleasure axis has replaced the pleasure-pain axis as the engine of civilised life. An information-theoretic perspective on life in Heaven 14) Effectively unlimited material abundance / molecular nanotechnology. Status goods long persisted in basement reality, as did relics of the cash nexus on the blockchain. Yet in a world where both computational resources and the substrates of pure bliss aren’t rationed, such ugly evolutionary hangovers first withered, then died. Blockchain – Wikipedia 15) Posthuman aesthetics / superhuman beauty. The molecular signatures of aesthetic experience have been identified, purified and overexpressed. Life is saturated with superhuman beauty. What passed for “Great Art” in the Darwinian era is no more impressive than year 2000 humans might judge, say, a child’s painting by numbers or Paleolithic daubings and early caveporn. Nonetheless, critical discernment is retained. Transhumans are blissful but not “blissed out” – or not all of them at any rate. Art – Wikipedia 16) Gender transformation. Like gills or a tail, “gender” in the human sense is a thing of the past. We might call some transhuman minds hyper-masculine (the “ultrahigh AQ” hyper-systematisers), others hyperfeminine (“ultralow AQ” hyper-empathisers), but transhuman cognitive styles transcend such crude dichotomies, and can be shifted almost at will via embedded AI. Many transhumans are asexual, others pan-sexual, a few hypersexual, others just sexually inquisitive. “The degree and kind of a man’s sexuality reach up into the ultimate pinnacle of his spirit”, said Nietzsche – which leads to (17). Object Sexuality – Wikipedia Empathizing & Systematizing Theory – Wikipedia 17) Physical superhealth. In 3000, everyone feels physically and psychologically “better than well”. Darwinian pathologies of the flesh such as fatigue, the “leaden paralysis” of chronic depressives, and bodily malaise of any kind are inconceivable. The (comparatively) benign “low pain” alleles of the SCN9A gene that replaced their nastier ancestral cousins have been superseded by AI-based nociception with optional manual overrides. Multi-sensory bodily “superpowers” are the norm. Everyone loves their body-images in virtual and basement reality alike. Morphological freedom is effectively unbounded. Awesome robolovers, nights of superhuman sensual passion, 48-hour whole-body orgasms, and sexual practices that might raise eyebrows among prudish Darwinians have multiplied. Yet life isn’t a perpetual orgy. Academic subcultures pursue analogues of Mill’s “higher pleasures”. Paradise engineering has become a rigorous discipline. That said, a lot of transhumans are hedonists who essentially want to have superhuman fun. And why not? 18) World government. Routine policy decisions in basement reality have been offloaded to ultra-intelligent zombie AI. The quasi-psychopathic relationships of Darwinian life – not least the zero-sum primate status-games of the African savannah – are ancient history. Some conflict-resolution procedures previously off-loaded to AI have been superseded by diplomatic “mind-melds”. In the words of Henry Wadsworth Longfellow, “If we could read the secret history of our enemies, we should find in each man’s life sorrow and suffering enough to disarm all hostility.” Our descendants have windows into each other’s souls, so to speak. 19) Historical amnesia. The world’s last experience below “hedonic zero” marked a major evolutionary transition in the evolutionary development of life. In 3000, the nature of sub-zero states below Sidgwick’s “natural watershed” isn’t understood except by analogy: some kind of phase transition in consciousness below life’s lowest hedonic floor – a hedonic floor that is being genetically ratcheted upwards as life becomes ever more wonderful. Transhumans are hyper-empathetic. They get off on each other’s joys. Yet paradoxically, transhuman mental superhealth depends on biological immunity to true comprehension of the nasty stuff elsewhere in the universal wavefunction that even mature superintelligence is impotent to change. Maybe the nature of e.g. Darwinian life, and the minds of malaise-ridden primitives in inaccessible Everett branches, doesn’t seem any more interesting than we find books on the Dark Ages. Negative utilitarianism, if it were conceivable, might be viewed as a depressive psychosis. “Life is suffering”, said Gautama Buddha, but fourth millennials feel in the roots of their being that Life is bliss. Invincible ignorance? Perhaps. Negative Utilitarianism – Wikipedia 20) Super-spirituality. A tough one to predict. But neuroscience can soon identify the molecular signatures of spiritual experience, refine them, and massively amplify their molecular substrates. Perhaps some fourth millennials enjoy lifelong spiritual ecstasies beyond the mystical epiphanies of temporal-lobe epileptics. Secular rationalists don’t know what we’re missing. 21) The Reproductive Revolution. Reproduction is uncommon in a post-aging society. Most transhumans originate as extra-uterine “designer babies”. The reckless genetic experimentation of sexual reproduction had long seemed irresponsible. Old habits still died hard. By year 3000, the genetic crapshoot of Darwinian life has finally been replaced by precision-engineered sentience. Early critics of “eugenics” and a “Brave New World” have discovered by experience that a “triple S” civilisation of superhappiness, superlongevity and superintelligence isn’t as bad as they supposed. 22) Globish (“English Plus”). Automated real-time translation has been superseded by a common tongue – Globish – spoken, written or “telepathically” communicated. Partial translation manuals for mutually alien state-spaces of consciousness exist, but – as twentieth century Kuhnians would have put it – such state-spaces tend to be incommensurable and their concepts state-specific. Compare how poorly lucid dreamers can communicate with “awake” humans. Many Darwinian terms and concepts are effectively obsolete. In their place, active transhumanist vocabularies of millions of words are common. “Basic Globish” is used for communication with humble minds, i.e. human and nonhuman animals who haven’t been fully uplifted. Incommensurability – SEoP Uplift (science_fiction) – Wikipedia 23) Plans for Galactic colonization. Terraforming and 3D-bioprinting of post-Darwinian life on nearby solar systems is proceeding apace. Vacant ecological niches tend to get filled. In earlier centuries, a synthesis of cryonics, crude reward pathway enhancements and immersive VR software, combined with revolutionary breakthroughs in rocket propulsion, led to the launch of primitive manned starships. Several are still starbound. Some transhuman utilitarian ethicists and policy-makers favour creating a utilitronium shockwave beyond the pale of civilisation to convert matter and energy into pure pleasure. Year 3000 bioconservatives focus on promoting life animated by gradients of superintelligent bliss. Yet no one objects to pure “hedonium” replacing unprogrammed matter. Interstellar Travel – Wikipedia Utilitarianism – Wikipedia 24) The momentous “unknown unknown”. If you read a text and the author’s last words are “and then I woke up”, everything you’ve read must be interpreted in a new light – semantic holism with a vengeance. By the year 3000, some earth-shattering revelation may have changed everything – some fundamental background assumption of earlier centuries has been overturned that might not have been explicitly represented in our conceptual scheme. If it exists, then I’ve no inkling what this “unknown unknown” might be, unless it lies hidden in the untapped subjective properties of matter and energy. Christian readers might interject “The Second Coming”. Learning the Simulation Hypothesis were true would be a secular example of such a revelation. Some believers in an AI “Intelligence Explosion” speak delphically of “The Singularity”. Whatever – Shakespeare made the point more poetically, “There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy”. As it stands, yes, (24) is almost vacuous. Yet compare how the philosophers of classical antiquity who came closest to recognising their predicament weren’t intellectual titans like Plato or Aristotle, but instead the radical sceptics. The sceptics guessed they were ignorant in ways that transcended the capacity of their conceptual scheme to articulate. By the lights of the fourth millennium, what I’m writing, and what you’re reading, may be stultified by something that humans don’t know and can’t express. Ancient Skepticism – SEoP OK, twenty-four predictions! Successful prophets tend to locate salvation or doom within the credible lifetime of their intended audience. The questioner asks about life in the year 3000 rather than, say, a Kurzweilian 2045. In my view, everyone reading this text will grow old and die before the predictions of this answer are realised or confounded – with one possible complication. Opt-out cryonics and opt-in cryothanasia are feasible long before the conquest of aging. Visiting grandpa in the cryonics facility can turn death into an event in life. I’m not convinced that posthuman superintelligence will reckon that Darwinian malware should be revived in any shape or form. Yet if you want to wake up one morning in posthuman paradise – and I do see the appeal – then options exist: p.s. I’m curious about the credence (if any) the reader would assign to the scenarios listed here. Why I think the Foundational Research Institute should rethink its approach by Mike Johnson I. What is the Foundational Research Institute? What I like about FRI: What is FRI’s research framework? II. Why do I worry about FRI’s research framework? Objection 1: Motte-and-bailey Objection 2: Intuition duels Objection 3: Convergence requires common truth Objection 5: The Hard Problem of Consciousness is a red herring Objection 6: Mapping to reality McCabe concludes that, metaphysically speaking, Objection 7: FRI doesn’t fully bite the bullet on computationalism Objection 8: Dangerous combination Three themes which seem to permeate FRI’s research are: (1) Suffering is the thing that is bad. III. QRI’s alternative But is it right? What we’ve built with QRI’s framework IV. Closing thoughts Mike Johnson Qualia Research Institute My sources for FRI’s views on consciousness: Flavors of Computation are Flavors of Consciousness: Is There a Hard Problem of Consciousness? Consciousness Is a Process, Not a Moment How to Interpret a Physical System as a Mind Dissolving Confusion about Consciousness Debate between Brian & Mike on consciousness: Max Daniel’s EA Global Boston 2017 talk on s-risks: Multipolar debate between Eliezer Yudkowsky and various rationalists about animal suffering: The Internet Encyclopedia of Philosophy on functionalism: Gordon McCabe on why computation doesn’t map to physics: Luke Muehlhauser’s OpenPhil-funded report on consciousness and moral patienthood: Scott Aaronson’s thought experiments on computationalism: My work on formalizing phenomenology: My colleague Andrés’s work on formalizing phenomenology: A parametrization of various psychedelic states as operators in qualia space: A brief post on valence and the fundamental attribution error: The Most Important Philosophical Question Albert Camus famously claimed that the most important philosophical question in existence was whether to commit suicide. I would disagree. For one, if Open Individualism is true (i.e. that deep down we are all one and the same consciousness) then ending one’s life will not accomplish much. The vast majority of “who you are” will remain intact, and if there are further problems to be solved, and questions to be answered, doing this will simply delay your own progress. So at least from a certain point of view one could argue that the most important question is, instead, the question of personal identity. I.e. Are you, deep down, an individual being who starts existing when you are born and stops existing when you die (Closed Individualism), something that exists only for a single time-slice (Empty Individualism), or maybe something that is one and the same with the rest of the universe (Open Individualism)? I think that is a very important question. But probably not the most important one. Instead, I’d posit that the most important question is: “What is good, and is there a ground truth about it?” In the case that we are all one consciousness maybe what’s truly good is whatever one actually truly values from a first-person point of view (being mindful, of course, of the deceptive potential that comes from the Tyranny of the Intentional Object). And in so far as this has been asked, I think that there are two remaining possibilities: Does ultimate value come down to the pleasure-pain axis, or does it come down to spiritual wisdom? Thus, in this day and age, I’d argue that the most important philosophical (and hence most important, period) question is: “Is happiness a spiritual trick, or is spirituality a happiness trick?” What would it mean for happiness to be a spiritual trick? Think, for example, of the possibility that the reason why we exist is because we are all God, and God would be awfully bored if It knew that It was all that ever existed. In such a case, maybe bliss and happiness comes down to something akin to “Does this particular set of life experiences make God feel less lonely”? Alternatively, maybe God is “divinely self-sufficient”, as some mystics claim, and all of creation is “merely a plus on top of God”. In this case one could think that God is the ultimate source of all that is good, and thus bliss may be synonymous with “being closer to God”. In turn, as mystics have claimed over the ages, the whole point of life is to “get closer to God”. Spirituality, though, goes beyond God: Within (atheistic) Buddhism the view that “bliss is a spiritual trick” might take another form: Bliss is either “dirty and a sign of ignorance” (as in the case of karma-generating pleasure) or it is “the results of virtuous merit conducive to true unconditioned enlightenment“. Thus, the whole point of life would be to become free from ignorance and reap the benefits of knowing the ultimate truth. And what would it mean for spirituality to be a happiness trick? In this case one could imagine that our valence (i.e. our pleasure-pain axis) is a sort of qualia variety that evolution recruited in order to infuse the phenomenal representation of situations that predict either higher or lower chances of making copies of oneself (or spreading one’s genes, in the more general case of “inclusive fitness”). If this is so, it might be tempting to think that bliss is, ultimately, not something that “truly matters”. But this would be to think that bliss is “nothing other than the function that bliss plays in animal behavior”, which couldn’t be further from the truth. After all, the same behavior could be enacted by many methods. Instead, the raw phenomenal character of bliss reveals that “something matters in this universe”. Only people who are anhedonic (or are depressed) will miss the fact that “bliss matters”. This is self-evident and self-intimating to anyone currently experiencing ecstatic rapture. In light of these experiences we can conclude that if anything at all does matter, it has to do with the qualia varieties involved in the experiences that feel like the world has meaning. The pleasure-pain axis makes our existence significant. Now, why do I think this is the most important question? IF we discover that happiness is a spiritual trick and that God is its source then we really ought to follow “the spiritual path” and figure out with science “what is it that God truly wants”. And under an atheistic brand of spirituality, what we ought to figure out is the laws of valence-charged spiritual energy. For example, if reincarnation and karma are involved in the expected amount of future bliss and suffering, so be it. Let’s all become Bodhisattvas and help as many sentient beings as possible throughout the eons to come. On the other hand, IF we discover (and can prove with a good empirical argument) that spirituality is just the result of changes in valence/happiness, then settling on this with a high certainty would change the world. For starters, any compassionate (and at least mildly rational) Buddhist would then come along and help us out in the pursuit of creating a pan-species welfare state free of suffering with the use of biotechnology. I.e. The 500 odd million Buddhists world-wide would be key allies for the Hedonistic Imperative (a movement that aims to eliminate suffering with biotechnology). Recall Dalai Lama’s quote: “If it was possible to become free of negative emotions by a riskless implementation of an electrode – without impairing intelligence and the critical mind – I would be the first patient.” [Dalai Lama (Society for Neuroscience Congress, Nov. 2005)]. If Buddhist doctrine concerning the very nature of suffering and its causes is wrong from a scientific point of view and we can prove it with an empirically verified physicalist paradigm, then the very Buddhist ethic of “focusing on minimizing suffering” ought to compel Buddhists throughout the world to join us in the battle against suffering by any means necessary. And most likely, given the physicalist premise, this would take the form of creating a technology that puts us all in a perpetual pro-social clear-headed non-addictive MDMA-like state of consciousness (or, in a more sophisticated vein, a well-balanced version of rational wire-heading). And now some subjective impressions about the conference… Psychedelic Ambiance The Gods 1. God As the Beginning 2. I Am God 3. God Out There 4. God As Him/Her/It 5. God As The Group 6. God As Orgasm and Sex 7. God As Death 8. God As Drugs 9. God As the Body 10. God As Money 11. God As Righteous Wrath 12. God As Compassion 13. God As War 14. God As Science 15. God As Mystery 16. God As the Belief, the Simulation, the Model 17. God As the Computer 18. God Simulating Himself 19. God As Consciousness-without-an-Object 20. God As Humor 22. The Ultimate Simulation 23. God As the Diad Psychedelic Gods – Timothy Leary The Crowd The look from the Sunset Cruise at the Psychedelic Science 2017 Conference *Even the bathroom urinals seemed to have sacred geometry: How Every Fairy Tale Should End “And even though the princess defeated the dragon and married the prince at the end of the story, the truth is that the hedonic treadmill and the 7-year itch eventually caught up to them and they were not able to ‘live happily ever after’. “Thankfully, the princess got really interested in philosophy of mind and worked really hard on developing a theory of valence in order to ‘sabotage the mill’ of affective ups and downs, so to speak. After 10 years of hard work, three book-length series of blog posts, a well founded team of 17 rational psychonauts, and hundreds of experiments involving psychedelics and brain computer interfaces, at last the princess was able to create a portable device capable of measuring what amounts to a reasonable proxy for valence at an individual level in sub-second timescales, which over time enabled people to have reliable and sustainable control over the temporal dynamics of valence and arousal. “Later on the prince developed a Moloch-aware and Singleton-proof economy of information about the state-space of consciousness, and thus kick-started the era of ethical wireheads; the world became a true fariy tale… a wondrous universe of enigmatic -but always blissful- varieties of ineffable qualia. After this came to pass, one could truly and sincerely say that the prince and the princess both became (functionally and phenomenally) happily ever after. The End.” Political Peacocks Extract from Geoffrey Miller’s essay “Political peacocks” The hypothesis Humans are ideological animals. We show strong motivations and incredible capacities to learn, create, recombine, and disseminate ideas. Despite the evidence that these idea-processing systems are complex biological adaptations that must have evolved through Darwinian selection, even the most ardent modern Darwinians such as Stephen Jay Gould, Richards Dawkins, and Dan Dennett tend to treat culture as an evolutionary arena separate from biology. One reason for this failure of nerve is that it is so difficult to think of any form of natural selection that would favor such extreme, costly, and obsessive ideological behavior. Until the last 40,000 years of human evolution, the pace of technological and social change was so slow that it’s hard to believe there was much of a survival payoff to becoming such an ideological animal. My hypothesis, developed in a long Ph.D. dissertation, several recent papers, and a forthcoming book, is that the payoffs to ideological behavior were largely reproductive. The heritable mental capacities that underpin human language, culture, music, art, and myth-making evolved through sexual selection operating on both men and women, through mutual mate choice. Whatever technological benefits those capacities happen to have produced in recent centuries are unanticipated side-effects of adaptations originally designed for courtship. The predictions and implications The vast majority of people in modern societies have almost no political power, yet have strong political convictions that they broadcast insistently, frequently, and loudly when social conditions are right. This behavior is puzzling to economists, who see clear time and energy costs to ideological behavior, but little political benefit to the individual. My point is that the individual benefits of expressing political ideology are usually not political at all, but social and sexual. As such, political ideology is under strong social and sexual constraints that make little sense to political theorists and policy experts. This simple idea may solve a number of old puzzles in political psychology. Why do hundreds of questionnaires show that men more conservative, more authoritarian, more rights-oriented, and less empathy-oriented than women? Why do people become more conservative as the move from young adulthood to middle age? Why do more men than women run for political office? Why are most ideological revolutions initiated by young single men? None of these phenomena make sense if political ideology is a rational reflection of political self-interest. In political, economic, and psychological terms, everyone has equally strong self-interests, so everyone should produce equal amounts of ideological behavior, if that behavior functions to advance political self-interest. However, we know from sexual selection theory that not everyone has equally strong reproductive interests. Males have much more to gain from each act of intercourse than females, because, by definition, they invest less in each gamete. Young males should be especially risk-seeking in their reproductive behavior, because they have the most to win and the least to lose from risky courtship behavior (such as becoming a political revolutionary). These predictions are obvious to any sexual selection theorist. Less obvious are the ways in which political ideology is used to advertise different aspects of one’s personality across the lifespan. In unpublished studies I ran at Stanford University with Felicia Pratto, we found that university students tend to treat each others’ political orientations as proxies for personality traits. Conservatism is simply read off as indicating an ambitious, self-interested personality who will excel at protecting and provisioning his or her mate. Liberalism is read as indicating a caring, empathetic personality who will excel at child care and relationship-building. Given the well-documented, cross-culturally universal sex difference in human mate choice criteria, with men favoring younger, fertile women, and women favoring older, higher-status, richer men, the expression of more liberal ideologies by women and more conservative ideologies by men is not surprising. Men use political conservatism to (unconsciously) advertise their likely social and economic dominance; women use political liberalism to advertise their nurturing abilities. The shift from liberal youth to conservative middle age reflects a mating-relevant increase in social dominance and earnings power, not just a rational shift in one’s self-interest. More subtley, because mating is a social game in which the attractiveness of a behavior depends on how many other people are already producing that behavior, political ideology evolves under the unstable dynamics of game theory, not as a process of simple optimization given a set of self-interests. This explains why an entire student body at an American university can suddenly act as if they care deeply about the political fate of a country that they virtually ignored the year before. The courtship arena simply shifted, capriciously, from one political issue to another, but once a sufficient number of students decided that attitudes towards apartheid were the acid test for whether one’s heart was in the right place, it became impossible for anyone else to be apathetic about apartheid. This is called frequency-dependent selection in biology, and it is a hallmark of sexual selection processes. What can policy analysts do, if most people treat political ideas as courtship displays that reveal the proponent’s personality traits, rather than as rational suggestions for improving the world? The pragmatic, not to say cynical, solution is to work with the evolved grain of the human mind by recognizing that people respond to policy ideas first as big-brained, idea-infested, hyper-sexual primates, and only secondly as concerned citizens in a modern polity. This view will not surprise political pollsters, spin doctors, and speech writers, who make their daily living by exploiting our lust for ideology, but it may surprise social scientists who take a more rationalistic view of human nature. Fortunately, sexual selection was not the only force to shape our minds. Other forms of social selection such as kin selection, reciprocal altruism, and even group selection seem to have favoured some instincts for political rationality and consensual egalitarianism. Without the sexual selection, we would never have become such colourful ideological animals. But without the other forms of social selection, we would have little hope of bringing our sexily protean ideologies into congruence with reality. Memetic Vaccine Against Interdimensional Aliens Infestation By Steve Lehar Alien Contact – it won’t happen the way you expect! When radio and television were first invented they were hailed as a new channel for the free flow of information that will unite mankind in informed discussion of important issues. When I scan the dial on my radio and TV however I find a dismal assortment of misinformation and emotionalistic pap. How did this come to be so? The underlying insight is that electronic media are exploited by commercial interests whose only goal is to induce me to spend my money on them, and they will feed me any signal that will bring about this end. (In Television, it is YOU that are the product, the customer is the advertiser who pays for your attention!) This is a natural consequence of the laws of economics, that a successful business is one that seeks its own best interests in any way it can. I don’t intend to discuss the morality of the ‘laws of economic nature’. But recognition of this insight can help us predict similar phenomena in similar circumstances. Indeed, perhaps this same insight can be applied to extraterrestrial matters. I make first of all the following assumptions, with which you may choose either to agree or disagree. • That there are other intelligences out there in the universe. • That ultimately all intelligences must evolve away from their biological origins towards artificially manufactured forms of intelligence which have the advantages of eternal life, unrestricted size, and direct control over the parameters of their own nature. • That it is in the best interests of any intelligence to propagate its own kind throughout the universe to the exclusion of competing forms, i.e. that intelligences that adopt this strategy will thereby proliferate. Acceptance of these assumptions, together with the above mentioned insight, leads to some rather startling conclusions. Artificially manufactured life forms need not propagate themselves physically from planet to planet, all they need to do is to transmit their ‘pattern’, or the instructions for how to build them. In this way they can disperse themselves practically at the speed of light. What they need at the receiving end is some life form that is intelligent enough to receive the signal and act on it. In other words, if some alien life form knew of our existence, it would be in their interests to beguile us into manufacturing a copy of their form here on earth. This form would then proceed to scan the skies in our locality in search of other gullible life forms. In this way, their species acts as a kind of galactic virus, taking advantage of established life forms to induce them to make copies of their own kind. Man on a psychedelic state experiencing a “spiritual message” (i.e alien infomercials for consciousness technology) coming from a civilization of known pure replicators who “promise enlightenment in exchange of followers” (the oldest trick in the book to get a planet to make copies of you). The question that remains is how are they to induce an intelligent life form (or in our case, a semi-intelligent life form) to perform their reproductive function for them? A hint of how this can be achieved is seen in the barrage of commercials that pollute our airwaves here on earth. Advertisers induce us to part with our money by convincing us that ultimately it is in our own best interests to do so. After convincing us that it is our baldness, overweight, bad breath etc. which is the root of all our personal problems, they then offer us products that will magically grow hair, lose weight, smell good etc. in exchange for our money. The more ruthless and blatant ones sell us worthless products and services, while more subtle advertisers employ an economic symbiosis whereby they provide services that we may actually want, in exchange for our money. This latter strategy is only necessary for the more sophisticated and discriminating consumers, and involves a necessary waste of resources in actually producing the worthwhile product. This is the way I see it happening. As the transmissions from the early days of radio propagate into space in an ever expanding sphere, outposts on distant planetary systems will begin to detect those transmissions and send us back carefully engineered ‘commercials’ that depict themselves as everything that we desire of an alien intelligence; that they are benevolent, wise, and deeply concerned for our welfare. They will then give us instructions on how to build a machine that will cure all the problems of the world and make us all happy. When the machine is complete, it will ‘disinfect’ the planet of competing life forms and begin to scan the skies from earth in search of further nascent planets. If we insist on completely understanding the machine before we agree to build it, then they may have to strike a bargain with us, by making the machine actually perform some useful function for us. Possibly the function will be something like a pleasure machine, so that we will all line up to receive our pleasure from them, in return for our participation in their reproductive scheme. (A few ‘free sample’ machines supplied in advance but with a built-in expiration time would certainly help win our compliance). A rich variety of life forms may bombard us with an assortment of transmissions, at various levels of sophistication. If we succumb to the most primitive appeals we may wind up being quickly extinguished. If we show some sophistication, we might enjoy a brief period of hedonistic pleasure before we become emotionally enslaved. If we wish to deal with these aliens on an equal basis however, we would have to be every bit as shrewd and cunning as they themselves are, trading for real knowledge from them in return for partial cooperation with their purposes. This would be no small feat, considering that their ‘pitch’ may have been perfected and tuned through many epochs of planetary conquest and backed with an intelligence beyond our imaginings. Its just a thought!
dcf931ccea77d6e1	Sunday, February 10, 2013 First Post: A Response to Awet on Philosophy and Character I hastily tried to post this entry as a casual comment and realized the formatting had been sterilized & changed past acceptable limits. So, I decided to make my own blog and start a dialogue/monologue about the problem of evil, atheism, and better consciousness, with a focus on the philosophy of Arthur Schopenhauer. Here is the original post from an excellent and thoughtful writer, Awet, who shares many of my philosophical interests and motivated me to take my contemplation online: Heterodoxia - Philosophy Can [Not] Change You The italicized remarks are Awet's and some are from 1 or 2 other blog posts from his site. The basic thrust of my post is to (1) push Awet on whether he represented Schopenhauer's theory of agency correctly, as well as Schopenhauer's character, since he sprinkled in a little ad hominem; (2) argue for the impotence of philosophy/ethics to change character as esse (and invite a dialogue on whether concepts like esse even hold water, since Awet believes metaphysics is totally illusory); (3) loosely assert some pessimistic conclusions about human nature and speculate about quantum agency, again, for the sake of future edifying dialogue. If free choice or an act of will is taken as an event, and all events take place in time, then the idea of an intelligible choice, or the act of choice that takes place in the timeless domain of the Kantian thing in itself is completely incoherent. Whatsoever is incoherent cannot sustain as a solution. Awet does not accept the intelligible/empirical solution to the free will / determinism problem and his understanding of Kant and Schopenhauer is = or > mine. However, Awet also seems to say that Schopenhauer believes philosophy cannot change our character because it cannot motivate us. This seems to misrepresent Schopenhauer's thought and Awet's own understanding of Schopenhauer: "Motives, however, can influence character through knowledge, and that is how a person’s manner can change while his character remains the same. Motives can influence the will, alter its direction, but not change the will. Therefore, pace Seneca, willing cannot be taught, and always remains inscrutable. Motives themselves are concepts, abstract representations of reason, and through the conflict of several motives, the strongest emerges and determines the will with necessity." Character + Motive = Action, is my understanding of Schopenhauer's teaching, i.e. motives do not change character, but the influence they bring changes our behavior; Awet seems to say that no influence is possible since our character does not change, in the linked blog post. I do believe that philosophy cannot teach morality/virtue in the sense that concepts do not change our being/essence (esse). So, in modern speech, people are guided by motives that they value more than others; the operari is affected by motives, which reveals the transcendentally free esse appearing fixed in time/phenomenal experience. The phenomenal conscious motives "direct" our transcendental choice through the ephemeral world of representation. I think Awet should treat this linkage in greater depth. Awet says: Schopenhauer argues that there is no bridge between the heart and the mind because all theoretical knowledge acquired from books or instruction cannot motivate — their concepts are dead. It seems to me that Schopenhauer is trying to hoist his own petard here, by prescribing how philosophy should be done. No? But Schopenhauer is arguing that prescriptive philosophy/ethics has no effect on the character of particular persons, which I take to be an incredibly insightful and true description of human nature. Schopenhauer is admitting that no person will become better or worse, morally, for reading his work, which is quite the opposite of hoisting his own petard. When Schopenhauer writes about why he writes (why the genius writes according to Schopenhauer), he does not give a very clear reason at all (an instinct of a unique sort): "The motive which moves genius to productivity is, on the other hand, less easy to determine (compared to the motive which moves talent, i.e. money and fame). It isn't money, for genius seldom gets any. It isn't fame: fame is too uncertain and, more closely considered, of too little worth. Nor is it strictly for its own pleasure, for the great exertion involved almost outweighs the pleasure. It is rather an instinct of a unique sort by virtue of which the individual possessed of genius is impelled to express what he has seen and felt in enduring works without being conscious of any further motivation. It takes place, by and large, with the same sort of necessity that a tree brings forth fruit, and demands of the world no more than a soil on which the individual can flourish. More closely considered, it is as if in such an individual the will to live, as the spirit of the human species, had become conscious of having, by a rare accident, attained for a brief span of time to a greater clarity of intellect, and now endeavors to acquire the products of this clear thought and vision for the whole species, which indeed is the intrinsic being of the individual, so that their light may continue to illumine the darkness and stupor of the ordinary human consciousness. It is from this that there arises that instinct which impels genius to labor in solitude to complete its work without regard for reward, applause or sympathy, but neglectful rather even of its own well-being. To make its work, as a sacred trust and the true fruit of its existence, the property of mankind, laying it down for a posterity better able to appreciate it: this becomes for genius a goal more important than any other, a goal for which it wears the crown of thorns that shall one day blossom into a laurel wreath. Its striving to complete and safeguard its work is just as resolute as that of the insect to safeguard its eggs and provide for the brood it will never live to see: it deposits its eggs where it knows they will find life and nourishment, and dies contented". --Vol. 2 "On Philosophy and the Intellect" as translated in Essays and Aphorisms (1970), as translated by R. J. Hollingdale. If Schopenhauer was hoisting his own petard, wouldn't he try to demonstrate a rational + prescriptive ethics of compassion, i.e. that we ought (can) to be good/virtuous by doing x, y, z? Schopenhauer believed the ultimate significance of life was moral, and he knowingly failed his own standard, by a vast breadth. Thinkers as great as Kierkegaard gleefully claimed "Schopenhauer is not who he thinks he is" (paraphrase) in an attempt to prove that Schopenhauer's metaphysics was wrong because Schopenhauer didn't become an ascetic/saint. Nietzsche did the same from the other way round, e.g. Schopenhauer negates God, and the world, but preserves morality; is this a pessimist? (paraphrase again, sorry). Critics love to talk about Schopenhauer's arrogance/pride, but I believe he humbly recognized his own moral imperfection, his distance from his own conception of salvation, his unbecoming attachment to life & lack of philosophical equanimity. His expression that the great sculptor need not be beautiful, nor the philosopher be a saint, seems to capture his sense of self-condemnation. Almost every philosopher attempts to portray himself as living up to the requirements of his own ethical knowledge, Schopenhauer wasn't even in the ballpark for his own standards. With regard to humanity, what should we make of the fact that almost nobody philosophizes; that the very idea of a person being in dead earnest about philosophy occurs to no one? (Schopenhauer paraphrase again, sorry) Rational arguments do not change what we believe about the meaning of life (they are all tautologies anyway, right?); everyone searches their feelings in light of arguments/experience to determine their ultimate disposition. Schopenhauer thinks that direct/intuitive understanding is the source of innate moral disposition which we graft rational dogma upon. This explains why asceticism is practiced very similarly (in terms of self-denial) in many (all?) world religions despite disparate dogmatic commitments. I think Schopenhauer is right that no philosophy/religion changes our character and, much more radically, that we are fundamentally evil (or life is meaningless). The insoluble problem of evil seems to be the only real problem of existence; for Schopenhauer, the original astonishment that anything exists followed by horror at the ubiquity of suffering and death, ground his fixation on the riddle of existence. But almost nobody gives a damn about the suffering and death inherent to life and its implications. Instead, the majority justifies egoism, and two minorities pursue (1) the well-being of all and (2) the woe of all, with an infinite spectrum in between. Yet Awet praises Schopenhauer for recognition of the primacy of the incoherent: Contra the dogma of philosophers, Arthur Schopenhauer realized that reason is not the basic essence of man. Greatest insight (separate post where Awet states this remark) Moreover, the incoherent is simply a fact of all attempts at complete explanation. Schopehauer does not believe he has proven that the intelligible character / empirical character theory is true, he simply speculates that metaphysically we are free transcendentally (or life has no moral significance) because empirically we appear determined by the principle of sufficient reason / causality. None of Schopenhauer's transcendental claims are put forth as certain truths; Schopenhauer is explicit that the "Will" is only a best guess and not exhaustive of ultimate reality; in fact, if salvation exists from apart from Will then Will cannot constitute the whole of ultimate reality. All transcendental language is necessarily incoherent; however, and I do not agree that because a proposition is incoherent that it cannot sustain a metaphysical solution. One suggestion Awet offered for the riddle of existence is to "establish your lucidity in the middle of what negates it", which hardly seems coherent, but I found it deeply meaningful. (Awet's remark was from a personal email, not his blog) Awet also claims that although causation was taken as universal and absolutely necessary during the heyday of Newtonian science, in our post-modern times, quantum indeterminacy provides an escape hatch [to the strict Kantian determinism used for the solution to free-will problem in transcendental idealism]. I do not believe that quantum indeterminacy necessarily provides an "escape hatch" for allowing empirical freedom from causation because physicists are focused on how to describe results rather explain why one result happened over another. In other words, the physical ontology isn't the focus of quantum physics because we are still struggling to develop resources to accurately say what is happening. Perhaps quantum physics affirms an epistemic/material determinism trapped within ontological freedom/uncertainty. Tim Maudlin's Distilling Metaphysics from Quantum Physics from the Oxford Handbook of Metaphysics has a section on determinism that may be illuminating: "Historically, the most widely remarked metaphysical innovation of quantum theory over classical physics is the rejection of determinism in favor of chance. Events such as the decay of a radioactive atom are typically held to be fundamentally random: there is no reason at all that the decay takes place at one time rather than another. Atoms that are physically identical in every respect may nonetheless behave differently. Einstein was resistant to the idea that God plays dice, and his insistence on determinism is taken to be a mark of a reactionary inability to accept the quantum theory. Things are not quite so simple. Does either the pragmatic formalism or empirical result of any experiment require us to abandom determinism? No. The pragmatic formalism requires an interpretation, and some interpretations posit deterministic laws while other employ fundamentally stochastic dynamics. Further, little can be said in the way of generalization." Maudlin goes on to note that the Schrödinger equation itself is deterministic, so any interpretation not employing wave collapse at a fundamental level must find its indeterminism apart from his theory, if at all. The main problem to be solved in quantum theory is not "an explanation of why one result happened rather than another (restoring determinism), but rather to have the theoretical resources to describe the experiment as having had one result rather than another. That problem is answered in the first place simply by having more than the wavefunction in the physical ontology, irrespective of the dynamics." Maudlin concludes the section by explaining that "the question of determinism is only tangential to the motives of the enterprise ... So we can't say that quantum theory forces indeterminism on us. Furthermore, the whole issue looks like a case of spoils to the victor than a fundamental point of contention: if some consideration militates in favour of a specific interpretation, the question of determinism will simply follow suit, and it seems very unlikely that determinism itself will be a decisive consideration." With the frontier of physics so foggy, we simply aren't in a position to talk about physical ontology yet. I would love to explore the possibility of reworking Schopenhauer's epistemology in light of quantum physics, and I am not convinced yet that his theory is obsolete/irrelevant. It is curious that both Schrödinger and Einstein were both avid readers and enthusiasts of Schopenhauer; perhaps his fundamental insights can survive emerging discoveries.
f454ee32bf2340fa	Quantum Mechanics Winter, 2012 Lectures in this Course 1. 1 Introduction to quantum mechanics 2. 2 The basic logic of quantum mechanics Professor Susskind introduces the simplest possible quantum mechanical system: a single particle with spin. He presents the fundamental logic of quantum mechanics in terms of preparing and measuring the direction of the spin. This fundamental... [more] 3. 3 Vector spaces and operators Professor Susskind elaborates on the abstract mathematics of vector spaces by introducing the concepts of basis vectors, linear combinations of vector states, and matrix algebra as it applies to vector spaces. He then introduces linear operators... [more] 4. 4 Time evolution of a quantum system Professor Susskind opens the lecture by presenting the four fundamental principles of quantum mechanics that he touched on briefly in the last lecture. He then discusses the evolution in time of a quantum system, and describes how the classical... [more] 5. 5 Uncertainty, unitary evolution, and the Schrödinger equation Professor Susskind begins the lecture by introducing the Heisenberg uncertainty principle and explains how it relates to commutators. He proves that two simultaneously measurable operators must commute. If they don't then the observables... [more] 6. 6 7. 7 Entanglement and the nature of reality This lecture takes a deeper look at entanglement. Professor Susskind begins by discussing the wave function, which is the inner product of the system's state vector with the set of basis vectors, and how it contains probability amplitudes for the... [more] 8. 8 Particles moving in one dimension and their operators 9. 9 Fourier analysis applied to quantum mechanics and the uncertainty principle Professor Susskind opens the lecture with a review of the entangled singlet and triplet states and how they decay. He then shows how Fourier analysis can be used to decompose a typical quantum mechanical wave function. 10. 10 The uncertainty principle and classical analogs Professor Susskind begins the final lecture of the course by deriving the uncertainty principle from the triangle inequity. He then shows the correspondence between the motion of wave packets and the classical equations of motion. The expectation... [more]
219721a84e0de762	Disclaimer: I do not know a whole terrible lot about the intricacies of either chaos theory or quantum mechanics, let alone the combination of the two, this is more a philosophical thing than a scientific one, I know I get a lot of things wrong (on both sides) 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. Anscombe (No publishing information provided).
4b537b06f5961fbb	Monday, June 15, 2015 A brief introduction to basis sets In order to compute the energy we need to define mathematical functions for the orbitals. In the case of atoms we can simply use the solutions to the Schrödinger equation for the $\ce{H}$ atom as a starting point and find the best exponent for each function using the variational principle. But what functions should we use for molecular orbitals (MOs)? The wave function of the $\ce{H2+}$ molecule provides a clue (Figure 1) Figure 1. Schematic representation of the wave function of the $\ce{H2+}$ molecule. It looks a bit like the sum of two 1$s$ functions centered at each nucleus (A and B), $${\Psi ^{{\text{H}}_{\text{2}}^ + }} \approx \tfrac{1}{{\sqrt 2 }}\left( {\Psi _{1s}^{{{\text{H}}_{\text{A}}}} + \Psi _{1s}^{{{\text{H}}_{\text{B}}}}} \right)$$ Thus, one way of constructing MOs is as a linear combination of atomic orbitals (the LCAO approximation), $${\phi _i}(1) = \sum\limits_{\mu = 1}^K {{C_{\mu i}}{\chi _\mu }(1)} $$ an approximation that becomes better and better as $K$ increases. Here $\chi _\mu$ is a mathematical function that looks like an AO, and is called a basis function (a collection of basis functions for various atoms is called a basis set), and $C_{\mu i}$ is a number (sometimes called an MO coefficient) that indicates how much basis function $\chi _\mu$ contributes to MO $i$, and is determined for each system via the variational principle. Note that every MO is expressed in terms of all basis function, and therefore extends over the entire molecule. If we want to calculate the RHF energy of water, the basis set for the two $\ce{H}$ atoms would simply be the lowest energy solution to the Schrödinger equation for $\ce{H}$ atom $${\chi _{{{\text{H}}_{\text{A}}}}}(1) = \Psi _{1s}^{\text{H}}({r_{1A}}) = \frac{1}{{\sqrt \pi }}{e^{ - \left\| {{{\bf{r}}_1} - {{\bf{R}}_A}} \right\|}}$$ For the O atom, the basis set is the AOs obtained from, say, an ROHF calculation on $\ce{O}$, i.e. $1s$, $2s$, $2p_x$, $2p_y$, and $2p_z$ functions from the solutions to the Schrödinger equation for the $\ce{H}$ atom, where the exponents ($\alpha_i$'s) have been variationally optimized for the $\ce{O}$ atom, $$\Psi _{1s}^{\text{H}},\;\Psi _{2s}^{\text{H}},\;\Psi _{2p}^{\text{H}} \xrightarrow{\frac{\partial E}{\partial \alpha_i}=0} \phi _{1s}^{\text{O}},\;\phi _{2s}^{\text{O}},\;\phi _{2p}^{\text{O}} \equiv \;\left\{ {{\chi _O}} \right\}$$ Notice that this only has to be done once, i.e. we will use this oxygen basis set for all oxygen-containing molecules. We then provide a guess at the water structure and the basis functions are placed at the coordinates of the respective atoms. Then we find the best MO coefficients by variational minimization, $$\frac{{\partial E}}{{\partial {C_{\mu i}}}} = 0 $$ for all $i$ and $\mu$. Thus, for water we need a total of seven basis functions to describe the five doubly occupied water MOs ($K = 7$ and $i = 1-5$ in Eq 5). This is an example of a minimal basis set, since it is the minimum number of basis functions per atoms that makes chemical sense. One problem with the LCAO approximation is the number of 2-electron integrals it leads to, and the associated computational cost. Let’s look at the part of the energy that comes from the Coulomb integrals \sum\limits_{i = 1}^{N/2} {\sum\limits_{j = 1}^{N/2} {2{J_{ij}}} } &= \sum\limits_{i = 1}^{N/2} {\sum\limits_{j = 1}^{N/2} {2\left\langle {\left. {{\phi _i}(1){\phi _j}(2)} \right\|\frac{1}{{{r_{12}}}}\left\| {{\phi _i}(1){\phi _j}(2)} \right.} \right\rangle } } \\ &= \sum\limits_i^{N/2} {\sum\limits_j^{N/2} {2\left\langle {\left. {{\phi _i}(1){\phi _i}(1)} \right\|\frac{1}{{{r_{12}}}}\left\| {{\phi _j}(2){\phi _j}(2)} \right.} \right\rangle } } \\ &= \sum\limits_i^{N/2} {\sum\limits_j^{N/2} {2\left\langle {{{\phi _i}{\phi _i}}} \mathrel{\left \| {\vphantom {{{\phi _i}{\phi _i}} {{\phi _j}{\phi _j}}}} \right. } {{{\phi _j}{\phi _j}}} \right\rangle } } \\ &= \sum\limits_\mu ^K {\sum\limits_\nu ^K {\sum\limits_\lambda ^K {\sum\limits_\sigma ^K {\sum\limits_i^{N/2} {\sum\limits_j^{N/2} {2{C_{\mu i}}{C_{\nu i}}{C_{\lambda j}}{C_{\lambda j}}\left\langle {{{\chi _\mu }{\chi _\nu }}} \mathrel{\left \| {\vphantom {{{\chi _\mu }{\chi _\nu }} {{\chi _\lambda }{\chi _\sigma }}}} \right. } {{{\chi _\lambda }{\chi _\sigma }}} \right\rangle } } } } } } \\ &= \sum\limits_\mu ^K {\sum\limits_\nu ^K {\sum\limits_\lambda ^K {\sum\limits_\sigma ^K {\tfrac{1}{2}{P_{\mu \nu }}{P_{\lambda \sigma }}\left\langle {{{\chi _\mu }{\chi _\nu }}} \right. } We have roughly $(N/2)^2$ Coulomb integrals involving molecular orbitals but roughly $1/8K^4$ Coulomb integrals involving basis functions (the factor of 1/8 comes from the fact that some the integrals are identical and need only be computed once). Using a minimal basis set $K$ = 80 for a small organic molecule like caffeine $\ce{(C8H10N4O2)}$ and results in ca 5,000,000 2-electron integrals involving basis functions! (That being said, you can perform an RHF energy calculation with 80 basis functions on your desktop computer in a few minutes. The problem with $K^4$-scaling is that a corresponding calculation with 800 basis functions would take a few days on the same machine. So you can forget about optimizing the structure on that machine.) This is one of the reasons why modern computational quantum chemistry requires massive computers. This is also the reason why the basis set size it a key consideration in a quantum chemistry project. The 2-electron integrals also pose another problem: the basis functions defined so far are exponential functions (also known as Slater type orbitals or STOs). 2-electron integrals involving STOs placed on four different atoms do not have analytic solutions. As a result, most quantum chemistry programs use Gaussian type orbitals (or simply Gaussians) instead of STOs, because the 2-electron integrals involving Gaussians have analytic solutions. Obviously, $${e^{ - \alpha {r_{1A}}}} \approx {e^{ - \beta r_{1A}^2}}$$ is a poor approximation, so a linear combination of Gaussians are used to model each STO basis function (Figure 2) $${e^{ - \alpha {r_{1A}}}} \approx \sum\limits_i^X {{a_{i\mu }}{e^{ - {\beta _i}r_{1A}^2}}} = {\chi _\mu }$$ Figure 2. (a) An exponential function is not well represented by one Gaussian, but (b) can be well represented by a linear combination of three Gaussians. Here the $a_{i\mu}$parameters (or contraction coefficients) as well as the Gaussian exponents are determined just once for a given STO basis function. $\chi_\mu$ is a contracted basis function and the $X$ individual Gaussian functions are called primitives. Generally, three primitives are sufficient to represent an STO, and this basis set is known at the STO-3G basis set. $p$- and $d$-type STOs are expanded in terms of $p$- and $d$-type primitive Gaussians [e.g. $({x_1} - {x_A}){e^{ - \beta r_{1A}^2}}$ and $({x_1} - {x_A})({y_1} - {y_A}){e^{ - \beta r_{1A}^2}}$]. An RHF calculation using the STO-3G basis set is denoted RHF/STO-3G. Unless otherwise noted, this usually also implies that the geometry is computed (i.e. the minimum energy structure is found) at this level of theory. Minimal basis sets are usually not sufficiently accurate to model reaction energies. This is due to the fact that the atomic basis functions cannot change size to adjust to their bonding environment. However, this can be made possible by using some the contraction coefficients as variational parameters. This will increase the basis set size (and hence the computational cost) so this must be done judiciously. For example, we’ll get most improvement by worrying about the basis functions that describe the valence electrons that participate most in bonding. Thus, for $\ce{O}$ atom we leave the 1$s$ core basis function alone, but “split” the valence 2$s$ basis function into linear combinations of two and one Gaussians respectively, {\chi _{1s}} &= \sum\limits_i^3 {{a_{i1s}}{e^{ - {\beta _i}r_{1A}^2}}} \\ {\chi _{2{s_a}}} &= \sum\limits_i^2 {{a_{i2s}}{e^{ - {\beta _i}r_{1A}^2}}} \\ {\chi _{2{s_b}}} &= {e^{ - {\beta _{2{s_b}}}r_{1A}^2}} \\ \end{split} $$ and similarly for the 2$p$ basis functions. This is known as the 3-21G basis set (pronounced “three-two-one g” not “three-twenty one g”), which denotes that core basis functions are described by 3 contracted Gaussians, while the valence basis functions are split into two basis functions, described by 2 and 1 Gaussian each. Thus, using the 3-21G basis set to describe water requires 13 basis functions: two basis functions on each $\ce{H}$ atom (1$s$ is the valence basis function of the H atom) and 9 basis functions on the $\ce{O}$ atom (one 1$s$ function and two each of 2$s$, 2$p_x$, 2$p_y$, and 2$p_z$). The $\chi _{2{s_a}}$ basis function is smaller (i.e., the Gaussians have a larger exponent) than the basis function. Thus, one can make a function of any intermediate size by (variationally) mixing these two functions (Figure 3). The 3-21G is an example of a split valence or double zeta basis set (zeta, ζ, is often used as the symbol for the exponent, but I find it hard to write and don’t use it in my lectures). Similarly, one can make other double zeta basis sets such as 6-31G, or triple zeta basis sets such as 6-311G. Figure 3. Sketch of two different sized $s$-type basis functions that can be used to make a basis function of intermediate size As the number of basis functions ($K$ in Eq 2) increase the error associated with the LCAO approximation should decrease and the energy should converge to what is called the Hartree-Fock limit ($E_{\text{HF}}$) that is higher than the exact energy ($E_{\text{exact}}$) (Figure 4). The difference is known as the correlation energy, and is the error introduced by the orbital approximation $$\Psi ({{\bf{r}}_1},{{\bf{r}}_2}, \ldots {{\bf{r}}_N}) \approx \left\| {{\phi _1}(1){{\bar \phi }_1}(2) \ldots {\phi _{N/2}}(N - 1){{\bar \phi }_{N/2}}(N)} \right\rangle $$ Figure 4. Plot of the energy as a function the number of basis functions. However, in the case of a one-electron molecule like $\ce{H2+}$ we would expect the energy to converge to $E_{\text{exact}}$ since there is no orbital approximation. However, if we try this with the basis sets discussed thus far we find that this is not the case (Figure 5)! Figure 5. Plot of the energy of $\ce{H2+}$ computed using increasingly larger basis sets. What’s going on? Again we get a clue by comparing the exact wave function to the LCAO-wave function (Figure 6). Figure 6. Comparison of the exact wave function and one computed using the 6-311G basis set. We find that compared to the exact result there is not “enough wave function” between the nuclei and too much at either end. As we increase the basis set we only add $s$-type basis functions (of varying size) to the basis set. Since they are spherical they cannot be used to shift electron from one side of the $\ce{H}$ atom to the other. However, $p$-functions are perfect for this (Figure 7). Figure 7. Sketch of the polarization of an s basis function by a p basis function So basis set convergence is not a matter of simply increasing the number of basis functions, it is also important to have the right mix of basis function types. Similarly, $d$-functions can be used to “bend” $p$-functions (Figure 8). Figure 8. Sketch of the polarization of a p basis function by a d basis function Such functions are known as polarization functions, and are denoted with the following notation. For example, 6-31G(d) denotes d polarization functions on all non-$\ce{H}$ atoms and can also be written as 6-31G. 6-31G(d,p) is a 6-31G(d) basis set where p-functions have been added on all $\ce{H}$ atoms, and can also be written 6-31G*. A RHF/6-31G(d,p) calculation on water involves 24 basis functions: 13 basis functions for the 6-31G part (just like for 3-21G) plus 3 $p$-type polarization functions on each H atom and 5 $d$-type polarization functions (some programs use 6 Cartesian d-functions instead of the usual 5). Anions tend to have very diffuse electron distributions and very large basis functions (with very small exponents) are often needed for accurate results. These diffuse functions are denoted with “+” signs: e.g. 6-31+G denotes one s-type and three $p$-type diffuse Gaussians on each non-$\ce{H}$ atom, and 6-31++G denotes the addition of a single diffuse $s$-type Gaussian on each $\ce{H}$-atom. Diffuse functions also tend to improve the accuracy of calculations on van der Waals complexes and other structures where the accurate representation of the outer part of the electron distribution is important. Of course there are many other basis sets available, but in general they have the same kinds of attributes as described already. For example, aug-cc-pVTZ is a more modern basis set: “aug” stands for “augmented” meaning “augmented with diffuse functions”, “pVTZ” means “polarized valence triple zeta”, i.e. it is of roughly the same quality as 6-311++G(d,p). “cc” stands for “correlation consistent” meaning the parameters were optimized for correlated wave functions (like MP2, see below) rather than HF wave function like Pople basis sets [such as 6-31G(d)] described thus far. Related blog posts Complete basis set extrapolation Monday, June 1, 2015 Computational Chemistry Highlights: May issue The May issue of Computational Chemistry Highlights is out. Table of content for this issue features contributions from CCH editors Steven Bachrach and Jan Jensen: Exploring the Accuracy Limits of Local Pair Natural Orbital Coupled-Cluster Theory Uthrene, a radically new molecule? 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21a0f68a5ebf3a32	Sunday, May 10, 2015 Quantum Computing and the Many-Worlds Interpretation of Quantum Mechanics In my last posting, The Software Universe as an Implementation of the Mathematical Universe Hypothesis we explored Max Tegmark’s proposal that our physical Universe, and the Software Universe that we IT professionals and end-users are all immersed in, is simply an unchanging eternal mathematical structure that has always existed in a Platonic sense. In that posting we discussed Max Tegmark’s proposal that there is a Level III multiverse comprised of an infinite number of Level I and Level II multiverses that are constantly splitting due to Hugh Everett’s Many-Worlds Interpretation of quantum mechanics. In this posting I would like to further explore the Many-Worlds Interpretation of quantum mechanics as it relates to quantum computing because many quantum computer researchers consider it key to the advancement of quantum computing. The concept of quantum computing goes back to some early work in 1982 by Richard Feynman and David Deutsch, but it was David Deutsch who carried the idea forward and came up with the very first theoretical design of a quantum computer, similar to Alan Turing’s 1936 theoretical description for classical computers . Here is a link to David Deutsch’s seminal 1985 paper describing quantum computers and contrasting them with the classical computers that we work with today: A more accessible outline of quantum computing can be found in David Deutsch’s book The Fabric of Reality (1997). Another very good book on quantum computing is Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos (2006) by Seth Lloyd. Seth Lloyd is currently working on quantum computers at MIT, and is the first quantum mechanical engineer in MIT’s Mechanical Engineering department. Seth Lloyd is recognized for proposing the very first technologically feasible design for a quantum computer. In his book he proposes that the physical Universe is a huge quantum computer calculating how to behave and generates what we observe in the physical Universe, along the lines of Max Tegmark’s Level III multiverse. A good online synopsis of this idea is available in The Computational Universe (2002), in which he calculates the computing power of the entire physical Universe treated as one large quantum computer. You can find this fascinating paper at: So why is the Many-Worlds Interpretation important to quantum computer research? Well, the whole point to quantum computation is that a quantum computer can perform a huge number of logical operations in parallel using a limited amount of hardware, while classical computers need dedicated hardware for each logical operation. For example, in a classical computer, like your laptop, a 1-bit memory location can hold a 1 or a 0, but in a quantum computer, a 1-qubit memory location can hold both a 1 and a 0 at the same time in a mixed quantum state! In a classical computer, when your code reads the 1 or 0 at the top of an if-then-else block, it will do one thing or the other by branching either into the then-block of code or into the else-block of code. But in a quantum computer, a 1-qubit memory location can be in a mixed quantum state of being both 1 and 0 at the same time, so when the quantum computer reads the 1-qubit memory location, it logically splits into two quantum computers. One of the twin quantum computers performs the then-block, while the other quantum computer performs the else-block at the same time and in parallel. So with a quantum computer, you can have a single computer behave like a nearly infinite number of computers all working in parallel on the same problem at the same time. In that sense, a quantum computer would behave very much like Mickey’s water-carrying brooms in The Sorcerer's Apprentice segment of Walt Disney’s Fantasia, constantly splitting in two to perform a task at each logical branch of your program: In his book, David Deutsch asks the very compelling question of where exactly are all of those computations being performed, if not in a huge number of parallel universes by a huge number of parallel quantum computers? That is why the Many-World’s Interpretation of quantum mechanics seems so natural to those working on quantum computers. In fact, it is rather difficult to picture how a quantum computer could operate using the standard Copenhagen Interpretation of quantum mechanics. So let me refresh your memory on the Copenhagen Interpretation of quantum mechanics before proceeding. In 1927, Niels Bohr and Werner Heisenberg proposed a very positivistic interpretation of quantum mechanics now known as the Copenhagen Interpretation. You see, Bohr was working at the University of Copenhagen Institute of Theoretical Physics at the time. The Copenhagen Interpretation contends that absolute reality does not really exist. Instead, there are an infinite number of potential realities, defined by the wavefunction ψ of a quantum system, and when we make a measurement of a quantum system, the wavefunction of the quantum system collapses into a single value that we observe, and thus brings the quantum system into reality (see Quantum Software for more on wavefunctions). This satisfied Max Born’s contention that wavefunctions are just probability waves. The Copenhagen Interpretation suffers from several philosophical problems though. For example, Eugene Wigner pointed out that the devices we use to measure quantum events are also made out of atoms which are quantum objects in themselves, so when an observation is made of a single atom of uranium to see if it has gone through a radioactive decay using a Geiger counter, the atomic quantum particles of the Geiger counter become entangled in a quantum superposition of states with the uranium atom. If the uranium has decayed, then the uranium atom and the Geiger counter are in one quantum state, and if the atom has not decayed, then the uranium atom and the Geiger counter are in a different quantum state. If the Geiger counter is fed into an amplifier, then we have to add in the amplifier too into our quantum superposition of states. If a physicist is patiently listening to the Geiger counter, we have to add him into the chain as well, so that he can write and publish a paper which is read by other physicists and is picked up by Time magazine for a popular presentation to the public. So when does the “measurement” actually take place? We seem to have an infinite regress. Wigner’s contention is that the measurement takes place when a conscious being first becomes aware of the observation. Einstein had a hard time with the Copenhagen Interpretation of quantum mechanics for this very reason because he thought that it verged upon solipsism. Solipsism is a philosophical idea from Ancient Greece. In solipsism, your Mind is the whole thing, and the physical Universe is just a figment of your imagination. So I would like to thank you very much for thinking of me and bringing me into existence! Einstein’s opinion of the Copenhagen Interpretation of quantum mechanics can best be summed up by his statement "Is it enough that a mouse observes that the Moon exists?". Einstein objected to the requirement for a conscious being to bring the Universe into existence, because in Einstein’s view, measurements simply revealed to us the condition of an already existing reality that does not need us around to make measurements in order to exist. But in the Copenhagen Interpretation, the absolute reality of Einstein does not really exist. Additionally, in the Copenhagen Interpretation, objects do not really exist until a measurement is taken, which collapses their associated wavefunctions, but the mathematics of quantum mechanics does not shed any light on how a measurement could collapse a wavefunction. The collapse of the wavefunction is also a one-way street. According to the mathematics of quantum mechanics a wavefunction changes with time in a deterministic manner, so like all of the other current effective theories of physics, they are reversible in time and can be run backwards. This is also true in the Copenhagen Interpretation, so long as you do not observe the wavefunction and collapse it by the process of observing it. In the Copenhagen Interpretation, once you observe a wavefunction and collapse it, you cannot undo the collapse, so the process of observation becomes nonreversible in time. That means if you fire photons at a target, but do not observe them, it is possible to reverse them all in time and return the Universe back to its original state. That is how all of the other effective theories of physics currently operate. But in the Copenhagen Interpretation, if you do observe the outgoing photons you can never return the Universe back to its original state. This can best be summed up by the old quantum mechanical adage - look particle, don’t look wave. A good way to image this in your mind is to think of a circular tub of water. If you drop a pebble into the exact center of a circular tub of water, a series of circular waves will propagate out from the center. Think of those waves as the wavefunction of an electron changing with time into the future according to the Schrödinger equation. When the circular waves hit the circular walls of the tub they will be reflected back to the center of the tub. Essentially, they can be viewed as moving backwards in time. This can happen in the Copenhagen Interpretation so long as the electron is never observed as its wavefunction moves forward or backward in time. However, if the wavefunction is observed and collapsed, it can never move backwards in time, so observation becomes a one-way street. In 1956 , Hugh Everett working on his Ph.D. under John Wheeler, proposed the Many-Worlds Interpretation of quantum mechanics as an alternative. The Many-Worlds Interpretation admits an absolute reality, but claims that there are an infinite number of absolute realities spread across an infinite number of parallel universes. In the Many-Worlds Interpretation, when electrons or photons encounter a two-slit experiment, they go through one slit or the other, and when they hit the projection screen they interfere with electrons or photons from other universes that went through the other slit! In Everett’s original version of the Many-Worlds Interpretation, the entire Universe splits into two distinct universes whenever a particle is faced with a choice of quantum states, and so all of these universes are constantly branching into an ever growing number of additional universes. In the Many-Worlds Interpretation of quantum mechanics, the wavefunctions or probability clouds of electrons surrounding an atomic nucleus are the result of overlaying the images of many “real” electrons in many parallel universes. Thus, according to the Many-Worlds Interpretation wavefunctions never collapse. They just deterministically evolve in an abstract mathematical Hilbert space and are reversible in time, like everything else in physics. While doing research for The Software Universe as an Implementation of the Mathematical Universe Hypothesis I naturally consulted Max Tegmark’s HomePage at: and I found a link there to Hugh Everett’s original 137-page Jan 1956 draft Ph.D. thesis in which he laid down the foundations for the Many-Worlds Interpretation. This is a rare document indeed because on March 1, 1957, Everett submitted a very compressed version of his theory in his final 36-page doctoral dissertation, "On the Foundations of Quantum Mechanics", after heavy editing by his thesis advisor John Wheeler to make his Ph.D. thesis more palatable to the committee that would be hearing his oral defense and also to not offend Niels Bohr, one of the founding fathers of the Copenhagen Interpretation and still one of its most prominent proponents. But years later John Wheeler really did want to know what Niels Bohr thought of Hugh Everett’s new theory and encouraged Everett to visit Copenhagen in order to meet with Bohr. Everett and his wife did finally travel to Copenhagen in March of 1959, and spent six weeks there. But by all accounts the meeting between Bohr and Everett was a disaster, with Bohr not even discussing the Many-Worlds Interpretation with Everett. Below is the link to Hugh Everett’s original 137-page Jan 1956 draft Ph.D. thesis: I have also placed his thesis on Microsoft One Drive at:!1437&authkey=!ADIm_WTYLkbx90I&ithint=file%2cpdf Since I love to read the original source documents for great ideas, like Copernicus’s On the Revolutions of the Celestial Spheres (1543), Galileo’s the Starry Messenger (1610) and Dialogue Concerning the Two Chief World Systems (1632), Newton’s Principia (1687), and Darwin’s On the Origin of Species (1859), I could not resist reading Hugh Everett’s original work too. So in this posting I would like to step through Hugh Everett’s original Ph.D. thesis with you page by page, with a little translation along the way. To do that, let’s focus on the introduction and the concluding chapter of his original Ph.D. thesis, where he outlines what he is trying to achieve, and then skip over most of the math in the intervening chapters. For those chapters I will only highlight his key findings as he builds his case for the Many-Worlds Interpretation. For the remainder of this posting, direct quotes from Hugh Everett’s original Ph.D. thesis will be in blue, while my comments will be in black. The Many-Worlds Interpretation Hugh Everett, III We begin, as a way of entering our subject, by characterizing a particular interpretation of quantum theory which, although not representative of the more careful formulations of some writers, is the most common form encountered in textbooks and university lectures on the subject. With the very first sentence of his Ph.D. thesis, Hugh Everett lays down the gauntlet and begins by discussing the Copenhagen Interpretation of quantum mechanics and classifying it as not one of the more “careful formulations”. He is correct about the textbooks of the day exclusively teaching the Copenhagen Interpretation. I took my very first quantum mechanics course in 1970, and in those days the Copenhagen Interpretation was taught as a quantum mechanical fact. In fact, the textbooks of the day did not even refer to the idea of the act of measurement collapsing wavefunctions as the Copenhagen Interpretation because that would infer that other interpretations were even possible. A physical system is described completely by a state function ψ, which is an element of a Hilbert space, and which furthermore gives information only concerning the probabilities of the results of various observations which can be made on the system. The state function ψ is thought of as objectively characterizing the physical system, i.e., at all times an isolated system is thought of as possessing a state function, independently of our state of knowledge of it. On the other hand, ψ changes in a causal manner so long as the system remains isolated, obeying a differential equation. Thus there are two fundamentally different ways in which the state function can change: Hugh Everett begins his dissertation stating what everybody already agrees upon in classical quantum mechanics. Every physical system, like a single electron, can be described by a wavefunction called ψ that is a solution to Schrödinger’s equation. Note that in his thesis Hugh Everett sometimes uses the term “state function” and sometimes the term “wave function”, rather than the term wavefunction ψ. All of these terms mean the same thing. They are just solutions to the Schrödinger wave equation, which sometimes Hugh Everett refers to simply as the “wave equation”. The wavefunction ψ is a wiggly line that extends over the whole Universe, but has the greatest amplitude near where the electron is most likely to be found. The wavefunction ψ is also a complex function with both a real and imaginary part, so it has both an amplitude and a phase (See The Foundations of Quantum Computing for details). The chief difference between quantum mechanics and classical mechanics is that in classical mechanics objects have definite properties, like a definite position or a definite velocity. This is not so in quantum mechanics. In quantum mechanics, objects can be in a mixture or superposition of states. For example, if you pin down the exact location of an electron in quantum mechanics, the electron is said to be in a certain state of position called an eigenstate and with a certain numerical position that is called an eigenvalue. In the Copenhagen Interpretation the act of measurement takes an object that is in a superposition of states and collapses its wavefunction down into a particular eigenstate with a particular eigenvalue. And this is a totally probabilistic process. The wavefunction itself does not specifically say where the object is located in advance. The wavefunction just tells you the probability of observing specific eigenstates with specific eigenvalues, and this probability is obtained by finding the square of the wavefunction's amplitude at a given position. For example, observing a hydrogen atom which initially is in a superposition of many states might determine that the hydrogen atom is in its ground state eigenstate at a known energy level that is its energy eigenvalue. Hugh Everett then goes on to define two ways the wavefunction ψ can change with time: Process 1: A discontinuous change brought on by observation. In the standard Copenhagen Interpretation this causes the wavefunction ψ of the electron which is spread out over the entire Universe with decreasing amplitude as you get further away from where the electron is likely to be found, to suddenly collapse so that the amplitude of the wavefunction ψ of the electron becomes huge where the electron is observed. Process 2: The electron is not observed, so its wavefunction ψ sort of smears out with time in a deterministic manner. The question of the consistency of the scheme arises if one contemplates regarding the observer and his object-system as a single (composite) physical system. Indeed, the situation becomes quite paradoxical if we allow for the existence of more than one observer. Let us consider the case of one observer A, who is performing measurements upon a system S, the totality (A + S) in turn forming the object-system for another observer, B. If we are to deny the possibility of B's use of a quantum mechanical description (wave function obeying wave equation) for A + S, then we must be supplied with some alternative description for systems which contain observers (or measuring apparatus). Furthermore, we would have to have a criterion for telling precisely what type of systems would have the preferred positions of "measuring apparatus" or "observer" and be subject to the alternate description. Such a criterion is probably not capable of rigorous formulation. On the other hand, if we do allow B to give a quantum description to A + S, by assigning a state function ψ A+S, then, so long as B does not interact with A + S, its state changes causally according to Process 2, even though A may be performing measurements upon S. From B's point of view, nothing resembling Process 1 can occur (there are no discontinuities), and the question of the validity of A's use of Process 1 is raised. That is, apparently either A is incorrect in assuming Process 1, with its probabilistic implications, to apply to his measurements, or else B's state function, with its purely causal character, is an inadequate description of what is happening to A + S. Basically, if I am observer A and I observe electron S using Process 1, according to the Copenhagen Interpretation I collapse the wavefunction ψ of the electron down to a single point in space. But I am just made up of a huge number of quantum particles too, just like the single electron that I observed. So if you, as observer B, do not watch me (as observer A) observing electron S, the wavefunction ψ A+S that describes me and the electron does not collapse and it continues to change in a deterministic manner according to Process 2. So either I, as observer A, do not really collapse the wavefunction of the electron with Process 1, or you, as observer B, do not let the combination of me and the electron evolve in time in an undisturbed manner according to Process 2. Thus, the Copenhagen Interpretation leads to a contradiction when more than one observer is involved. To better illustrate the paradoxes which can arise from strict adherence to this interpretation we consider the following amusing, but extremely hypothetical drama. Isolated somewhere out in space is a room containing an observer, A, who is about to perform a measurement upon a system S. After performing his measurement he will record the result in his notebook. We assume that he knows the state function of S (perhaps as a result of previous measurement), and that it is not an eigenstate of the measurement he is about to perform. A, being an orthodox quantum theorist, then believes that the outcome of his measurement is undetermined and that the process is correctly described by Process 1. In the meantime, however, there is another observer, B, outside the room, who is in possession of the state function of the entire room, including S, the measuring apparatus, and A, just prior to the measurement. B is only interested in what will be found in the notebook one week hence, so he computes the state function of the room for one week in the future according to Process 2. One week passes, and we find B still in possession of the state function of the room, which this equally orthodox quantum theorist believes to be a complete description of the room and its contents. If B's state function calculation tells beforehand exactly what is going to be in the notebook, then A is incorrect in his belief about the indeterminacy of the outcome of his measurement. We therefore assume that B's state function contains non-zero amplitudes over several of the notebook entries. At this point, B opens the door to the room and looks at the notebook (performs his observation). Having observed the notebook entry, he turns to A and informs him in a patronizing manner that since his (B's) wave function just prior to his entry into the room, which he knows to have been a complete description of the room and its contents, had non-zero amplitude over other than the present result of the measurement, the result must have been decided only when B entered the room, so that A, his notebook entry, and his memory about what occurred one week ago had no independent objective existence until the intervention by B. In short, B implies that A owes his present objective existence to B's generous nature which compelled him to intervene on his behalf. However, to B's consternation, A does not react with anything like the respect and gratitude he should exhibit towards B, and at the end of a somewhat heated reply, in which A conveys in a colorful manner his opinion of B and his beliefs, he rudely punctures B's ego by observing that if B's view is correct, then he has no reason to feel complacent, since the whole present situation may have no objective existence, but may depend upon the future actions of yet another observer. Clearly, in a Universe with more than one observer, the opening hypothesis of his thesis that wavefunctions change in time by either Process 1 or Process 2 cannot be right. Otherwise, nothing in the Universe would ever “really” happen until its very last sentient being took a peek into the room above and collapsed its very complicated wavefunction. Hugh Everett next proposes several alternative explanations. It is now clear that the interpretation of quantum mechanics with which we began is untenable if we are to consider a universe containing more than one observer. We must therefore seek a suitable modification of this scheme, or an entirely different system of interpretation. Several alternatives which avoid the paradox are: Alternative 1: To postulate the existence of only one observer in the universe. This is the solipsist position, in which each of us must hold the view that he alone is the only valid observer, with the rest of the universe and its inhabitants obeying at all times Process 2 except when under his observation. This view is quite consistent, but one must feel uneasy when, for example, writing textbooks on quantum mechanics, describing Process 1, for the consumption of other persons to whom it does not apply. If we try to limit the applicability so as to exclude measuring apparatus, or in general systems of macroscopic size, we are faced with the difficulty of sharply defining the region of validity. For what n might a group of n particles be construed as forming a measuring device so that the quantum description fails? And to draw the line at human or animal observers, i.e., to assume that all mechanical aparata obey the usual laws, but that they are somehow not valid for living observers, does violence to the so-called principle of psycho-physical parallelism, and constitutes a view to be avoided, if possible. To do justice to this principle we must insist that we be able to conceive of mechanical devices (such as servomechanisms), obeying natural laws, which we would be willing to call observers. Alternative 3: To admit the validity of the state function description, but to deny the possibility that B could ever be in possession of the state function of A + S. Thus one might argue that a determination of the state of A would constitute such a drastic intervention that A would cease to function as an observer. The first objection to this view is that no matter what the state of A + S is, there is in principle a complete set of commuting operators for which it is an eigenstate, so that, at least, the determination of these quantities will not affect the state nor in any way disrupt the operation of A. There are no fundamental restrictions in the usual theory about the knowability of any state functions, and the introduction of any such restrictions to avoid the paradox must therefore require extra postulates. The second objection is that it is not particularly relevant whether or not B actually knows the precise state function of A + S. If he merely believes that the system is described by a state function, which he does not presume to know, then the difficulty still exists. He must then believe that this state function changed deterministically, and hence that there was nothing probabilistic in A's determination. It is assumed that the correct complete description, which would presumably involve further (hidden) parameters beyond the state function alone, would lead to a deterministic theory, from which the probabilistic aspects arise as a result of our ignorance of these extra parameters in the same manner as in classical statistical mechanics. This brief list of alternatives is not meant to be exhaustive, but has been presented in the spirit of a preliminary orientation. We have, in fact, omitted one of the foremost interpretations of quantum theory, namely the position of Niels Bohr. The discussion will be resumed in the final chapter, when we shall be in a position to give a more adequate appraisal of the various alternate interpretations. For the present, however, we shall concern ourselves only with the development of Alternative 5. Alternative 5 is Hugh Everett’s Many-Worlds Interpretation of quantum mechanics. In this interpretation of quantum mechanics he completely eliminates Process 1 as a way for wavefunctions to change with time. Instead, he plans to bring in the acts of measurement and observation under Process 2, and simply let the wavefunctions evolve with time according to the wave equation. In this interpretation of quantum mechanics wavefunctions are the fundamental thing and provide all that can be known of the Universe. In fact, the whole Universe can be considered to be one single very complex wavefunction evolving with time. That is why he calls his theory the theory of the universal wavefunction. We shall be able to introduce into the theory systems which represent observers. Such systems can be conceived as automatically functioning machines (servomechanisms) possessing recording devices (memory) and which are capable of responding to their environment. The behavior of these observers shall always be treated within the framework of wave mechanics. Furthermore, we shall deduce the probabilistic assertions of Process 1 as subjective appearances to such observers, thus placing the theory in correspondence with experience. We are then led to the novel situation in which the formal theory is objectively continuous and causal, while subjectively discontinuous and probabilistic. While this point of view thus shall ultimately justify our use of the statistical assertions of the orthodox view, it enables us to do so in a logically consistent manner, allowing for the existence of other observers. At the same time it gives a deeper insight into the meaning of quantized systems, and the role played by quantum mechanical correlations. In order to bring about this correspondence with experience for the pure wave mechanical theory, we shall exploit the correlation between subsystems of a composite system which is described by a state function. A subsystem of such a composite system does not, in general, possess an independent state function. That is, in general a composite system cannot be represented by a single pair of subsystem states, but can be represented only by a superposition of such pairs of subsystem states…. there is no single state for Particle 1 alone or Particle 2 alone, but only the superposition of such cases. In fact, to any arbitrary choice of state for one subsystem there will correspond a relative state for the other subsystem, which will generally be dependent upon the choice of state for the first subsystem, so that the state of one subsystem is not independent, but correlated to the state of the remaining subsystem. Such correlations between systems arise from interaction of the systems, and from our point of view all measurement and observation processes are to be regarded simply as interactions between observer and object-system which produce strong correlations. Let one regard an observer as a subsystem of the composite system: observer + object-system. It is then an inescapable consequence that after the interaction has taken place there will not, generally, exist a single observer state. There will, however, be a superposition of the composite system states, each element of which contains a definite observer state and a definite relative object-system state. Furthermore, as we shall see, each of these relative object-system states will be, approximately, the eigenstates of the observation corresponding to the value obtained by the observer which is described by the same element of the superposition. Thus, each element of the resulting superposition describes an observer who perceived a definite and generally different result, and to whom it appears that the object-system state has been transformed into the corresponding eigenstate. In this sense the usual assertions of Process 1 appear to hold on a subjective level to each observer described by an element of the superposition. We shall also see that correlation plays an important role in preserving consistency when several observers are present and allowed to interact with one another (to "consult" one another) as well as with other object-systems. In order to develop a language for interpreting our pure wave mechanics for composite systems we shall find it useful to develop quantitative definitions for such notions as the "sharpness" or "definiteness" of an operator A for a state ψ, and the "degree of correlation" between the subsystems of a composite system or between a pair of operators in the subsystems, so that we can use these concepts in an unambiguous manner. The mathematical development of these notions will be carried out in the next chapter (II) using some concepts borrowed from Information Theory. We shall develop there the general definitions of information and correlation, as well as some of their more important properties. Throughout Chapter II we shall use the language of probability theory to facilitate the exposition, and because it enables us to introduce in a unified manner a number of concepts that will be of later use. We shall nevertheless subsequently apply the mathematical definitions directly to state functions, by replacing probabilities by square amplitudes, without, however, making any reference to probability models. Having set the stage, so to speak, with Chapter II, we turn to quantum mechanics in Chapter III. There we first investigate the quantum formalism of composite systems, particularly the concept of relative state functions, and the meaning of the representation of subsystems by noninterfering mixtures of states characterized by density matrices. The notions of information and correlation are then applied to quantum mechanics. The final section of this chapter discusses the measurement process, which is regarded simply as a correlation-inducing interaction between subsystems of a single isolated system. A simple example of such a measurement is given and discussed, and some general consequences of the superposition principle are considered. This will be followed by an abstract treatment of the problem of Observation (Chapter IV). In this chapter we make use only of the superposition principle, and general rules by which composite system states are formed of subsystem states, in order that our results shall have the greatest generality and be applicable to any form of quantum theory for which these principles hold. (Elsewhere, when giving examples, we restrict ourselves to the non-relativistic Schrödinger Theory for simplicity.) The validity of Process 1 as a subjective phenomenon is deduced, as well as the consistency of allowing several observers to interact with one another. Chapter V supplements the abstract treatment of Chapter IV by discussing a number of diverse topics from the point of view of the theory of pure wave mechanics, including the existence and meaning of macroscopic objects in the light of their atomic constitution, amplification processes in measurement, questions of reversibility and irreversibility, and approximate measurement. The final chapter summarizes the situation, and continues the discussion of alternate interpretations of quantum mechanics. With that Hugh Everett ends the introduction of his thesis. Basically, he is proposing that Process 1, in which an external observer A observes a quantum system like an electron, and causes a discontinuous change to the electron’s wavefunction ψ is an illusion. Instead, the wavefunction of observer A becomes “correlated” with the wavefunction of the electron into a composite wavefunction of both observer A and the electron. This composite wavefunction then evolves in time according to the wave equation. Thus, Process 1 really does not exist. Everything in the Universe just evolves in time according to Process 2. In the following chapters, Hugh Everett goes on to explain how this evolution of a correlated composite wavefunction can produce all of the strange quantum mechanical things we observe in the lab. The present chapter is devoted to the mathematical development of the concepts of information and correlation. As mentioned in the introduction we shall use the language of probability theory throughout this chapter to facilitate the exposition, although we shall apply the mathematical definitions and formulas in later chapters without reference to probability models. We shall develop our definitions and theorems in full generality, for probability distributions over arbitrary sets, rather than merely for distributions over real numbers, with which we are mainly interested at present. We take this course because it is as easy as the restricted development, and because it gives a better insight into the subject. The first three sections develop definitions and properties of information and correlation for probability distributions over finite sets only. In section four the definition of correlation is extended to distributions over arbitrary sets, and the general invariance of the correlation is proved. Section five then generalizes the definition of information to distributions over arbitrary sets. Finally, as illustrative examples, sections seven and eight give brief applications to stochastic processes and classical mechanics, respectively. Now the really heavy math begins once we leave the Introduction and proceed into the main body of Hugh Everett’s Ph.D. thesis. He first goes into the mathematics of probability distributions that would be familiar to anybody who deals with statistics. He is mainly concerned with joint probabilities and conditional distributions because he is focusing upon what happens when observer A observes electron S. He reminds the reader about what independence means because it is important to his proposal. If observer A never interacts with electron S it means that they are statistically independent and cannot affect each other, but if observer A does observe electron S it means they are correlated and that is a whole different story. Independence means that the random variables take on values which are not influenced by the values of other variables with respect to which they are independent. That is, the conditional distribution of one of two independent variables, Y, conditioned upon the value xi for the other, is independent of xi, so that knowledge about one variable tells nothing of the other. Next he applies Claude Shannon’s concept of Information (1948) to the analysis (see Some More Information About Information for details) for joint distributions that are correlated and not independent. Remember, Claude Shannon’s formulation for the concept of Information hinges upon the amount of “surprise” there is in a signal composed of 1s and 0s, while in softwarephysics we use Leon Brillouin’s concept of Information as a form of negative entropy (see The Demon of Software for details). Hugh Everett goes on to conclude that if two things are not dependent upon each other, like me observing electron S1 and you observing a different electron S2 that the total amount of Information obtained is equal to the sum of the Information I get about electron S1 and you get about electron S2. For independent random variables X, Y, ... ,Z, the following relationship is easily proved: (2.4) IXY... Z = IX + IY + ... + IZ (X, Y, ... ,Z independent) , so that the information of XY... Z is the sum of the individual quantities of information, which is in accord with our intuitive feeling that if we are given information about unrelated events, our total knowledge is the sum of the separate amounts of information. …. But what if there is some correlation when measuring two things that depend upon each other? Like measuring the market value of your home and your annual salary? Those two numbers are not independent of each other, so knowing one tells you something about the other. ….which we are told, the natural thing to do to arrive at a single number to measure the strength of correlation is to consider the expected change in information about X, given that we are to be told the value of Y. This quantity we call the correlation information, or for brevity, the correlation, of X and Y, and denote it by {X, Y}…. ….Thus the correlation is symmetric between X and Y, and hence also equal to the expected change of information about Y given that we will be told the value of X. Furthermore, according to (3.3) the correlation corresponds precisely to the amount of "missing information" if we possess only the marginal distributions, i.e., the loss of information if we choose to regard the variables as independent. Here he is saying that suppose you have a box full of interacting molecules and you know their individual positions and velocities. That information he defines as the marginal information about the marginal distributions of the molecules. But if molecule X bounces off molecule Y, then their positions and velocities will no longer be independent because molecule X has interacted with molecule Y, so their marginal distributions are correlated. Hugh Everett calls the information tied up with that correlation the correlation information {X,Y} and it is symmetric {X,Y} = {Y,X}. He then goes on to take his definition of correlation information to derive the conservation of information in classical mechanics (see The Demon of Software for details). Remember that physicists get very nervous about the idea of destroying information because then they cannot reverse the effective theories of physics in time. we have proved that….and the total information is conserved. Now it is known that the individual (marginal) position and momentum distributions tend to decay, except for rare fluctuations, into the uniform and Maxwellian distributions respectively, for which the classical entropy is a maximum. This entropy is, however, except for the factor of Boltzman's constant, simply the negative of the marginal information (7.4) Imarginal = IX1 + IY1 + IZ1 + ... + Ipxn + Ipyn + Ipzn which thus tends towards a minimum. But this decay of marginal information is exactly compensated by an increase of the total correlation information (7.5) {total} = Itotal - Imarginal Thus Hugh Everett ends Chapter II of his thesis by deriving a concept of Information that is very similar to Leon Brillouin’s concept of Information as being a form of negative entropy. Remember, Hugh Everett defined marginal information as the information about the individual molecules, and {total} as the total amount of correlation information that is created by the molecules bouncing off each other and becoming correlated. So we can rewrite equation (7.5) as: Itotal = Icorrelation + Imarginal which says that the total amount of Information in classical mechanics does not change and that information is conserved. For example, let’s say you start off with a box that initially only has molecules on the left side of the box. So you begin with lots of marginal information about the individual molecules because you know they are all in the left side of the box. But as time progresses, the molecules will bounce off each other and begin to scatter into the right side of the box, until the molecules finally become smoothly spread throughout the entire box, as the second law of thermodynamics predicts. In the process, the marginal information of the molecules will decrease with time, but at the same time the correlation information of the molecules will increase as the molecules bounce off each other and become correlated, so the total amount of information remains constant. Having mathematically formulated the ideas of information and correlation for probability distributions, we turn to the field of quantum mechanics. In this chapter we assume that the states of physical systems are represented by points in a Hilbert space, and that the time dependence of the state of an isolated system is governed by a linear wave equation. It is well known that state functions lead to distributions over eigenvalues of Hermitian operators (square amplitudes of the expansion coefficients of the state in terms of the basis consisting of eigenfunctions of the operator) which have the mathematical properties of probability distributions (non-negative and normalized). The standard interpretation of quantum mechanics regards these distributions as actually giving the probabilities that the various eigenvalues of the operator will be observed, when a measurement represented by the operator is performed. A feature of great importance to our interpretation is the fact that a state function of a composite system leads to joint distributions over subsystem quantities, rather than independent subsystem distributions, i.e., the quantities in different subsystems may be correlated with one another. The first section of this chapter is accordingly devoted to the development of the formalism of composite systems, and the connection of composite system states and their derived joint distributions with the various possible subsystem conditional and marginal distributions. We shall see that there exist relative state functions which correctly give the conditional distributions for all subsystem operators, while marginal distributions can not generally be represented by state functions, but only by density matrices. In Section 2 the concepts of information and correlation, developed in the preceding chapter, are applied to quantum mechanics, by defining information and correlation for operators on systems with prescribed states. It is also shown that for composite systems there exists a quantity which can be thought of as the fundamental correlation between subsystems, and a closely related canonical representation of the composite system state. In addition, a stronger form of the uncertainty principle, phrased in information language, is indicated. The third section takes up the question of measurement in quantum mechanics, viewed as a correlation producing interaction between physical systems. A simple example of such a measurement is given and discussed. Finally some general consequences of the superposition principle are considered. Hugh Everett then goes on to discuss composite systems consisting of several parts. For example, suppose we have a system S composed of two electrons S1 and S2. ….It is well known that if the states of a pair of systems S1 and S2 are represented by points in Hilbert spaces H1 and H2 respectively, then the states of the composite system S = S1 + S2 (the two systems S1 and S2 regarded as a single system S are represented correctly by points of the direct product of H1 and H2…. After a great a great deal of math Hugh Everett concludes: ….Therefore there exists in general no state for S1 which correctly gives the marginal expectations for all operators in S1…. However, even though there is generally no single state describing marginal expectations, we see that there is always a mixture of states, .... which does yield the correct expectations. which means that when two systems S1 and S2 interact with each other, there is no wavefunction for just S1 or S2 alone, but there is a wavefunction for the composite system of S1 and S2 together. That means that when you as system S1 observe an electron S2 there are no longer separate wavefunctions for you and the electron. Instead, you and the electron become entangled into a single composite wavefunction for both you and the electron. He goes on to summarize this as: In summary, we have seen in this section that a state of a composite system leads to joint distributions over subsystem quantities which are generally not independent. Conditional distributions and expectations for subsystems are obtained from relative states, and subsystem marginal distributions and expectations are given by density matrices. There does not, in general, exist anything like a single state for one subsystem of a composite system. That is, subsystems do not possess states independent of the states of the remainder of the system, so that the subsystem states are generally correlated. One can arbitrarily choose a state for one subsystem, and be led to the relative state for the other subsystem. Thus we are faced with a fundamental relativity of states, which is implied by the formalism of composite systems. It is meaningless to ask the absolute state of a subsystem - one can only ask the state relative to a given state of the remainder of the system…. Next he discusses the marginal information of individual particles and the correlated information due to the particles interacting with each other by bouncing off each other in terms of operators acting upon their wavefunctions ψ. Remember in quantum mechanics, the wavefunction ψ of a particle is the whole deal and contains all of the information there is about the particle, like its position and velocity. In quantum mechanics that information is determined by applying mathematical operators to the wavefunction ψ. For example, if you want to know how much energy a particle has, there is a mathematical operator that you can apply to its wavefunction ψ that will give you an actual number. It’s like if you want to know how much money somebody has on them, you can apply an operation to them that frisks them down and checks all of their pockets for wallets, billfolds, loose bills and change and then adds it all up. We wish to be able to discuss information and correlation for Hermitian operators A, B, ... , with respect to a state function ψ. These quantities are to be computed, through the formulas of the preceding chapter, from the square amplitudes of the coefficients of the expansion of ψ terms of the eigenstates of the operators. Finally Hugh Everett notes that quantum mechanics is very much like classical mechanics in regards to the relationship between the entropy and Information that is obtained by applying the above mathematical operators. Remember, in classical mechanics we saw that the correlation information is the information that arises from particles interacting with each other by bouncing off each other. The chief difference for quantum mechanics is that instead of particles bouncing off each other, we have mathematical operators operating on their wavefunctions instead: ….It is also interesting to note that the quantity - Trace(ρ ln ρ ) is (apart from a factor of Boltzman's constant) just the entropy of a mixture of states characterized by the density matrix ρ. Therefore the entropy of the mixture characteristic of a subsystem S1 for the state ψS = ψS1 + S2 is exactly matched by a correlation information {S1 ,S2}, which represents the correlation between any pair of operators A, B, which define the canonical representation. The situation is thus quite similar to that of classical mechanics. Next Hugh Everett takes up the thorny issues of measurement in quantum mechanics. Recall that in the Copenhagen Interpretation measuring the wavefunction ψ of an electron causes it to mysteriously collapse to a single point and that is where you will find the electron. We now consider the question of measurement in quantum mechanics, which we desire to treat as a natural process within the theory of pure wave mechanics. From our point of view there is no fundamental distinction between "measuring apparata" and other physical systems. For us, therefore, a measurement is simply a special case of interaction between physical systems - an interaction which has the property of correlating a quantity in one subsystem with a quantity in another…. ….Nearly every interaction between systems produces some correlation however. Suppose that at some instant a pair of systems are independent, so that the composite system state function is a product of subsystem states ψS = ψS1 ψS2 . Then this condition obviously holds only instantaneously if the systems are interacting - the independence is immediately destroyed and the systems become correlated. We could, then, take the position that the two interacting systems are continually "measuring" one another, if we wished…. ….Suppose that we have a system of only one coordinate, q, (such as position of a particle), and an apparatus of one coordinate r (for example the position of a meter needle)…. ….This principle has the far reaching implication that for any possible measurement, for which the initial system state is not an eigenstate, the resulting state of the composite system leads to no definite system state nor any definite apparatus state. The system will not be put into one or another of its eigenstates with the apparatus indicating the corresponding value, and nothing resembling Process 1 can take place…. ….Thus in general after a measurement has been performed there will be no definite system state nor any definite apparatus state, even though there is a correlation. It seems as though nothing can ever be settled by such a measurement. Furthermore this result is independent of the size of the apparatus, and remains true for apparatus of quite macroscopic dimensions…. Suppose, for example, that we coupled a spin measuring device to a cannonball, so that if the spin is up the cannonball will be shifted one foot to the left, while if the spin is down it will be shifted an equal distance to the right. If we now perform a measurement with this arrangement upon a particle whose spin is a superposition of up and down, then the resulting total state will also be a superposition of two states, one in which the cannonball is to the left, and one in which it is to the right. There is no definite position for our macroscopic cannonball! This behavior seems to be quite at variance with our observations, since macroscopic objects always appear to us to have definite positions. Can we reconcile this prediction of the purely wave mechanical theory with experience, or must we abandon it as untenable? In order to answer this question we must consider the problem of observation itself within the framework of the theory. To understand the above section we need a little background in experimental physics. Electrons have a quantum mechanical property called spin. You can think of an electron’s spin like the electron has a little built-in magnet. In fact, it is the spin of the little electron magnets that add up to make the real magnets that you put on your refrigerator. When you throw an electron through a distorted magnetic field that is pointing up the electron will pop out in one of two states. It will either be aligned with the magnetic field (called spin up) or it will be pointing 1800 in the opposite direction of the magnetic field (called spin down). Both the spin up and spin down conditions are called an eigenstate. Prior to the observation of the electron’s spin, the electron is in a superposition of states and is not in an eigenstate. Now if the electron in the eigenstate of spin up is sent through the same magnetic field again, it will be found to pop out in the eigenstate of spin up again. Similarly, a spin down electron that is sent through the magnetic field again will also pop out as a spin down electron. Now here is the strange part. If you rotate the magnetic field by 900 and send spin up electrons through it, 50% of the electrons will pop out with a spin pointing to the left, and 50% will pop out with a spin pointing to the right. And you cannot predict in advance which way a particular spin up electron will pop out. It might spin to the left, or it might spin to the right. The same goes for the spin down electrons – 50% will pop out spinning to the left and 50% will pop out spinning to the right. Figure 1 - In the Stern-Gerlach experiment we shoot electrons through a distorted magnetic field. Classically, we would expect the electrons to be spinning in random directions and the magnetic field should deflect them in random directions, creating a smeared out spot on the screen. Instead, we see that the act of measuring the spins of the electrons puts them into eigenstates with eigenvalues of spin up or spin down and the electrons are either deflected up or down. If we rotate the magnets by 900, we find that the electrons are deflected to the right or to the left. In the above section, Hugh Everett is proposing that when a device, like our magnets above, measures the spin of an electron that is in an unknown state, and not in a spin up or spin down eigenstate, the device does not put the electron into a spin up or spin down eigenstate as the Copenhagen Interpretation maintains. Instead the device and the electron enter into a correlated composite system state or combined wavefunction with an indeterminate spin of the electron. In the next chapter Hugh Everett explains how this new worldview can be used to explain what we observe in the lab. In fact, he will propose that from the perspective of the measuring magnets and the electron, two independent observational histories will emerge, one with the measuring magnets finding a spin up electron and one with the measuring magnets finding a spin down electron, and both of these will be just as “real” as the other. For them, the Universe has essentially split in two, with each set in its own Universe. That is where the “Many-Worlds” in the Many-Worlds Interpretation of quantum mechanics comes from. We shall now give an abstract treatment of the problem of observation. In keeping with the spirit of our investigation of the consequences of pure wave mechanics we have no alternative but to introduce observers, considered as purely physical systems, into the theory. We saw in the last chapter that in general a measurement (coupling of system and apparatus) had the outcome that neither the system nor the apparatus had any definite state after the interaction - a result seemingly at variance with our experience. However, we do not do justice to the theory of pure wave mechanics until we have investigated what the theory itself says about the appearance of phenomena to observers, rather than hastily concluding that the theory must be incorrect because the actual states of systems as given by the theory seem to contradict our observations. Recall that in Chapter III Hugh Everett demonstrated that when an observer O observes an electron, the wavefunction ψ of observer O and whatever apparatus that is used to observe the electron become entangled or “correlated” with the wavefunction ψ of the electron into a total state function ψ of the Observer and the electron together, and that neither the observer O nor the electron have separate wavefunctions after the observation is made. We shall see that the introduction of observers can be accomplished in a reasonable manner, and that the theory then predicts that the appearance of phenomena, as the subjective experience of these observers, is precisely in accordance with the predictions of the usual probabilistic interpretation of quantum mechanics. We are faced with the task of making deductions about the appearance of phenomena on a subjective level, to observers which are considered as purely physical systems and are treated within the theory. In order to accomplish this it is necessary to identify some objective properties of such an observer (states) with subjective knowledge (i.e., perceptions). Thus, in order to say that an observer O has observed the event a, it is necessary that the state of O has become changed from its former state to a new state which is dependent upon a. It will suffice for our purposes to consider our observers to possess memories (i.e., parts of a relatively permanent nature whose states are in correspondence with the past experience of the observer). In order to make deductions about the subjective experience of an observer it is sufficient to examine the contents of the memory. As models for observers we can, if we wish, consider automatically functioning machines, possessing sensory apparata and coupled to recording devices capable of registering past sensory data and machine configurations. We can further suppose that the machine is so constructed that its present actions shall be determined not only by its present sensory data, but by the contents of its memory as well. Such a machine will then be capable of performing a sequence of observations (measurements), and furthermore of deciding upon its future experiments on the basis of past results. We note that if we consider that current sensory data, as well as machine configuration, is immediately recorded in the memory, then the actions of the machine at a given instant can be regarded as a function of the memory contents only, and all relevant experience of the machine is contained in the memory. Now remember this is 1956! There really weren’t many computers running around in 1956. I know because I was there. Still, Hugh Everett is now proposing to take human observers out of the equation and replace them with computers using Artificial Intelligence instead. This is a wise move because human observers use consciousness to record observations, and we still do not understand what consciousness is. By taking human observers out of the analysis he avoids that complication. No wonder that the Many-Worlds Interpretation seems to naturally lend itself to quantum computers. Computers were part of the analysis from the very beginning. When dealing quantum mechanically with a system representing an observer we shall ascribe a state function, ψO to it. When the State ψO describes an observer whose memory contains representations of the events A,B, ... ,C we shall denote this fact by appending the memory sequence in brackets as a subscript, writing: ψO[A,B, ... ,C] The symbols A,B, ... ,C, which we shall assume to be ordered time wise, shall therefore stand for memory configurations which are in correspondence with the past experience of the observer. These configurations can be thought of as punches in a paper tape, impressions on a magnetic reel, configurations of a relay switching circuit, or even configurations of brain cells. We only require that they be capable of the interpretation "The observer has experienced the succession of events A,B, ... ,C." (We shall sometimes write dots in a memory sequence, [. .. A,B, ... ,C], to indicate the possible presence of previous memories which are irrelevant to the case being considered.) Our problem is, then, to treat the interaction of such observer-systems with other physical systems (observations), within the framework of wave mechanics, and to deduce the resulting memory configurations, which we can then interpret as the subjective experiences of the observers. The machine with Artificial Intelligence is going to make a series of observations A, B, C…. and record them in its memory. Hugh Everett concludes with this summary: In the language of subjective experience, the observer which is described by a typical element, ψ'i,j...k, of the superposition has perceived an apparently random sequence of definite results for the observations. It is furthermore true, since in each element the system has been left in an eigenstate of the measurement, that if at this stage a redetermination of an earlier system observation Sl takes place, every element of the resulting final superposition will describe the observer with a memory configuration of the form [... ,ai1 ,... , ajl , ... , akr,ajl ] in which the earlier memory coincides with the later – i. e., the memory states are correlated. It will thus appear to the observer which is described by a typical element of the superposition that each initial observation on a system caused the system to "jump" into an eigenstate in a random fashion and thereafter remain there for subsequent measurements on the same system. Therefore, qualitatively, at least, the probabilistic assertions of Process 1 appear to be valid to the observer described by a typical element of the final superposition. So when you throw an electron through a nonuniform magnetic field, the machine with Artificial Intelligence is going to record that the electron randomly “jumps ” into a spin up eigenstate or a spin down eigenstate and then continues to remain a spin up or spin down electron. Hugh Everett then proceeds to summarize all of this and explain how such an observer O that becomes entangled or “correlated” with an electron will leave behind in its memory a sequence of events that is exactly what we observe in the lab. The electron will seem to behave in a random manner until it is observed and put into a particular eigenstate, and then the electron will remain in that eigenstate until it is perturbed again. Since the latest observation supplies all of the possible information about the relative system state of the observer O and the electron, and previous observations are not correlated with it, the Heisenberg Uncertainty Principle is not violated either. If the observer O measures the electron’s velocity, a further measure of its position will blur its velocity. We can therefore summarize the situation for an arbitrary sequence of observations, upon the same or different systems in any order, and for which the number of observations of each quantity in each system is very large, with the following result: Except for a set of memory sequences of measure nearly zero, the averages of any functions over a memory sequence can be calculated approximately by the use of the independent probabilities given by Process 1 for each initial observation, on a system, and by the use of the transition probabilities (2.23) for succeeding observations upon the same system. In the limit, as the number of all types of observations goes to infinity the calculation is exact, and the exceptional set has measure zero. This prescription for the calculation of averages over memory sequences by probabilities assigned to individual elements is precisely that of the orthodox theory (Process 1). Therefore all predictions of the usual theory will appear to be valid to the observer in almost all observer states, since these predictions hold for almost all memory sequences. In particular, the uncertainty principle is never violated, since, as above, the latest measurement upon a system supplies all possible information about the relative system state, so that there is no direct correlation between any earlier results of observation on the system, and the succeeding observation. Any observation of a quantity B, between two successive observations of quantity A (all on the same system) will destroy the one-one correspondence between the earlier and later memory states for the result of A. Thus for alternating observations of different quantities there are fundamental limitations upon the correlations between memory states for the same observed quantity, these limitations expressing the content of the uncertainty principle. In conclusion, we have described in this section processes involving an idealized observer, processes which are entirely deterministic and continuous from the over-all viewpoint (the total state function is presumed to satisfy a wave equation at all times) but whose result is a superposition, each element of which describes the observer with a different memory state. We have seen that in almost all of these observer states it appears to the observer that the probabilistic aspects of the usual form of quantum theory are valid. We have thus seen how pure wave mechanics, without any initial probability assertions, can lead to these notions on a subjective level, as appearances to observers. So if an observer throws lots of electrons through a nonuniform magnetic field the observer will perceive the electrons popping out randomly in spin up and spin down eigenstates, but what really is happening is that the observer and the electrons are splitting off into their own universes each time an electron goes through the magnetic field. One observer-electron pair splits off into a spin up universe, while another observer-electron pair splits off into a spin down universe. 3 Several Observers We shall now consider the consequences of our scheme when several observers are allowed to interact with the same systems, as well as with one another (communication). In the following discussion observers shall be denoted by O 1, O2,..., other systems by S1, S2,...,and observables by operators A, B, C,.... We shall also wish to allow communication among the observers, which we view as an interaction by means of which the memory sequences of different observers become correlated. (For example, the transfer of impulses from the magnetic tape memory of one mechanical observer to that of another constitutes such a transfer of information.) Case 1: We allow two observers to separately observe the same quantity in a system, and then compare results. After a bit of math, he concludes: This means that observers who have separately observed the same quantity will always agree with each other. For example, suppose a spin up electron pops out of our measuring magnets and observer O1 measures it with a set of magnets and finds it to be a spin up electron. If observer O2 observes the very same spin up electron, he will also measure it to be a spin up electron, and when observers O1 and O2 compare results they will agree. This is in agreement with the finding that the Copenhagen Interpretation predicts. Case 2: We allow two observers to measure separately two different, noncommuting quantities in the same system. For Case 2 we could have one observer measure the spin of an electron, while the other observer measures the velocity of the electron. The spin of an electron and its velocity are noncommuting quantities. Commuting quantities are quantities that fall under the Heisenberg Uncertainty Principal, like the position and velocity of the electron, where the measurement of one affects the measurement of the other. For this case Hugh Everett mathematically demonstrates that again the same results are obtained as predicted by the Copenhagen Interpretation. Case 3: We suppose that two systems S1 and S2 are correlated but no longer interacting, and that O1 measures property A in S1 and O2 property B in S2. It is therefore seen that one observer's observation upon one system of a correlated, but non-interacting pair of systems, has no effect on the remote system, in the sense that the outcome or expected outcome of any experiments by another observer on the remote system are not affected. Paradoxes like that of Einstein-Rosen-Podolsky which are concerned with such correlated, non-interacting, systems are thus easily understood in the present scheme. Case 3 is the basis for the infamous EPR (Einstein-Rosen-Podolsky) paradox that has caused so much grief for the Copenhagen Interpretation. The EPR Paradox goes like this. Suppose we prepare many pairs of quantum mechanically “entangled” electrons that conserve angular momentum. Each pair consists of one spin up electron and one spin down electron, but we do not know which is which at the onset. Now let the pairs of electrons fly apart and let two observers measure their spins. If observer A measures an electron there will be a 50% probability that he will find a spin up electron and a 50% chance that he will find a spin down electron, and the same goes for observer B, 50% of observer’s B electrons will be found to have a spin up, while 50% will be found with a spin down. Now the paradox of the EPR paradox, from the perspective of the Copenhagen Interpretation, is that when observer A and observer B come together to compare notes, they find that each time observer A found a spin up electron, observer B found a spin down electron, even though the electrons did not know which way they were spinning before the measurements were performed. Somehow when observer A measured the spin of an electron, it instantaneously changed the spin of the electron that observer B measured . Einstein hated this “spooky action at a distance” feature of the Copenhagen Interpretation that made physics nonlocal, meaning that things that were separated by great distances could still instantaneously change each other. He thought that it violated the speed of light speed limit of his Special Theory of Relativity that did not allow information to travel faster than the speed of light. Einstein thought that the EPR paradox was the final nail in the coffin of quantum mechanics. There had to be some “hidden variables” that allowed electrons to know if they “really” were a spin up or spin down electron. Hugh Everett solves this problem by letting the electrons be in all possible spin states in a large number of parallel universes. When observers measure the spin of an electron, they really do not measure the spin of the electron. They really measure in which universe they happen to be located in, and since everything in the Many-Worlds Interpretation relies on “correlated” composite wavefunctions, it should come as no surprise that when observer A and observer B come together, they find that their measurements of the electron spins are correlated. We have now completed the abstract treatment of measurement and observation, with the deduction that the statistical predictions of the usual form of quantum theory (Process 1) will appear to be valid to all observers. We have therefore succeeded in placing our theory in correspondence with experience, at least insofar as the ordinary theory correctly represents experience. We should like to emphasize that this deduction was carried out by using only the principle of superposition, and the postulate that an observation has the property that if the observed variable has a definite value in the object-system then it will remain definite and the observer will perceive this value. This treatment is therefore valid for any possible quantum interpretation of observation processes, i.e., any way in which one can interpret wave functions as describing observers, as well as for any form of quantum mechanics for which the superposition principle for states is maintained. Our abstract discussion of observation is therefore logically complete, in the sense that our results for the subjective experience of observers are correct, if there are any observers at all describable by wave mechanics. In this chapter we shall consider a number of diverse topics from the point of view of our pure wave mechanics, in order to supplement the abstract discussion and give a feeling for the new viewpoint. Since we are now mainly interested in elucidating the reasonableness of the theory, we shall often restrict ourselves to plausibility arguments, rather than detailed proofs. 1. Macroscopic objects and classical mechanics In the light of our knowledge about the atomic constitution of matter, any "object" of macroscopic size is composed of an enormous number of constituent particles. The wave function for such an object is then in a space of fantastically high dimension (3N, if N is the number of particles). Our present problem is to understand the existence of macroscopic objects, and to relate their ordinary (classical) behavior in the three dimensional world to the underlying wave mechanics in the higher dimensional space. Let us begin by considering a relatively simple case. Suppose that we place in a box an electron and a proton, each in a definite momentum state, so that the position amplitude density of each is uniform over the whole box. After a time we would expect a hydrogen atom in the ground state to form, with ensuing radiation. We notice, however, that the position amplitude density of each particle is still uniform over the whole box. Nevertheless the amplitude distributions are now no longer independent, but correlated. In particular, the conditional amplitude density for the electron, conditioned by any definite proton (or centroid) position, is not uniform, but is given by the familiar ground state wave function for the hydrogen atom. What we mean by the statement, "a hydrogen atom has formed in the box," is just that this correlation has taken place - a correlation which insures that the relative configuration for the electron, for a definite proton position, conforms to the customary ground state configuration. The wave function for the hydrogen atom can be represented as a product of a centroid wave function and a wave function over relative coordinates, where the centroid wave function obeys the wave equation for a particle with mass equal to the total mass of the proton-electron system. Therefore, if we now open our box, the centroid wave function will spread with time in the usual manner of wave packets, to eventually occupy a vast region of space. The relative configuration (described by the relative coordinate state function) has, however, a permanent nature, since it represents a bound state, and it is this relative configuration which we usually think of as the object called the hydrogen atom. Therefore, no matter how indefinite the positions of the individual particles become in the total state function (due to the spreading of the centroid), this state can be regarded as giving (through the centroid wave function) an amplitude distribution over a comparatively definite object, the tightly bound electron-proton system. The general state, then, does not describe any single such definite object, but a superposition of such cases with the object located at different positions. In the above section Hugh Everett proposes putting an electron and a proton in a box with each particle given a known initial momentum. Then according to the Heisenberg Uncertainty Principle we cannot know anything about their positions, so they must be uniformly smeared out over the insides of the whole box, and they should stay that way forever. However, eventually the electron and proton will interact and form a hydrogen atom, giving off a photon in the process. The two particles will then be defined by a composite correlated wavefunction that corresponds to the ground state of a hydrogen atom. This composite correlated wavefunction can be viewed as the product of a centroid wavefunction with the mass of a hydrogen atom and a relative wavefunction spread over coordinates relative to the proton. If we then open the box and release the hydrogen atom this centroid wavefunction will spread out all over the place as the hydrogen atom diffuses away from the box, but there still will be a relative component of the total composite wavefunction that represents the relative location of the electron with respect to the proton. In a similar fashion larger and more complex objects can be built up through strong correlations which bind together the constituent particles. It is still true that the general state function for such a system may lead to marginal position densities for any single particle (or centroid) which extend over large regions of space. Nevertheless we can speak of the existence of a relatively definite object, since the specification of a single position for a particle, or the centroid, leads to the case where the relative position densities of the remaining particles are distributed closely about the specified one, in a manner forming the comparatively definite object spoken of. Suppose, for example, we begin with a cannonball located at the origin, described by a state function: where the subscript indicates that the total state function ψ describes a system of particles bound together so as to form an object of the size and shape of a cannonball, whose centroid is located (approximately) at the origin, say in the form of a real gaussian wave packet of small dimensions, with variance σ02 for each dimension. If we now allow a long lapse of time, the centroid of the system will spread in the usual manner to occupy a large region of space.... It is not true that each individual particle spreads independently of the rest, in which case we would have a final state which is a grand superposition of states in which the particles are located independently everywhere. The fact that they are in bound states restricts our final state to a superposition of "cannonball" states. The wave function for the centroid can therefore be taken as a representative wave function for the whole object. Similarly, in the above section Hugh Everett mathematically demonstrates that if we have a large number of particles that constitute a cannonball with a composite wavefunction ψ[cj(0,0,0)] defined upon the coordinates (0,0,0) that this composite wavefunction will indeed spread out with time, just like the wavefunction for a single unbound electron will spread out with time, but the individual particles will not spread out all over the place causing the cannonball to essentially evaporate. Thus large objects composed of bound particles will continue to behave as large objects composed of bound particles as time progresses. Next, he describes what an observer would record when observing the cannonball move through space. The observer would become correlated into a superposition of his wavefunction with that of the centroid cannonball wavefunction, and the cannonball will then appear to behave in a manner conforming to classical mechanics: Let us now consider the result of an observation (considered along the lines of Chapter IV) performed upon a system of macroscopic bodies in a general state. The observer will not become aware of the fact that the state does not correspond to definite positions and momenta (i.e., he will not see the objects as "smeared out" over large regions of space) but will himself simply become correlated with the system - after the observation the composite system of objects + observer will be in a superposition of states, each element of which describes an observer who has perceived that the objects have nearly definite positions and momenta, and for whom the relative system state is a quasi-classical state in the previous sense, and furthermore to whom the system will appear to behave according to classical mechanics if his observation is continued. We see, therefore, how the classical appearance of the macroscopic world to us can be explained in the wave theory. Since Hugh Everett has eliminated Process 1 from his theory, he next addresses what happens when an observation is made. For example, suppose observer A uses magnets to measure the spin of an electron and finds that the electron is a spin up electron. In the Copenhagen Interpretation the act of observing the electron will collapse its wavefunction into a spin up eigenstate and this is an irreversible process that cannot be reversed in time. Hugh Everett goes through some more mathematics using what he has already discussed above to come to a different conclusion: 3. Reversibility and irreversibility So instead of the observer collapsing the wavefunction of the electron with his magnets, the observer splits into two observers. One observer sees a spin up electron and the other observer sees a spin down electron. These two observers are totally unaware of each other and are completely cut off from each other with no possibility to interact. This is how when a quantum computer reads a 1-qubit memory location that is in a superposition of 1 and 0 at the top of an if-then-else block, one quantum computer will execute the then-block, while the other instance of the quantum computer will execute the else-block. ….We take this opportunity to caution against a certain viewpoint which can lead to difficulties. This is the idea that, after an apparatus has interacted with a system, in "actuality" one or another of the elements of the resultant superposition described by the composite state-function has been realized to the exclusion of the rest, the existing one simply being unknown to an external observer (i.e., that instead of the superposition there is a genuine mixture). This position must be erroneous since there is always the possibility for the external observer to make use of interference properties between the elements of the superposition. In the present example, for instance, it is in principle possible to deflect the two beams back toward one another with magnetic fields and recombine them in another inhomogeneous field, which duplicates the first, in such a manner that the original spin state (before entering the apparatus) is restored. This would not be possible if the original Stern-Gerlach apparatus performed the function of converting the original wave packet into a non-interfering mixture of packets for the two spin cases. Therefore the position that after the atom has passed through the inhomogeneous field it is "really" in one or the other beam with the corresponding spin, although we are ignorant of which one, is incorrect. Shooting a beam of electrons through an inhomogeneous magnetic field will cause two beams to seemingly emerge, one with spin up electrons and one with spin down electrons. But according to Hugh Everett each electron will end up in each beam, but in two separate universes, and each electron will be just as “real” as the other. For example, Hugh Everett maintains that this must be so because theoretically it is possible to reflect the electrons coming out of a Stern-Gerlach device back through the device to return the spin up and spin down electrons back into being electrons in a mixed state of spin up and spin down. Essentially, this is what our circular tub of water would do to the circular waves arising from dropping a pebble into the center of the circular tub of water. It is therefore improper to attribute any less validity or "reality" to any element of a superposition than any other element, due to this ever present possibility of obtaining interference effects between the elements. All elements of a superposition must be regarded as simultaneously existing. Below is Hugh Everett’s final chapter in its entirety where he nicely sums things up, without any mathematics at all. Because the theory gives us an objective description, it constitutes a framework in which a number of puzzling subjects (such as classical level phenomena, the measuring process itself, the inter-relationship of several observers, questions of reversibility and irreversibility, etc.) can be investigated in detail in a logically consistent manner. It supplies a new way of viewing processes, which clarifies many apparent paradoxes of the usual interpretation - indeed, it constitutes an objective framework in which it is possible to understand the general consistency of the ordinary view. We shall now resume our discussion of alternative interpretations. There has been expressed lately a great deal of dissatisfaction with the present form of quantum theory by a number of authors, and a wide variety of new interpretations have sprung into existence. We shall now attempt to classify briefly a number of these interpretations, and comment upon them. In its unrestricted form this view can lead to paradoxes like that mentioned in the introduction, and is therefore untenable. However, this view is consistent so long as it is assumed that there is only one observer in the universe (the solipsist position - Alternative 1 of the Introduction). This consistency is most easily understood from the viewpoint of our own theory, where we were able to show that all phenomena will seem to follow the predictions of this scheme to any observer. Our theory therefore justifies the personal adoption of this probabilistic interpretation, for purposes of making practical predictions, from a more satisfactory framework While undoubtedly safe from contradiction, due to its extreme conservatism, it is perhaps overcautious. We do not believe that the primary purpose of theoretical physics is to construct "safe" theories at severe cost in the applicability of their concepts, which is a sterile occupation, but to make useful models which serve for a time and are replaced as they are outworn. Another objectionable feature of this position is its strong reliance upon the classical level from the outset, which precludes any possibility of explaining this level on the basis of an underlying quantum theory. (The deduction of classical phenomena from quantum theory is impossible simply because no meaningful statements can be made without pre-existing classical apparatus to serve as a reference frame.) This interpretation suffers from the dualism of adhering to a "reality" concept (i.e., the possibility of objective description) on the classical level but renouncing the same in the quantum domain. There is some political maneuvering going on in the above passage. The “popular” interpretation really is the Copenhagen Interpretation, but Niels Bohr was still a living giant of quantum theory at the time, and it would not be wise for this Ph.D. thesis to be seen as a direct attack on the Copenhagen Interpretation and Niels Bohr. So Hugh Everett breaks apart the Copenhagen Interpretation into two parts. Part 1 he calls the “popular” interpretation in which wavefunctions mysteriously collapse when an observation is made. Part 2 he calls the Copenhagen Interpretation where wavefunctions are just a mathematical tool used to perform calculations after you set up a macroscopic experiment to make a quantum mechanical measurement. For example, the wavefunctions of electrons passing through a nonuniform magnetic field could be used to calculate that 50% will be observed to be spin up electrons, while 50% will be observed to be spin down electrons. So now we know that the “popular” interpretation that Hugh Everett has been attacking from the very first line of his thesis: is really not Niels Bohr’s sacred Copenhagen Interpretation at all. It is really the “popular” interpretation that he has been attacking all along. I think we all would have practiced a similar maneuver in his shoes. The ψ function is therefore regarded as a description of an ensemble of systems rather than a single system. Proponents of this interpretation include Einstein, Bohm, Wiener and Siegal. Einstein hopes that a theory along the lines of his general relativity, where all of physics is reduced to the geometry of space-time could satisfactorily explain quantum effects. In such a theory a particle is no longer a simple object but possesses an enormous amount of structure (i.e., it is thought of as a region of space-time of high curvature). It is conceivable that the interactions of such "particles" would depend in a sensitive way upon the details of this structure, which would then play the role of the "hidden variables". However, these theories are non-linear and it is enormously difficult to obtain any conclusive results. Nevertheless, the possibility cannot be discounted. Bohm considers ψ to be a real force field acting on a particle which always has a well-defined position and momentum (which are the hidden variables of this theory). The ψ-field satisfying Schrödinger equation is pictured as somewhat analogous to the electromagnetic field satisfying Maxwell's equations, although for systems of n particles the ψ -field is in a 3n-dimensional space. With this theory Bohm succeeds in showing that in all actual cases of measurement the best predictions that can be made are those of the usual theory, so that no experiments could ever rule out his interpretation in favor of the ordinary theory. Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that if is a real field then the associated particle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory. Wiener and Siegal have developed a theory which is more closely tied to the formalism of quantum mechanics. From the set N of all nondegenerate linear Hermitian operators for a system having a complete set of eigenstates, a subset I is chosen such that no two members of I commute and every element outside I commutes with at least one element of I . The set I therefore contains precisely one operator for every orientation of the principal axes of the Hilbert space for the system. It is postulated that each of the operators of I corresponds to an independent observable which can take any of the real numerical values of the spectrum of the operator. This theory, in its present form, is a theory of infinitely many "hidden variables," since a system is pictured as possessing (at each instant) a value for everyone of these "observables" simultaneously, with the changes in these values obeying precise (deterministic) dynamical laws. However, the change of anyone of these variables with time depends upon the entire set of observables, so that it is impossible ever to discover by measurement the complete set of values for a system (since only one "observable" at a time can be observed). Therefore, statistical ensembles are introduced, in which the values of all of the observables are related to points in a "differential space," which is a Hilbert space containing a measure for which each (differential space) coordinate has an independent normal distribution. It is then shown that the resulting statistical dynamics is in accord with the usual form of quantum theory. It cannot be disputed that these theories are often appealing, and might conceivably become important should future discoveries indicate serious inadequacies in the present scheme (i.e., they might be more easily modified to encompass new experience). But from our viewpoint they are usually more cumbersome than the conceptually simpler theory based on pure wave mechanics. Nevertheless, these theories are of great theoretical importance because they provide us with examples that "hidden variables" theories are indeed possible. A stochastic theory which emphasizes the particle, rather than wave, aspects of quantum theory has been investigated by Bopp. The particles do not obey deterministic laws of motion, but rather probabilistic laws, and by developing a general "correlation statistics" Bopp shows that his quantum scheme is a special case which gives results in accord with the usual theory. (This accord is only approximate and in principle one could decide between the theories. The approximation is so close, however, that it is hardly conceivable that a decision would be practically feasible.) Bopp's theory seems to stem from a desire to have a theory founded upon particles rather than waves, since it is this particle aspect (highly localized phenomena) which is most frequently encountered in present day high-energy experiments (cloud chamber tracks, etc.). However, it seems to us to be much easier to understand particle aspects from a wave picture (concentrated wave packets) than it is to understand wave aspects (diffraction, interference, etc.) from a particle picture. Nevertheless, there can be no fundamental objection to the idea of a stochastic theory, except on grounds of a naked prejudice for determinism. The question of determinism or indeterminism in nature is obviously forever undecidable in physics, since for any current deterministic [ probabilistic] theory one could always postulate that a refinement of the theory would disclose a probabilistic [deterministic] substructure, and that the current deterministic [probabilistic] theory is to be explained in terms of the refined theory on the basis of the law of large numbers [ignorance of hidden variables). However, it is quite another matter to object to a mixture of the two where the probabilistic processes occur only with acts of observation. This view also corresponds most closely with that held by Schrödinger. However, this picture only makes sense when observation processes themselves are treated within the theory. It is only in this manner that the apparent existence of definite macroscopic objects, as well as localized phenomena, such as tracks in cloud chambers, can be satisfactorily explained in a wave theory where the waves are continually diffusing. With the deduction in this theory that phenomena will appear to observers to be subject to Process I, Heisenberg's criticism of Schrödinger’s opinion - that continuous wave mechanics could not seem to explain the discontinuities which are everywhere observed - is effectively met. The "quantum jumps" exist in our theory as relative phenomena (i.e., the states of an object-system relative to chosen observer states show this effect), while the absolute states change quite continuously. The wave theory is definitely tenable and forms, we believe, the simplest complete, self-consistent theory. We should like now to comment on some views expressed by Einstein. Einstein's criticism of quantum theory (which is actually directed more against what we have called the "popular" view than Bohr's interpretation) is mainly concerned with the drastic changes of state brought about by simple acts of observation (i.e., the infinitely rapid collapse of wave functions), particularly in connection with correlated systems which are widely separated so as to be mechanically uncoupled at the time of observation. At another time he put his feeling colorfully by stating that he could not believe that a mouse could bring about drastic changes in the universe simply by looking at it. In the case of observation of one system of a pair of spatially separated, correlated systems, nothing happens to the remote system to make any of its states more "real" than the rest. It had no independent states to begin with, but a number of states occurring in a superposition with corresponding states for the other (near) system. Observation of the near system simply correlates the observer to this system, a purely local process - but a process which also entails automatic correlation with the remote system. Each state of the remote system still exists with the same amplitude in a superposition, but now a superposition for which element contains, in addition to a remote system state and correlated near system state, an observer state which describes an observer who perceives the state of the near system. From the present viewpoint all elements of this superposition are equally "real." Only the observer state has changed, so as to become correlated with the state of the near system and hence naturally with that of the remote system also. The mouse does not affect the universe - only the mouse is affected. This is Hugh Everett’s solution to the EPR paradox. Recall that if we prepare many pairs of quantum mechanically “entangled” electrons that conserve angular momentum, initially each electron will be in a mixture of spin states because it has not been measured yet. In the Copenhagen Interpretations these electrons “really” do not know what their spins are at this point, but when observer A and observer B later measure their spins with a Stern-Gerlach device and then compare notes, they will find that whenever observer A measured a spin up electron, observer B measured its twin as a spin down electron. Since the electrons “really” did not know what their spins were before being measured, somehow measuring the spin of an electron “here” instantaneously determined the spin of its twin over “there”, and the “here” and “there” can be on the opposite ends of the visible Universe. In 1982 Alain Aspect actually conducted an experiment that validated this finding using photons instead of electrons, so this is not just a thought experiment. The Universe actually behaves like this! Here is Hugh Everett’s solution. Each near electron is in a number of superposition states with its twin electron that is over “there”, and each of those superpositions must conserve angular momentum because that is the law, meaning that one electron is a spin up and the other is a spin down. When observer A “measures” a near electron, observer A becomes correlated with the near electron, and also with its twin electron over “there” because both of those electrons were already in a correlated superposition state to begin with. When observer B measures the twin electron over “there”, he becomes correlated with the twin electron, and consequently with the correlated superposition state of observer A and both electrons. That is why when observer A finds a spin up electron, observer B finds a spin down electron. Basically, observer A and observer B are not really measuring the spins of the electrons. Instead, they have really put together a very complex experiment that always places them into the same universe amongst many parallel universes. That is why the mouse does not affect the universe. Our theory in a certain sense bridges the positions of Einstein and Bohr, since the complete theory is quite objective and deterministic ("God does not play dice with the universe"), and yet on the subjective level, of assertions relative to observer states, it is probabilistic in the strong sense that there is no way for observers to make any predictions better than the limitations imposed by the uncertainty principle. In conclusion, we have seen that if we wish to adhere to objective descriptions then the principle of the psycho-physical parallelism requires that we should be able to consider some mechanical devices as representing observers. The situation is then that such devices must either cause the probabilistic discontinuities of Process 1, or must be transformed into the superpositions we have discussed. We are forced to abandon the former possibility since it leads to the situation that some physical systems would obey different laws from the rest, with no clear means for distinguishing between these two types of systems. We are thus led to our present theory which results from the complete abandonment of Process 1 as a basic process. Nevertheless, within the context of this theory, which is objectively deterministic, it develops that the probabilistic aspects of Process 1 reappear at the subjective level, as relative phenomena to observers. One is thus free to build a conceptual model of the universe, which postulates only the existence of a universal wave function which obeys a linear wave equation. One then investigates the internal correlations in this wave function with the aim of deducing laws of physics, which are statements that take the form: Under the conditions C the property A of a subsystem of the universe (subset of the total collection of coordinates for the wave function) is correlated with the property B of another subsystem (with the manner of correlation being specified). For example, the classical mechanics of a system of massive particles becomes a law which expresses the correlation between the positions and momenta (approximate) of the particles at one time with those at another time. All statements about subsystems then become relative statements, i.e., statements about the subsystem relative to a prescribed state for the remainder (since this is generally the only way a subsystem even possesses a unique state), and all laws are correlation laws. The theory based on pure wave mechanics is a conceptually simple causal theory, which fully maintains the principle of the psycho-physical parallelism. It therefore forms a framework in which it is possible to discuss (in addition to ordinary phenomena) observation processes themselves, including the inter-relationships of several observers, in a logical, unambiguous fashion. In addition, all of the correlation paradoxes, like that of Einstein, Rosen, and Podolsky, find easy explanation. While our theory justifies the personal use of the probabilistic interpretation as an aid to making practical predictions, it forms a broader frame in which to understand the consistency of that interpretation. It transcends the probabilistic theory, however, in its ability to deal logically with questions of imperfect observation and approximate measurement. Since this viewpoint will be applicable to all forms of quantum mechanics which maintain the superposition principle, it may prove a fruitful framework for the interpretation of new quantum formalisms. Field theories, particularly any which might be relativistic in the sense of general relativity, might benefit from this position, since one is free to construct formal (non-probabilistic) theories, and supply any possible statistical interpretations later. (This viewpoint avoids the necessity of considering anomalous probabilistic jumps scattered about space-time, and one can assert that field equations are satisfied everywhere and everywhen, then deduce any statistical assertions by the present method.) By focusing attention upon questions of correlations, one may be able to deduce useful relations (correlation laws analogous to those of classical mechanics) for theories which at present do not possess known classical counterparts. Quantized fields do not generally possess pointwise independent field values, the values at one point of space-time being correlated with those at neighboring points of space-time in a manner, it is to be expected, approximating the behavior of their classical counterparts. If correlations are important in systems with only a finite number of degrees of freedom, how much more important they must be for systems of infinitely many coordinates. Finally, aside from any possible practical advantages of the theory, it remains a matter of intellectual interest that the statistical assertions of the usual interpretation do not have the status of independent hypotheses, but are deducible (in the present sense) from the pure wave mechanics, which results from their omission. For the more mathematically gifted, I encourage you to try reading the full text of Hugh Everett’s original draft Ph.D. thesis. One reason John Wheeler had Hugh Everett heavily edit his original 137-page document down to his final 36-page doctoral dissertation was that he was afraid the departmental physicists on Hugh Everett’s dissertation committee would not understand the material and that the oral defense of his Ph.D. thesis would not go well, and could possibly even lead to his failure to pass his thesis defense. So do not feel too badly if the mathematics goes way over your head. Comments are welcome at To see all posts on softwarephysics in reverse order go to: Steve Johnston
f888792f92c0bbd4	University of Minnesota School of Physics & Astronomy Home > People > Michel Janssen The Trouble with Orbits: The Stark effect in the Old and the New Quantum Theory Anthony Duncan and Michel Janssen, Studies in History and Philosophy of Modern Physics. 48 (2014): 68–83. Download from The old quantum theory and Schrödinger's wave mechanics (and other forms of quantum mechanics) give the same results for the line splittings in the first-order Stark effect in hydrogen, the leading terms in the splitting of the spectral lines emitted by a hydrogen atom in an external electric field. We examine the account of the effect in the old quantum theory, which was hailed as a major success of that theory, from the point of view of wave mechanics. First, we show how the new quantum mechanics solves a fundamental problem one runs into in the old quantum theory with the Stark effect. It turns out that, even without an external field, it depends on the coordinates in which the quantum conditions are imposed which electron orbits are allowed in a hydrogen atom. The allowed energy levels and hence the line splittings are independent of the coordinates used but the size and eccentricity of the orbits are not. In the new quantum theory, this worrisome non-uniqueness of orbits turns into the perfectly innocuous non-uniqueness of bases in Hilbert space. Second, we review how the so-called WKB (Wentzel-Kramers-Brillouin) approximation method for solving the Schrödinger equation reproduces the quantum conditions of the old quantum theory amended by some additional half-integer terms. These extra terms remove the need for some arbitrary extra restrictions on the allowed orbits that the old quantum theory required over and above the basic quantum conditions.
80244c25fbc1f5d4	Sunday, January 20, 2008 Last time we explored the development of quantum mechanics in the early part of the 20th century and saw how it led to the concept that the wavelike characteristics of particles could be expressed in terms of a complex wavefunction ψ(x) consisting of real and imaginary parts. We saw that, at first, physicists had a hard time figuring out what these complex wavefunctions really meant. Then in 1928, Max Born came up with the idea that the wavefunction solutions to Schrödinger’s equation could be thought of as probability waves, and that the probability of finding a particle at some point along the x-axis could be obtained by multiplying the particle’s wavefunction by its complex conjugate ψ(x)ψ(x) at each point along the x-axis. This was a key insight. In classical mechanics, we also use equations to figure things out, but in all cases, we try to manipulate the equations to solve for the desired quantity that we are interested in. We always try to end up with an equation that looks like: E = ½mv² which is the classical equation for the kinetic energy of a particle. Then all we have to do is the old “plug ‘n chug” to get the kinetic energy E by plugging in the mass m and velocity v of the particle into the above formula. But in Born’s interpretation of the wavefunction, we did not do that to obtain the position of a particle. Instead, we performed a mathematical operation on the wavefunction ψ(x) itself by multiplying it by its complex conjugate ψ(x)ψ(x), and we did not get an exact answer either, just an exact probability. Some additional mathematical thought shows that the wavefunction is the whole deal, meaning that everything that can be known about a particle, such as its position, energy, momentum, and angular momentum, is encapsulated within the wavefunction itself, and to obtain values for these quantities you have to perform strange mathematical operations upon the wavefunction. For example, going back to the time independent Schrödinger equation for a particle moving along the x-axis: -ħ² d²ψ(x) + V(x) ψ(x) = E ψ(x) ── ────── 2m dx² We see that if we define a mathematical operation H as: H = -ħ² d² + V(x) ── ─── 2m dx² then we can rewrite the Schrödinger equation simply as: H ψ(x) = E ψ(x) The wavefunctions ψ(x) that satisfy the above equation are called eigenfunctions and the corresponding measured values of E are called eigenvalues. “Eigen” roughly means “characteristic” in German (you may have noticed that nearly all of the early 20th century physicists I have mentioned in this blog were Germans, and that is where this terminology came from). So for the solutions to Schrödinger’s equation for a particle in a box, the eigenfunctions are: ψn(x) = √ 2/L sin(nπx/L) n = 1, 2, 3, ... and the eigenvalues are: En = n²h² n = 1, 2, 3, ... m = mass of the particle (electron in this case) L = width of the box h = Planck’s constant n = quantum number In general, the way you solve problems in quantum mechanics is to first solve Schrödinger’s equation for the problem at hand to obtain the desired wavefunctions (eigenfunctions). Then you apply mathematical operators to the eigenfunctions to obtain eigenvalues, which are the quantized answers you are seeking for your problem: O ψ(x) = o ψ(x) where O is some mathematical operator and o is a measured quantized value. The above analysis can be applied to the hydrogen atom using Dirac’s equation, which is Schrödinger’s equation augmented by special relativity to take into account the fact that the electron orbiting the proton of a hydrogen atom is moving at a relativistic velocity. The result is a series of eigenfunction wavefunctions with associated eigenvalues, both defined by quantum numbers n, l, m, and s. The quantum number n defines the energy levels of the electron in the hydrogen atom and the l and m quantum numbers define the angular momentum of the electron as it orbits around the proton in the nucleus. The quantum number s is the strange quantum number that defines the inherent angular momentum of the electron itself, even though the electron is currently depicted as a fundamental particle with a dimension of zero and should not have any intrinsic angular momentum from a classical perspective. The result is that the electron in a hydrogen atom can exist as a series of electron wavefunctions (eigenfunctions) which are 3-dimensional “probability clouds” about the central proton of the hydrogen atom. Like the particle in a box, which only had one quantum number - n, the quantum numbers for the hydrogen atom n, l, m, and s are just eigenvalues for the eigenfunction solutions to the Dirac or Schrödinger equation. For example, Figure 2 of Quantum Software shows the eigenfunction solutions for Schrödinger’s equation for the particle in a box for the eigenvalues n = 1, 2, and 3. These quantum numbers are hard to grasp mentally because our common sense is based upon our experiences with relatively large objects, so we do not have any quantum mechanical intuition. A helpful, but somewhat misleading, model is to relate the quantum numbers of the hydrogen atom to a classical system like the Earth orbiting the Sun. Such a model has its limitations, but I would bet that most physicists secretly harbor it deep down in their subconscious minds. The chief difference is that for the classical Earth-Sun system, the items below can take on continuous values, while their quantum counterparts can only take on fixed quantized values. n - The approximate distance of the electron from the proton nucleus of hydrogen, like the distance of the Earth from the Sun. Now all the electron wavefunctions of the hydrogen atom are actually spread out over the entire Universe, so no matter where your hydrogen atom might be, there is a small chance that its electron is in Peoria. However, the most likely location of the electron will be close in near the proton nucleus of its hydrogen atom. The larger n is, the further out will be the electron’s maximum probability of existence. l – The total amount of angular momentum of the electron, like the angular momentum of the Earth orbiting the Sun. m – The direction in which the angular momentum vector points, like an arrow perpendicular to the orbital plane of the Earth’s orbit about the Sun s – The inherent spin angular momentum of the electron, like the Earth spinning on its axis. To make matters more confusing, chemists use the term “orbital” for these eigenfunctions or wavefunctions, probably because they too need to relate the quantum electron-proton system of hydrogen to a classical Earth-Sun system, in order to try to make sense of it all. The terms orbital, eigenfunction, or wavefunction all mean the same thing, they are just 3-dimensional probability clouds, so for the sake of clarity going forward, I will simply refer to these eigenfunctions as wavefunctions or orbitals. The rules for the quantum numbers fall out of the solutions to Schrödinger’s equation for the hydrogen atom and go like this, for any given energy level n, there can be one or more wavefunctions or orbitals based upon the following rules: For any n: l = n – 1 m = the range of integers from –l to +l, like -2, -1, 0, +1, +2 s = ± ½ The first energy level has only one wavefunction or orbital: n = 1 l = 0 because n – 1 = 0 m = 0 because the range of -l to +l = -0 to +0 = 0 s = ± ½ Chemists call this wavefunction the 1s orbital, which can hold 2 electrons, one with spin up ↑ and one with spin down ↓, and is denoted as 1s2. Notice that because l = 0, the 1s orbital has no quantized angular momentum, which is rather strange. You would think that electrons orbiting a proton should have some angular momentum, they certainly would according to Newtonian mechanics, but the 1s2 electrons do not have any. The second energy level has 4 wavefunctions or orbitals: 1. n = 2, l = 0, m = 0, s= ± ½ Chemists call this the 2s orbital, which again, can hold 2 electrons, one with spin up ↑ and one with spin down ↓, denoted by 2s2. Again, because l = 0 the 2s orbital has no quantized angular momentum. The other 3 orbitals do have some quantized angular momentum because l = 1: 2. n = 2, l = 1, m = -1, s= ± ½ 3. n = 2, l = 1, m = 0, s= ± ½ 4. n = 2, l = 1, m = +1, s= ± ½ Chemists call these three orbitals the 2p orbitals, each of which can also hold 2 electrons, one with spin up ↑ and one with spin down ↓. The 2p orbitals are denoted by 2px, 2py,2pz, corresponding to m = -1, m = 0, and m = +1. Figure 1 – The n=1 and n=2 Orbitals (Eigenfunctions or Wavefunctions) for the hydrogen atom (click to enlarge) Notice that the 1s and 2s orbitals, with no quantized angular momentum because l = 0 for them, have spherically-shaped orbitals or wavefunctions. Again, these orbitals or wavefunctions are just probability clouds as depicted in the speckle plots of Figure 1. The 2p orbitals do have some quantized angular momentum because l = 1 for these orbitals. As we shall see, this is a key point, the 2p orbitals, and all orbitals with non-zero quantized angular momentum (meaning l > 0), have a 3-dimensional shape with preferred directions for electron existence, rather than being spherically-shaped orbitals like the 1s and 2s orbitals, with no preferred direction for electron existence. The 2p orbitals, on the other hand, each consist of two lobes that bulge in the x, y, and z directions, all oriented 900 to each other. So as you can see from the speckle plots of the 2p orbitals in Figure 1, the electrons of the 2p orbital have preferred zones of existence pointing in different directions, whereas the 1s and 2s orbitals do not. This is key to the formation of molecules with a 3-dimensional shape, like the organic molecules in living things. In fact, if it were not for the quantized angular momentum of electrons, carbon-based organic molecules would not have complex 3-dimensional shapes, and you would not be here contemplating the marvels of the quantum mechanics of electrons! This is all very impressive, but as you can see, the math gets pretty heavy even for the simple hydrogen atom, which consists of a single electron orbiting a nucleus consisting of a single proton. The problem is that the next atom in the periodic table, helium, presents even greater mathematical challenges. In fact, nobody has ever exactly solved the Schrödinger equation for the helium atom, much less Dirac’s equation, because helium has a nucleus composed of two protons and two neutrons, and consequently, has two electrons orbiting its nucleus. The problem is that the two electrons interact with each other, and this complication requires that some approximations be made in order to solve Schrödinger’s equation for helium and the other 92 naturally occurring elements in the periodic table. Physicists call these approximations chemistry. Yes, from the perspective of physics, the entire science of chemistry is just an approximate extension of the effective theory of quantum mechanics. This might sound a bit arrogant, especially since, like an old married couple, the chemists had to nag the physicists for more than 100 years before the physicists finally came up with the brilliant insight that atoms really did exist after all. In retaliation, chemists call quantum mechanics P-chem (physical chemistry). Most medical doctors do not start out in pre-med programs as physics majors, but many doctors do have a pre-med major in chemistry or biology. So if you are ever aggravated by one of your care givers, just tell them that lately your P-chem has been bothering you. There is a good chance that you will trigger a devastating Post-Traumatic Stress Disorder flashback. To extend quantum mechanics to chemistry, we are faced with the daunting challenge of trying to find the wavefunctions for the electrons orbiting a molecule instead of orbiting the single nucleus of a single atom. Recall that a molecule is simply a combination of two or more atoms that are chemically bound together. In a molecule, we end up with multiple electrons orbiting multiple atomic nuclei containing protons and neutrons, all interacting with each other via the electromagnetic force between the electrons and protons. The wavefunctions of electrons orbiting the molecular nuclei are known as molecular orbitals, just as the wavefuntions of the electrons orbiting the nucleus of a single atom are called atomic orbitals. The concept of molecular orbitals was first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928, very shortly after the development of quantum mechanics by Heisenberg and Schrödinger in 1926. The first simple approximation for the solution of molecular orbitals was introduced in 1929 by Sir John Lennard-Jones as a linear combination of the atomic orbitals of the individual constituent atoms of the molecule. What Lennard-Jones did was to mix the wavefunctions of the individual atoms together to come up with the combined molecular orbitals of the electrons in molecules, like mixing together yellow paint with blue paint to come up with green paint. Figure 2 depicts the resulting molecular orbitals surrounding the atomic nuclei of a molecule. This figure is a little misleading, in that if the wavefunctions of the electron probability clouds surrounding a single atom were blown up to the size of a football stadium, the protons and neutrons of the atomic nucleus would be about the size of shelled peanuts on the 50 yard line. Figure 2 - Molecular orbitals surrounding atomic nuclei (click to enlarge) The electrons in a molecule are subject to two additional constraints. First of all, they cannot violate the Pauli exclusion principle that each electron has to have a unique combination of quantum numbers. Secondly, the electrons will arrange themselves in molecular orbitals to minimize their free energy in accordance with the second law of thermodynamics. Thus, the second law causes the electrons to fill the molecular orbitals with the lowest energy levels first, and the Pauli exclusion principle prevents the electrons from all occupying the same orbital with the lowest energy level. The result is that the electrons pile up into a hierarchy of molecular orbitals, just as they do in the atomic orbitals of an atom. We have already seen that the second law of thermodynamics can be expressed in many ways, and here is another. Recall that the second law states that the total amount of entropy (disorder) in the Universe must always increase whenever a change is made. Entropy is a measure of the depreciation of the Universe. Another expression of the second law is that systems naturally tend to minimize their free energy, the energy available to do work. Here is an old bar trick that illustrates this effect. Take out a book of paper matches and rip out one of the matches. Now offer to buy the next round of beers if anybody in your party can drop the match from a height of one foot onto the bar and have the match land on an edge. After several failed attempts, make the following counter offer. Turn to one of your companions and offer to buy the next round of beers if you cannot successfully drop the match from a height of one foot onto the bar and have it land on an edge. However, if you do succeed, then your companion must buy the next round. Now take the match and simply fold it into a “V” shape. When you drop the match, it will naturally land on an edge. What is happening here is that the match is seeking a state of maximum entropy and minimum free energy. For the folded match, the state of minimum free energy is when the match is on an edge, while for the unfolded match, the state of minimum free energy is when the match is lying flat. Note that a flat match will not stand on its edge because, by falling over, it can release potential energy into kinetic energy. On the other hand, the folded match lying on its edge cannot fall over to a lower state of free energy. As they taught you in high school, atoms like to combine into molecules by sharing electrons in covalent bonds. This is accomplished through shared molecular orbitals between atoms. The first atomic orbital of atoms 1s2 can hold 2 electrons, the next atomic orbital can hold 8 electrons, 2 in the 2s2 orbital and 6 in the 2px2, 2py2, and 2pz2 orbitals, and so forth down through the rest of the periodic table. So for hydrogen H, we can have two hydrogen atoms combine into a molecule of diatomic hydrogen by having each atom of hydrogen share its single electron with the other hydrogen atom to form H2 displayed as: What happens from a molecular orbital point of view is that the 1s atomic orbitals of each hydrogen atom combine to form a sigma σ molecular bond which has a lower energy than the two 1s atomic orbitals combined together, so this σ bond holds the two hydrogen atoms together as a diatomic hydrogen molecule. There is also a σ* molecular orbital at a higher energy level than the 1s atomic orbitals. The σ molecular bond has lower energy because the two electrons of the hydrogen molecule have a high probability of being located between the two positively charged protons of the hydrogen nuclei to which they are attracted. On the other hand, for the σ* molecular orbital, the two hydrogen electrons spend most of their time further away from the two positively charged protons of the hydrogen molecule, which takes more potential energy and puts the σ* molecular orbital at a higher energy level than the σ bond. Figure 3 – The Molecular Orbitals of a Diatomic Hydrogen Molecule (click to enlarge) For the remainder of this post, let’s focus on carbon C, because the carbon atom is the basis for nearly all the molecules used by living things, and, in fact, the chemists have honored carbon with its own branch of chemistry called organic chemistry, because organic chemistry so dominates the field due to its biological and commercial significance. Carbon has a nucleus containing six protons and usually six neutrons and thus has 6 electrons. The first two electrons fit into the first atomic orbital of carbon 1s2, leaving four electrons left over for molecular bonding. These four electrons are called valence electrons. Remember, carbon will try to share four additional electrons with its four valence electrons to reach the magic number of 8 electrons to completely fill its 2p orbital. Figure 4 – Carbon Tries To Share 4 Additional Electrons To Complete Its 2p Shell (click to enlarge) This makes carbon very unique in that it can form very complex organic molecules, since each carbon atom can bind to up to four additional atoms. For example, methane can be depicted as: Carbon can also form very long molecules by chaining together many carbon atoms along a carbon backbone: H H H H H H H H \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| H H H H H H H H Similarly, nitrogen N has a nucleus composed of seven protons and seven neutrons and consequently has seven electrons. Again, two electrons fit into its lowest atomic orbital leaving five left over for bonding. Thus nitrogen would like to share three additional electrons to get to the magic number of 8 for its second orbital. Oxygen O has a nucleus of 8 protons and 8 neutrons with 8 surrounding electrons. After two electrons fill its lowest atomic orbital, there are six electrons left over, leaving oxygen looking for two additional electrons. When we combine carbon, hydrogen, oxygen, and nitrogen together we can form complex organic molecules like the amino acid serine: H H O \| \| \|\| Notice that in serine, carbon has managed to share two electrons with one of the oxygen atoms in a double bond. Serine has a carbon backbone of only two carbon atoms, but as you can imagine, it is possible to form very complicated and very large organic molecules by hanging all sorts of side group atoms off a very long carbon chain backbone. This is all accomplished through Lennard-Jones’ concept of hybridized molecular orbitals. For example, carbon’s four valence electrons can occupy four sp3 hybridized molecular orbitals formed by mixing together the s and px, py, and pz atomic orbitals of carbon. This yields a tetrahedral-shaped set of sp3 hybridized molecular orbitals for carbon, and this is the most common molecular orbital configuration for carbon. Figure 5 – The sp3 Hybridized Orbitals Are a Combination of s and p Atomic Orbitals (click to enlarge) Methane is formed by pairing up the single electron in the 1s atomic orbital of four hydrogen H atoms with the four sp3 orbitals of carbon C, forming σ bonds between the carbon C and hydrogen H atoms. Figure 6 – Methane Forms a Tetrahedral Shape Because of sp3 Hybridized Carbon Orbitals (click to enlarge) One of the misconceptions that can easily arise when you study chemistry is that when you look at all the chemical formulas and molecular models in your course work, your eye is naturally drawn to the symbols for the atomic elements such as C, H, N, and O. This naturally makes you think of atomic nuclei, composed of protons and neutrons, binding together via their valence electrons to form molecules. As we have seen above, this is a bit of a distortion. Chemistry is really all about electrons in molecular orbitals. The atomic nuclei of atoms are really just dead weight, providing positive charge via their protons, but not really performing anything chemically significant beyond that. So when you look at a chemical formula or model for a molecule, you should really think of it as a collection of electron wavefunctions surrounding some highly concentrated positive charge in the nuclei of the atoms. The electrons really do all the work in chemistry, creating the microscopic chemical behaviors of substances, such as their chemical reactivity, acidity, and ability to oxidize other substances, and also the macroscopic characteristics of substances such as their melting and vaporization temperatures, specific heat, color, rigidity, ductility, and tensile strength. Most of everyday life is just electrons doing their thing in different quantum states; with the protons and neutrons of atomic nuclei just along for the ride. Carbon can also form a hybridized orbital called sp2, which takes on a triangular shape and also an sp hybridized orbital which has a linear shape. The sp2 and sp orbitals lead to another kind of molecular bond called a π bond via a π molecular orbital. In Figure 7, we see the molecular bonding for ethane, ethene, and ethyne, which highlights this kind of π bonding. In ethane, each carbon atom forms σ bonds with three hydrogen atoms and also with the other carbon atom in ethane, using its four valence electrons in sp3 orbitals, as we have already seen with methane. In ethene, there are only two hydrogen atoms for each carbon atom to bind with, so there is a double bond between the carbon atoms denoted as: What happens is that each of the carbon atoms have three of their four valence electrons in sp2 hybridized orbitals bound to the 1s orbitals of the two hydrogen atoms and also with the other carbon atom, forming σ bonds with all. The last valence electron of each carbon remains in a lobe-shaped atomic p orbital of each carbon atom. These two p orbital electrons form a π bond between the carbon atoms via a π molecular orbital, which is much weaker than the σ bonds between the carbon atoms. Chemists say that the electrons in the π bond are “delocalized”, meaning they are kind of floating above and below the plane of the carbon atoms. This is just the chemists’ way of expressing the quantum weirdness of electrons not knowing exactly where they are. Ethyne is even stranger. Each carbon has two valence electrons in linearly shaped sp hybridized orbitals. One valence electron is bound to a hydrogen atom and the other valence electron is bound to the other carbon atom via σ bonds. The remaining two valence electrons of each carbon are in p orbitals of the carbon atoms, and form two π bonds between the carbons. Thus there are three bonds between the carbon atoms, one σ bond and two π bonds. Chemists denote a triple bond as: Figure 7 – Ethene and Ethyne Form σ and π Bonds (click to enlarge) The key point is that it is the quantized angular momentum of electrons that is the chief element of chemistry. Because the electron wavefunctions or orbitals of carbon with quantized angular momentum have complex 3-dimensional shapes, organic molecules also have complex 3-dimensional shapes. And because carbon can combine with so many different atoms and has four valence electrons, organic molecules can become huge affairs with very complicated 3-dimensional shapes. When we study softwarebiology, we will see that large complex organic molecules with very complicated 3-dimensional structures are key to living things. These large organic molecules have very complicated molecular orbitals with strange shapes that can fit together like a lock and key to perform biological functions. The fitting together of organic molecules is accomplished via the electromagnetic force. Remember that plots of molecular orbitals are just the probability clouds or wavefunctions of the molecular electrons. When you plot the electron probability cloud for a molecule, frequently you will find that the electrons have a higher probability of being found near one part of the molecule compared to the other parts. This part of the molecule will then have a net negative charge, while the other portions will have a net positive charge. Such molecules are called polar molecules, and the positive portion of a polar molecule will be attracted to the negative portion of other polar molecules. In Figure 8, we see polar water molecules attracting each other. The molecular electrons of a water molecule have a higher probability of being near the oxygen atom, compared to the two hydrogen atoms of the molecule, so water molecules have a net negative charge near the oxygen side of the molecule and a net positive charge near the hydrogen side of the molecule. The negative oxygen portion of water molecules are attracted to the positive hydrogen portion of water molecules, forming what are known as hydrogen bonds. These hydrogen bonds in water form a weak lattice of water molecules even when water is in a liquid state. This highly polar nature of water is what gives water very high melting and boiling point temperatures because the water molecules like to stick together due to the electromagnetic attraction between molecules. The electrical attraction between water molecules allows water molecules to come together in a crystal latice (ice) at a much higher temperature than a non-polar molecule of a similar weight. Similarly, the electrical stickiness of water molecules prevents them from boiling away unless they are jiggled by a lot of thermal energy. The polar nature of water and its tendency to form a lattice of water molecules bound together by hydrogen bonds is very important in biology. In SoftwareBiologywe will see that this is a necessary condition for the formation of cellular membranes. Figure 8 – Water Molecules Are Polar and the Positive Parts Attract the Negative Parts (click to enlarge) Notice that the bonding angle between the oxygen atom and the two hydrogen atoms is 104.45 0 and that all three do not line up in a straight line. Again this is due to the strange geometry of the molecular orbitals of water, like the sp3 molecular orbitals of methane. This geometry all goes back to the strange lobe-like probability cloud, or wavefunction, of the p orbital of electrons that have an angular momentum quantum number of l = 1. If the wavefunction for the p orbital electrons did not have this lobe-like shape, water molecules would be linear and would not be polar molecules because the negative oxygen atom would be sandwiched between to positive hydrogen atoms, and we would not be here marveling at water molecules because life in this Universe would probably be impossible. Because organic molecules can be polar and quite large with very complicated 3-dimensional structures, they can form large intermeshing affairs, that fit together like a lock and key. Because the shapes of the organic molecules have to be just so and the charge patterns just right for organic molecules to fit together like a lock and key, they offer a bit of specificity – only certain organic molecules can fit into a locking position with another. In SoftwareBiology, we will see this is a key requirement for living things. Figure 9 – Only Organic Molecules of the Correct Shape Can Fit Together in a Locking Position (click to enlarge) So here is the strange thing. If electrons did not have quantized angular momentum, all atomic orbitals would be spherically-shaped like the 1s and 2s orbitals in Figure 1, and the most complicated molecule you could make would be a very long linear molecule with σ bonds between atoms, like the diatomic hydrogen molecule of Figure 3. Clearly, living things could not exist in such a universe. Living things need large complex 3-dimensional molecules in order to exist. This is an example of the weak Anthropic Principle in action, which will be covered in a future posting on SoftwareBiology. I have been focusing on molecules composed of carbon, hydrogen, oxygen, and nitrogen atoms because these are the atoms of life and will come in handy when we switch our focus to the biological aspects of softwarephysics. Fully 96% of your body weight is due to carbon, hydrogen, oxygen, and nitrogen atoms with the remaining 4% coming from traces of other atoms such as sulfur (S) and phosphorous (P). High School Chemistry is Vindicated The end result of all this quantum mechanics is to confirm all of the chemistry you learned in high school which was empirically discovered by chemists in the 19th century. Chemistry is all about electrons and the electromagnetic force between the electrons and protons. This is rather strange, since the electrons in an atom represent an insignificant amount of the mass of an atom. Protons have 1836 times as much mass as electrons, and neutrons are just slightly more massive than protons, with a mass that is equal to 1.00138 times that of a proton. Thus, a 200 pound man consists of about 100 pounds of protons, 100 pounds of neutrons, but only about 0.87 ounces of electrons! Yet all of your interactions with the Universe are performed with this small mass of electrons. Everything you see, hear, smell, taste, and feel results from the interactions of less than one ounce of electrons. And all of the biochemical reactions that keep you alive, and even your thoughts at this very moment are all accomplished with this small mass of electrons! This all stems from the fact that, although electrons are very light relative to protons and neutrons, for some unknown reason, they pack a whopping amount of electrical charge. In fact, the light electrons have the same amount of electrical charge as the much heavier protons, just with the opposite sign, so it is the electromagnetic force that really counts in chemistry, not the electrons themselves. In that regard, chemistry can really be considered to be the study of the electromagnetic force, and not the study of matter, since electrons are nearly massless particles. Let us adopt the physicist’s perspective, in which all of chemistry can be seen as simply an extension of the effective theory of quantum mechanics. With that in mind, let us explore the corresponding implications for softwarechemistry. Recall that the individual characters in a sample line of source code: discountedTotalCost = (totalHours * ratePerHour) - costOfNormalOffset; C = 01000011 = ↓ ↑ ↓ ↓ ↓ ↓ ↑ ↑ H = 01001000 = ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↓ N = 01001110 = ↓ ↑ ↓ ↓ ↑ ↑ ↑ ↓ O = 01001111 = ↓ ↑ ↓ ↓ ↑ ↑ ↑ ↑ We may then think of each character in the above line of code as an atom in an organic molecule. Thus, each variable in the line of code becomes an organic molecule in a chemical reaction with the other variables or organic molecules in the line of code, and ultimately produces a macroscopic software effect. The 8 quantized bits for each character are the equivalent of the spins of 8 electrons in 8 electron shells that may be either in a spin up↑ or spin down ↓ state. And the chemical characteristics of each character (atom) are determined by the arrangements of the spin up ↑ or spin down ↓ state of the bits (electrons) in the character. The characters (atoms) in each variable come together to form an organic molecule, in which the spins of all the associated characters form molecular orbitals for the variable, giving the variable its ultimate softwarechemical characteristics. As a programmer, your job is to assemble characters (atoms) into variables (molecules) that interact in lines of code to perform the desired functions of the software under development. Living things have evolved very reliable methods to do the same thing. Just take a look at any college textbook on biochemistry. In it you will find very complicated flow charts of biosynthetic and metabolic pathways that put to shame the multithreaded logic found in a typical java EJB. And the fact that these pathways are constantly being run in a multithreaded manner, trillions upon trillions of times, within each of the 100 trillion cells in your body, just boggles the mind. For example, Figure 10 depicts the famous Krebs cycle. In 1937, Hans Krebs proposed the Krebs cycle for organisms that have an oxygen-based metabolism. The Krebs cycle is a programming loop that controls the breakdown of proteins, fats and carbohydrates into smaller molecules. The loop results in the liberation of carbon dioxide and electrons that are used to form high-energy phosphate bonds in the form of adenosine triphosphate (ATP) - the chemical energy reservoir of cells. Krebs discovered how certain individual reactions are linked to each other in a do-loop and how energy is released by this process for use by the cell for all its activities. He proposed the steps in this loop in 1937, and was awarded the 1953 Nobel Prize in Physiology or Medicine for this work. The Krebs cycle is like a processing loop that transforms the value of a share of IBM stock into physical cash coming out of an ATM. The share of IBM stock begins as set of bits stored somewhere in cyberspacetime on the computers run by your online stockbroker. When you sell the share of stock, the bits storing the share of stock are debited from your account and some bits over in the IBM portion of cyberspacetime are deleted too. The resulting cash value is credited to the money market bits of your online stockbroker account. From there the cash value bits can be transferred to your local bank, and ultimately, you can punch in some numbers into an ATM and out pops some physical cash that can be used to buy a cup of coffee. In this example, the share of IBM stock is like an energy rich carbohydrate molecule and the cash popping out of the ATM is like a molecule called ATP, which is the biochemical energy equivalent of cash. In a capitalistic economic system, money is the equivalent of energy in a biochemical sense, since, as everybody knows, it makes the world go round. There is an old joke in thermodynamics that energy is the ability to do work, while money is the ability not to do work. Figure 10 – The Krebs cycle (click to enlarge) Your body uses the energy in ATP to build the complex organic molecules necessary to perform the functions of life. Taking simple atoms and producing complex organic molecules from them is clearly a violation of the second law of thermodynamics because we are taking disordered atoms and creating highly ordered organic molecules from them. As a programmer, you are well aware of the equivalent problem of assembling characters into lines of code that actually work. The only way around this problem is to degrade the low entropy chemical energy in carbohydrates and fats into disordered heat energy, and that is what your body does. Using the Krebs cycle, your body converts the chemical energy stored in carbohydrates and fats into chemical energy stored in ATP. The cells in your body then degrade the low entropy chemical energy stored in ATP into heat energy in order to create complex organic molecules. In this way, the second law is not violated. Your body heat is a way for your body to excrete entropy, while increasing its internal information content at the same time in the form of information rich organic molecules. The moment you die, your body begins to cool off, and you begin to disintegrate as the second law of thermodynamics runs wild. I will close with that sobering thought in mind. Next time, as promised, we will continue on with exploring the really strange implications of quantum mechanics, in an effort to combat the objection that equating the characters in a line of code with physical atoms is a bit of a stretch. You will learn that, thanks to 20th century physics, there really isn’t much tangible stuff left in the physical Universe, so equating the bits of information in source code in the Software Universe with physical atoms in the physical Universe, is really not such a stretch after all. Comments are welcome at To see all posts on softwarephysics in reverse order go to: Steve Johnston
df75a0fdfbde2143	Long-Standing Space Disk Mystery Solved by Basic Quantum Physics Quantum mechanics is concerned with the behavior of the tiniest of particles, and usually the mathematics behind it is relegated to this tiny realm. Now, a researcher from the California Institute of Technology has used a fundamental quantum physics equation to understand huge self-gravitating space disks. Konstantin Batygin, an assistant professor at Caltech, has discovered that the changing shapes of spinning disks of matter around massive astronomical objects like black holes can be described by the Schrödinger equation. The evolution of these disks has stumped astrophysicists for many years. Swarming matter 3_6_Quantum Space Disks An artist's impression of the research, published in Monthly Notices of the Royal Astronomical Society. James Tuttle Keane/California Institute of Technology From the satellites that fly around Earth to the the planets that swarm around the sun, gravitational forces create huge rotating disks of matter throughout the universe. Over time, these flat circular disks can become warped and distorted, but astrophysicists don’t really know why. Batygin decided to use a mathematical scheme called perturbation theory to try and explain why these spinning disks lost their shape. The model, frequently used in astronomy, blended individual bits of matter traveling on particular orbital trajectories into wires. These concentric loops of matter slowly spread angular momentum between each other. These wires can mirror the real orbital evolution over millions of years, resulting in a fairly accurate approximation of the changing disk. Batygin’s mathematics, however, revealed an unexpected result. A fundamental quantum physics equation was hiding in his model. "Eventually, you can approximate the number of wires in the disk to be infinite, which allows you to mathematically blur them together into a continuum,” he explained. “When I did this, astonishingly, the Schrödinger Equation emerged in my calculations." The Schrödinger equation 3_6_Schrödinger's Cat Dead or alive? Schrödinger is famous for his quantum physics experiment in which a cat in a box may or may not have been poisoned. Dhatfield/Wikimedia Commons The Schrödinger equation—named for creator and Nobel Prize-winning physicist Erwin Schrödinger—forms the basis of quantum physics. It describes the strange behavior of atomic and subatomic systems that evolve over time. On a tiny scale, particles behave more like waves than individual particles. Batygin’s research suggests this "wave-particle duality" could also describe the large warps of an astronomical disk. "The Schrödinger equation is an unlikely formula to arise when looking at distances on the order of light years," said Batygin. "I was fascinated to find a situation in which an equation that is typically used only for very small systems also works in describing very large systems." Finding such an important and well-studied equation deep within self-gravitational disk models should shed light on the strange and mysterious phenomena. The link between the two spheres of science, Batygin says, in hindsight “seems like an obvious connection.” “Fundamentally, the Schrödinger equation governs the evolution of wave-like disturbances." he added. "In a sense, the waves that represent the warps and lopsidedness of astrophysical disks are not too different from the waves on a vibrating string, which are themselves not too different from the motion of a quantum particle in a box." Join the Discussion
4933990ef63136af	Dismiss Notice Join Physics Forums Today! Math suggestions for learning QM 1. Mar 21, 2013 #1 Someone asked me which math topics to study in order to learn quantum information theory. I thought it was a good question, so here's my answer. Warning: this is off the top of my head, so it probably needs additions and/or corrections. Most of this advice applies to anyone doing quantum mechanics. If Q Info isn't your subject, you might want to focus more on calculus and PDEs and less on density matrices, entropy, and Markov processes. I think the biggest problem with quantum mechanics is that almost every statement is either 0) ambiguous or 1) full of math jargon. So it's very important to know how to translate the math jargon. Here are some examples: • A finite-dimensional density matrix is a convex combination of rank-1 projection operators, each of which acts on the Hilbert space ##\mathbb{C}^N##. • The generator of time evolution is ##-\frac{\imath}{\hbar}\hat{H}(t)##, where the Hamiltonian ##\hat{H}## is a self-adjoint linear operator. • The set of all traceless ##N \times N## self-adjoint complex matrices forms a real Lie algebra with the commutator as its Lie bracket. This algebra is isomorphic to ##\mathfrak{su}(N)##. • The von Neumann entropy of a density matrix is the Shannon entropy of its eigenvalues. The first step in QM is figuring out what the hell that stuff says. For example, a Hilbert space is an abstract vector space with a definition of inner product that satisfies certain rules for convergence of infinite series. ##\mathbb{C}^N## is a Hilbert space which can be used to represent state vectors for ##N##-level systems. For most practical purposes, I think of each vector in this space as a column of ##N## complex numbers. (So does MATLAB.) A good start is to look for books/classes/websites with these words in them: • Linear algebra, vector space, inner product • Eigenvalues, eigenvectors, the spectral theorem • Random variable, probability space, probability distribution • Statistical physics, Shannon entropy, Markov process, density matrix • Multivariable linear ordinary differential equations (The finite-dimensional Schrödinger equation is a multivariable linear ODE.) • Partial differential equations, diffusion, Laplacian (The infinite-dimensional Schrödinger equation is closely related to diffusion PDEs.) • Group theory, Lie algebra A huge amount of QM consists of manipulating matrices and matrix-like things. (Dirac notation suggests treating infinite-dimensional linear operators as if they were matrices, sort of.) So it's good to know lots of matrix tricks. My favorite "matrix cheat sheet" is available here. If you're already good at matrix algebra, then a little bit of Lie group theory goes a long way in QM. I'm not an expert at it, but I know what Lie meant by "infinitesimal generator." It helps that my advisor is an expert, so he can correct my dumb mistakes before I publish them. The next steps depend on exactly what topic you're studying. I learned stochastic calculus, which is important for my thesis. Most Q Info people probably don't know much of that, but they often know a lot more than me about logic circuits and binary algorithms. People who actually build qubits need to learn the specific physics of their design, e.g. Josephson junctions or quantum optics or crystal defects. Good luck! Or, if you think there's no such thing as luck: may the gradient of potential be against you. 2. jcsd 3. Mar 21, 2013 #2 User Avatar Science Advisor But F=-∇ϕ 4. Mar 21, 2013 #3 So if ##\nabla \Phi## is against you, then The Force must be with you. (rimshot) 5. Mar 22, 2013 #4 Staff: Mentor Gee mate I have a degree in math and even I didn't do group theory with Lie Algebra's and stuff - had to learn it later after reading some QM books - but did two courses on functional analysis and Hilbert Spaces which was a help. My view is if you have most of the stuff above you are good to go - you can pick up the rest as you go. Last edited: Mar 22, 2013 6. Mar 26, 2013 #5 User Avatar Science Advisor Great intro to QM too.
647618c2deee3122	How does the connectedness of a quantum graph affect the localization of eigenfunctions? Math Physics Seminar Friday, January 26, 2018 - 15:00 1 hour (actually 50 minutes) Skiles 202 Georgia Tech Quantum theory includes many well-developed bounds for wave-functions, which can cast light on where they can be localized and where they are largely excluded by the tunneling effect. These include semiclassical estimates, especially the technique of Agmon, the use of "landscape functions," and some bounds from the theory of ordinary differential equations. With A. Maltsev of Queen Mary University I have been studying how these estimates of wave functions can be adapted to quantum graphs, which are by definition networks of one-dimensional Schrödinger equations joined at vertices.
63654a0a22c6e65a	Skip to content BMC Structural Biology Open Access Clustering and percolation in protein loop structures BMC Structural Biology201515:22 Received: 7 April 2015 Accepted: 13 October 2015 Published: 29 October 2015 High precision protein loop modelling remains a challenge, both in template based and template independent approaches to protein structure prediction. We introduce the concepts of protein loop clustering and percolation, to develop a quantitative approach to systematically classify the modular building blocks of loops in crystallographic folded proteins. These fragments are all different parameterisations of a unique kink solution to a generalised discrete nonlinear Schrödinger (DNLS) equation. Accordingly, the fragments are also local energy minima of the ensuing energy function. We show how the loop fragments cover practically all ultrahigh resolution crystallographic protein structures in Protein Data Bank (PDB), with a 0.2 Ångström root-mean-square (RMS) precision. We find that no more than 12 different loop fragments are needed, to describe around 38 % of ultrahigh resolution loops in PDB. But there is also a large number of loop fragments that are either unique, or very rare, and examples of unique fragments are found even in the structure of a myoglobin. Protein loops are built in a modular fashion. The loops are composed of fragments that can be modelled by the kink of the DNLS equation. The majority of loop fragments are also common, which are shared by many proteins. These common fragments are probably important for supporting the overall protein conformation. But there are also several fragments that are either unique to a given protein, or very rare. Such fragments are probably related to the function of the protein. Furthermore, we have found that the amino acid sequence does not determine the structure in a unique fashion. There are many examples of loop fragments with an identical amino acid sequence, but with a very different structure. Loop modelingProtein backboneC α trace problem Protein taxonomy [15] reveals that crystallographic protein structures have surprisingly little conformational diversity. It might be that the majority of different conformations have already been found [6, 7]. This apparent convergence in protein structure provides the rationale for the development of comparative modelling or threading techniques [812]. These approaches try to predict the tertiary structure of a folded protein using libraries of known protein structures as templates. According to the community-wide Critical Assessment for Structural Prediction (CASP) tests [13], at the moment this kind of methods have the best predictive power to determine a folded conformation. In the loop regions, comparative modelling approaches still continue lacking in their precision [14, 15]. It is not uncommon that there are gaps in the loop regions that need to be filled by various insertion techniques. The success in loop modelling is also often limited to super-secondary structures where α-helices and β-strands are connected to each other by relatively short twists and turns [16, 17]. In the case of a very short loop, with no more than three residues, the shape can be determined by a combination of geometrical considerations and stereochemical constraints [18]. In the case of longer loops, both template based and template independent methods are being developed to predict their shapes [1921]. The underlying assumption is that the number of loop conformations which can be accommodated by a given sequence should be limited. The different fragments which are already available in the Protein Data Bank (PDB) [22] database could then be used like Lego bricks, as structural building blocks in constructing the loops. A given amino acid sequence is simply divided into short fragments, and the shape of the ensuing loop is deduced using homologically related fragments that have known structures. The entire protein is then assembled by joining these fragments together. For the process of joining the fragments, both all-atom energy functions and comparisons with closely homologous template structures in the Protein Data Bank can be utilised [8, 9, 12, 14]. In the present article we propose a new systematic, purely quantitative method to identify and classify the modular building blocks of PDB loops; we identify a loop following the DSSP [23] convention. Our approach is based on a first-principles energy function [2429]. It is built on the concept of universality [3036] to model the fragments of even long protein loops in terms of different parameterisations of a unique kink that solves a variant [37, 38] of the discrete nonlinear Schrödinger (DNLS) equation [39, 40]. Our starting point is the observation made in [41] that over 92 % of loops in those PDB structures that have been measured with better than 2.0 Å resolution, can be composed from 200 different parameterisations of the kink profile, with better than 0.65 Ångström RMSD (root-mean-square-distance) accuracy. Here we refine this observation, with the aim to develop it into a systematic loop fragment classification scheme. For this we consider only those ultrahigh precision PDB structures that have been measured with better than 1.0 Å resolution. This ensures that the B-factors in the loop regions are small, and in particular that the structures have not been subjected to extensive refinement procedures. Indeed, two loop fragments should be considered different only, when the average interatomic distance is larger than the average Debye-Waller B-factor fluctuation distance. If the B-factors are large, any systematic attempt to identify and/or distinguish two fragments becomes ambiguous. In the case of these intra-high resolution structures we can aim for the RMSD precision of 0.2 Å. We estimate this to be the extent of zero point fluctuations i.e. a distance around 0.2 Å corresponds to the intrinsic uncertainty in the determination of heavy atom positions along the protein backbone. Thus any difference less than 0.2 Å between average atomic coordinates is essentially undetectable. By explicit constructions, we show how in the case of this subset of ultrahigh resolution PDB protein structures, the loops can be systematically modeled using combinations of the unique kink of the generalised DNLS equation. As such, our approach provides a foundation for a general approach to classify loops in high precision crystallographic PDB structures, in terms of an energy function based first-principles mathematical concept. C α based Frenet frames Let r i (i=1,…,N) be the coordinates of the protein backbone α-carbon (C α) atoms. The indexing starts from the N terminus. At each r i we introduce the discrete Frenet frame (t i ,n i ,b i ) shown in Fig. 1 following the method in reference [42]. Fig. 1 Discrete Frenet frame. (Color online) Discrete Frenet frame vectors t,n,b are shown in arrows From the Frenet frames, we define the virtual C α backbone bond (κ) and torsion (τ) angles shown in Fig. 2 as follows, $$ \cos\kappa_{i+1} \ = \ \mathbf t_{i+1} \cdot \mathbf t_{i} $$ Fig. 2 Bond and torsion angles. (Color online) Bond (κ i ) and torsion (τ i ) angles with the definitions as Eqs. (1) and (2) are noted in the figure $$ \cos\tau_{i+1} \ = \ \mathbf b_{i+1} \cdot \mathbf b_{i} $$ We identify the bond angle κ [ 0,π] with the latitude angle of a two-sphere which is centered at the C α carbon; the tangent vector t points towards the north-pole where κ=0. The torsion angle τ [ −π,π) is the longitudinal angle on the sphere. We have τ=0 on the great circle that passes both through the north pole and through the tip of the normal vector n, and the longitude increases in the counterclockwise direction around the tangent vector. We stereographically project the sphere onto the complex (x,y) plane from the south-pole $$ z=x+iy \ \equiv \ \sqrt{x^{2} + y^{2}} \, e^{i\tau} \ = \ \tan\left(\kappa/2 \right) \, e^{i\tau} $$ as shown in Fig. 3; the north-pole where κ=0 becomes mapped to the origin (x,y) =(0,0) while the south-pole κ=π is sent to infinity. Fig. 3 Stereogrphic projection. (Color online) Stereographic projection of two sphere with latitude κ and longitude τ We often find it convenient to extend the range of the latitude κ from positive to arbitrary real values. We compensate for this double covering of the sphere by introducing the following discrete $\mathbb Z_{2}$ gauge transformation $$ \begin{array}{llll} \ \ \ \ \ \ \ \ \ \kappa_{k} & \to & - \ \kappa_{k} & \quad\text{for \ \ all} \ \ k \geq i \\ \ \ \ \ \ \ \ \ \ \tau_{i} & \to & \quad\tau_{i} - \pi \end{array} $$ This transformation has no effect on the backbone coordinates r i , and it leaves the C α backbone intact. The C α trace visualization, loops and kinks The C α map We visualise the backbone C α trace of a given protein in terms of a trajectory on the stereographically projected two-sphere, as follows [4345]: At the location of each C α we introduce the corresponding discrete Frenet frames (t i ,n i ,b i ). The base of the i t h tangent vector t i is located at the position r i of the i t h C α carbon, it coincides with the centre of the two-sphere and the vector t i points towards the north-pole. We translate the sphere from the location of the i t h C α to the location of the (i+1) t h C α, without introducing any rotation of the sphere with respect to the i t h Frenet frames. We identify the direction of t i+1, i.e. the direction towards the C α carbon at site r i+2 from the site r i+1, on the surface of the sphere in terms of the ensuing spherical coordinates (κ i ,τ i ). We repeat the procedure for all the backbones in PDB. To enhance statistics, for visualisation purposes we use here those protein structures that have been measured with better than 2.0 Å resolution, which gives us the map shown in Fig. 4 a; see also Figure S1 in Additional file 1. The color intensity correlates directly with the statistical distribution of the (κ i ,τ i ): red is large, blue is small and white is none. The map describes the direction of the C α carbon at r i+2 as it is seen at the vertex r i+1, in terms of the Frenet frames at r i . Fig. 4 C α stereographical projection map and folding index. (Color online) a The stereographically projected Frenet frame map of backbone C α atoms, with major secondary structures identified. Also shown is the directions of the Frenet frame normal vector n; the vector t points upwards and colour coding corresponds to the number of PDB entries with red large, blue small and white none. b An example of a loop (kink) trajectory, starting (a) and ending (e) in α-helical position Note how the statistical distribution in Fig. 4 concentrates within an annulus, roughly between the latitude angle values (in radians) κ1 and κπ/2. The exterior of the annulus is a sterically excluded region. The entire interior is in principle sterically allowed, but it is very rarely occupied in the case of folded proteins. The four major secondary structure regions, α-helices, β-strands, left-handed α-helices and loops, are identified according to their PDB classification. For example, (κ,τ) values (in radians) for which $$ \left\{ \begin{array}{lll} \kappa_{i} & \approx & \frac{\pi}{2} \\ \tau_{i} & \approx & 1 \end{array} \right. $$ describes a right-handed α-helix, and values for which $$ \left\{ \begin{array}{lll} \kappa_{i} & \approx & 1 \\ \tau_{i} & \approx & \pm \pi \end{array} \right. $$ describes a β-strand. We note that the Fig. 4 a is akin the Newman projection of stereochemistry: The vector t i which is denoted by the red dot at the center of the figure, points along the backbone from the proximal C α at r i towards the distal C α at r i+1, and the colour intensity displays the statistical distribution of the r i+2 direction. We also note that the Fig. 4 provides non-local information on the backbone geometry; the information content extends over several peptide units. This is unlike the Ramachandran map, which can only provide localised information in the immediate vicinity of a single C α carbon. As shown in [46], the C α backbone bond and torsion angles (κ i ,τ i ) are sufficient to reconstruct the entire backbone, while the Ramachandran angles are not. In Fig. 4 b we visualise as an example a path made by a generic protein loop that connects two right-handed α-helical structures. A notable property of the trajectory drawn in Fig. 4 b is that it encircles the north-pole of the two-sphere. It turns out that this kind of encircling is quite generic for loops, even entire folded proteins [47]. Consequently, we assign to each loop a winding number which we term folding index that we denote I n d f [47] and define as follows, $$\begin{array}{{20}l} {Ind}_{f}&=&\left[\frac{\Gamma}{\pi}\right] \end{array} $$ $$\begin{array}{{20}l} \Gamma&=&\sum\limits_{i=n_{1}+2}^{n_{2}-2} \left\{ \begin{array}{ll} \tau_{i}-\tau_{i-1}-2\pi & ~~\text{if}~~ \tau_{i}-\tau_{i-1} > \pi\\ \tau_{i}-\tau_{i-1}+2\pi & ~~\text{if}~~ \tau_{i}-\tau_{i-1} < -\pi\\ \tau_{i}-\tau_{i-1} & ~~\text{otherwise} \end{array}\right. \end{array} $$ Here [x] denotes the integer part of x, and Γ is the total rotation angle (in radians) that the projections of the C α atoms of the consecutive loop residues make around the north pole. The folding index is a positive integer when the rotation is counterclockwise, and a negative integer when the rotation is clockwise. The folding index can be used to detect and classify loop structures and entire folded proteins, in terms of its values. The value is equal to twice the number of times the ensuing pathway encircles the north-pole in the map of Fig. 4; for the trajectory shown in Fig. 4 b the folding index is +2. The discrete nonlinear Schrödinger equation The virtual bond length between two neighboring C α atoms is essentially constant, with the value 3.8 Å. Accordingly the Helmholtz free energy for the C α trace backbone can be expressed in terms of the virtual bond angles κ i and dihedral angles τ i only. To the leading order in the infrared limit the result coincides with $$ \begin{aligned} F &= - \sum\limits_{i=1}^{N-1} 2\, \kappa_{i+1} \kappa_{i} + \sum\limits_{i=1}^{N} \left\{2 {\kappa_{i}^{2}} + c \, \left({\kappa_{i}^{2}} - m^{2}\right)^{2}\right.\\ &\quad\left.+\, b \, {\kappa_{i}^{2}} {\tau_{i}^{2}} + d \, \tau_{i} + e \, {\tau^{2}_{i}} + q \, {\kappa_{i}^{2}} \tau_{i} \vphantom{\left\{2 {\kappa_{i}^{2}} + c \, \left({\kappa_{i}^{2}} - m^{2}\right)^{2}\right.}\right\} \end{aligned} $$ This is essentially the Hamiltonian of the discrete nonlinear Schrödinger equation [39, 40]; for a detailed derivation we refer to [2429]. Remarkably, the free energy (9) supports a kink (topological soliton) as a classical solution [37, 38]. An excellent approximation of a kink can be obtained by naively discretising the kink solution of the continuum nonlinear Schrödinger equation [37, 38, 48] $$ \kappa_{i} = \frac{\mu_{1} \exp\left[ \sigma_{1} (i-s) \right] + \, \mu_{2} \exp\left[ - \sigma_{2} (i-s)\right]} {\exp\left[ \sigma_{1} (i-s) \right] + \exp\left[- \sigma_{2} (i-s)\right]} $$ The torsion angles τ are then expressed as functions of the bond angles κ $$ \tau_{i} [\!\kappa] \ = \ - \frac{1}{2} \, \frac{d + q {\kappa_{i}^{2}}}{e + b{\kappa_{i}^{2}}} $$ For the torsion angles, from (11) we conclude that the overall scale of the parameters (d,q) and (e,b) cancel in the expression (11). This leaves us with only three independent parameters. In (10) there are four parameters when we use translation invariance to remove s. Thus the profile of a single kink becomes fully determined in terms of seven independent parameters. This coincides exactly with the number of independent coordinates along a C α backbone segment, with six residues. For this, we may always place the first residue to coincide with the origin of a Cartesian (xyz) coordinate system. We can always place the second residue along the z-axis, and we can always place the third residue on the x=0 plane. Thus, there is only one independent coordinate for the three first residues. Since the remaining three residues can each be placed to arbitrary angular directions, there are six additional independent coordinates. Accordingly, a segment with six residues indeed engages seven independent parameters. Clustering and percolation We shall classify the loop structures in PDB in terms of the following clustering algorithm: • We define a cluster to be a set of loop fragments such that for each fragment in a given cluster there is at least one other fragment within a prescribed RMS cut-off distance. Two clusters are disjoint, when the RMSD between any fragment in the first cluster and any fragment in the second cluster exceeds this prescribed RMS cut-off distance. • We define the initiator of a cluster to be an a priori random loop fragment which defines the cluster by completion, as follows: We start with the initiator. We identify all those fragments in our entire data set which deviate from the initiator by less than the given RMS cut-off distance. We continue the process by identifying all those fragments, that deviate from the fragments that we have identified in the previous step, by less than the RMS cut-off distance. We repeat the procedure until we find no additional fragments in PDB, within the RMS cut-off distance from any of those fragments that have been identified in the previous steps. The cluster is clearly independent of its initiator, any element of the cluster could be used as the initiator. But the cluster depends on the RMS cut-off distance. Moreover, if the RMS cut-off distance is too large, no clear clustering is observed. According to [49] for a PDB protein structure which is measured with resolution 2.0 Å or better, the characteristic values of the thermal B-factors are mostly less than around $$ B_{max} \ \buildrel < \over \sim \ 35 \ \text{\r{A}}^{2} $$ From the Debye-Waller relation we then obtain the following estimate for the one standard deviation error in the atomic coordinates $$ \sqrt{<x^{2} >}_{max} \ = \ \sqrt{\frac{B_{max}}{8\pi^{2}}} \ \approx 0.65 { \text{\r{A}}} $$ Thus, two loop fragments that have been measured with 2.0 Å resolution should be (in average) considered different only, when their RMS distance exceeds 0.65 Å. The construction of PDB loop fragments in terms of the kink profile (10), (11) in those crystallographic protein structures which have been measured with resolution 2.0 Å or better, has been addressed in [41]. There, it was found that over 92 percent of loops can be covered in a modular fashion by 200 explicit kink profiles (10), (11), with RMSD accuracy that matches (13) i.e. with less than 0.65 Å RMSD deviation from the crystallographic structure. Thus 0.65 Å RMS distance is the appropriate RMS cut-off value, to search for for the more refined clustering patterns in those crystallographic structures which have been measured with resolution 2.0 Å. However, we find that the value 0.65 Å is too large, to observe clear clustering patterns. Accordingly, we shall search for clustering by considering only those PDB structures that have been determined with the ultrahigh resolution 1.0 Å or better. For these ultrahigh resolution structures, a precision better than the value (13) can be expected. To determine an appropriate value, we display in Fig. 5 the number of all C α atoms in all currently available PDB structures, that have been measured with resolution 1.0 Å or better, as a function of their Debye-Waller fluctuation distance. For most of the structures, the fluctuation distance is clearly below the upper bound (13); the maximum of the curve is located at around 0.3 Å. We also observe the (essential) absence of structures with a fluctuation distance less than 0.1 Å; historically this distance is considered as the boundary wavelength between x-rays and γ-rays. Fig. 5 Debye-Waller fluctuations for PDB structures. Number of C α entries in PDB measured with resolution under 1.0 Å vs. the Debye- Waller fluctuation distance. The blue line denotes the Debye-Waller fluctuation distance distribution for β-sheets, black for α-helices, and red for loop. The entries near 0 correspond to the PDB structures 1ETL,1ETM and 1ETN. Note the logarithmic scale Using a combination of Fig. 5 with various tests that we have performed, we have arrived at the conclusion that 0.2 Å in RMS distance can be currently adopted as a reasonable estimate for the minimal zero-point fluctuation distance in ultra-high resolution structures, those that have been measured with better than 1.0 Å resolution. Thus we shall try and see, to what extent loops in these protein structures can be classified in terms of elemental fragments, such that two fragments are considered different when their RMS distance exceeds 0.2 Å. According to Fig. 5, over 99 % of individual C α carbons that have been measured with below 1.0 Å resolution, have a B-factor fluctuation distance which is larger than 0.2 Å; our choice of cut-off distance is close to the 3- σ level. We note that other cut-off values can be introduced; the ultimate appears to be 0.1 Å. But our qualitative conclusions are fairly independent of the value chosen, provided it is small enough to provide a clustering pattern. In this article our goal is to present a proof-of-concept. To our knowledge, no related analysis has been previously attempted, to systematically classify the loop structures in ultra-high resolution crystallographic protein conformations, in a quantitative fashion using an energy function. In particular, no commonly accepted experimental standard exist, that we could rely on, to infer the “most preferred” cut-off value. We hope that such a value can be eventually inferred, from careful experimental measurements. Thus, at the moment we have no criterion to prefer any other particular value, 0.2 Å i.e. around 3- σ appears to be a reasonable choice at this point. We start the identification of loop fragments, using the set of 200 fragments constructed in [41]. But our results are independent of the starting point, quite similar results can be obtained using a fairly generic set of loop fragments as a starting point. We note that the fragments in [41] have between five and nine residues, and most of them (116 out of 200) have six residues. We have already argued that six is the optimal number of residues in a loop fragment, as it matches the number of independent parameters in the kink profile (10), (11). Thus, we shall consider only fragments that have six residues, in the clustering algorithm. In this manner, we find that we can classify all PDB fragments into clusters, each determined by their initiator. We have found that there are clusters that have a very large number of fragments. But we also find that there are clusters with only a single, or very few fragments. It is natural to expect that those clusters which are large, contain mostly fragments that are structurally important. On the other hand, those clusters which are small should include mainly fragments that are functionally important. Furthermore, we find several examples of amino acid sequences that are included in different clusters: The sequence does not define the structure, in a unique fashion. This leads us to address the concept of cluster percolation: Given the sequence of a loop fragment in a cluster, percolation means that there are other, possibly new clusters where the same sequence appears but with a different structure. We have constructed our clusters by starting with the 200 loop fragments that were introduced in [41]. Around 92 % of all loops in those PDB structures that have been measured with resolution better than 2.0 Å, are within a 0.65 Å RMS distance from some of the 200 loop fragments. However, when we decrease the RMSD cut-off distance to 0.2 Å, which is the cut-off distance used in the present article, the coverage drops to below 2 % [41]. We remark that the authors of reference [41] did not investigate clustering, as the concept is defined here. In [41] all the RMS distances were evaluated from the fixed set of 200 loop fragments, and the coverage of PDB loop structures was determined in terms of these fixed loop fragments. When we specify to the present subset of PDB structures in [41] that have been measured with better than 1.0 Å resolution, we find that a total of 102 out of the 200 fragments in [41] have been measured with this resolution. We use these 102 loop fragments as the initiators, to start our clustering construction. The 102 loop fragments in [41] that have been measured with better than 1.0 Å resolution, have between five and nine residues. Here we have argued that a loop fragment modelled by (10), (11) has six residues. There are 70 such clusters among the 200, but only 14 of them contain more than 30 fragments. Moreover, two of these merge together into an α-helical structure, when we subject them to our clustering algorithm; we call them bends instead of kinks. The remaining 12 loop fragments determine clusters which cover around 38 % of the 1.0 Å protein loop structures, when we use our 0.2 Å RMSD cut-off. These loop fragments are our final initiators. In Table 1 we list the PDB entry codes and residue numbers of these initiators. Table 1 The list of 12 initiators for clusters that have 6 residues and give rise to 30 or more entries in the ensuing clusters (PDB code, chain, PDB sites), together with the number of entries Cluster # # entries 1vyr_A (174–179) 1g4i_A (56–61) 1gkm_A (163–168) 4f18_A (1244–1249) 1a6m_A (18–23) 1cex_A (140–145) 1a6m_A (56–61) 1iee_A (47–52) 1brf A (5–10) 1ixh_A (200–205) 2o7a_A (62–67) 1gkm_A (9–14) We proceeded to describe some of the major features of the ensuing 12 clusters. Additional details including a breakdown according to amino acid constituents in each cluster, are presented in Figure S2 of Additional file 1. The Figs. 6 and 7 show the (κ,τ) distribution in each of the 12 clusters on the stereographically projected two-sphere of Fig. 4; note that the definition of bond angle takes three residues while the definition of torsion angle takes four. Thus for a 6 residue loop fragment there are three (κ,τ) pairs. The fourth κ-value could be used to refine the loop classification, but here this possibility is not considered. Fig. 6 The stereographic maps of 12 clusters I-VI. The clusters I-VI in Table 1 are shown on the stereographic map like Fig. 4 a; In each panel the order along the C α backbone is r e db l u ey e l l o w Fig. 7 The stereographic maps of 12 clusters VII-XII. The clusters VII-XII in Table 1 are shown on the stereographic map like Fig. 4 a; The ordering along the C α backbone is r e db l u ey e l l o w In Figs. 8 and 9 we show the three dimensional pictures of the initiators of the twelve clusters. Fig. 8 The initiators of the 12 clusters I-VI. The initiators I-VI listed in Table 1 are shown in their three dimensional backbone environment. The (dark) red color identifies the initiator, and the (light) yellow color identifies the immediate backbone environment Fig. 9 The initiators of the 12 clusters VII-XII. The initiators VII-XII listed in Table 1 are shown in their three dimensional backbone environment. The (dark) red color identifies the initiator, and the (light) yellow color identifies the immediate backbone environment A detailed inspection reveals that except for IV, all the initiators have the canonical structure of a single kink, in terms of the folding index (8). Moreover, the initiator I is part of a short loop connecting an α-helix and a β-strand. However, the bond and torsion angle spectrum which we display in Fig. 10 a shows that this loop is actually a pair of two kinks which are very close to each other, and the initiator I is the second kink along the backbone. Fig. 10 The (κ,τ) spectrum of initiator I and IV. The figure a shows the $\mathbb Z_{2}$ gauge transformed spectrum of bond and torsion angles in the case of the initiator I. This reveals that the initiator is a two-kink configuration that forms a loop between α-helical and β-stranded regular secondary structures. The figures b and c show the bond and torsion angle spectra of the bend-like initiator IV prior and subsequent to the $\mathbb Z_{2}$ gauge transformation, respectively On the other hand, a comparison with (8) suggests that the initiator IV exhibits a somewhat small variation in the values of the torsion angles, for a kink. This can be seen in Fig. 6. The torsion angle values suggest that the initiator IV resembles more a bent α-helix than a kink. In Fig. 10 b, c we show the spectrum of the bond and torsion angles of the initiator IV, both before and after we have implemented the $\mathbb Z_{2}$ gauge transformation. Since this bent structure determines an isolated cluster according to our 0.2 Å cut-off criteria, it is included among our loop fragments. In Figs. 11 and 12 we show the three dimensional figures of each of the twelve clusters, including all the entries. Fig. 11 The 3D superimposed structures for 12 clusters I-VI. The clusters I-VI in Table 1 are superimposed in three dimensions. The colour ranges from red (initiator) to blue (the entry with largest RMSD distance from initiator) Fig. 12 The 3D superimposed structures for 12 clusters VII-XII. The clusters VII-XII in Table 1 are superimposed in three dimensions. The colour ranges from red (initiator) to blue (the entry with largest RMSD distance from initiator) Finally, we have also investigated how the coverage of the 12 clusters increases, when we increase the cut-off distance. The results are shown in Table 2. Table 2 The coverage of the 12 clusters obtained using the initiators in Table 1, as a function of the cut-off distance Cut-off (Å) Coverage (%) Cluster elongation and completion In addition of the 12 initiators listed in Table 1, among the 102 loop fragments of [41] that we have considered, there is also one initiator that has only five residues. The PDB code is 1p1x_A (80–84). The ensuing cluster with five residue long elements is very large: There are a total of 42618 entries. The reason for the occurrence of such a large cluster is that the RMSD clustering criteria 0.2 Å is too large for revealing clustering patterns in five-residue-long loop segment: The five-residue-long loop fragment covers all the five-residue-long loops, within the chosen cut-off criterion. In Fig. 13 we show the distribution of (κ,τ) values in this cluster. Fig. 13 The stereographic map generated by cluster 1p1x_A (80–84). In a the distribution of the first (κ,τ) and in b the distribution of the second (κ,τ). Note the widely spreaded distributions of this cluster There is also an overlap with each of the 12 clusters that we obtained previously. Together the 13 clusters cover around 96.1 % of all PDB loop structures. It is apparent that an initiator with only five residues is too short to identify a clustering pattern of PDB loops, even with 0.2 Å precision. Consequently we have elongated this initiator. For this, we have systematically added residues at the beginning and at the end of the individual elements in its cluster, to search for clustering patterns. For example, we may take the element 1p1x_A (80–84), elongate it to 1p1x_A (80–85) and 1p1x_A (79–84), and then use these two elongated ones as initiators to do the clusterings: We denote by H, S and L a residue which is located in a helix, strand and loop respectively, according to the PDB classification. The five residue long cluster which is generated by 1p1x_A (80–84) consists of several different elements, such as for example LLLLL, HLLLL, LLLLS etc. As an example, we have selected the pattern LLLLL which has the largest number of elements; there are a total of 7901 elements. We have elongated each of these 7901 elements into a protein loop fragment with six residues, by incorporating the corresponding PDB residue which is either right before the first L residue, or immediate after the last L residue. In this manner we find 15802 different loop fragments with six residues each. We have investigated the corresponding clustering patterns: There are 30 new clusters with more than 30 elements, bringing the total number of the clusters with more than 30 elements, to 42. We list these 30 additional clusters in Table 3. In Figs. 14, 15 and 16 we display the (κ,τ) distributions of these 30 clusters. A visual inspection of these clusters reveals, that at the level of the (κ,τ) distribution the cluster 26 appears to display additional sub-clustering. But the present cut-off value 0.2 Å is not sufficiently refined to detect this sub-clustering, at the level of RMS distance. Furthermore, the clusters 29 and 30 both appear to merge with the regular β-strand. In Fig. 17 we show the corresponding initiators: The cluster 29 is clearly a loop, while the cluster 30 consist of the regular β-strand and thus we exclude it from our set of loop fragments. This leaves us with a total of 41 clusters, with 30 or more loop fragments. These clusters cover around 52 % of all loop structures in PDB. Fig. 14 The stereographic map of the first 10 clusters in Table 3. The ordering along the C α backbone is r e db l u ey e l l o w Fig. 15 The stereographic map of the clusters 1120 in Table 3. The ordering along the C α backbone is r e db l u ey e l l o w Fig. 16 The stereographic map of the clusters 2130 in Table 3. The ordering along the C α backbone is r e db l u ey e l l o w Fig. 17 The initiators 29 (left) and 30 (right) in Table 3. The cluster 29 consists of loops, while the cluster 30 consist of regular β-strands Table 3 The 30 clusters with six residues, obtained by elongation of the LLLLL subset of the cluster which is generated by 1p1x_A (80–84) Cluster # Match # Cluster # Match # 1kwf_A (324–329) 1xg0_A (15–20) 1byi_A (123–128) 2pve_A (23–28) 4iau_A (78–83) 1vyr_A (23–28) 2o9s_A (841–846) 1j0p_A (54–59) 4ayo_A (233–238) 2rh2_A (48–53) 1pwm_A (171–176) 3p8j_A (240–245) 1gdq_A (123–128) 4gda_B (62–67) 2wur_A (30–35) 7a3h_A (232–237) 3zsj_A (190–195) 1n55_A (31–36) 4kxu_A (257–262) 1f94_A (40–45) 1n4u_A (121–126) 2pfh_A (305–310) 1nls_A (155–160) 1ab1_A (41–46) 3dk9_A (356–361) 1gci_A (188–193) 1o7j_C (119–124) 3ne0_A (1094–1099) 4hen_A (169–174) 3hyd_A (1–6) By completing the elongation process we have identified 3240 different clusters with 0.2 Å cut-off. These clusters cover around 85 % of all those PDB loop sites in our set of resolution better than 1.0 Å proteins. Among these clusters there are 1677 unique ones, in the sense that the cluster has only single element. Thus, around 14 % of all loop structures in PDB appear to be unique, to the given protein. In addition, there are 1531 rare clusters with two or more, but less than 32 elements. Thus, there are 32 clusters with 32 or more elements. The remaining 15 % of loop fragments that are not covered by the 3240 clusters, can be constructed by completion. For example, we can search for novel clusters by using the patterns other than LLLLL in the five residue cluster generated by 1p1x_A (80–84). But when the four patterns HLLLL, LLLLH, SLLLL and LLLLS are included the coverage increases no more than around one per cent. Cluster percolation We have also investigated the relation between the sequence and the structure, using the 42 clusters listed in Tables 1 and 3. Here we only describe some of the major features, more details can be found in Figure S3 in Additional file 1. There are several examples of identical sequences that correspond to different structures in different proteins. Accordingly a sequence clearly does not determine a unique structure. When a given sequence gives rise to multiple structures, we have a phenomenon we call cluster percolation. These sequences with multiplet structures may be utilised to try and introduce novel clusters. For example, in Table 4 those sequences that are found both in Cluster VIII and outside of it, are listed, together with their PDB identifications and RMSD to the initiator of Cluster VIII. Table 4 Sequences that appear both in and outside of cluster VIII; only the entry outside of the cluster is identified. The RMSD is evaluated from the initiator of cluster VIII; H stands for helix, L for loop and S for strand PDB entry PDB structure 2vb1_A (47–52) 3lzt_A (47–52) 4lzt_A (47–52) 3odv_A (20–25) 2agt_A (126–131) 2pzn_A (126–131) 3u2c_A (126–131) 4hen_A (54–59) 1g2y_B (18–23) 1mn8_B (47–52) 4a7u_A (91–96) 1iee_A (100–105) 2vb1_A (100–105) 4b4e_A (100–105) 4lzt_A (100–105) 3akq_A (161–166) 3akt_A (161–166) 3akt_B (161–166) As an example, in Fig. 18 a we compare the four PDB structures that have the identical sequence SDGNGM in the Table 4. The difference between the two mutually similar structures 2vb1 A (100–105) and 4lzt A (100–105) to the two equally mutually similar structures 1iee A (100–105) and 4b4e A (100–105) is visually apparent. A visual comparison with the Cluster VIII in Fig. 12 also reveals that both 1iee A (100–105) and 4b4e A (100–105) are clearly outside of this cluster. Fig. 18 Examples of percolation in Cluster VIII, listed in Table 4. In a the SDGNGM entries 2vb1 A (100–105) (blue), 4lzt A (100–105) (green), 1iee A (100–105) (yellow) and 4b4e A (100–105) (cyan) with with the initiator 1iee A (47–52) (red). In b the ADGKPV entry 4hen A (54–59) (blue) with the initiator 1iee A (47–52) (red) Figure 18 b shows the comparison of the sequence ADGKPV to the initiator. The difference between the structures of 4hen A (54–59) and the initiator is again clear. The structure of 4hen A (54–59) is also quite different from the structures in Fig. 18 a, and from the Cluster VIII shown in Fig. 12. In Table S1 of Additional file 1 we list those sequences that appear both in the 12 clusters of Table 1 and in protein structures which are not contained in any of the clusters. We have investigated these structures, and found 454 new clusters. But most of them have very few elements, only two of them have more than 30 elements. With these new clusters the coverage becomes increased to 88 %. In Fig. 19 we show the (κ,τ) distributions on the stereographically projected two-sphere of the two clusters with more than 30 elements; the initiators are 1ix9_A (133–138) and 3aj4_B (73–78) correspondingly. These two clusters are found by considering the sequences LKGDKL in cluster III and KDCMLQ in cluster XI, respectively. Fig. 19 The (κ,τ) distributions of the two clusters with more than 30 entries obtained by percolation. Clusters with initiators a 1ix9_A (133–138) and b 3aj4_B (73–78) Example: Myoglobin Myoglobin is a widely studied protein, thus we have analysed its loop structure from the present perspective. We have chosen the crystallographic oxymyoglobin structure 1A6M [50] which is one of the few myoglobin structures that have been measured with resolution better than 1.0 Å, for our comparative study. We have located in 1A6M four putative kink segments with six residues each, that are either unique or very rare in PDB, with our 0.2 Å RMSD cut-off. These kinks are located between helices C and D, and between helices E and F. The two putative kinks between helices C and D correspond to the residue sites (41–46) and (48–53). The two putative kinks between helices E and F correspond to residue sites (77–82), and the practically overlapping (78–83). In Fig. 20 we show how in our PDB set, the number of matches for each of these four kinks depends on the RMS cut-off distance. Fig. 20 The number of matches for different kinks in myoglobin. In each panel, x-axis is the different RMSD cut-off value (r rmsd ) while y-axis is the number of the entries whose RMSD values compared with the initiator are in the range [r rmsd ,r rmsd +0.05]. Panels ad are for different kinks of myoglobin: a (41–46), b (48–53), c (77–82) and d (78–83) The 1A6M is closely related to the PDB entries 1A6G, 1A6K and 1A6N; they represent four different ligation states of the same protein. Each of the three 1A6G, 1A6K and 1A6N have been measured with resolution above 1.0 Å, thus they do not appear in our data set. In Table 5 the RMS distance of the four rare kinks of 1A6M are compared to the corresponding kinks in 1A6G, 1A6K and 1A6N. All the RMSD values are below the cut-off 0.2 Å. Table 5 RMS Distance between the four kinks in 1A6M and the corresponding segments in the three other ligation states (in Å ngströms) We conclude that the four kinks are stable, in the sense that they do not change their conformation when the ligation state changes. Chain inversion Finally, the operation of local chain inversion along a protein segment is defined as a mapping, that sends a sequence with C α coordinates $$\left\{ \ \mathbf r(i), \ \mathbf r(i+1), \ \ldots \, \ \mathbf r(i+k-1), \ \mathbf r(i+k) \ \right\} $$ into a sequence with C α coordinates $$\{ \ \mathbf r(i+k), \ \mathbf r(i+k-1), \ \ldots \, \ \mathbf r(i+1), \ \mathbf r(i) \ \} $$ We note that a regular secondary structure such as an α-helix becomes mapped onto itself i.e. remains invariant under chain inversion. But we have found that the 12 clusters that we have constructed are not inversion invariant; the inversion does not map a cluster onto itself. Thus one might expect that new clusters could be found by inversion of these clusters. However, surprisingly we have found only one single example of a PDB segment by inversion. This is the segment (1115–1120) in the PDB structure 1MC2. Thus local chain inversion is apparently a broken symmetry, in the case of protein loops. This sets the loops apart from the regular structures like α-helices and β-strands. We have introduced the concept of loop clustering to analyse those ultrahigh resolution crystallographic protein structures in PDB, that have been measured with resolution 1.0 Å or less. We have chosen these structures since we expect, that in the case of a ultrahigh resolution measurement there should be less need to introduce structure validation. Thus there should also be less bias towards a priori chemical knowledge and stereochemical paradigms, in this subset of PDB proteins. Moreover, our investigation of 2.0 Å subset shows that high resolution is necessary to reveal the clustering structure in the case of protein crystals. We have inquired to what extent the protein structures can be constructed in a modular fashion. For the modular building blocks we have chosen different parameterisations of the unique kink solution to a generalised discrete nonlinear Schrödinger equation. The precision we have used as a criterion in making a difference between two structures is 0.2 Å in RMSD. We have concluded that this should be the shortest meaningful RMS distance that can be introduced, at the moment, to classify different modular protein components. We have identified a set of 12 different kink parameterisations, which cover around 38 % of all PDB loop structures. Accordingly, these are loop patterns that are abundantly present in the folded proteins. It appears to us, that these kinks are often located in such protein segments that are structurally important, as opposed to those that are functionally important. We have introduced various techniques to extent the initial set of 12 kinks, and we have found that around 52 % of loop regions become covered when we introduce a set of 29 additional kinks. But in order to cover the remaining 48 % of protein loops, we need to substantially increase the number of kinks. For example, we need to introduce over 1000 kinks to cover over 88 % of loops. In particular, we have concluded that there are several kinks that are very rare, even unique, in PDB when we use the present cut-off value. We propose that a rare or even unique kink should have a an important functional rôle, in a protein. This can be exemplified by the myoglobin 1A6M segments (41–46), (48–53) and (78–83) which are all rare. These segments also constitute the CD corner and EF corner in myoglobin, which have been argued to be closely related to the ligand migration process [51, 52]. Protein loops are built in a modular fashion, in terms of various parametrisations of the kink solution to a generalised version of the discrete nonlinear Schrödinger equation. Most loops can be built from a very small number of modular components, these loops are most likely important for the overall structure of the protein. However, there are also several unique, or very rare loops, which are most likely related to the function. The amino acid sequence does not define the structure uniquely, instead a given sequence can give rise to several different conformations. Availability of supporting data The datasets supporting the result of this article are available in Protein Data Bank (PDB) by confining the resolution better than 1.0 Å ( Discrete Nonlinear Schrö dinger; PDB: Protein Data Bank Critical Assessment for Structural Prediction AJN acknowledges support from Vetenskapsrådet, Carl Trygger’s Stiftelse för vetenskaplig forskning, and Qian Ren Grant at BIT. 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65b0775c0ba2cbca	Viewpoint: Reconnecting to superfluid turbulence • Joseph J. Niemela, The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy Physics 1, 26 Images of vortex motion in superfluid helium reveal connections between quantum and classical turbulence and may lead to an understanding of complex flows in both superfluids and ordinary fluids. Illustration: Alan Stonebraker, adapted from [25]. Figure 1: (Left) A vortex reconnection in which two vortex lines meet, the lines break and then reform by swapping the broken lines. A model employing such reconnections was able to account for turbulence in a superfluid [15]. (Right) These reconnections can create Kelvin waves in which a vortex line oscillates about an equilibrium position like a plucked string (the rightmost wavy vortex compared to the undisturbed filament to its left). These waves, in turn can produce phonons in the superfluid that dissipate turbulent energy.(Left) A vortex reconnection in which two vortex lines meet, the lines break and then reform by swapping the broken lines. A model employing such reconnections was able to account for turbulence in a superfluid [15]. (Right) These reconnections can c... Show more Superfluid flows are interesting playgrounds, where hydrodynamics confronts quantum mechanics. One of the more important and interesting questions is what a complex turbulent flow would look like in a superfluid that was prevented from rotational motion except for circulation about individual, discrete vortex filaments, each having single quanta of circulation about a core of atomic dimensions. This is a great simplification when compared to ordinary turbulence, in which vortices and eddies can have any strength and size. A number of recent works, which have substituted superfluids for ordinary fluids in standard turbulence experiments, have suggested that turbulence in the two fluids is nearly indistinguishable. However, in a recent paper in Physical Review Letters, M. S. Paoletti, M. E. Fisher, and D. P. Lathrop at the University of Maryland, and K. R. Sreenivasan of the Abdus Salam International Center for Theoretical Physics, Trieste, have probed turbulent superfluid flow at small enough scales to see a clear difference [1]. This was achieved by dressing the quantized vortices in turbulent superfluid liquid 4He with small clusters or particles of frozen hydrogen, formed by injecting a small amount of H2 diluted with helium gas into the liquid helium, and then optically tracking their motion. The difference is not only dramatic—strongly non-Gaussian distributions of velocity replacing the near-Gaussian statistics in classical homogeneous and isotropic turbulence—but it appears to also have a simple explanation. Reconnections between quantized vortices occurring at the microscopic level of the core can give rise to the same statistical signature that these authors have observed. Such events, established experimentally here as a robust feature, are necessary to fully explain turbulence in superfluids and fundamental to understanding how a pure superfluid like 4He at absolute zero can shed its turbulent energy in the complete absence of viscosity. The reconnections we have in mind can be roughly described as follows: two vortex filaments that approach each other closely, attempt to cross, forming sharp cusps at the point of closest approach. At this point they can break apart, so that part of one vortex reconnects with part of the other, and so forth, significantly changing topology. Reconnections, which are a significant feature of superfluid turbulence, are not unique to it, and can occur in ordinary fluids [2], magnetized plasmas [3], and perhaps even between cosmic strings [4]. Reconnection between broken magnetic field lines in the sun is a relatively common occurrence leading to solar flares. However, there is a fundamental difference: classical reconnections are related to energy dissipation through viscosity, whereas in quantum fluids they take place due to a quantum stress acting at the scale of the core without changes of total energy [5]. Liquid 4He becomes superfluid below about 2.2 K, resulting from a type of Bose condensation as the de Broglie wavelength of the individual helium atoms becomes comparable to the average spacing between them. It then behaves as if it were composed of two intermingling and independent fluids: a superfluid with zero viscosity and zero entropy, and a viscous normal fluid, each having its own velocity field and density, where the ratio of superfluid to normal fluid density varies from 0 at the transition to 1 at absolute zero. From this model it follows that the superfluid component must also be irrotational (the curl of velocity must be zero) and this would have seemed to rule out turbulence altogether were it not for the peculiar vortices that are at the “core” of this story. These vortices, first proposed by Onsager [6] and Feynman [7], can easily be seen [8] in solutions of the nonlinear Schrödinger equation (NLSE) for the condensate wave function of an ideal Bose gas. For these vortex solutions, a coherence length gives the distance over which the amplitude of the wave function rises radially from zero to some constant value. Since the superfluid density is given by the squared modulus of the wave function, this approximately defines the size of the vortex core, which for superfluid 4He is extremely small, on the order of one angstrom. The vortex circulation is obtained by integrating the superfluid velocity around a loop enclosing the superfluid-free core (thus avoiding the irrotational condition of the two fluid model) and the solitary stable value that results, namely Planck’s constant divided by the mass of a single helium atom, yields singly quantized line vortices [9]. M. S. Paoletti et al. [1] Video 1: individual reconnection events are annotated by white circles and evidenced by groups of hydrogen clusters rapidly separating from one another. The clusters that are trapped on the vortices enable the authors to measure the separation as the vortices approach and retract from one another. Feynman [7] suggested a model for turbulence in the superfluid, which he envisioned as a tangle of such quantized line vortices. But how could a collection of these vortices, having just one quanta of circulation each, resemble classical turbulence in a viscous fluid with all its swirls from large to small? More specifically, would the statistical properties of a turbulent superfluid match those of classical turbulence? For this, we start with the following picture for ordinary fluids: energy injected into a flow at some large scale is transferred without dissipation by a cascade process to smaller and smaller scales, until it is finally dissipated into heat at the smallest scale where viscosity becomes important. In the 1940s, a dimensional analysis by Kolmogorov [10] corresponding to this picture of turbulence produced the well-known spectral energy density E(k)=cϵ2/3k-5/3 for wave numbers k between those of energy injection and dissipation, where c is a constant and ϵ the energy dissipation rate per unit mass. This spectral distribution should be independent of how the turbulence was generated in the first place. With this as background, Maurer and Tabeling [11] showed that for the turbulent flow between two counter-rotating discs, the same Kolmogorov energy spectrum with wave-number exponent -5/3 could be observed above and below the transition temperature in liquid 4He. Similar experiments with moving grids [12] also showed this quantum mimicry of classical turbulence. What is going on here? These experiments had at least two things in common: the fraction of normal nonsuperfluid was small but not negligible, and the measurements were sensitive to scales much larger than that of individual vortex lines in the turbulent state. About the first, note that motion of a quantized vortex relative to the normal fluid produces a mutual friction force [13], coupling the two fluids at large scales (as well as providing dissipation at small ones), so it is not unthinkable then that both normal and superfluid act together to produce a Kolmogorov spectrum. This may take place [14] as a result of a partial or complete polarization, or local alignment of spin axes, of a large number of vortex filaments that mimics the range of eddies we see in classical flows. A simple example of such polarization under nonturbulent conditions is the well-known mimicking of solid body rotation in a rapidly rotating container filled with superfluid helium, which results from the alignment of a large array of quantized vortices all along the axis of rotation [8]. At the scale of individual vortices, Schwarz [15] developed numerical simulations of superfluid turbulence, based on the assumption that vortex filaments approaching each other too closely will reconnect (see the left panel of Fig. 1). Using entirely classical analysis, he was able to account for most of the experimental observations in the commonly studied thermal counterflow, a flow in which the normal fluid carries thermal energy away from a heater and a mass-conserving counter-current of superfluid is produced. Koplik and Levine [8], using the nonlinear Schrödinger equation, showed that Schwarz’ assumptions about reconnections were correct. Even this flow, which unlike the other experiments mentioned above, has no classical analog, also exhibits a classical decay when probed on length scales that are large compared to the average intervortex line spacing [16]. Vortex reconnections should be frequent in superfluid turbulence [17] and this is a fundamental difference from the classical case. At absolute zero, where there is neither viscosity nor mutual friction to dissipate energy, reconnections between vortices are expected [18] to lead to Kelvin waves along the cores (see right panel of Fig. 1), allowing the energy cascade to proceed beyond the level of the intervortex line spacing. Kelvin waves are defined as helical displacements of a rectilinear vortex line propagating along the core. When a vortex reconnection occurs, the cusps or kinks at the crossing point (see above) can relax into Kelvin waves and subsequent reconnections in the turbulent regime generate more waves whose nonlinear interactions lead to a wide spectrum of Kelvin waves extending to high frequencies. At the highest frequencies (wave numbers) these waves can generate phonons, thus dissipating the turbulent kinetic energy. The bridge between classical and quantum regimes of turbulence [19, 20], it seems, must be provided by numerous reconnection events. In the work of Paoletti et al. [1], a thermal counterflow as described above is allowed to decay and then probed at the level of discrete vortex lines by illuminating the hydrogen particles moving with the vortices with a laser light sheet. Viewing the scattered light at right angles to the sheet with a CCD camera allows the motion of the vortices to be tracked (see Video 1). This relies on previous work showing that hydrogen tracers could be trapped on the vortices [21, 22]. Large velocities of recoil associated with reconnection events have recently been observed experimentally [23] and in simulations [24]. Paoletti et al. [1] are able to show that the observed, strongly non-Gaussian distributions of velocity due to these atypically large velocities are quantitatively consistent with the frequent reconnection of quantized line vortices. To the extent that turbulent flows are necessarily characterized by their statistical properties, this work provides a clear experimental foundation for a bridge connecting the classical and quantum turbulent regimes. While insights from the well-studied turbulence problem in ordinary flows have allowed us to move forward in understanding quantum turbulence, the reverse might be said as well: the knowledge we gain there may well yield new insights into classical turbulence, a problem of immense interest in both engineering and large natural flows in fluids and plasmas, and for which a satisfying theoretical framework has yet to be found. Just as in the classical problem, experiments and simulations play a large role, and this leads to many challenges, especially as the temperature is lowered to a pure helium superflow regime. The work of Paoletti et al. [1] is a large step in this direction, allowing us to experimentally confirm our picture of how quantum turbulence proceeds. Going to very low temperatures will require different and more difficult techniques of generating the turbulence than these authors used (in the almost complete absence of the normal component) but ultimately the freely vibrating vortices there may give us the best opportunity to listen clearly to the strange and complex sounds emitted from an “instrument” whose quantum strings are plucked by reconnections. 1. M. S. Paoletti, M. E. Fisher, K. R. Sreenivasan, and D. P. Lathrop, Phys. Rev. Lett. 101, 154501 (2008) 2. S. Kida, M. Takaoka, and F. Hussain, J. Fluid Mech. 230, 583 (1991) 3. E. R. Priest and T. G. Forbes, Magnetic Reconnection: MHD Theory and Applications[Amazon][WorldCat] (Cambridge University Press, 2007) 4. A. Hanany and K. Hashimoto, arXiv:hep-th/0501031v2 (2005) 5. M. Leadbeater, T. Winiecki, D. C. Samuels, C. F. Barenghi, and C. S. Adams, Phys. Rev. Lett. 86, 1410 (2001); C. F. Barenghi, Physica D 237, 2195 (2008) 6. R. J. Donnelly, Quantized Vortices in Helium II[Amazon][WorldCat] (Cambridge University Press, 1991) 7. R. P. Feynman, in Progress in Low Temperature Physics, Vol. 1, edited by C. J. Gorter (North-Holland, Amsterdam, 1955) 8. J. Koplik and H. Levine, Phys. Rev. Lett. 71, 1375 (1993) 9. W. F. Vinen, Proc. Roy. Soc. Lond. A Mat. 260, 218 (1961) 10. A. Kolmogorov, Dokl. Acad. Nauk SSSR 30, 301 (1941) 11. J. Maurer and P. Tabeling, Europhys. Lett. 43, 29 (1998) 12. S. R. Stalp, L. Skrbek, and R. J. Donnelly, Phys. Rev. Lett. 82, 4831 (1999) 13. H. E. Hall and W. F. Vinen, Proc. Roy. Soc. A238, 215 (1956) 14. W. F. Vinen and J. J. Niemela, J. Low Temp. Phys. 128, 167 (2002) 15. K. W. Schwarz, Phys. Rev. B 31, 5782 (1985) 16. L. Skrbek in Vortices and Turbulence at Very Low Temperatures[Amazon][WorldCat], edited by C. F. Barenghi and Y. A. Sergeev (Springer, New York, 2008), p. 91 17. M. Tsubota, T. Araki, and S. K. Nemirovskii, Phys. Rev. B 62, 11751 (2000) 18. B. V. Svistunov, Phys. Rev. B 52, 3647 (1995) 19. W. F. Vinen, J. Low Temp. Phys. 145, 7 (2006) 20. E. Kozik and B. V. Svistonov, arXiv:cond-mat/0703047v3 (2007) 21. D. R. Poole, C. F. Barenghi, Y. A. Sergeev, and W. F. Vinen, Phys. Rev. B 71, 064514 (2005) 22. G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, Nature 441, 588 (2006) 23. G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan and D. P. Lathrop, Proc. Natl. Acad. Sci. U.S.A. (to be published) 24. S. Nazarenko, J. Low Temp. Phys. 132, 1 (2003) 25. C. F. Barenghi, in Vortices and Turbulence at Very Low Temperatures[Amazon][WorldCat], edited by C. F. Barenghi and Y. A. Sergeev (Springer, New York, 2008), p.1 About the Author Image of Joseph J. Niemela Joseph J. Niemela is a member of the permanent scientific staff at the Abdus Salam International Center for Theoretical Physics in Trieste, Italy, where he conducts research in fluid dynamics and low-temperature physics, and coordinates activities in optics and lasers as well as science education. He is also a member of the faculty of the Doctoral School of Environmental and Industrial Fluid Mechanics of the University of Trieste. Read PDF Subject Areas SuperfluidityFluid Dynamics Related Articles Synopsis: Racing to the Bottom Fluid Dynamics Synopsis: Racing to the Bottom Focus: Bumblebees In Turbulence Biological Physics Focus: Bumblebees In Turbulence Synopsis: Twisted Fluid Flows Fluid Dynamics Synopsis: Twisted Fluid Flows More Articles
0f4913b51c339995	Essay:Rebuttal to Counterexamples to Relativity From Conservapedia Jump to: navigation, search 2. Quasars are disappearing, contrary to the theory of relativity. Relativity does not predict quasars; relativity was formulated in 1905, and quasars weren't discovered until the 1950s. The word "relativity" does not occur anywhere in the cited article. While quasars are associated with black holes, no serious scientist believes that black holes don't exist or that relativity is wrong. That astronomers have stopped looking for this phenomenon, after finding more than 10 examples of it, may or may not have been a wise decision regarding the allocation of telescope and satellite time. This could be a counterexample to both GR and Newtonian gravity—in both, the radius is defined in terms of conserved quantities. 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 1970s 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. 4. The Pioneer anomaly. 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.[3] 6. Quantum entanglement near the event horizon of a black hole .... 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. 8. 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. 9. 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. 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.[6][7] The claims of faster-than-light neutrinos have now been refuted very thoroughly.[8][9] 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 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. Update: Another, much more commonplace observation of gravitational wave emission has been reported.[11] 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. For a massless particle, the speed is always c. Furthermore, this discontinuity in momentum does exist in Newtonian physics with the formula . If we let v approach c while m approaches 0 we get , while light has a nonzero momentum. 17. 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. 19. The observed lack of curvature in overall space. 21. The action-at-a-distance of quantum entanglement. Scientists' choosing not to try to explain "plausible media" for things described in the Bible is not evidence that Relativity is wrong. Gravitons are a prediction of Quantum mechanics, 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. 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 (inertial or non-inertial) frames, as the 'counterexample' implies. The Dirac equation, which gives rise to antiparticles and the theory of spinors, was an early example of introducing relativity into the Schrödinger equation. 28. The uniformity in temperature throughout the universe. Since the 70's much work has been done on the subject of black hole thermodynamics,[13][14] 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.[15] The Counterexamples to Relativity article has labelled this work in a footnote as "[c]ontrived explanations", with no explanation for this characterization given. The data are not diverging from the predictions. The Global Positioning System (GPS) uses general relativity to achieve greater accuracy.[16] cosmic inflation parity violation in the weak force the Chandrasekhar limit for white dwarf stars where c is the speed of light. With a photon of zero rest mass, this gives: where is the photon's wavelength.[18] The cited paper does not refute relativity. 48. 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. 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 1990s. See also 4. 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. 11. BBC article
518a5a6d0380e835	Friday, March 11, 2016 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere Measurement isn't a violation of unitarity In the mid 1920s, thanks to the conservative yet revolutionary work by giants such as Heisenberg, the foundations of physics have switched from the old framework of classical physics to the postulates of quantum mechanics. The new general rules have been completely understood. Since that time, only the search for the "right Hamiltonians" and "their implications" was open. The new philosophical underpinnings were shown to be consistent, complete, nothing has changed about them since 1925 (or we might put the threshold to 1927 so that the papers clarifying the uncertainty principle etc. are included), and all the evidence suggests that there's no reason to expect any change to these basic philosophical foundations of physics in the future. Florin Moldoveanu doesn't like these facts because much of his work (and maybe most of it) is based on the denial of the fact that quantum mechanics works and it works remarkably well. So he wrote, among other things: Apparently to show that something hasn't been clear about the basic rules of the game since the 1920s, he wrote a blog post dominated by the basic introduction to the Leibniz identity, the Jacobi identity, and tensor products. Are you joking, Florin? While the universal postulates of quantum mechanics have been known since the 1920s, the "fancy new hi-tech topics" that you discussed now have been known at least since the 19th century! Moldoveanu wants to impress you by the (freshman undergraduate algebra) insight that the Hermiticity is related to unitarity and the Leibniz identity, and so on. The precise "equivalences" he describes are confused – it is the Hermiticity (of the Lie algebra generators), and not the Leibniz identity, that is equivalent to the unitarity (of Lie group elements). The Leibniz identity works even for non-unitary operations and it is how the differential generators always act on product representations (e.g. composite systems in quantum physics), according to the dictionary between the Lie groups and Lie algebras. But I don't want to analyze his technical confusion which is intense. It would be easier to make him forget about everything he knows and teach him from scratch than to try to fix all his mistakes. I want to focus on the big picture – and the historical picture. To argue that something hasn't been settled since the 1920s, he talks about the Leibniz identity, the Jacobi identity, the rules of the Lie groups and the Lie algebras. Are you listening to yourself, Florin? Leibniz lived between 1646 and 1716 so one should be able to figure out that the Leibniz identity probably wasn't born after 1925. Even more relevant is the history of Lie groups and Lie algebras. Sophus Lie did most of this work between 1869 and 1874. The Lie algebra commutators were understood to obey the Jacobi identity which has been known before Lie did his key contributions. Most of Jacobi's work was published in the 1860s but the publication was posthumous: Jacobi lived from 1804 to 1851. Killing and Cartan added their knowledge about the Cartan subalgebra and maximal tori etc. in the 1880s. All this mathematical apparatus was ready decades before physicists made their new insights about the foundations of physics in the 1920s. In the same way, mathematicians understood the representation theory. For example, if there are two independent objects, their properties are described by two sets of operators. The two sets commute with one another – this is clearly needed for the two objects to exist independently. The Hilbert space is a representation which means that the minimum Hilbert space transforming under both sets of observables has to be a tensor product. The relevance of the tensor product ${\mathcal H}_A\otimes {\mathcal H}_B$ for the quantum description of a composite system has been immediately obvious when quantum mechanics was presented. The mathematical underpinnings had been known for decades – and, in fact, Heisenberg had no trouble to rediscover the matrix calculus when he needed it. The tensor product Hilbert space appears because it's a representation of the group $G_A\times G_B$, a direct product that is needed to describe the observables of two parts of the composite system. Florin, are you really serious when you present these basic things as justifications of your claim that something fundamental about the general rules of quantum mechanics hasn't been clear from the 1920s? Even though most of his blog post is dedicated to these basic, mostly 19th century, mathematical insights, the title reads Why are unitarity violations fatal for quantum mechanics? Indeed, unitarity violations would be fatal for a quantum mechanical theory – they would prevent the sum of all probabilities of mutually exclusive outcomes from being equal to 100 percent. However, there is just no violation of unitarity in the theories we actually use to describe Nature. Unitarity is indeed a universal rule – it is the quantum counterpart of some axioms in the usual probability calculus (where the sum of probabilities of different options is always 100 percent). Why does Moldoveanu think otherwise? He thinks otherwise because he believes that the measurement introduces non-unitarity to quantum mechanics. The word "non-unitarity" only appears in the following sentence of his text: Sadly, this critical sentence is completely wrong. This has some implications. For example, this wrongness invalidates almost all papers by Moldoveanu that use the word "unitarity" because he just doesn't know what this condition is, when it holds, and whether it holds. The unitarity is a condition in quantum mechanics that imposes the rule that "probabilities add up to 100 percent" within the quantum formalism. But what is unitarity more accurately? By unitarity, quantum physicists mean exactly the same thing as the 19th century mathematicians. So the matrix $U$ with the matrix entries $U_{ij}$ – and similarly for an operator that may be defined without a choice of a basis i.e. without indices – is the condition\[ U U^\dagger={\bf 1}\quad {i.e.} \quad \sum_j U_{ij}U^\dagger_{jk} = \delta_{ik}. \] This is the unitarity. In quantum mechanics, it holds whenever $U$ is an evolution operator (by a finite or infinite time i.e. the S-matrix is included) – or the operator of any finite transformation, for that matter (e.g. the rotation of a system by an angle). The evolution operators in non-relativistic quantum mechanics, Quantum Electrodynamics, the Standard Model, and string theory (among many other theories) perfectly obey this condition. That's why we say that all these quantum mechanical theories are unitary – they pass this particular health test. Moldoveanu and tons of other anti-quantum zealots want to contradict this statement by pretending that the measurement of a quantum system is a modification of Schrödinger's equation that deviates from the action of the unitary evolution operators above, and is therefore non-unitary. But that's a result of a completely wrong and sloppy thinking about all the concepts. The collapse doesn't mean that there is a violation of unitarity. To understand this simple sentence, we must be careful and look "what are the objects that are unitary". The answer is that the unitary matrices such as $U$ above are the matrices whose entries are the probability amplitudes. The general postulate in quantum mechanics that we have referred to is that the matrices of evolution operators' probability amplitudes – between the basis of possible initial states and the basis of the possible final states – are unitary. And be sure that they are and the measurements don't change anything about it. Why don't they change anything about it? Because the "sudden collapse of the wave function" that the measurement induces isn't a modification of the evolution operator or a deformation of Schrödinger's equation. Instead, the "sudden collapse" is an interpretation of the wave function. Quantum mechanics says that after the measurement, one of the possible outcomes becomes true. It "even" allows us to calculate the probabilities of the individual outcomes. But the very fact that quantum mechanics says something about the probabilities of the outcomes implicitly means that one of the outcomes will become the truth after the measurement. This simple claim is implicitly included in all the rules of quantum mechanics. We may obviously add it explicitly, too. When we measure whether a cat is dead or alive, and quantum mechanics predicts the probabilities to be 36% and 64%, there can't be any "vague mixed semi-dead semi-alive" state of the cat after the measurement. This claim logically follows from the statement that the "probabilities of dead and alive are 36% and 64%" and it doesn't need any additional explanation. If it were possible for the measurement of the cat to yield some vague "semi-dead semi-alive" outcome, the probabilistic statement would have to allow this option. To do so, quantum mechanics would have to predict that the "probability is 30% for dead, 60% for alive, and 10% for some semi-dead semi-alive fuzzy mixture". But when the laws of quantum mechanics omit the third option, it means that this option's probability is 0% which means that it is impossible for the post-measurement state to be semi-dead, semi-alive. If you need some extra explanations or repetitions of this fact, that the ill-defined post-measurement outcomes are banned by quantum mechanics, then it is because you are retarded, Florin, not because the foundations of quantum mechanics need some extra work. The ultimate reason why Moldoveanu and others refuse to understand simple points like that – e.g. the point that there is no non-unitarity added by the measurement – is that they are refusing to think quantum mechanically. When we say that the matrix entries of an evolution operator are probability amplitudes, we understand it but the likes of Moldoveanu don't. They may hear the words but they ignore their content. They totally overlook the fact that the matrix entries that decide about the unitarity are probability amplitudes. They just think that they are some classical degrees of freedom (that objectively exist and don't require observers), Schrödinger's equation is a classical evolution equation, and the measurement must be "modelled" as an exception for the Schrödinger's equation or its deformation of a sort. But all these assumptions are completely wrong. The wave function is not a classical wave. It is not a set of classical degrees of freedom. Schrödinger's equation isn't an example of a classical evolution equation. And the measurement isn't described by anything that looks like an equation for the evolution at all. The measurement yields sharp outcomes because quantum mechanics postulates that there are sharp outcomes – the spectrum of an operator lists all the a priori possible outcomes – and it tells you how to calculate their probabilities from the complex probability amplitudes. It's only the probability amplitudes that may be meaningfully organized into linear operators and therefore matrices. If you want to engineer some "action on a wave function that also visualizes the collapse", then you are trying to construct a classical model describing the reality. You are not doing a proper quantum analysis of the problem. And if you created such a model where the wave function is a classical field that "collapses" according to some equations, the "operation" wouldn't even be linear, so it wouldn't make sense to ask whether it's unitary. In fact, the operation on the initial wave function wouldn't even be a map because even if the initial state is exactly the same twice, the final outcomes may be different – because of the quantum indeterminacy or randomness. Because this operation assigning the final state isn't even a map (because the final outcomes of the measurements aren't uniquely determined by the initial state), it makes absolutely no sense to talk about its being unitary. Of course it can't be correctly shown to be unitary. It can't be unitary if it is not even a map! Only for maps, and yes, you need linear maps, you can meaningfully talk about their being unitary. For other "processes", the adjective is ill-defined (like "whether the number five is green"). The "operation of the collapse" on the wave function isn't unitary but it isn't non-unitary, either. It isn't a map so it's meaningless to talk about its being unitary. And if you managed to "redefine" the transformations in some way so that the act of the measurement would count as "non-unitary evolution", despite its randomness (failure to be a map) and nonlinearity, then it wouldn't be a problem, anyway. What's needed for consistency of the theory is the unitarity of the pre-measurement probability amplitudes (because the unitarity plays the same role as the conditions for probabilities that should add to 100 percent etc.), not some probability amplitudes modified by random-generator-dependent "collapses". So even if the collapse were redefined as a "non-unitary evolution of a sort", it just wouldn't mean that there is a problem to worry about or to solve. Again, in the normal approach, the object whose unitarity is a meaningful question is the matrix/operator of the probability amplitudes (defining an evolution or a transformation). Those don't contain any "collapses" because the very meaning of the word "probability" is that we substitute the "widespread" distributions "before" we know a particular outcome i.e. without any collapses. And the matrices of probability amplitudes for evolution operators must be unitary in all logically consistent quantum mechanical theories. Even if you are a bit confused about the logic, you should be able to understand that there is almost certainly "nothing intelligent and deep" waiting to be found here. Moldoveanu's and similar people's "work on the foundations" is just an artifact of their inability to understand some very simple logical arguments fully described above – and at many other places. They're crackpots but like most crackpots, they work with the assumption that they can never be wrong. That's not a good starting point to understand modern physics. Pedagogic bonus: from classical physics to quantum mechanics I am afraid that I have written very similar things to this appendix in the past. But even if it is the case and the text below fails to be original, repetition may sometimes be helpful. Here's a way to see in what way quantum mechanics generalizes classical physics – and why it's foolish to try to look for some "problems" or "cure to problems" in the process of the measurement. A theory in classical mechanics may be written in terms of the equations for the variables $x(t),p(t)$\[ \frac{dx}{dt} = \frac{\partial H}{\partial p}, \quad \frac{dp}{dt} = -\frac{\partial H}{\partial x} \] for some Hamiltonian function $H(x,p)$, OK? Now, classical physics allows the objective state at every moment i.e. the functions $x(t),p(t)$ to be fully determined. But you may always switch to the probabilistic description which is useful and relevant if you don't know the exact values of $x(t),p(t)$ – everything that may be known. Introduce the probability distribution $\rho(x,p)$ on the phase space that is real and normalized,\[ \int dx\,dp\, \rho(x,p)=1. \] It's trivial to have many copies of $x,p$, just add an index, and rename some of the variables etc. Fine. What is the equation obeyed by the probability distribution $\rho(x,p;t)$? We are just uncertain about the initial state but we know the exact deterministic equations of motion. So we may unambiguously derive the equation obeyed by the probability distribution $\rho$. The result is the Liouville equation of statistical mechanics. How do we derive and what it is? The derivation will be addressed to adult readers who know the Dirac delta-function. If the initial microstate is perfectly known to be $(x,p)=(x_0,p_0)$, then the distribution at that initial moment is\[ \rho(x,p) = \delta (x-x_0) \delta(p-p_0). \] With this initial state, how does the system evolve? Well, the $x,p$ variables are known at the beginning and the evolution is deterministic, so they will be known at all times. In other words, the distribution will always be a delta-function located at the right location,\[ \rho(x,p;t) = \delta [x-x(t)] \delta[p-p(t)] \] What is the differential equation obeyed by $\rho$? Calculate the partial derivative with respect to time. You will get, by the Leibniz rule and the rule for the derivative of a composite function,\[ \frac{\partial \rho (x,p;t)}{\partial t} &= \delta'[x-x(t)] \dot x(t) \delta[p-p(t)]+\\ &+ \delta[x-x(t)] \delta'[p-p(t)] \dot p(t) \] or, equivalently (if we realize that $\rho$ is the delta-function and substitute it back),\[ \frac{\partial\rho}{\partial t} = \frac{\partial \rho}{\partial x}\dot x(t)+\frac{\partial \rho}{\partial p}\dot p(t). \] This is the Liouville equation for the probabilistic distribution on the phase space, $\rho$. The funny thing is that this equation is linear in $\rho$. And because every initial distribution may be written as a continuous combination of such delta-functions and because the final probability should be a linear function of the initial probabilities, we may just combine all the delta-function-based basis vectors $\rho(x,p;t)$ corresponding to the classical trajectories $x(t),p(t)$, and we will get a general probability distribution that behaves properly. In other words, because of the linearity in $\rho$ and because of the validity of the equation for a basis of functions $\rho(x,p;t)$, the last displayed equation, the Liouville equation, holds for all distributions $\rho(x,p;t)$. Excellent. I emphasize that this Liouville equation is completely determined by the deterministic equations for $x(t),p(t)$. Aside from the totally universal, mathematical rules of the probability calculus, we didn't need anything to derive the Liouville equation. Nothing is missing in it. But when we measure an atom's location to be $x_1$, then the distribution $\rho(x,p;t)$ "collapses" because of Bayesian inference. We have learned some detailed information so our uncertainty has decreased. But this collapse doesn't need any "modifications" of the Liouville equation or further explanations because you may still assume that the underlying physics is a deterministic equation for $x(t),p(t)$ and all the $\rho$ stuff was only added to deal with our uncertainty and ignorance. The form of the Liouville equation is exact because it was the probabilistic counterpart directly derived from the deterministic equations for $x(t),p(t)$ which were exact, too. What changes in quantum mechanics? The only thing that changes is that $xp-px=i\hbar$ rather than zero. This has the important consequence that the deterministic picture beneath everything in which $x(t),p(t)$ are well-defined $c$-number functions of time is no longer allowed. But the equation for $\rho$ is still OK. Before we switch to quantum mechanics, we may substitute the Hamilton equations to get\[ \frac{\partial\rho}{\partial t} = \frac{\partial \rho}{\partial x}\frac{\partial H}{\partial p}-\frac{\partial \rho}{\partial p}\frac{\partial H}{\partial x} \] and realize that this form of the Liouville equation may be written in terms of the Poisson bracket\[ \frac{\partial \rho(x,p;t)}{\partial t} = \{\rho(x,p;t),H(t)\}_{\rm Poisson}. \] That's great (up to a conventional sign that may differ). This equation may be trusted even in quantum mechanics where you may imagine that $\rho$ is written as a function (imagine some Taylor expansion, if you have a psychological problem that this is too formal) of $x,p$. However, $x,p$ no longer commute, a technical novelty. But the density matrix $\rho$ in quantum mechanics plays the same role as the probability distribution on the classical phase space in classical physics. You may imagine that the latter is obtained from the former as the Wigner quasiprobability distribution. Because of the usual, purely mathematically provable relationship between the Poisson brackets and the commutator, we may rewrite the last form of the Liouville equation as the von Neumann equation of quantum mechanics\[ \frac{d\rho(t)}{dt} = i\hbar [H,\rho(t)] \] that dictates the evolution of the density matrix or operator $\rho$. (Thankfully, people agree about the sign conventions of the commutator.) It can no longer be derived from a deterministic starting point where $x(t),p(t)$ are well-defined $c$-numbers – they cannot be sharply well-defined because of the uncertainty principle (i.e. nonzero commutator) – but the probabilities still exist and no modifications (let alone "non-unitary terms" etc.) are needed for the measurement. The measurement is just a version of the Bayesian inference. It's still basically the same thing but this inference must be carefully described in the new quantum formalism. If you like Schrödinger's equation, it is not difficult to derive it from the von Neumann equation above. Any Hermitian matrix $\rho$ may be diagonalized and therefore written as a superposition\[ \rho = \sum_j p_j \ket{\psi_j}\bra{\psi_j} \] Because the von Neumann equation was linear in $\rho$, each term in the sum above will evolve "separately from others". So it is enough to know how $\rho=\ket\psi \bra\psi$ evolves. For this special form of the density matrix, the commutator\[ [H,\rho] = H\rho - \rho H = H\ket\psi \bra \psi - \ket\psi \bra \psi H \] and these two terms may be nicely interpreted as two terms in the Leibniz rule assuming Schrödinger's equation\[ i\hbar \frac{d\ket\psi}{dt} = H\ket\psi \] and its Hermitian conjugate\[ -i\hbar \frac{d\bra\psi}{dt} = \bra\psi H. \] So if the wave function $\ket\psi$ obeys this equation (and its conjugate), then the von Neumann equation for $\rho=\ket\psi\bra\psi$ will follow from that. The implication works in the opposite way as well (Schrödinger's equation follows from the von Neumann equation if we assume the density matrix to describe a "pure state") – except that the overall phase of $\ket\psi$ may be changed in a general time-dependent way. The pure state $\ket\psi$ corresponds to the "maximum knowledge" in the density matrix $\rho=\ket\psi\bra\psi$. In quantum mechanics, it still leads to probabilistic predictions for most questions, because of the uncertainty principle. Mixed states are superpositions of terms of the form $\ket{\psi_i}\bra{\psi_i}$. The coefficients or weights are probabilities and this way of taking mixtures is completely analogous (and, in the $\hbar\to 0$ limit, reduces) to classical probability distributions that are also "weighted mixtures". Because we have deduced the quantum equations from the classical ones, it's as silly as it was in classical physics to demand some "further explanations" of the measurement, some "extra mechanisms" that allow the unambiguous result to be produced. In classical physics, it's manifestly silly to do so because we may always imagine that the exact positions $x(t),p(t)$ have always existed – we just didn't know what they were and that's why we have used $\rho$. When we learn, the probability distribution encoding our knowledge suddenly shrinks. End of the story. In quantum mechanics, we don't know the exact values $x(t),p(t)$ at a given time. In fact, we know that no one can know them because they can't simultaneously exist, thanks to the uncertainty principle. But the probabilistic statements about $x,p$ do exist and do work, just like they did in classical statistical physics. But the Schrödinger or von Neumann equation is "as complete" and "as perfectly beautiful" as their counterpart in classical physics, the Liouville equation of statistical physics. The latter was ultimately derived (and no adjustments or approximations were needed at all) from the deterministic equations for $x(t),p(t)$ that the critics of quantum mechanics approve. We just allowed some ignorance on top of the equations for $x(t),p(t)$ and the Liouville equation followed via the rules of the probability calculus. So the Liouville equation just can't be "less satisfactory" than the classical deterministic laws for $x(t),p(t)$. Nothing is missing. And the von Neumann and Schrödinger equations are exactly analogous equations to the Liouville equation – but in systems where $xp-px=i\hbar$ is no longer zero. So the von Neumann or Schrödinger equations must unavoidably be complete and perfectly satisfactory, too. They still describe the evolution of some probabilities – and, we must admit because of the imaginary nonzero commutator, complex probability amplitudes. Because of the uncertainty principle, some ignorance and uncertainty – and probabilities strictly between 0 and 100 percent – are unavoidable in quantum mechanics. But the system of laws is exactly as complete as it was in classical statistical physics. No special explanation or mechanism is needed for the measurement because the measurement is still nothing else than a process of the reduction of our ignorance. In this process, $\rho$ suddenly "shrinks" because it's one step in Bayesian inference. It has always been. In classical physics, this Bayesian inference may be thought of as our effort of learning about some "objectively existing truth". In quantum mechanics, no objective truth about the observables may exist because of the uncertainty principle. But the measurement is still a process analogous to the Bayesian inference. It improves our subjective knowledge – shrinks the probability distribution – as a function of the measured quantity. But because of the nonzero commutator, the measurement increases the uncertainty of the observables that "maximally" fail to commute with the measured one. So the measurement reduces (well, eliminates) our uncertainty about the thing we measure, but it affects other quantities and increases our uncertainty about other quantities. In quantum mechanics, our measurements are not informing us about some "God's and everyone's objective truth" (as in classical physics) because none exists. But they're steps in learning about "our subjective truth" that is damn real for us because all of our lives will depend on the events we perceive. In most practical situations, the truth is "approximately objective" (or "some approximate truth is objective"). Fundamentally, the truth is subjective but equally important for each observer as the objective truth was in classical physics. But just try to think about someone who says that a "special modification of the Liouville equations of motion" is needed for the event when we look at a die that was tossed and see a number. The probability distribution $\rho$ collapses. Well, there is nothing magic about this collapse. We are just learning about a property of the die we didn't know about – but we do know it after the measurement. The sudden collapse represents our learning, the Bayesian inference. In classical physics, we may imagine that what we're learning is some "objective truth about the observables" that existed independently of all observers and was the "ultimate beacon" for all observers who want to learn about the world. In quantum mechanics, no such "shared objective truth" is possible but it's still true that the measurement is an event when we're learning about something and the collapse of the wave function (or density matrix) is no more mysterious than the change of the probabilities after the Bayesian inference that existed even in classical physics. I am confident – and I saw evidence – that many of you have understood these rather crystal clear facts about the relationship between classical physics, quantum mechanics, measurements, and probabilities. But maybe people like Florin Moldoveanu don't want to understand. Maybe it's natural to expect them not to understand these simple things because their jobs often depend on their continued ignorance, confusion, and stupidity. Add to Digg this Add to reddit snail feedback (0) :
888c6e375e0b068b	by John Holbo on January 21, 2014 I find this confusing. (via Gizmodo.) This earlier video provides a nice introduction as well. MPAVictoria 01.21.14 at 3:27 pm “I find this confusing.” You ain’t kidding. These sort of things always make my brain hurt. Probably why I dropped calculus in my first year of university. dn 01.21.14 at 3:32 pm This has been making the rounds on Facebook, and it’s silly. The sleight of hand is more fundamental than any “step” in the math – they’re pretending that the sum in question is convergent when it isn’t. 1-1+1-1+1-1+… doesn’t add up to 1/2 either. It adds up to nothing, not even zero. Not convergent. Kaveh 01.21.14 at 3:35 pm When he says that 1-1+1-1… = 0.5, he’s applying a particular definition of the value of the infinite series that is different from ‘what limit does it approach’. ‘What limit does it approach’ is the more common definition of the value of a series like this. And 1-1+1… doesn’t approach any limit, neither does 1+2+3+4… It’s like those proofs where they prove 2+2=5 by dividing by 0. Except that while it never makes sense to divide by 0, apparently the assumption that 1-1+1-…=0.5 actually takes you someplace interesting, so the video isn’t bogus, but it is sneaky. Or, it’s a bit like if this were a geometry problem, and they used a non-Euclidean geometry without telling you (we tend to assume things are Euclidean…). Somebody who knows more math could give more details. dn 01.21.14 at 3:37 pm Good explanation here. Z 01.21.14 at 3:42 pm Oh! I never thought that CT would one day post on my area of research! You guys taking guest post (who wouldn’t die to know how one could have guessed the result beforehand)? David Steinsaltz 01.21.14 at 3:50 pm There is no such thing as 1+2+3+… This is the kind of nonsense that physicists love, taking some formal rules and giving it a name that sounds like something people are familiar with (e.g., “strings”). What they are talking about is that you can define a function that is equal to (1/1)^k+(1/2)^k+(1/3)^k+… for k bigger than 1, but also has values defined for other k. If you look at what it is for k=-1, the result is -1/12. But this has nothing to do with anything a reasonable person might call 1+2+3+… Even if that person were a physicist. It’s not that they did anything wrong — though you could make an equally convincing video to prove that it’s a different number — but it’s basically just mystification. Petar 01.21.14 at 3:54 pm The arguments are not sleight of hand. They aren’t entirely rigorous either, to be fair, but they could be made rigorous with a bit of effort. It would just alienate the viewer it’s meant to entertain. Anyway, they use a different notion of a ‘convergent series’ from the standard analysis text. Whenever standard analysis says a series is convergent, this definition will agree and give you the same limit. But this definition is more general and allows you to define a limit for a larger class of series. dn 01.21.14 at 3:56 pm Wikipedia shows that the video is in fact even wronger than Kaveh is saying; if you scroll down, it indicates that the Cesaro sum of 1+2+3+4+5+… is also not convergent, i.e. it equals nothing. The creators of the video pretended that you could equate the Cesaro sum with the actual sum and then do algebra with it, which is not the case. (Again, 1-1+1-1+… does not equal 1/2 either. You can’t equivocate between “does not exist” and “is 1/2”.) Belle Waring 01.21.14 at 3:57 pm I found that by shifting the sets along a different number of places when adding the two identical sets I got very different results. Say, none, such that it was: +1 -2 +3 -4 +5 -6… If I then choose to group these together variously it seems I start with 1 and can imagine everything else to be -2? Oder, two places rather than one: +1 -2 +3 -4 +5 -6 +7 -8… +1 -2 +3 -4 +5 -6 Now, if I fancy, we will begin with -1, but go on with -2, hypothetically? But what if I should decide to group them the other way around, such that the larger positive number befriended its adjacent lower negative buddy? And how did we ever get to 1/2? Belle Waring 01.21.14 at 3:58 pm I mean, I know this is intended to be some sort of helpful illustration, and not anything like a proof, but it just seems so totally confusing that I don’t know at all what to think. Ed Herdman 01.21.14 at 4:04 pm What I got out of it was that you can ask a simple question: Does a set converge? And if it does you might be able to compare it to another set, if it converges also. If one set doesn’t converge, then the right side of the equation is not useful for that application. Likewise, you don’t get from increasing positive numbers to a negative sum. AcademicLurker 01.21.14 at 4:06 pm Bloggers have been all over this. In addition to the link in 4, this is pretty helpful. I think part of what irritated people is that the reason the folks in the original video ended up misrepresenting things is that they were overly eager to arrive at a facile “Ha! Look how crazy and counter intuitive math is!” moment. Sort of like the way the writers at Slate would teach math, if they were to try. Kaveh 01.21.14 at 4:07 pm dn – that link was helpful. An exchange in the comments says that it is possible, w/ important assumptions that also weren’t stated in the video (i.e. if the series is in the complex plane), to assign a Cesaro sum to a series that ‘goes to infinity’, but even if that is correct, they still go wrong with the ‘shift’. elm 01.21.14 at 4:08 pm This presentation is pretty deeply misleading. The misdirection starts when they arrive at “1 – 1 + 1 – 1 + 1 …” = 1/2 That is not accurate. It’s wholly inappropriate to use equality for the process that they employed. My own two semesters of university calculus tells me that the series “1 – 1 + 1 – 1 + 1…” does not converge, so its sum has no particular value. Any process that builds on a misstep like that in mathematics is simply incorrect and flawed. Belle @8,9: You’re absolutely correct. Shifting terms and adding series in that way is a tricky business and the presenters in the video play very fast and loose with how they do it. There is a sense in which you can get the result they present — that the sum of natural numbers is -1/12 — but it involves analytic continuation and taking a series outside of its circle of convergence. It’s also true that some elements of quantum physics, like QED, rely on similar mathematically-nonsensical operations. However, if they had arranged their presentation somewhat differently, they could have arrived at any answer whatsoever, so their presentation has nothing to do with that bit of useful nonsense. Rob in CT 01.21.14 at 4:09 pm Hah. That’ll leave a mark (or should, but yeah, no). David Steinsaltz 01.21.14 at 4:13 pm #9 is exactly right. There is a true mathematical fact at issue here, but this is the sort of thing that gives “formal proof” a bad reputation. It’s not a procedure that can reliably give the right answer, and it doesn’t clarify any important principles. On the contrary, it seems intentionally obfuscatory. The intention seems to be primarily to provoke a reaction like that of #1 here: You must be pretty smart if you can make sense of this kind of weird shit. Aaron 01.21.14 at 4:15 pm The point is that is a string of numbers, and you can choose how to assign a value to the ‘sum’. The most common way is to take the series of partial sums, i.e. s1 = 1, s2 = 1+2 = 3, s3= 1+2+3 = 6, … and see if that sequence has a limit in the traditional sense. In this case, the sequence clearly doesn’t have a limit, so the sum does not exist. However, there are many other ways you could assign a ‘sum’ to the original sequence. One might also require that this assignment satisfy a number of nice properties. Examples of this are things like Cesaro summation and Abelian summation. Those two, for example, assign a value of 1/2 to the ‘sum’ 1-1+1-1+1 … Even more, you can show that any assignment that has some nice properties must give a value equal to 1/2. These two don’t work for the ‘sum’ however, but you can do zeta-function regularization or Ramanujan summation to get the value -1/12. The cool thing is that this isn’t just mathematical sleight of hand. These values are telling you something about the original series. One (very) mathematical explanation is given by Terry Tao here. Hardy wrote an entire book about this stuff called, “Divergent Series”. But, being a physicist, the cool thing for me is that you can actually see this in the laboratory as it’s related to the Casimir effect. dn 01.21.14 at 4:15 pm What AcademicLurker said. I’m not a mathematician, but I do enjoy math. It frustrates me when people come up with misleading sht like this to try and portray math as some deep mystery beyond the ken of mere mortals. Math is challenging, but it’s not out to tear down all your intuitions. dn 01.21.14 at 4:19 pm Kaveh @12 – ah, that comment makes sense. I, too, was slightly wrong. Thanks for that. Warren 01.21.14 at 4:19 pm There is a theorem in calculus/real analysis, the Riemann Series Theorem or Riemann Rearrangement Theorem. If you take one of these series with infinitely many positive terms and infinitely many negative terms, strictly speaking a “conditionally convergent” series, you can rearrange them to get any real number which you want. Good for paradox but not good mathematics. mattski 01.21.14 at 4:23 pm Speaking from ignorance here but aren’t imaginary/irrational numbers extremely useful in physics, and aren’t they fundamentally inconceivable? Walt 01.21.14 at 4:37 pm This video is the worst thing ever to happen. Yes, worse than the Iraq war or the series finale of Battlestar Galactica. The argument in the video is 100% bullshit. There are settings where you can assign a value to divergent series, but there is no unique universal way of doing so. For example, in some settings if you assign a value to 1 + 2 + 3 +…, it’s -1/12. In other settings it’s obviously +infinity. (What’s weird about the video is that they use Cesaro summability for 1 – 1 + 1 – 1…, and if you apply Cesaro summability to 1 + 2 +3 +…, you get +infinity.) elm 01.21.14 at 4:43 pm mattski @ 19: Yes, they are extremely useful, no they are not fundamentally inconceivable. Imaginary numbers are a straightforward extension of real numbers, motivated by including the element i (the square root of -1). Irrational numbers (and hence the real numbers) are necessary as soon as you want to define the square root of 2 and in a bunch of other operations. Both are a bit counter intuitive and involve some weirdness, but they are still approachable. dn 01.21.14 at 4:44 pm mattski – Imaginary numbers (complex numbers) and irrational numbers are not the same. Irrational numbers, at least, are very conceivable and you can see them every day, not just in weird physics. The square root of 2, for example, or pi, both of which can be easily illustrated geometrically. Complex numbers are a little more tricky, but they don’t really replace your intuitions about real numbers; they’re just a different class of number that you can also work with in well-defined and often useful ways. This demonstration, on the other hand, plays fast and loose with definitions/assumptions to produce a result that conflicts with our intuitions about natural numbers. I vaguely recall reading of an exchange between Wittgenstein and Russell, in which Wittgenstein essentially criticized Russell and Whitehead’s attempt to ground arithmetic in logic by arguing that they were reasoning the wrong way: if the logic produced an arithmetic that conflicted with our ordinary understanding of how natural numbers behave, we would take this not as a demonstration that we were doing the math wrong, but as an indication that there was something wrong with the logic. (I may be mangling Wittgenstein here. I’m not any more a philosopher than I am a mathematician.) Anonymous 01.21.14 at 4:47 pm I see what you mean about imaginary numbers but I don’t think irrational numbers are inconceivable. Draw a square with sides of one inch and the lengths of that square’s diagonals will be irrational. MPAVictoria 01.21.14 at 4:50 pm I love that there are people out there who actually understand this. I am quite jealous. mattski 01.21.14 at 4:50 pm Yes, I was aware that irrational and imaginary numbers not the same category. But isn’t the square root of -1 fundamentally inconceivable? Seems to me that it is. elm 01.21.14 at 4:56 pm mattski @ 24: To paraphrase Richard Feynman — How can you say it is inconceivable when you have already conceived the idea? It may be counter intuitive, or you may not like it, or it may be hard to visualize it, but you have already conceived it. Katherine 01.21.14 at 5:02 pm Speaking as a very visual person – who did maths up to 18 mostly by imagining pictures in my head – yes, imaginary numbers are difficult, nay, impossible to visualise. So you don’t. Inconceivable? No. Like the name says – it’s imaginary and used to work other things out. Kaveh 01.21.14 at 5:04 pm @21 Ditto that, I found complex numbers more approachable than a lot of other things I studied in math, back in the day. I was really sold when I learned that you can use them to get the mathematical function for the bell curve. Kaveh 01.21.14 at 5:05 pm Katherine @26, they’re actually not that hard to visualize at all, once you get past just the basic concept of sqrt(-1) = i and start building up structures with them, like the complex plane. dn 01.21.14 at 5:08 pm mattski – depends what you mean by “inconceivable”. A person might not, in ordinary life, know where to look for a complex number, or how to “picture” it. But “the square root of -1” is, in another sense, entirely conceivable; It’s just a number that, when multiplied by itself, yields -1. (For that matter: can you conceive of -1 itself? How would you draw me a picture of it? Even the concept of zero took hundreds of years for mathematicians to arrive at.) The imaginary unit may strike us as a strange number, but as I see it, it behaves as consistently as any other number and it doesn’t contradict my intuition about what 2+2 equals. Futility 01.21.14 at 5:09 pm dn 01.21.14 at 5:16 pm In another sense you CAN picture a complex number; you just reimagine the “number line” as a plane, a two-dimensional space. You can draw geometric figures in this space and use them to illustrate some conclusions about complex numbers that would seem odd if simply notated symbolically. A unit circle drawn in the complex plane, for example, provides a visualization for the close relationship between exponential and trigonometric functions which can initially blow your mind if you’re only shown the equations. Another lurker 01.21.14 at 5:19 pm @belle #8 Regarding regrouping and shifting series, there is a very interesting result called Riemann series theorem [1] that states that given a convergent series that is not absolutely convergent [2] you can get to sum to any number you want by rearranging the terms. [1] https://en.wikipedia.org/wiki/Riemann_series_theorem [2] a_1+a_2+…< ∞, but \|a_1\|+\|a_2\|+… = ∞ Another lurker 01.21.14 at 5:21 pm An example of a non absolutely convergent series is: mattski 01.21.14 at 5:23 pm I appreciate the responses. Thanks all. I do find this sort of thing fascinating, and lament the deterioration in my gray-matter. dn, depends what you mean by “inconceivable” Yes. So, I’m skeptical of the idea that simply slapping a label on something makes it conceivable. The word “god” makes for a great example. For my money, that is an attempt to conceive the inconceivable and the result is quite a bit of confusion. As far as the square root of minus 1, doesn’t it violate what we understand as the rules of multiplication? How is this hurdle overcome by the pasting on of a label? The question of whether simple negative numbers are conceivable is very interesting. I’m not sure about it, but visualizing a negative number as a process seems to satisfy my intuition… (it might also be a helpful reminder to think of positive numbers in a similar, instrumental way.) kent 01.21.14 at 5:26 pm I am not a mathematician, but: Take 1+2+3+4+5+6+… , and call it S1 Then subtract it from itself, but just “move it along a little bit,” as follows: 0+1+2+3+4+5+… 0+1+2+3+4+5 …. 1-0 = 1 2-1 = 1 3-2 = 1 Result: S1 – S1 = 1+1+1+1+1 … As a result, either S1 does not equal S1 [a clear contradiction] — or else 1+1+1+1… = 0 [which is stupid]. I conclude that if you are allowed to “just shift this along,” you can get pretty much any result you want. Thus “just shift this along” is not a valid mathematical procedure in this type of case. And thus the whole thing is bogus. Mario 01.21.14 at 5:38 pm The main reason this confuses people is that, as stated, it is bullshit, and there is nothing to understand. A smart mind that knows basic math cannot see much but gobbledygook in that. It is not impossible for it to make sense, but then you have to redefine the meaning of things (like equality and sum) rather fundamentally and do so explicitly. I hope the guys in the video are aware of that, otherwise that would be quite shocking. People tend to get immediately better at maths once they realize that it is entirely legitimate to NOT understand things, and to insist on the details. MattF 01.21.14 at 5:47 pm Here’s the important point– what you get when you ‘sum’ an infinite series is a matter of definition. One definition may give you the answer “You can’t do that”. But another definition may give a different answer, e.g., -1/12. One way to look at it is to ask “What do those three little dots mean?” They cannot mean “Do some operation an infinite number of times”, since that would take forever– and we don’t have forever to wait for an answer. One better possibility is that the three little dots mean “Seek professional advice”, e.g., ask a mathematician. In this case, a mathematician may think something technical like “Well, maybe analytic continuation could make this meaningful” and then attempt to explain that in a non-technical manner. elm 01.21.14 at 5:48 pm I think that these guys are trying to popularize mathematics with videos like this, but they probably harm that cause more than they help it. They could have used similar processes to “prove” that 1=0. A conclusion like that in a direct proof means that you made a mistake somewhere. In a proof by contradiction, it may mean that you made a mistake or may mean that your proof by contradiction was successful. I’m also curious to know what explanatory text appeared around that equation in their string theory textbook. One would think that the accompanying text would go a long way to explaining the intricacies of that non-standard result along with cautions about where and how to apply such things. mattski 01.21.14 at 5:54 pm Also, back to the OP. The following is impressionistic and strictly amateurish but, isn’t there a symmetry between the distance from 0 to 1, and the distance from 1 to infinity? So, can we visualize numbers in general as a sort of oscillation on between these two “regions?” And does that take some sting out of the paradox? bianca steele 01.21.14 at 6:07 pm You all know about heat rays. But have you seen my proof of the existence of coolth rays? Nine 01.21.14 at 6:12 pm Not sure if this will convince mattski but equation is Euler’s identity dn 01.21.14 at 6:18 pm In addition to what MattF says: everything in math gets more complicated when you start throwing in such notions as “infinity” or “infinitesimal”, which are intuitively conceived as indefinite. This is the very reason why the differential and integral calculus were somewhat controversial when first introduced; the concept of the “limit” was developed after the fact as a way to get the infinitesimals out of the infinitesimal calculus. More rigorous that way. (Abraham Robinson in the 1960s did take a shot at a non-standard calculus which revived the use of infinitesimals themselves in a more rigorous way; his method has its admirers but has never really caught on in the mainstream.) bianca steele 01.21.14 at 6:32 pm It supposed could be read as a satire on the difference between literary and scientific cultures. Except it reminds me more English physicist piss-taking in the vein of the coolth ray experiment (which I’ve never heard mentioned by an American but actually don’t remember well enough to repeat here). bianca steele 01.21.14 at 6:33 pm “It suppose” s.b. “I suppose it” Billikin 01.21.14 at 6:34 pm 1/2 = 0. S = 1 – 1 + 1 – 1 + . . . = 1/2 (Already proven.) = (1 – 1) + (1 – 1) + (1 – 1) + . . . = 0 + 0 + 0 + 0 + . . . = 0 S = 1/2 = 0 QED. OCS 01.21.14 at 6:44 pm I get that the square root of -1 is an extremely useful concept that lets you do all sorts of mathematics, and I heartily endorse it. But I’ve never understood what kind of a number we’re imagining it is. We have rules that say the square root of a number is a number which multiplied by itself gives that number. We have rules that say that two positives or two negatives multiplied by one another give a positive number. So are we imagining that the imaginary unit is neither positive nor negative, or maybe both at the same time? Or is the answer just that its useful, and we don’t need to worry about it? Billikin 01.21.14 at 6:49 pm Belle Waring: “I know this is intended to be some sort of helpful illustration” No, it is not. It is intended, as Castaneda’s Don Juan put it, to astound the Indians. I suppose that there is some interpretation of 1 + 2 + 3 + . . . = -1/12 which makes sense in physics. They did show a text in which that equation appears. However, they did not then provide any such interpretation. Instead, they went off into Fallacy Land. MattF 01.21.14 at 6:54 pm OCS: You’re correct to say that the imaginary unit is neither positive nor negative. This just means that ‘positive’ and ‘negative’ are properties that have limited usefulness. C’est la vie. dn: I’m not sure that anyone who has not reached the age of ‘mathematical maturity’ should actually try to read this, but here’s Terry Tao’s attempt to make nonstandard analysis seem reasonable to working analysts: christian_h 01.21.14 at 7:12 pm It should be pointed out that the first computation of the value of the Zeta function at -1 (which is really what these guys mean when they say “1 + 2 + 3 + … = -1/12, as others have pointed out before) is due to Leonhard Euler; and that his calculation (which is heuristic, as no complex analysis was available at the time) can on the one hand be turned into nonsense as in the video – and on the other can be made into a proof by re-thinking it using 19th century methods. It would have been educational to make a video explaining this – without the context it is misleading and as has been said, a mystification of maths. I have to run, but I will try to find Euler’s original argument later today. Here is a link sketching it in modern terms: Z 01.21.14 at 7:19 pm Well, Belle, as Barbie famously (but apocryphally) proclaimed: Math is hard. It took mathematicians two millennia to understand how to compute with infinite sums so you shouldn’t feel diminished if your are unable to rediscover all that lore starting from scratch. Nevertheless, the identity ζ(-1)=-1/12 which forms the title of this post and its close cousin ζ(0)=-1/2 (so 1+1+1+1+1+…=-1/2), an even superior one in fact, are such gems of human knowledge that their propagation in popular culture can only be deemed a good thing. dn 01.21.14 at 7:19 pm OCS @44: The imaginary unit is, in a sense, “positive” in that there is also such a thing as -i, whose relationship to i is analogous to the relationship of -1 to 1. [More specifically, -i=(-1)i.] The mistake is to try to locate both 1 and i on the same axis, because 1 is real and i is not. They’re perpendicular. When people say imaginary numbers are unintuitive, their intuitions would best be characterized as “not even wrong”. Your intuition is fine; it’s just not precise, because it ordinarily doesn’t have to deal with such objects as complex numbers. Eventually you train yourself not to privilege the real axis and the distinction no longer worries you. They are both just axes, and they both behave pretty much exactly the same, except that one has an extra symbol attached to it and the other doesn’t. mud man 01.21.14 at 7:41 pm Imagine you had a relay that switched between 1 and 0 volts. So you measure the voltage at the output. As it increase how fast the relay switches, the needle on your meter can’t move fast enough and settles on .5 volts, quivering slightly. Faster and faster: eventually the capacitance of the output wire acts like a “weight” that prevents the actual voltage from changing very fast, and the real instantaneous voltage stays close to .5 volts. I don’t know whether this has anything to do with physics. I am an engineer. It does occur to me, tho, that the observable universe isn’t big enough to contain any infinite sequences, so if such things are important, we need them to fold up into something definite. And if you are trying to describe continuous chaotic functions accurately, you need infinite sequences like the Taylor series. The Universe is a stranger place than you thot, bro. Belle: it doesn’t matter how you pair up the numbers. Arithmetic says all those sums must be the same because “commutativity”. He picked that particular pairing because he knew it was going to work, is all. mbw 01.21.14 at 7:42 pm @49 It’s pretty misleading to say that the real and imaginary axes are “pretty much exactly the same”. The real axis forms a field, you get a real number when you multiply two reals. The imaginary axis doesn’t. They’re very different. That particular side discussion reminds me of when mathematicians in the early 60’s wanted to set a whole new curriculum. IIRC ordinary 12 year olds were supposed to learn the complex numbers as the ring of polynomials mod (x^2+1). Slower kids would just have to learn them as the set of all ordered pairs with a particular multiplication rule. Walt 01.21.14 at 7:50 pm There is a geometric interpretation of imaginary and complex numbers. This interpretation is important historically because it made it clear that complex numbers were a well-defined thing. At this point, not only are the complex numbers are well-understood, but many other systems with more exotic multiplications, such as the quaternions (where the order of multiplication matters) octonions (where multiplication is not even associative), or the p-adics (which contain rational numbers, but not all real numbers, and where many weird infinite sums hold, such as 1 + 2 + 4 + 8 + … = -1). Odm 01.21.14 at 7:54 pm mud man: As people have pointed out upthread, when an infinite series is divergent (the partial sums do not approach a number), commutativity no longer holds. From what I remember, when a series is convergent, then you can rearrange as you please. elm 01.21.14 at 7:58 pm mud man: Finite and infinite sequences obey different sorts of rules. Procedures that are acceptable with finite sequences are not necessarily allowable with infinite sequences. That’s even more true with non-convergent sequences, or even with certain classes of convergent sequences (http://en.wikipedia.org/wiki/Riemann_series_theorem). Commutativity does not excuse his particular choice of pairings, the presenters operations with 1 – 1 + 1 -1 … are unjustifiable. Z 01.21.14 at 7:58 pm I’m afraid that’s not true, as has been pointed out before in the thread already: commutativity does not extend to infinite sums. He picked this particular pairing because he wanted to arrive at the correct result, and he did, though by an entirely wrong derivation, but Belle’s pairing was as justified (and in fact it is not hard to prove Walt’s assertion above that one could have arrived at any real value by choosing carefully a rearranging). As an aside, though it is true that the fact that the result is correct has a (not so easy) physical interpretation, I want to insist that this is a result of pure math, that it predates the physical interpretation by centuries and that the investigations of such results is ongoing and still considered very much central (or so says someone who earns his living doing it anyway). If I were to make a comparison, I would say this identity is to math what Newton’s apple is to physics (in particular, it is much more important than the more well-known e^(iπ)+1=0 which, though cute, is completely trivial). Phil Koop 01.21.14 at 8:14 pm @mattski, you ask “doesn’t [root -1] violate what we understand as the rules of multiplication?” No! The whole point is that it doesn’t; if you assume it is a number, then everything works out algebraically. That is its only meaning – there is nothing else to it. Of course it is true, as you note, that no one can show you what root -1 of an apple (say) looks like. But why get bent out of shape about “imaginary” numbers? No one can show you an example of a typical real number either; that is, if you randomly select with uniform probability a number from a finite real interval, then with probability one you will select a number that cannot be identified numerically with finite information. Numbers like 3 or 1/2 or pi or root 2, which all can be quantified with finite information (i.e. a computer program), are vanishingly rare. Real numbers are only needed because it is convenient to say that the limit of a sequence, if it exists, is a number (this is how real numbers are constructed.) But once you assume they exist, they follow all the rules. You can’t visualize a typical real number, but you were still happy to accept them; imaginary numbers are just an extension of this principle. bianca steele 01.21.14 at 8:51 pm Really, all that happened is that someone read a technical book under the assumption that it’s unnecessary to have any theoretical or technical vocabulary, or to read books in one field differently from books in another. The person of technical bent runs up against some confusion, and stops to figure out what he’s missing. The non-technical person feels the text should be written in plain language and interprets everything in that light regardless of context. The results are very different. When the latter type of person finds a contradiction, he blames the text, and that’s all. But I’ve found another proof that 1-1+1…=0.5. S is the sequence. Rewrite as 1-(1-1+1…). Then S=1-S, and S=0.5. Z 01.21.14 at 8:53 pm It should be pointed out that the first computation of the value of the Zeta function at -1 […] is due to Leonhard Euler; and that his calculation (which is heuristic, as no complex analysis was available at the time) Because a CT thread should include controversy and because Mao found it in himself to troll divergent series at 43, I will note that I disagree with the statement that Euler’s proof was heuristic. It was correct to the standard of proofs of the time and Euler could have, with no doubt, proved it to today’s standard if he had been required to. To be more specify (and for Christian benefit), the computation of negative zeta values does not require any non-trivial complex analysis, just a clever real analytic manipulation of the Taylor series of tdlog(1-t)/dt, it is the meromorphic continuation to the whole of C, which is much harder, which had to await Riemann and the assorted technology. Just some commenter 01.21.14 at 9:00 pm @mattski, @Phil Koop: For me, the key to understanding i was to see a geometric explanation of addition and multiplication on the complex plane. I can see how if you’re told only that there’s a defined quantity called i which equals the root of -1, but you’ve never grasped the mechanics of how it works to do arithmetic with it, it might seem incomprehensible that one can do arithmetic with it. The key is to understand (as others have explained) that i exists on a second axis, defined to be at right angles to the real number line, with both axes together defining the complex plane. Any point on that plane is a complex number, with a real and imaginary part, and multiplication of the numbers in this plane proceeds exactly as before for the real numbers, all lying on the real number line, but via a rotational* procedure for complex numbers (even better, the real number multiplication is just the trivial case of this rotation — a rotation of zero degrees). Working through some examples, and learning how you can do the multiplication either through a strictly algebraic procedure, or though an exactly equivalent rotational procedure based on the angles and magnitudes of vectors can make this somewhat more intuitive again. Once you understand how that works, and you can see that it’s an expansion of the concept of multiplication that’s wholly consistent with everything you learned about operations on real numbers, it seems like a brilliant and useful — but not mysterious or imponderable — expansion of the concept of numbers, because it allows for arithmetic with previously undefined quantities, and solutions to previously unsolvable equations. Just some commenter 01.21.14 at 9:07 pm Whoops, and of course, even better, you may then recognize you are already familiar with this rotational procedure in the case of multiplying by a negative number, which does entail a 180 degree rotation. So with real numbers, the rotation is either 0 degrees or 180, with purely imaginary numbers, 90 degrees, or 270, and for complex numbers in general, any possible rotation. Seeing all this way back when translated i, for me, from something that looked like a definitional trick to a part of an expanded conceptual framework. elm 01.21.14 at 9:14 pm I’m not familiar with the technical lingo in this language, but colloquially, is 1-1… really ‘divergent’? it seems more like ‘oscillating’ It’s not appropriate to intermix colloquial and technical language. In the technical sense, that series is divergent. And if some value is oscillating, then the center of it seems like a reasonable approximation of its value. Mathematical reasoning does not stand on how things seem. This is especially so when working with infinite or infinitesimal quantities. Furthermore, there’s a world of difference between equivalence and approximation. P O'Neill 01.21.14 at 9:16 pm This may be the first time CT has been facilitator of trolling an entire profession. Bruce Wilder 01.21.14 at 9:29 pm bianca steele @ 55 I just flashed on CT comment threads being replaced by a programmed loop spiralling robotically into an infinite future, to end only with the final dimming of the sun . . . Theophylact 01.21.14 at 9:35 pm It’s too damn bad that mathematicians were responsible for picking annoying and misleading terms for concepts that are neither irrational nor imaginary, but the mathematical sense of humor is as difficult to appreciate as math itself. Physicists aren’t much better, with their quarks and gluons and strings and half-dead cats. But you do have to give entities names or you can’t talk about them. dn 01.21.14 at 9:52 pm Re: terminology, I think calculus would be a lot more fun if we’d stuck with Newton’s terminology: not differentials, but “fluxions”. bourbaki 01.21.14 at 9:53 pm Counterpoint. One of the most overused adjectives in the field is normal. On a tangential note, any else here read Mathematics Made Difficult ? I feel it is very apropos. Collin Street 01.21.14 at 9:55 pm Rational <- ratio, no? a number that isn't a ratio, then, a non-ratio number, must be…? And "imaginary" came in contrast to the label "real". What else would you suggest calling either of them? Billikin 01.21.14 at 10:01 pm mathbabe has a cool blog post on this at http://mathbabe.org/2014/01/21/if-its-hocus-pocus-then-its-not-math/ “If it’s hocus pocus then it’s not math.” Just some commenter 01.21.14 at 10:06 pm @Collin Street, hmm, what about “second dimensional”? Of course, that might seem to include all complex numbers, not just the imaginaries, but perhaps the reals could be called the “first dimensional” numbers, the imaginaries could be called the “second dimensional” numbers, and complex numbers generally could be “two dimensional”, or planar numbers. Fairly descriptive, I think. Of course, “surreal numbers” and “hyperreal numbers” are already taken. elm 01.21.14 at 10:22 pm The topic is one of series, not sequences. They are different things. Mathworld’s definition of divergent series. mud man 01.21.14 at 10:22 pm Thank you all, I am corrected. elm 01.21.14 at 10:25 pm Additionally, the site “vitutor.com” is not a place that I’d look for mathematical definitions. dn 01.21.14 at 10:35 pm elm @78: I love that quote from Abel from your link: “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” Ed Herdman 01.21.14 at 10:38 pm #71 and “half dead cats,” meet #68. Ed Herdman 01.21.14 at 10:39 pm elm: I would also stipulate the attitude “Mao Cheng Ji in a math discussion? HA! HA! HA!” places monocle maidhc 01.21.14 at 11:01 pm What mud man said is correct, but applying the same reasoning to an infinite stream of pulses -1,1,-1,1… gives an integrated value of 0, not 0.5. That’s a different type of analysis than whether a series converges, though. Complex numbers are very useful in electrical engineering for calculating both the magnitude and phase of a signal. Nothing really imaginary about it. Just a conversion from polar to cartesian coordinates. However the cartesian coordinates happen to be in the complex plane. ezra abrams 01.21.14 at 11:09 pm 1) assume the universe is infinite 2) in any given volume (say our solar system) there are so many ways of arranging all of the matter and energy – a large, but finite number; call in X (I’m neglecting transport in and out of hte volume – perhaps a fatal error) X is the number of ways all the matter and energy in our solar system can be arranged. 3) If (1) is true, then there exists another volume of space with X identical to our own. 4) infact there are an infinite number of places where x is identical – an infinite number of solar systems exactly like this one 5) let Xa be a slightly different arrangement of matter and energy; say where in English the letter E is written as square C 6) there are an infinite number of places where there is a solar system exactly like our own, except E is written as C Infinity is so much fun Katherine 01.21.14 at 11:34 pm Anyway, as any fule kno, all numbers are equal to 47. christian_h 01.22.14 at 12:37 am Z @64 (8:53): I do not disagree with anything you write there – just expressed myself badly being in a hurry and all… so no controversy, sorry ;) The point I was trying to make is that while the video gives an incorrect argument, it is an argument based on a bad understanding of Euler’s proof obtained by substituting 1 in for t in the logarithmic differentials. So the video is a chance missed, not a random assault on reason. mattski 01.22.14 at 1:03 am Phil Koop, Thanks for the response, I appreciate it! Who said I was bent out of shape? :^) OK. I think I see what you’re getting at here. But if I understand you correctly what you are claiming seems problematic. I think you are saying that if we randomly put our “stylus” on a “line” representing the real numbers we are going to have a problem of infinite or unknowable decimal places. But that would raise an objection. In what sense have you pointed to a number if you cannot identify it? Wouldn’t it be reasonable to say, “you have not identified any specific number.” At least with numbers like Pi we have a means of identifying the number (as a ratio of two numbers) and the ability to produce the decimal series. But additionally, we can intuitively understand fractional numbers as volumes or weights. How many pounds does this bowling ball weigh? Well, how many decimal places do you want to go? How sensitive is your equipment? And as a benefit (?) thinking this way diminishes the importance of “excessive” decimal places, which might be useful when calculating atomic weights but for mundane purposes we don’t need them. Now, back to the root of -1. Yes, I can understand it as a “stipulation” or a “rule”. But I can’t map it onto my experience the way I can map regular multiplication. So it seems to me that there is an important sense in which I can never conceive it. mattski 01.22.14 at 1:09 am Also, Katherine. I think you fucked up. It is 17. You will find the proof encrypted here. Just some commenter 01.22.14 at 1:50 am Mattski @88: I attempted to answer this before, but if you see complex numbers as points on the plane and multiplication as multiplication of the magnitudes (distance of the point from the origin) and addition of the angles from the positive number line, then multiplication of complex numbers makes sense. It completely encompasses multiplication of real numbers you have an intuitive sense for, but includes another dimension of multiplication, quite literally. With a little “playing around” with this, you can develop an intuition for it, too: it is not just a definition, but has a geometric representation that can come to seem fairly natural: add the angles, and multiply the magnitudes. bourbaki 01.22.14 at 1:57 am One thing that might help (if you are interested) is to recognize that numbers have many different way of being thought of that preserve there underlying algebraic structure. I actually think that saying i is the square root of -1 obscures things and a more geometric perspective helps (though I study geometry so am biased). [warning this is a bit pedantic] For instance, one way to think of the real numbers is as an “action” on the set of points on the real line itself (this may seem pointless at first but I think the analogy is useful). Under this action multiplication by a positive number x acts on a point p on the real axis at distance \|p\| from 0 by stretching it out so it is at distance x\|p\| from 0. Multiplication by a negative number does the same thing, only it also reflects across the origin. Multiplication by 0 sends everything to 0. What one observes that acting by xy is the same as acting by y and then by x. Now you could just as easily imagine this action occurring on the plane. In this case multiplying a point p in the plane at distance \|p\| from the origin by a real number x gives you a point that is distance x\|p\| from the origin and lives on the line through the point and the origin (negative x again reflects while positive does not and 0 maps to the origin). I.E. we again scale. Now in the plane we also can think about rotations. In this picture the action by i the squareroot of -1 is by rotation by 90 degrees about the origin. The reason for this is if we act twice then we get rotation by 180 degrees about the origin which is the same as multiplying by -1 that is ii =-1. In general, any transformation of the plane that is a composition of a scaling and a rotaiton about the origin can be thought of as a complex number — i.e. a number of the form x+i y where x and y are real. The algebraic multiplication of complex numbers then corresponds to composing the transformations. Obviously one has to do some work to make sure this is completely justified but it is not too hard. What is interesting is that something along the same lines happens also in dimension 3 and one can construct an “extension” of the complex numbers to something 4 dimensional (but where one no longer has commutativity). These are called the quaternions. Just some commenter 01.22.14 at 1:58 am mattski: Have a look at http://www.mathsisfun.com/algebra/complex-number-multiply.html and scroll down to “Now For Some More Multiplication”. Once you get this, you have the central insight that makes complex multiplication intuitive. mattski 01.22.14 at 2:15 am @ 90, 91, 92 Many thanks! Alex K. 01.22.14 at 2:16 am The Youtube video is amateur hour. They need not just one, but several errors to obtain the result. Instead, I can show you a correct proof that 1=0, using geometry. The proof works very well on napkins and steamy windows, but there is an online version too. First we prove that all triangles are equilateral. The proof is here . Then, using a triangle with two sides of length 3 and 4 respectively, which we just proved to be equilateral, we have that 3=4. Subtracting 3 we get our result, along with the result that all numbers are equal to zero. Jason 01.22.14 at 2:16 am Terry Tao is awesome. To the proof that 1/2 = 0 by Billikin above: you are assuming associativity in infinite series, which is false. Also the -1/12 is the reason bosonic string theory is only consistent in 26 dimensions; the 12 becomes the 24 in the condition (D – 2)/24 = 1 [or something like that]. If you want pretend you can add an infinite number of things, weirdness is what you get. Belle Waring 01.22.14 at 3:00 am I always liked both irrational numbers and imaginary numbers I was happy when they came along in math. I was talking about the video last night with John just after he posted it, and I was thinking that the claim, “this is important for string theory! This is how we know there are 27 dimensions rolled up inside all the regular stuff!” is not of necessity compelling. I mean, may people seem to believe that etc. but is it the case that they know about the various additional dimensions? I am unsure. Belle Waring 01.22.14 at 3:15 am Ed Herdman, elm. No need for monocles. We have banned Mao, but unlike Hector who has quite politely excused himself when asked… Chris Warren 01.22.14 at 3:34 am Alex K Just because a chosen angle is bisected – does not mean that AP = PB for all triangles. In fact, AP = PB if and only if the triangle is isoceles. In other words, it is proving something by assuming it in the first place. Just like Arrow and Debreu. Alex K. 01.22.14 at 3:56 am “Just because a chosen angle is bisected – does not mean that AP = PB for all triangles.” Actually, P is chosen precisely as the midpoint of AB — that’s how P is defined. There is no trickery there. PHB 01.22.14 at 4:01 am Another way to spot the sleight of hand is to consider what is happening out at the infinite end of the series. infinities are being subtracted from infinities. Which can produce any result you like. It seems that the reason the astronomers have fallen for it is that they have a tendency to confuse models with reality. Superstring theory is not reality, nor does reality have 24 dimensions (or whatever the number is this week). At best the physicists have a MODEL that has 24 dimensions that is consistent with empirical observation. Most times they just have a model… Basically it is maybe possible to create a calculus of divergent series that is consistent if you restrict the transformations on the series that are permitted and the sum of the series is not the ‘total’ it is more like a characteristic index. And the resulting indexes are the sort of thing that can be then given meaning in some other calculus… So its not necessarily complete nonsense but showing that it means anything requires more effort than the shell game version… Chris Warren 01.22.14 at 4:04 am Alex K If P is the midpoint, then the perpendicular will not intersect with the ray from the angle opposite AB. The perpendicular bisector of AB will pass to the left of the ray if angle A angle B. The bisector of angle C can cut a perpendicular to AB, but there is no reason why this should pass through the midpoint of AB. If you draw a exaggerated scalene triangle you will see that any bisector is always biased towards the sides with the greater angles. Alex K. 01.22.14 at 4:08 am As I said, the proof works very well on napkins. Alex K. 01.22.14 at 4:09 am There is also no mistake in reasoning, given the drawing. Chris Warren 01.22.14 at 4:13 am Alex K Previous post did not accept mu greater than and less than symbols. … if angle A less than < angle B. … if angle A greater than > angle B. Try drawing an triangle with sides 1, 8, 10. and playing with angle bisectors. Alex K. 01.22.14 at 4:15 am “Try drawing an triangle with sides 1, 8, 10.” An unfortunate choice of sides surely? But, you don’t need to overexplain the thing. Nine 01.22.14 at 4:24 am I can’t believe Chris Warren is wasting time arguing this – it is one of the most famous fallacies there is. Chris Warren 01.22.14 at 4:35 am Alex K. There is also no mistake in reasoning, given the drawing. Yea – and that is how they teach economics at university to students who then go on to run the country!!!! Lee A. Arnold 01.22.14 at 4:36 am I think everybody should start dividing by zero. What the hell! Chris Warren 01.22.14 at 4:37 am I blame you – why didn’t you mention this earlier ???????? Eric H 01.22.14 at 4:44 am @OP and Belle, #9, and others Nah, the “shift” is very useful, but you must be careful about what is contained in the “…”. Here are two simple examples of the utility of shifting and grouping: What is the sum of all of the first 100 numerals, i.e. 1+2+3+…97+98+99+100? If you rearrange them in a convenient way, you have 100+(1+99)+(2+98)+…+(49+51)+50 which is now a much easier problem (5050). When trying to convert infinitely repeating decimals to fractions, i.e. 0.7777…: – 10.7777…=0.777… 90.7777…=7.000000 = 7 divide both sides by 9 –> Even this is a little dubious since you will get a result that 0.9999…=9/9=1. In this case, the sleight is not the shift, it is the 1-1+1-1… = 1/2. Lucie Rie Mann 01.22.14 at 4:44 am When I was a budding string theorist this used to be the Riemann Zeta function, and Zeta of -1 was indeed -1/12. Has that ceased to be the case? Is analytical continuation now considered Bad Math? Nine 01.22.14 at 4:47 am Heh … i saw the exchange just now, work and all. After scanning the days comments, I’ve to admit to a tremendous increase in respect for Mao’s trolling skillz … how does on do that on a math thread ?!!! Alex K. 01.22.14 at 5:15 am Maybe it’s about time that this thread turns into a discussion of socialism. Robert 01.22.14 at 6:10 am dn @ 18: “Math is challenging, but it’s not out to tear down all your intuitions.” I find it strange that this evolved into a discussion of complex numbers. Math has plenty of counterintuitive results. The bit about a randomly selected real number is good. What about the existence of non-measurable sets, given the axiom of choice? That shows that one cannot extend what seems obvious to the infinite sets without question. Or how about the existence of, at least, a countably infinite number of infinities? Or the Banach Tarski paradox? Lee A. Arnold 01.22.14 at 6:23 am I think there is a conceptual problem in the step of adding an infinite sequence to itself, calling it 2S, then dividing though by the 2 later. Because if you add infinity to itself, I don’t think you get 2x infinity. If you think you can do that, then it seems to me that you are claiming to have disproved Cantor’s continuum hypothesis. peter ramus 01.22.14 at 6:38 am Sir, I have the unshakeable belief that 0.9999… does indeed equal 1. I just can’t remember why anymore. The Raven 01.22.14 at 7:27 am peter ramus@115: indeed that series converges to 1. This can be seen by noting that, for any partial sum of the series, there will always be another partial sum further on in the series closer to 1, and no matter how small a number one chooses (denote that number by δ) there will always be a partial sum in that series that will be closer to 1 than 1-δ. The “trick,” if one can call it that, is to never make a direct statement about infinite values, but instead to observe that the partial values of the sums, always the sum of a finite number of terms, get ever closer to the limit value. For the rest, this is in fact very difficult mathematics, and it kept brilliant mathematicians happily occupied for around two centuries. The best mathematicians were able intuitively to arrive at valid conclusions, but it took a long time for them to justify those conclusions with rigorous logic. Niall McAuley 01.22.14 at 9:21 am One interesting thing about the “inconceivable” imaginary numbers is that real, physical things behave in ways which can be represented by imaginary numbers, especially things with frequencies. For example, the electrical thingy often called Resistance in simple physics texts is characterized by a real number, but when dealing with alternating current (like what comes out of a plughole/outlet), the corresponding thingy is Impedance, which is a complex value with both a magnitude (real) and a phase (imaginary). If you put an AC voltage across a load with a certain impedance, the current you get is given by dividing two complex numbers. And because of wave/particle duality, everything has a frequency, and the basic equations of quantum theory (such as the Schrödinger equation) are written in complex numbers with imaginary parts. dk 01.22.14 at 11:02 am Mattski @ 87, you might be interested in the idea of ” computable numbers”. Katherine 01.22.14 at 11:28 am Mattski @88 – nope, 47 Belle Waring 01.22.14 at 2:17 pm I meant to say, Z, with regard to your comment above (“well, Belle, as Barbie famously (but apocryphally) proclaimed: Math is hard. It took mathematicians two millennia to understand how to compute with infinite sums so you shouldn’t feel diminished if your are unable to rediscover all that lore starting from scratch”), what in the ever-loving blue-eyed fuck? Are you trying to provoke my righteous wrath for some reason? Why, when you can just wait around ten or fifteen minutes? You ain’t got to tip the jar onto your own head. The jar that’s full of wasps, all of them shiny new, glistening yellow and black, just crawled out of their paper hexagons. Mathematically and structurally they’re not really any good like what bees would have, because wasps don’t need to save things for later. More like when you find a snakeskin that a copperhead shed, and you hope it left it far behind. Thin. Used to be full of poison. Safe to look at now, hell, it’s kind of fascinating, isn’t it? If you hold it up to the light you can see the paleness of the old unstriped places, and the wide crawling plates along the belly. That’s what lets them s-curve up a tree or a brick wall, 90 damn degrees, and at just the same dignified pace they would go through a field of dead grass. But who would want to just go and pull a thing like that right over onto his own head, when he didn’t have any call to? It’s not as if you were Stephenson-quoter-kun and had license to call me Belle-chan all the time. There’s no love here. I mean, if you want me to get pissed as all hail I guess, sure, but I’m unclear on your motivations. I like to have a backstory. Did I drive your father to suicide with my ruinous informal lending practices? Rate Shoes more highly than they merited? Adore the Wu Tang Clan more than is appropriate for anyone who is not, per Straightwood, reliving the days of her youth in a way that is both unseemly and embarrassing for all to watch, like that time you brought wicked-strong heroin home one time and your dad got low for the first time since he was in his 20s, since he’d just been tripping and smoking weed that whole time in between and it was all [makes waaak waaak noise from TV]? I need, like, something to go on besides ‘apparently sexist math dick who grants a few comments later that my grouping in the sets in question was as good as any other, which is to say, all options can be employed for evil in the manner in the video.’ Maybe fill me in on some person details? Or get a more ludicrous nym? What are you, Zorro? Katherine 01.22.14 at 2:19 pm I was wondering about taking on the Barbie comment, but I figured you’d do it so much better Belle. And boy, I was right. Mao Cheng Ji 01.22.14 at 2:23 pm Excuse me, why are my comments disappearing? Have I missed something? What’d I do? Z 01.22.14 at 2:47 pm Belle, I can see you’re angry. Please please take in account that English is not my native language when I say I’m not 100% sure about what exactly. Math is hard. That’s a fact. So you shouldn’t feel bad if you find computations about divergent series difficult to make sense of, especially when they are performed randomly as in the video, seeing it took centuries to the best mathematical mind to make sense of it. I wrote this as a word of consolation. Not how you took it, most clearly. Also, I did not exactly grant that your ways of doing it was equally good a few comments later; I always thought so: you watched a confusing video about hard math, you felt confused. Nothing wrong about it. The contrary would have suggested some lack of appreciation of what math is. That’s what I meant. Now I believe that the source of your wrath is the Barbie quotation. OK, I guess I should have selected the other synonymous quote I enjoy about this topic (to recall, the topic of being confused about math): “Young man, in mathematics you don’t understand things. You just get used to them” from Von Neumann. Perhaps this one can also be interpreted in a sexist way, so I apologize in advance if it is, as it is not my intention. You ask for personal details. Rhetorically, I guess. Nevertheless, I will point out that if you follow the link on my name and delve a bit, you should find relatively ample evidence of the fact that I’m quite (like just a tiny little bit) involved in the promotion of mathematics (you know, to the tune of several hundred of hours a year) among young students who desires to start again scientific studies after having chosen a different path earlier. I’ll let you guess what is the overwhelming gender of these students (I mean, I recognize fully your right to write anything you like about anyone on your own blog, but I won’t hide that “sexist math dick” hurt a bit). Belle Waring 01.22.14 at 2:51 pm You’re banned Mao, I thought I had told you this earlier, but now I’ve extra double-dog told you with a giant post about it. GO. AWAY. Belle Waring 01.22.14 at 2:52 pm More accurately, I thought Henry had told you when he told Hector. Z 01.22.14 at 2:53 pm Also, in the interest of full disclosure, I append the post I had written before noticing Belle’s comment at 118. Spot on! Justifying this computation by appealing to String theory is exactly having it backwards. This is because mathematicians have found a way to make sense (and beautiful sense) of these computations that (many centuries later) physicists could envision such exotic (and in fact still largely controversial) model of space-time, not the other way round. And there are far, far fewer people who can give any reasonable argument about the existence of these extra-dimensions that people who can either give a rigorous proof of these identities or explain in lay terms why there are true. Belle Waring 01.22.14 at 2:57 pm How on earth is starting out with “as Barbie said ‘math is hard'” in a discussion with a woman who has just said, “I think the way they added these two sets looks like bullshit, also I don’t see how on earth the first set equaled 1/2 in the first place” going to equal anything other than ‘sexist math dick’, in any language? You know how badly math studies are skewed, gender ratio-wise. Do you want to know how badly ‘political blogger’ gender ratios are skewed? Do you want to know how badly ‘active commenters on this blog’ gender ratios are skewed? 20%, 20%, 20%, fuck all, ain’t nobody, and fuck all, stated variously. Why piss me off like that when there’s no cause? Z 01.22.14 at 3:00 pm I promise I shut up after this one. Belle, please consider that the following question is absolutely completely sincere. Is that insulting your mathematical ability or your intelligence (damn, I just checked the OED so fearful I was that “diminish” did not mean the same thing in English and in French)? I honestly intended it as an encouragement. Or is it like completely and totally the Barbie quote doing the bad work here? Belle Waring 01.22.14 at 3:01 pm You are talking to a woman whom you do not know, who is confused about a genuinely confusing thing. You choose a gender-specific statement, that originally (if apocryphally in your view) came from a plastic, highly sexualized doll intended for 10-year-olds. Can you imagine any possible world in which that is not sexist or offensive? Belle Waring 01.22.14 at 3:03 pm You might have gotten away with the latter (though it still smacks of ‘nice try, kiddo! someday you might even learn trigonometry!’) had you not included the former. I’m going to bed now too so I don’t wish to argue about it. Consider carefully though, is this how you treat the female students whom you wish to encourage in math? Belle Waring 01.22.14 at 3:08 pm You don’t actually need to shut up after that one, I’m genuinely curious to hear your response and I’ll read it in the morning. Z 01.22.14 at 3:14 pm Belle 125. Ok! I understood. Here is, to be crystal clear, what I intended. 1) The video is a jumble of confused thing. 2) You, Belle, saw this and correctly remarked that if you were allowed the same power as the guy in the video, you could get different results. 3) You also expressed confusion about it all. 4) I meant to congratulate you on your step 2) and to reassure you that it is normal to have troubles with step 3). Contrary to many fallacies (like the divide by zero one), this one is not easily corrected and the right way to argue with these objects is not obvious at all. I don’t think any human being could devise it from scratch. 5) But what I managed was an insult on your step 2 (like Poor Belle, she cannot even see why her reasoning is wrong at step 2) which makes my comment later on the thread that your step 2) was correct all the more unfair. My apologies. Please believe me that this what happened. The Barbie quote is quasi-legendary among people involved in math education, as is xkcd 385, and I honestly did not think that you could have thought I was quoting it in some literal way. But I guess we commenters are not as well-known that maybe we wished we were (I had perhaps the illusion that after 9 years or so of commenting on CT, I had somehow managed to show that I wasn’t a complete asshole). Belle Waring 01.22.14 at 3:29 pm Then I should have been paying closer attention to your non-assholishness, probably. It’s maybe the single letter problem! It’s much harder to remember a single letter as a pseudonym I find. And the reason I asked, “do you want me to get mad at you or what?” was because you otherwise seemed totally reasonable. In your subsequent comment also. OK, fine, you didn’t pull a jar of wasps onto your head with all the wasps inside. Nonetheless I’d have to say that unless your interlocutor is a personal friend who is also a set theorist, I’d just never, ever, ever…(now we may imagine)… bust out the Barbie thing. Ever. Belle Waring 01.22.14 at 3:34 pm Except now! Now, ironically, you can say the Barbie thing to me all day long and I won’t care. No, we do remember our long-time commenters, but you should consider that there are a ton of trolls who burn out like brief candles, often being remarkably awful on the way. I mean, I was joking in the above post but IRL we have about 450,000 comments on this blog… Lee A. Arnold 01.22.14 at 4:33 pm I think that math should not be that hard. I conclude that it is usually poorly taught! Math has only a few basic types of actions: collecting, counting, comparing, computing (add-subtract-multiply-divide), ordering, rearranging, closeness and estimating, etc. Teachers ought to find a way to present them all at the beginning of study, in a single simple symbolic format. I do not think that “category theory” does the trick, but it is close. (That list is a rearrangement of the very great Saunders MacLane, Mathematics: Form and Function, p.35. Almost a unique book; there are a couple of books in the same vein by Martin H. Krieger.) One pedagogic problem is in the types of students. The smart students who immediately understand math are already locked into the context of this list of basic actions with simple objects (like numbers). They are “mathematically inclined”. The smart students who don’t get math easily are usually more facile in changing to other contexts of life, in which these basic actions do not easily apply, or can be found ridiculous. They may be better at emotions, of at formulating higher logical types of concepts, or at assimilating humor or paradox. One thing teachers could have done for me personally would have been to start by explaining the areas where math can never apply, or is highly unlikely to. I was interested in the health of the ecosystem, and in the sources of new ideas and innovations. If teachers had explained to me at the beginning that math will never precisely predict complex systems, and that math will never precisely predict the emergence of new properties in biological and social systems, then I wouldn’t have found the constant limitations of mathematics in these areas to make the subject of math so repulsive in my student years. GiT 01.22.14 at 5:05 pm For accuracy’s sake, I believe it is Malibu Stacey who says, “Math is hard. Let’s go bake cookies for the boys.” The (recalled) barbie doll satirized says, “Math class is tough.” alkali 01.22.14 at 5:40 pm At the risk of beating a dead horse, I would restate the necessary insight here as: Mathematical concepts may have alternative definitions. For example, suppose you put two dots on a piece of graph paper such that one dot is three over and four up from the other dot. What is the distance between the two dots? You could compute that distance as “taxi cab” distance (3 blocks over plus 4 blocks up = 7 blocks, as a taxi would drive between two locations in Manhattan) or “as the crow flies” distance (5 blocks, which you can measure with a ruler or calculate with the Pythagorean theorem). Both concepts of distance are sensible and valid if used consistently, and indeed there are many other meaningful definitions of distance. We only get into trouble if we mix and match definitions in the same discussion. (“Alice says the two dots are 7 blocks apart, but Bob says they are 5 blocks apart. IT’S A LOGICAL PARADOX! Mathematics is disproved!” No, it isn’t.) Likewise, the concept of “sum of an infinite series” can have multiple definitions. One of those definitions (Ramanujan summation) gives 1 + 2 + 3 + … = -1/12. This result is unexpected in the context of this video because this definition is being slipped in under the radar. If the video had said, “Here is a highly specialized and counterintuitive definition of how the sum of an infinite series should be calculated, and that counterintuitive definition produces this counterintuitive result,” no one would have been surprised, and we wouldn’t be having this discussion. mattski 01.22.14 at 5:44 pm bourbaki @ (what is now) 88 Yes! I find this helpful. And indeed @ (what is now) 37 I offered a similar idea. Thinking of numbers as actions or processes is in a sense de-mystifying. Better not to think of numbers as ‘things.’ mattski 01.22.14 at 5:45 pm Katherine @ 117 Well, I am temporarily stymied. But give me some time. mattski 01.22.14 at 5:48 pm Can’t resist: Alex K @ 102 That was wicked! elm 01.22.14 at 6:14 pm alkali @ 135: That touches on one of the other annoying parts of the video (which gets to a bit of mathematical shorthand as well). (The notation of this post will be a bit lumpy, hopefully it’s not too confusing) Outside of specialized contexts, when a someone with a moderate math education sees notation like: a[1] + a[2] + a[3] + a[4] + a[5] + … = b It’s typical to assume that this is shorthand for: Limit(n->infinity) { Sum{m in 1..n} (a[m]) } = b Evaluating this (commonly-assumed) convention for the given series (1+2+3+4+5+…) shows that the limit is +infinity — exactly as a non-mathematically-inclined person would expect. So one sleight of hand in their result is substituting the Riemann zeta function (evaluated at -1) for the series (1+2+3+4+5…) and/or substituting Ramanujan summation for the more-typical limit process. Another very large issue, of course, is that the process they use to present that result is bogus and much more likely to confuse than to enlighten. It’s somewhat like the following well-known “proof”: 16/64 = 1/4 Start with: 16/64 Cancel the 6s: 16 / 64 Result: 1/4 Ragweed 01.22.14 at 8:13 pm Belle @ 122-3 -I understand why you made that announcement sans comments. But let me say that I had a momentary flash of a standing ovation at the end of it. Though, really, disemvowling is much more fun. Bloix 01.22.14 at 8:13 pm The video is not just an ordinary bit of sleight of hand. It exploits the seeming paradox that arises when you treat infinity as if it were just a very big number that acts like any other number. It isn’t. Infinity behaves in ways that appear to be paradoxical, as here, but in fact work out very neatly once you accept that infinity is not just another number. The best known version of the paradox is “The Paradox of the Grand Hotel,” or “Hilbert’s Paradox,” after the mathematician David Hilbert, who devised it to help explain the theory of transfinite numbers, discovered by Georg Cantor. But before we get to the Grand Hotel, let’s take a look at the series. The series 1 – 1 + 1 – 1 + 1 … would have a definite value if it were not infinite – that is, if it stopped at any point. If it stopped 100 billion trillion gazillion, the value would be zero. And if it stopped at 100 billion trillion gazillion and one, the value would be one. But because the series is infinite and does not approach a limit, it’s value is neither zero nor one, nor anything in between. We can make the value appear to be as big or small as we like, positive or negative, all the way up to infinity. The video tells us that the value of the series is 1/2. But by the same logic we can make the value 1. Here’s how: Write the series as 1 + -1 + 1 + -1 + 1 … Now, put in some parens: 1 + (-1 + 1) + (-1 + 1) … Reverse the order of the numbers in the parens: 1 + (1+ – 1) + [(1+ -1) … Drop the parens: 1+1 + -1 + 1 + -1 … Add the first two 1’s together: 2 + -1 + 1 + -1 … Put the minus signs back: 2 -1+1 -1 … Which by the video’s logic is 1. And you can do this again and again, and move another 1, and then another 1, and then another 1, up to the front an infinity number of times, and there will always be an infinity number of 1’s left to pair up with the infinity number of -1’s that you didn’t move up. So the series can be made to appear to have any value, all the way up to infinity. That can’t be right, can it? And you could do the reverse, and move a -1 to the front, and then another and another, and there will still be an infinity number of -1’s to pair up with the infinity number of 1’s that you didn’t move. So you can make the value appear to be a large a negative number as you’d like, up to – infinity. That can’t be right either, can it? That’s what infinity means. It’s not the same as “a really big number,” it’s an inexhaustible supply of numbers. To our minds, it’s genuinely inconceivable, hence it can be used to trick us, as the video does. Yet it turns out that there’s whole branch of mathematics devoted to it. When I was a wee boy, I read George Gamow’s wonderful little book called “1,2,3 … Infinity” (published 1947), which has a chapter on the mathematician Georg Cantor’s insights into the nature of infinity and his discovery of transfinite numbers – that is, that there is an orderly series of infinities, from smallest to largest. There are an infinity of them, and that they can be the subject of operations like any other numbers – except that they behave rather oddly based on our intuitions. In explaining the very first, smallest infinity – the one that’s being exploited in the video – Gamow makes use of Hilbert’s Paradox of The Grand Hotel. He asks you to imagine an ordinary hotel, with 100 rooms and 100 guests. A traveler looking for a place to stay is turned away – no vacancy. But now imagine The Grand Hotel, with infinity rooms and infinity guests. A traveler arrives. Every single room is occupied. No problem, says the manager, you can stay in Room No. 1. We will ask the guest in room No. 1 to move to room No. 2, and we move the guest from No. 2 to No. 3, and from No. 3 to No. 4, and …. And if two travelers arrive separately, we move the guests in Nos. 1 and 2 to Nos. 3 and 4, and the guests from 2 and 4 to 5 and 6 …. So how many vacant rooms are there at the Grand Hotel? Well, there are none. But how many rooms does the hotel have for additional guests ? Well, an infinity of rooms. And this works because infinity plus one is infinity. So is infinity plus 2. That’s pretty obvious. But what is infinity plus infinity? That’s also infinity. And since infinity plus infinity is infinity, what is infinity minus infinity? It must be infinity. And that’s the value of the series in the video – an infinity of ones, minus an infinity of ones, which equals infinity. Not zero, not one – infinity. If this is at all interesting, I do recommend Gamow’s 65-year old book – the relevant part is the first chapter, and it is available as a pdf is here. There’s a little bit of literally accurate but objectionable discussion of the mathematical ability of a Hottentot, but otherwise it’s as fun and informative as I remember it. robotslave 01.22.14 at 8:34 pm Lucie Rie Mann @108 If more budding string theorists were a bit more attentive in their pure mathematics courses, we might have more pure mathematicians, and fewer former string theorists. Any attentive student of complex analysis will tell you that Riemann’s analytic continuation of the zeta function extends the domain to the entire complex plane except s=1, where it is undefined. There are of course several methods of calculation, which happen to be convenient for various mathematical models in physics, that will produce a “value” of -1/12 for the zeta continuation as it approaches s=1, but this does not change the fact that the continuation is undefined at that point. Walt 01.22.14 at 9:49 pm I think Belle’s initial reaction demonstrates why this video is the worst thing ever to happen. An intelligent non-mathematician like Belle can see that the argument in the video lends itself to all kinds of answers. But of course any non-megalomaniac will think not that the argument in the video is gibberish but they must be confused about something. The effect of the video is actually remove mathematical knowledge from the world. Math is hard, but it’s not that hard. And it’s obviously ridiculous to say that 1 + 2 + 3 + 4 + = -1/12 is “the” correct answer, the hallowed names of Euler, Riemann, and Ramanujan notwithstanding. The obvious non-mathematician objections to this equation are all correct. The partial sums are all positive numbers and steadily increase, so the limit sure seems like it should be positive. It’s not Cesaro summable: even if you buy the idea that if the series is oscillating you should take the average, here the average is +infinity. Intuitively, -1/12 sounds like the wrong answer, and that intuition is perfectly sound. Here’s another ridiculous consequence (inspired by robotslave’s comment above). According to the zeta function argument, the sum 1 + 1/2 + 1/3 + … diverges: the zeta function doesn’t have an analytic continuation to s=1. But using the analytic continuation argument, the obviously larger sum 1 + 2 + 3 + 4 + … magically converges. But it’s not magic. It’s misapplying totally comprehensible mathematics to produce mystical stoner mathematics. In the hands of once-in-a-generation geniuses like Euler and Ramanujan mystical stoner mathematics can produce surprising patterns that future generations will want to explain, but for the rest of us all we’ll get out of mystical stoner mathematics is the munchies. Walt 01.22.14 at 9:56 pm robotslave: 1 + 2 + 3 + … is zeta(-1), not zeta(1). js. 01.22.14 at 10:08 pm This thread is kinda awesome (if often over my head), but wanted to note that dn’s link waaaaay up at #4 is super helpful for anyone still scratching their heads. Trader Joe 01.22.14 at 10:10 pm Bloix @141′ Thank’s for the reminder about Gamow (I’d never have remembered the name)…I’m 100% sure that I have that book and have kept it all these years as something I know intuitively is too good to part with even though it surely must be >30 but < infinity years since I've last consulted it. What little I remember about topics such as these (which I find interesting in that brain teaser sort of way) were learned from that book…. Larry Gonick 01.22.14 at 10:53 pm The one to listen to here is Steinsaltz. The Scientific American explains it fully at http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/, and Terry Tao has a more general treatment using something called “smoothed sums” at http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/. Bloix 01.22.14 at 11:37 pm #143 – “The effect of the video is actually remove mathematical knowledge from the world.” Yeah, I completely agree with this. Even worse – the whole attitude seems to be intended to make people conclude that math is for assholes. So why should we listen to experts about global warming? Evolution? GMO? Better to stick to the Bible. #146 – My father, who was a generation or so younger than Gamow, met him several times and had tremendous respect for him. He gave me “1,23 …” (among other books like “Electronics for Boys and Girls” and “Flatland, by A. Square”) in hopes that I would be a scientist, too. Alas, I am a mere lawyer, but at least I don’t have the fear of science and math that many of my compatriots do. Ed Herdman 01.23.14 at 1:43 am I know George Gamow only from a nice edition of Mr. Tompkins stories…and an anecdote about a science demonstration gone awry. It’s unfortunate that you find page after page of results in Google when you look for ‘Carl Sagan overrated’ (and that nasty theme had started even back at the height of Sagan’s fame in the early ’80s), and you also quickly find out that people think Stephen Hawking and Neil DeGrasse Tyson are overrated too. I bet Bill Nye is overrated, also! Somebody thinks Albert Einstein is overrated. When you get down to it, though, many people have learned over the years from the popularizers of science, and I hope there will be many more men and women to follow in their footsteps. Some people have criticized Neil DeGrasse Tyson should be spending his time fighting theism…I don’t see that. He’s been wise not just to avoid that, but I think he’s also been wise not to go far from his core competency just in the way the Bad Astronomer has here: We want more people talking up science, but we don’t want them crowding out the best people and the best information with quick and easy contrarianisms. That annoys people here on CT and it’s just as useless in the public sphere at large. TM 01.23.14 at 5:41 pm I found one comment at http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/ worth repeating: “To frequently these days smart people are taken in by the “Malcolm Gladwell” effect. A desire to explain something outside of their field of interest with a simple counter-intuitive solution.” I also tend to agree with 143, except I must note that worse things than stupid youtube videos have happened*. What I agree with is that the video promotes the widespread view that Math is mostly arbitrary and useless. Now, if you are a math teacher, how are you going to convince your student that that result they came up with really makes no sense and here’s why? They might just say, duh, if 1+2+3+…=-1/12, then why shouldn’t my percentage be more than 100, or my probability negative, or this sum of squares come out negative? Some of us try hard to teach students how to distinguish a meaningful question from a pointless one, and how to recognize when a result obviously makes no sense. The sum of all natural numbers is a meaningless question for normal purposes, and a negative number is a nonsense result for a sum of positives, unless you operate in a specific theoretical context that needs to be defined and explained before you can use it meaningfully. What is displayed in this video takes common sense out of math. JimV 01.23.14 at 6:50 pm It was a fun video, with links in the comments to deep and rigorous explanations. As the presenter said in the video, he tried to think of a semi-intuitive way to present a mathematical result which has actual, real-world applications, and what he did in five minutes was the best he could think of. It raises, or should raise, questions in people’s minds for deeper study and for some humility in appreciation that the universe is a very strange place; and maybe human beings aren’t as smart as we would like to think. I for one enjoyed it. mattski 01.24.14 at 2:06 am I enjoyed it too, and not being expert in math, wasn’t offended by it’s sloppiness. Well, except for the ‘shift’, which did seem difficult to justify. That looked analogous to phase change, which–duh!–can either amplify or cancel out. But I come back to the idea that numbers aren’t about the physical universe, despite the fact that they’re useful for describing it. Numbers, I’m thinking, are a human activity, more like baseball than gravity. Belle Waring 01.24.14 at 2:22 am No, man. Numbers are real and would exist even if humans never existed. Facts about squares are just true facts about squares. I’m a Platonist about math. Possibly a stoner Platonist about math, but nonetheless. IME lots of mathematicians are, though of course many (most) are not. My friend who did algebraic topology was like, ‘naw, we pretty clearly just made this shit up.’ Set theorist friend by contrast? Secret Platonist but embarrassed to tell other mathematicians. otpup 01.24.14 at 3:36 am Belle, I think math people (especially that branch of applied mathematics called physics) are more platonists than you might think. There is that commonplace due to Wigner of the “unreasonable” success of heavily mathematical theories predicting things in the real world that were too wild to imagine. Now maybe that view point is wrong in some way, but it does effect the perspective of many, many people in the math and physical sciences. And many math people may eschew the platonist stance despite what they might really believe or suspect because they tend to be more humble than physics people and/or don’t want to take on the moral baggage that physics people inherited with nuclear weapons. “Math is just our little sandbox, not part of the real world and no effect on it.” Yeah, right. john c. halasz 01.24.14 at 4:08 am The concepts (or is it objects?) of mathematics are pure formal operations. They are neither empirical objects, nor substances: when one counts such, one doesn’t find the numbers inside them. (One can add apples and oranges; just reclassify them as fruit). That’s why Bertrand Russell, an empiricist, who thought that “knowledge by acquaintance” is the primary form of knowledge, remarked that when we do math, we don’t know what we’re talking about. The only requirement is that systems or sets of mathematical inferences be self-consistent. So the old question is: are mathematical “truths” invented or discovered? My sense of this is that when a new domain of mathematics is opened up, (because mathematicians have somehow intuited that it is not just operationally doable, but makes mathematical “sense”), then the basic rules that establish that domain also function as meta-rules, i.e. rules for the generation of further rules. Which gives to the development of further implications and operations in that domain the feel of something being discovered “out there”. But what exactly does mathematical “sense” consist in? Obviously, it depends on the evolved/inherited state of mathematical problematics. My sense of the matter, (and, mind, I only got to the level of basic calculus and statistics and that was IIRC a billion years ago), is that beyond the formal axiomatics of “proof”, (which famously have now been shown formally to have its limits), it tends to boil down, as others have indicated above, to the projection of point systems in n dimensions. And curiously, the development of new mathematics has mostly, though not always, preceded any (thought of) empirical application. An interesting, if speculative, question is to what extent the intuitions of mathematical “sense” are influenced by general cultural forms of sense-making, the “metaphysics” of different cultures. Greco-Roman math had no concept of zero and indeed the culture also conceived of being as substance, as what is unchanged in all change, thus “eterrnal”. It was the Indians, whose meditative practices strove to remove attachment from the sensory world and achieve “oneness”, who invented the concept of zero and the Arabs who believed in a creation ex nihilo who transmitted and further develop its implications. And then there is the story about a medieval Chinese mathematician who wrote the most advanced algebraic treatise of that time, demonstrating a wide variety of methods all of which resulted in the same answer or solution, (rather than, as a presumably Western mindset, reducing a wide variety of solutions to a single method). Another curious question is how are brains, which evolved largely as analog pattern-matching devices, and are not at all like digital computers, nevertheless can develop the ability to do advanced mathematics. Some of the mysteriousness and confusion attaching to the business and its status might derive from its “unnaturalness”. But I don’t think a foundationalist account could be considered at all credible anymore. Belle Waring 01.24.14 at 9:39 am The Mayans independently discovered zero. I actually considered writing about Sanskrit mathematics in grad school, not because I’m a math hot shot or anything, just because lots of things are both untranslated and not paid attention to by the non-mathematician Sanskrit readers in India itself. There are millions there, obviously, many with what’s pretty much native-level fluency, but they read it for religious reasons, not for historical mathematical research reasons. I would have had to learn Pali too, eh. My thinking was that I would translate it, ask mathematician friends what it was about if I really couldn’t determine it, and then see whether there were unknown results. Not unknown in the world, but more like, previously discovered by mathematicians writing in Sanskrit and then forgotten as the language user base switched so heavily to religious and ritual purposes. I discarded the idea as too practical. P.M.Lawrence 01.24.14 at 10:02 am Belle Waring, for what it’s worth, I just came to this page and read through the comments, and when I saw that Barbie reference I had no idea that the author was addressing a woman, or that it could easily be taken as sniping rather than something that had been pastiched in the Simpsons. I only realised otherwise when I saw your reaction. It’s quite possible that he (should I assume he?) was simply doing it recklessly and negligently rather than wilfully, either not knowing who you were or (if he had bumped into you in other contexts), just not making the connection, not putting two and two together as it were. You’d have to ask him (?) to find out if he (?) was even giving it a moment’s thought, at the time. Me, when I see “Barbie” (in any spelling), I tend to think “Klaus Barbie” and not “symbol of sexism in popular culture”, simply because I’m not a product of U.S. culture (apart from some influence at my very first school, which had a U.S. teacher – in Iraq). Belle Waring 01.24.14 at 10:52 am P.M. Lawrence, stop trolling me. Had you actually read the thread, rather than just pretended to have done so so that you could come down here to the bottom and insult me you would know that the commenter is a man (121); that he knows I am a woman (passim); that he works to educate young scientists math and strives to be non-sexist and thus was pained by the idea that he would be seen otherwise (121 ); that he has been commenting here for nine years (130); that, in his words, “[t]he Barbie quote is quasi-legendary among people involved in math education, as is xkcd 385, and I honestly did not think that you could have thought I was quoting it in some literal way” (130). So that far from failing to know the quote was offensive to a woman, Z rather thought that it was so obviously offensive that no one who thought him a person of any good will could ever think he was serious, and that he had hoped after nine years we would know he was a person of good will (130). So no, bitch, I don’t “have to ask him (?) to find out if he (?) was even giving it a moment’s thought, at the time.” P.M.Lawrence 01.24.14 at 11:39 am Belle Waring, you’ve just jumped in and accused me of offending you. Clearly I must really have offended you, but you appear to have proceeded on the basis of jumping to conclusions:- – You accused me of not actually reading any of this. Actually, I read all the comments in sequence, in full, but without memorising the cast of characters, rather paying attention to the matters raised. When I saw the Barbie reference, I literally did not recall that the person being addressed was female. For some reason, I hadn’t been paying much attention to whether readers were male or female until then; I misguidedly thought that the subject matter was more important. Since I hadn’t been paying any attention to that until then, and only then looked back to get the details straight, it occurred to me that the original writer might also not have been paying attention to that specific issue; attention, that is, to whether it would push your buttons, rather than being the very pastiche reference that I took it for, that was used in the Simpsons (Lisa versus Malibu Stacy, or something like that). I drew that possibility to readers’ attention, further suggesting that the only way to be sure was to ask. – You accused me of being deliberately insulting, on the basis that you don’t need any fact checking to just know the truth of the matter, and that I knew it too. Well, I may have succeeded in offending you, but I can assure readers that I sincerely believed that my intention was only ever to raise another possibility, based on my no doubt faulty impression of how it had struck me. But having been told in no uncertain terms that it is offensive enough to warrant insulting me for just suggesting asking if it could possibly have been the inadvertent result of oversight, I see that that also is proof enough that I cannot have meant to advance enquiry but can only have been deliberately provocative; it seems I do not know my own intentions as well as others do. And that is my Apologia pro sua. If it serves to condemn me yet further, for venturing to reply to righteous indignation, then I will know that the only acceptable reply to rage is to validate it by confirming its righteousness, and I will let others learn from my fate at your tongue and pen. mattski 01.24.14 at 1:31 pm But there aren’t any squares! Outside of our minds that is. Jim Buck 01.24.14 at 5:30 pm Once someone drew a square it was there, surely; and with its own telos? TM 01.24.14 at 5:59 pm Philosophy of Math on CT! That’s nice. I would so like to hear a really satisfactory account of what Math is. Most accounts are negative, dwelling on what math isn’t. “They are neither empirical objects, nor substances: when one counts such, one doesn’t find the numbers inside them.” (jch) While that sounds convincing, I would object that the whole concept of “empirical objects” is pretty screwed up. We think we know what empirical objects are but as soon as we look closer, the physics run into trouble. In that sense, maybe numbers aren’t even so different from “empirical objects”. I dimly remember somebody (I think I read it in Russell but he was quoting somebody else) asking what we mean when we refer to the “North Sea”. We think that’s an empirical physical entity right? But really it’s a concept or a classification invented by humans. There is a sense in which the North Sea, or the matter of which it is composed, exists objectively independently of humans perceiving it, and there’s a sense in which really there is no North Sea unless somebody draws a map and names it. The other example I find highly instructive is that of color (that I think is from Russell). When you have a red flower with five petals, how is the property of redness fundamentally different from the property of fiveness? It is difficult to justify color but not number as an objective physical essence. My interest now is mostly in education and there I prefer to stress that Math is indeed about the real world. At least that kind of Math that non-mathematicians should be familiar with. elm 01.24.14 at 6:20 pm P.M.Lawrence: The exchange you’re commenting on was an actual conversation occurring between actual people. It also featured moderately-heavy, poor-quality trolling, which has since been removed. At the time of the conversation, the participants were paying attention to each others’ identities. It’s not surprising that you, as a non-participant who is reading the history of that conversation, take it differently. You can afford an ignorance of who-said-what. Additionally, it’s an issue that has since been resolved and your posts look like ordinary trolling. mattski 01.24.14 at 6:58 pm @ 162 I love what you wrote. I love the way you put your finger on the problem of ‘scale’ or ‘resolution’ for lack of better terms. I think it is absolutely valid to say that “objects” are a function of our ‘degree of resolution.’ How far are we zooming in or zooming out in space… and in time? Because as we do so objects come and go. So “objects” in the physical world (and yes, even “physical world” gets dubious the closer we look) are provisional. The world is in flux and formations come and go. ‘Identity’ is an appearance, but often a valid and useful way (a necessary way!) to think about the contents of reality. But if we want to be really rigorous in our observations wouldn’t we have to conclude that every object we encounter is drained of its identity by impermanence? Mathematics though… beguilingly, has the appearance of being untouched by transience. Because it is nothing more than a collection of “rules.” And a rule can be thought of as an action. Indeed, math is a human game like language. We came up with it for the express purpose of describing the world. We ‘drape’ it over the world. But its relationship to reality is necessarily approximate. And a number, rather than resembling a “thing,” is more of an “instruction” to take a certain action. Nine 01.24.14 at 7:38 pm Lawrence@159 – “and I will let others learn from my fate at your tongue and pen.” My goodness, that sounds dire ! I can picture it verse – Some, from fallen Lawrence learning, lashed themselves to the mast while others succumbed insane to Belles’s soft siren song … TM 01.24.14 at 7:44 pm [158, 163, 165: Why oh why can’t people stop pursuing sidetracks?] Nine 01.24.14 at 7:47 pm Sorry, I just couldn’t resist – Lawrence’s response was too funny. TM 01.24.14 at 8:15 pm Sigh [Please delete] Belle Waring 01.25.14 at 8:21 am P.M. Lawrence: women who are insulted in any way on the internet are nearly always called upon to prove, not only that the insult occurred (and one gets plenty of pushback there) but also that the person intended to insult her and in precisely that way. Since this is, absent confession, almost impossible to obtain, women’s complaints can nearly always be dismissed with what has a surface appeal to neutral standards. “Maybe you misunderstood him?” “Who knows what he was thinking?” and so on. The important thing to remember is that, for the purpose of driving women out of public fora, it doesn’t matter what the various people intend by their comments. If the net effect is that after 50 dismissive comments along the Barbie lines a woman gives up, and stops asking questions about math, while an equally ignorant man will go on asking those same questions, it is a pernicious, awful thing–even if those 50 negative questions are the incredible result of a chain of improbable, totally innocent, unintended consequences and forgettings. It is also self-reinforcing: the fewer women there are who are willing to participate in comments on blogs, the fewer women will want to join up and start doing so as a n00b. Here at CT it is difficult to forget I am a woman. I am not a random commenter of whom one might easily lose track but in fact an active front page poster. I am additionally mentioned by name IN THE POST. Your claims to be an open-minded, helpful skeptic whose assistance is being spurned by your poisonous hostess fail in various ways. 1) you say I “have” to do something. To review, a woman has been told not to feel stupid about not rediscovering set theory from scratch (sure, who would). But she has also been, as I noted, addressed with terms used by a highly-sexualized doll intended for 10-year-olds, which had to be recalled by the company after public outcry. This woman doesn’t have to do anything before taking offense. She can just plain take offense. You cannot require me to do anything further, such as get into a conversation with the other person and learn exactly what he was thinking at the time, before deeming it a situation in which it was acceptable to feel pique. I was named in the comment, with my name, Belle, which close to zero human males have as a given name, BTW. Hard to imagine how the whole “who was being addressed?” thing kept going so, so wrong for you. 2) Had you actually read the thread, as I noted above, you would have seen that Z was genuinely upset to think that–even for a moment–I might imagine he was saying that to me in all seriousness. He took it to be so offensive a joke that it was obviously, of necessity satirical, given that I probably knew who he was due to his long tenure as a commenter. He also apologized, and I said, ‘OK fine, I understand and am not angry with you, but I recommend that you only ever say this to women who are both personal close friends and set theorists.’ This was an exaggeration. They need not be set theorists. It’s not so cutting edge anymore, is it? Now, after the discussion, his explanation, his apology, and my acceptance of his apology, you come along and decide to call me on the carpet because I am making assumptions about his thought processes, and tell me I ‘have’ to check what he intended. a) I don’t. b) Per impossible, if I did, I would just re-read Z’s comment 130, in which Z explains in considerable detail exactly what was going through his mind when he wrote his comments. Is there some reason that this step-by-step list of intentions fails to satisfy you, if I may ask? It’s a fair accusation that I am easily provoked to anger and when provoked I say things that are more vehement and unpleasant than they need to be. This is a personal failing on my part and I should try to be a better person. However, it is also true that I am deliberately insulted all the time, on purpose, by men who are sexist. I know this sounds like really meager motivation, but, for real. So much of the happens. Now, of people who actively comment at our site, I have made a number of random polls at various times (I do know people’s gender in most cases) and I’ve never come up with anything more than 20% of comments by women in a given thread. Usually it’s worse. If I sit around and let random commenters employ the classic “just askin’ questions. Why are feminists so afraid of rational inquiry?” strategy I will doom our threads to worse. Do you see how Katherine, above, says she sort of wanted to respond to the Barbie comment, but didn’t? Maybe there were 20 Katherines reading the thread who wanted to respond to the Barbie comment too, but then never commented at all. This is the tragedy of the commons. Bad commenters will drive out good ones. If someone says something racist, we don’t need to “look into their heart” and see if they are “a racist” or “truly hate brown people.” We just need to acknowledge that that thing they said was racist. So when people say something sexist, it also doesn’t matter what they intended. In this case, I asked Z “what the fuck?” and he said, “oh God, I can’t believe you thought I really meant that; I’m sorry,” and I said, “OK.” Why did you then feel the need to muddy the waters, exactly, when they had been distilled in a way so unusual and pleasant and clear, if I may ask precisely what you intended, P.M. Lawrence? Since a full discussion had gone on and everything had been settled and everyone’s motives and misunderstandings had been made crystal clear, what made you want to tell me that I ‘had’ to do something before getting offended? If you did read the thread, why did you pretend not to know whether Z was a man, when he himself said that he was? Do you think someone named “Belle” is likely to be a man, given that you know it is not a pseudonym of any kind? If you think this is plausible, could you please produce some evidence for the assertion? If not, why did you pretend to have been unable to discern whether the Barbie comment was addressed to a man or a woman? I have more, more detailed questions about your motivations, and whether I may have misunderstood you, and why you said precisely the things you did, but if you would just answer all these questions it would be very elucidating. I know you only commented from the start in a helpful spirit, so I don’t imagine you’ll mind answering? I really can’t imagine what you were thinking, so give me a hand, please. John Holbo 01.26.14 at 6:52 am Well, we still don’t know what math is, but I hope Lawrence learned something! Saurs 01.27.14 at 10:25 am Thank you, thank you, thank you for going after this kind of shit, swiftly, precisely, and without fail. It must be exhausting having to shoulder this kind of niggling burden each and every time you post or comment, but it really is much appreciated and makes this space safer for other women. Z 01.27.14 at 1:51 pm This thread is so painful to me: my favorite topic (the math sub-discipline called special values of L-function) discussed on my favorite blog; and all I managed was to provoke a clusterfuck of comments. I guess each man trolls the things he loves. My apologies again to John, Belle, Katherine and now P.M Lawrence for dragging him in this melee. Belle Waring 01.27.14 at 3:42 pm Saurs: I appreciate the support. It is a pain in the ass sometimes. All the times. Z: It’s cool man. There were plenty of good comments. Don’t apologize to P.M. Lawrence, though, he’s trolling like a mofo. P.M.Lawrence 01.28.14 at 1:23 am Readers, I have been forbearing to follow these things up, because my earlier attempt seemed to make matters worse and I preferred to give matters a chance to settle. But I see that I keep being accused of deliberate trolling, of lying about what I did and did not read and understand and when, and so on – and in other posts, with ripples that may keep spreading. So I just want readers to register that I have in fact noticed all this, and that I am being patient in the hope that, at some point, I can make my peace with anybody who might have been inadvertently offended, as well as clarifying matters. I regret if this reply is taken as fuel for flames; I realise the possibility, and I regret it if that puts off a settlement, but I also really want readers to notice that I have been holding off patiently rather than accepting the mistaken accusations of ill intent and so on. So now I will go back to waiting, unless someone wants to suggest some quicker yet safe and honourable way of making peace. Yama 01.28.14 at 12:02 pm Sorry Lawrence, I am just a long time lurker, but I am embarrassed at how folks are getting treated here lately. Good luck. Lucie Rie Mann 01.29.14 at 6:57 am Well roboslave, at least I remember something, albeit vague, from 17 years and countless moles of kiln fumes ago. Thanks for the reminder and now I’ll go back to tweaking my firing schedule so that my turquoise matte stops pinholing. I still don’t see why this discussion belongs here. Also it seems that someone insulted someone else which seems to happen in such pointless threads. Comments on this entry are closed.
e1f682859e1cb395	in Buddhadharma Wayfaring : the Tao, Emptiness & Process Theology by Wim van den Dungen Contents SiteMap of Philosophy SiteMap of Ancient Egyptian Sapience SiteMap "The Way in its absolute reality has no 'name'. It is (comparable to) uncarved wood. (...) Only when it is cut out are there 'names'." Lao-Tzŭ : Tao-te Ching, chapter 32. "The Way gathers in emptiness alone. Emptiness is the fasting of the mind.". Chuang-Tzŭ : Chuang-Tzŭ, section 4. "... all I do is put in motion the heavenly mechanism in me, I'm not aware of how the thing works." Chuang-Tzŭ : Chuang-Tzŭ, section 17. "No man is an island ..." Donne, J. : Meditation XVII. This is God in his function of the kingdom of heaven." Whitehead, A. N. : Process and Reality (PR), § 531. Taoism ("Tao-chia") or Wayfaring is part of the daily life of the Chinese people and a enigmatic, pervasive & ubiquitous aspect of their long culture. It nevertheless lacks a clear profile. To approach it, we need to study the techniques of Tending Life, the Way of the Immortals, but also Taoist liturgy, mythology, alchemy & mysticism. As an institution, Taoism never had a governing authority, canonical doctrines or dogmas. In China, there was no formal separation between religion and social activity. The Taoist masters were integrated in lay society and enjoyed no special status. Ordinary people would never call themselves "Taoists", or "Wayfarers", for this implied initiation into the Mysteries, reserved for masters & local sages. Broadly speaking, Taoism is a spiritual practice acting as the natural bond between all things, but one without doctrinal creed, profession of faith or dogma. This natural, spontaneous bond, based on nonresistance, is called -by lack of a better name- "Tao" (pronounced "dow"), the "Way". This concept is indefinable, at once transcendent & immanent, unnameable, ineffable and apprehended only in its multiple aspects, but present in all things ... The "Tao", the most fundamental concept of Wayfaring, is indicative of something underlying the change characterizing all things, the natural, spontaneous process regulating the cycles of the universe. Wayfaring is thus the pursuit of natural laws. Along this "way", in this process the universe finds its unity. The Tao makes whole, but is not itself the Whole. Table of Contents Shamanism : the Substratum of Taoism. 2 Very Short History of Taoism. 2.1 Classical Period. (a) Period of Spring and Autumn. (b) Period of the Warring States. 2.2 Taoist Religion. 2.3 Taoist Mysticism. 2.4 Taoist Alchemy. 2.5 Synthesis. 3 Against Substantialism : Brother Buddhism & Sister Taoism. 3.1 Buddhism in China. (a) Pure Land Buddhism. (b) Marks-of-Existence Buddhism. (c) Celestial Platform Buddhism. (d) Flower Garland Buddhism. (e) Ch'an Buddhism. 3.2 Emptiness and Dependent Arising. (a) Simultaneity in Wisdom-mind according to Tsongkhapa. (b) Six Instantiations explaining Emptiness. (c) The Cognitive Activity of a Buddha. (d) Dependent Arising. (e) The View in the Heart-sûtra. (f) The View in Hua-yen & T'ien-tai. 3.3 Absence of Essentialism in Classical Taoism. (a) The Nameless for Lao-tzŭ. (b) The Negation-of-Negation-of-Negation of Chuang-tzŭ. (c) Classical Taoism and Śûnyatâ. (d) The Non-Essentialist & Non-Conceptual Absolute Tao. 3.4 Brother Buddhism & Sister Taoism. 4 The Tao : the Way in Absolute & Relative Terms. (a) The Absolute Tao - Uncreated and Creating. (b) WU : the One - Created Potential Non-Being. (c) YU : The Two - Created Potential Being. 5 Taoist Metaphysics : Objective & Subjective Considerations. (a) The Cosmological Approach of Lao-tzŭ. (b) The Epistemological Approach of Chuang-tzŭ. 6 Ontological Tradition of the West. (a) Ancient Egyptian Heliopolitanism. (b) Hellenism. (c) Abrahamic Tradition. (d) The Renaissance and Modern Scientific Thought. 7 A New Theology. (a) Reasons to Resuscitate God ... (b) Desubstantializing Western Theology. 8 The God of Process Theology. (a) The Fundamental Categories of Process Philosophy. (b) The Primordial Nature of God. (c) The Consequent Nature of God. 9 Towards a Synthesis. (a) Rationality & Experience of Emptiness. (b) Dependent Arising. (c) The One. (d) Towards a Synthetic Ontological Scheme. "Humanity follows Earth, Earth follows Heaven, Heaven follows the Way, the Way follows Nature." Taoist proverb. What is Taoism ? Difficult to answer, this question points to a diverse variety. One common point can be isolated though : an emphasis on the natural cycle at work in all things ; the notion of a constant change having as background the nameless, undifferentiated and unifying primordial super force called "Tao", "the Way". And if we may believe Lao-tzŭ, the traditional & legendary fountainhead of Taoism, this super force is benevolent, tending towards the Greatest Possible Harmony. Through Chinese history, Taoism manifested in a multitude of phenomena, touching nearly all facets of this grand civilization : science, politics, religion, medicine, psychology, art, music, literature, drama, dance, design and warfare. Taoists used numerous formats, such as cosmology, history, mythology, fiction, humor, alchemy, magic, etc. The methods employed were also diverse : physical, psychosomatic & mental, including meditation, modes of movement, breathing, sexual yoga, imagination, dreaming, gazing & visualization ... The earliest known Taoist text is the I ching, composed in a time when divination was an integral part of government. The second most famous and popular text is the Tao-te ching, presented as advice to rulers, written in a time of the social and political decay of the ancient order. The third famous text is the Chuang-tzŭ, featuring an air of humorous abandon, anarchy, satire and foolish wisdom ... Around the same time, the Sun-tzŭ was compiled. While pacifist, it truly recognizes the realities of war and instead of articulating morals against it, it tries to induce preventive strategies to avoid conflict & warfare or palliative techniques, minimizing the trauma inflicted by actual war. Until recently trapped by Confucian bias, Western Orientalism has been reluctant to attend to Taoism. Sinologists and comparative religionists did not take into consideration what Taoism precisely covers. Although rooted in prehistoric Shamanism, with its "classical" authors (Lao-tzŭ and Chuang-tzŭ) at work centuries before the common era, organized Taoist religion emerged roughly around the second century CE and continued to influence the Chinese mentality until the first part of the twentieth century. Its presence is often totally ignored. Together with various forms of Buddhism & Confucianism, this Taoist religion fashioned the most important expressions of traditional Chinese scriptural truths, spiritual values and ritual practices. Eventually, these three were integrated in the Complete Reality School. Taoism may therefore be considered a very significant part of the native national religion of the vast majority of the Chinese people, and this at least for nearly two millennia. It is therefore strange to witness how, until the last few decades and this despite the rich abundance of textual and other materials, the study of Chinese civilization almost completely ignored or trivialized the historical, anthropological, sociological and religious complexities of the Taoist tradition. Thanks to the tradition of French academic sinology, pioneered, in the first part of the previous century, by Henri Maspero's studies of the Tao-tsang or Taoist Canon (issued under the Ming in 1445 and containing more than a disparate thousand works), the past thirty-five years have witnessed a revolution in the scholarly understanding of Chinese civilization. Unfortunately, these studies have not yet filtered down to the general scholarly and lay public, still identifying Taoism exclusively with enigmatic sages like Lao-tzŭ and Chuang-tzŭ, and totally oblivious of Taoist religion, with its ceremonies, theologies, meditations & alchemy. Although it is true these two sages were the forerunners of the Taoist "tradition" as it self-consciously emerged toward the end of the Han Dynasty (206 BCE - 219 CE), many other sublime authors, sects & schools participated in the gradual formation of the distinctive Chinese micro and macrocosmic ecological worldview. In the fashion of a massive accumulation of documents lacking any detailed inventory, the Taoist Canon, besides gathering together the "classical" works of Lao-tzŭ & Chuang-tzŭ, also contains pharmacological treatises, medical texts, hagiographies, ritual & magical texts, imaginary geographies, dietic & hygienic precepts, anthologies, hymns, speculations on the I ching, meditation techniques, alchemical texts, moral tracts, etc. Although the best and the worse can be found within this canon, this diversity constitutes its richness, showing the heterogeneous nature of Taoism often neglected or unknown in the West. Given this vastness, the present paper will have to make a limiting choice. Given the Buddhist perspective fostered here, attention will be focused on the role of emptiness in Classical Taoism as found in the works of Lao-tzŭ and Chuang-tzŭ. When this has been established, I will try to understand the macrocosmic worldview implied, in particular the ontological role played by the absolute Tao & the One. The microcosmic dimension of "preserving the One" (as found in the Su-ling ching) will not be addressed. This approach is the topic of a forthcoming comparative study on Taoist Meditation, Buddhist Meditation (Calm Abiding & Insight Meditation) & Buddhist Tantra. Finally, I try to interrelate the Eastern & Western appreciation of the Divine. By establishing points of comparison between the Taoist worldview and Process Theology, both based on a nonsubstantial concept of the Divine rooted in emptiness, this integrated approach intents to bring about a universal view on the One. 1 Shamanism : the Substratum of Taoism. "When the spirit is not focused externally, that is called spirituality ; to keep the spirit intact is called integrity." Wen-tzŭ, quoted in Clearly, Th. (transl), Practical Taoism, p. 24. Five thousand years ago, tribes settled along the banks of the Yellow River in the North of China. They did not possess national identity and lived along the river. Their activities were Neolithic : fishing, herding and the cultivation of crops. These tribes had leaders who had fought with wild animals and who were deemed to possess extraordinary powers. One of them was the legendary Yü, who could shapeshift into the form of a bear and who had no mother, but who had directly sprung from the body of his father ! In later works, the features attributed to Yü are in accord with what Mircea Eliade found to be the universal characteristics of Shamanism : heavenly flight, subterranean journeys, ecstatic states revealing the secrets of life, power over the elements of nature, healing abilities and knowledge about plants and their use. The shaman is able to enter trance-states at will and communicate with the spirits. The latter do not possess him, but are controlled by him. His altered states of consciousness do not befall on him, but he enters and exits them as he pleases. As a phenomenon, Shamanism can be found in all Neolithic communities and is rooted in Upper Paleolithic cave-spirituality. Yü was a "wu", a shaman. In his society, these shamans were very important members of the community. His father too had been a shaman possessing the power to shapeshift into the form of a bear. The tribal kings too were often shamans, able to ascend to heaven at will. In China, Shamanism entered a new stage when writing and reading emerged, i.e. with the advent of history. In the 12th century BCE, at the beginning of the Chou Dynasty (1122 - 225 BCE), kings and nobleman had shamans in their service as advisors, diviners and healers. Often, when the shaman was no longer able to serve his lord properly, he was put to death. It is during this period the tenets found in the I ching emerged. Hence historically, the fundamental intuitions of Wayfaring are over three millennia old, making it the oldest surviving spiritual practice on the planet. The activities of these shamans can be summarized as follows : summoning spirits : the shaman would cause the spirits to descend to the Earthly plane, offering his own body as temporary housing. Starting with a sacred dance, the spirit entered the body of the shaman who went into trance. The altered state of consciousness of the shaman was the precondition for the spirit to enter his body, making this trance-experience different from possession and the activity of magicians who's personal consciousness makes way for another state. In the case of the shaman, the spirits are subdued to his consciousness ; reading the signs : by observing the changing conditions of the natural world, the shaman was able to predict coming events ; interpreting dreams : by interpreting dreams, seen as vehicles for special signs, the shaman could understand the messages of the spirits. As "dream-masters" the shaman could be fully conscious (lucid) in his own dream states and visit the invisible worlds of the spirits and the deceased. He could enter the dreams of the living or influence these dreams ; causing rain : by doing certain ritual action (ceremonies), the shaman was able to control the weather, essential in rural communities. Causing rain to fall became the icon of spiritual activity per se, as we can see in the following sign : "Ling", translated as "rain-making" has three parts. The upper portion is "rain", the middle stands for three open mouths and the lower means "shaman" or "sorcerer". These are the three parts of his being : Heaven, Mind and Earth. In the classical texts, "ling" or "spiritual quality" points to a force moving & forming material structures in harmony with Heaven ; healing : as disease was deemed caused by invading spirits, the shaman (often women) controlled these spirits by using herbs and exorcism ; divining the future : by studying heaven, one could predict what would happen on Earth (cf. astrology). During and at the end of the Chou Dynasty, Shamanism lost its influence and retired in isolated areas on both sides of the Yang-tze and on the South-East coast of China. In these three feudal kingdoms (Ch'u, Wu and Yüeh) Shamanism prevailed. In the history of China, long after these feudal kingdoms had disappeared, this regional culture continued to exercise its influence on the philosophy, religion and spirituality of Chinese culture at large. 2 Very Short History of Taoism. The history of Taoism can be divided in four distinct phases : Classical, Mystical, Alchemical & Synthetical. 2.1 The Classical Period (770 - 220 BCE). In 770 BCE, the political unity of the Chou Dynasty collapsed. The next five hundred years were times of political chaos and civil war. This era is subdivided in the Period of Spring and Autumn (770 - 476 BCE), followed by the Period of the Warring States (475 - 221 BCE), during which seven feudal states constantly waged war. This chaos ended in 221 BCE, when one of the seven, the Ch'in, subdued their rivals and reunited China. During this strife and chaos, we find Confucius (551 - 479 BCE), Lao-tzŭ (6th century BCE), Mo-tzŭ (ca. 470 - 391 BCE), Sun-tzŭ (ca. 400 - 320 BCE), Mencius (372 - 289 BCE), Chuang-tzŭ (4th century BCE), Lieh-tzŭ (ca. 400 BCE) and Han-fei-tzŭ (ca. 280 - 233 BCE). These sages articulated the fundamental tenets of Chinese civilization, and it is important to note Classical Taoism emerged in the darkest hour of Chinese history, when unity had been lost for nearly half a millennium ! Lao-tzŭ and "his" Tao-te ching was probably written during the Period of Spring and Autumn, while Chuang-tzŭ composed the inner chapters of his Chuang-tzŭ during the Period of the Warring States. Confucius and Mo-tzŭ were concerned with moral philosophy, not with individual freedom and destiny (transformation). The term "Taoism" did not yet exist in Chuang-tzŭ's time, and these Mysteries were known as "Huang-lao chih Tao" or the "Way of the Yellow Emperor and the Old Master", referring to Lao-tzŭ. (a) the Period of Spring and Autumn : Semi-autonomous states emerged as a result of the increasing power of the warlords who helped erect the Chou Dynasty. Not unlike the Pharaohs at the end of the Old Kingdoim, the Chou emperors had given these men too much power and this resulted in open conflict. Five great families emerged : the Ch'in, the Ch'in, the Sung, the Chin and the Ch'u. They increased their military powers and tried to subdue each other. At the beginning, ca. 140 feudal warlords were active. At the end, only 44 remained. Endless wars and civil unrest were the order of the day ... These warlords realized a strong state was not only the outcome of military power. Diplomacy and statesmanship were needed too. This favored the rise of a new class of political and military advisors, emerging for the first time at the end of the Chou Dynasty (225 BCE). These men were itinerant and offered their services to various warlords. Some of these advisors at work during the Period of Spring and Autumn truly wanted to erect a better society and incited the rulers to practice virtue and benevolence. To this class of men belonged Confucius and Lao-tzŭ. The latter probably lived in the sixth and fifth centuries BCE, but, according to some scholars, his book -the Tao-te ching- did not obtain its present form until the third century BCE, while others disagree, considering it a work composed by one individual. Whether literary criticism reliably identifies it as composed by more than one author (representing a tradition of successive Taoists) or not, remains an open question ... "... the Tao-te ching as a whole is a unique piece of work distinctly colored by the personality of one unusual man, a shaman-philosopher." Izutsu, T. : Op.cit., p.292. Although Lao-tzŭ is considered as the father of Taoism, his historical reality indeed remains "in the clouds". Some scholars even claim the Old Master lived after Chuang-tzŭ ! His name was "Li Erh" and he came from the Southern feudal state of Ch'u. As a librarian working in the state archives, the Old Master belonged to the literate upper class. He resigned and vanished. Mythical tales have it he reached enlightenment, traveled to the Western border and disappeared as immortals do. But before doing so, he left a treatise of five thousand words to Wen-shih, the guard of the Western border and his first pupil. This text, the Tao-te ching, is probably the first Taoist text. The text reveals an original stance towards the Tao. By living in accordance with the Tao, individuals change and because the latter do, society changes. For Confucius, the nature of things was not important. A harmonious society was the outcome of correct ritual and following the ethical code. For the Taoists, knowing the natural order of things was the precondition. For only if individuals change in accordance with the natural order can society change for the better. For Lao-tzŭ, "wu wei", or "not doing" was not a state of being unconcerned (as was the case in the Chuang-tzŭ). For the "Old Master", the Tao was benevolent. Let me summarize the teachings of the Tao-te ching : 1. The Tao is the source of all things. It is nameless, invisible and not to be comprehended by conventional sources of knowledge (senses & reason). The Tao is limitless and inexhaustible. All things exist thanks to the Tao, for it is the background of all changes. The Tao, source of the "Ten Thousand Things", i.e. every actual thing, is not a Deity nor a spirit. Nor is it merely the whole (pantheism). It is present in all things (the whole), but is more than this totality (panentheism). On this cosmological point, Lao-tzŭ departs from Shamanism. The latter focused on a multitude of deities and spirits, while he intended unity by way of an impersonal "super force". Heaven and Earth are part of a unifying power, the Tao, at work behind all changes happening in the universe as an impersonal and nameless "way". But this super force is not neutral, for the heavenly way, following the way of the Tao, brings benefit to others and never causes harm. In the Tao-te ching, the Tao is an active, benevolent power. The sage, who embodies the Tao, is likewise engaged and participating. He is not withdrawn and uninterested in politics and the affairs of the world. 2. The Taoist sage is a member of society and concerned with its welfare. In the passages dealing with the Taoist sage, Shamanism returns. The wise has powers comparable to the legendary Yü, was immune to poison, talked to animals and had a body as soft as a baby. His sexual energy was very powerful and he practiced longevity. For Lao-tzŭ, "wu wei" is not inaction. This is very important to note. Indeed, for the Old Master, this crucial term had not yet degenerated to mere absence of activity. In the Tao-te ching, it points to not harming. The wise ruler is someone who is concerned with his people. He is active and so does not refrain from action, quite on the contrary. He practices benevolence ! This facet brings Lao-tzŭ close to Confucius. 3. To cultivate life is applying bodily techniques and acquiring the correct mental attitude. Regulating breath, applying postures and practicing methods to retain sexual energy aim to cultivate youth and restore vitality. In terms of life-style and mental attitude, the text teaches how cravings, passions, attachment to material things stimulate the senses & the mind, generate emotions, exhaust the body and are detrimental to one's health. The sage is concerned, offers help in a non-intrusive way, withdrawing as soon as the job is done. The teachings of the Tao-te ching represent the transition from a purely Shamanistic worldview, establishing a variety of spirits, towards a philosophical worldview uniting all elements of reality by the Tao, the sage and the cultivation of life. In the figure of the sage, it maintained some elements of Shamanism. This Taoism is optimistic, thinking it possible to change society for the better. It is engaged, not escapist. It intends to end strife and harmonize the world. But as the Period of Spring and Autumn ended in more violence, conflict, war & trauma, Wayfarers lost this optimism ! The historical context made them become escapist, pessimist and disillusioned in politics and the affairs of the world ... In this context the "crazy wisdom" of Chuang-tzŭ emerged ... (b) the Period of the Warring States : Around 390 BCE, the 44 feudal states had been reduced to seven large ones and three smaller ones. Because the latter served as buffer-states between the former, territorial expansion automatically meant military superpowers would confront each other. Because the conflicts had been ongoing for over three centuries, Taoists like Chuang-tzŭ considered it impossible to form stable ruling systems. Crooked noblemen and unscrupulous ministers were seen everywhere. Hence, the pursuit of power and wealth was deemed fundamentally in conflict with health and longevity. Those who adhered to the administration were openly criticized. "Wu-wei" was no longer benevolence, but identified with "non-action" and withdrawal from public life ... Conventions and society were deemed the greatest enemies of personal freedom and integrity ! This form of Taoism lost the appetite to reform society by way of the individual. It remained only interested in the latter, advised to turn his or her back to worldly affairs. In this period, Taoism entered a new stage. Politics was deemed mean, dangerous, while fame and wealth sacrificed liberty and longevity. Confucianism, and benevolent rulers like Yao and Sun became objects of scorn. Offering advise to rulers no longer interested Taoists. Political interest and longevity could not be reconciled. Hence, "wu wei" no longer implied non-harming, but non-involvement, letting things go as they go, radical nonresistance. The sage had no worldly preoccupation. Clearly this reduced the scope of Taoist practice, placing Taoists outside society ... As a result, their ideas about the Tao also changed. The Tao was seen as a neutral power, still the impersonal, nameless, implicate reality behind all things, but in no way benevolent. It had no influence on events, for what happened occurred and nothing could prevent or temper events (predestination). The Tao however remained nameless, invisible and impossible to grasp with thought. He who intuited its way, was an enlightened sage. The Tao was the origin of all things (as it had been in the Tao-te ching), but the notion all things in the universe were of equal value was added. Nothing was more important than something else and there were no "higher" or "lower" species, humankind included. Good and evil were equaled. Proper and bad politics evened. A kind of a-morality became fashionable, and this to the point of absurdity. Just as in the previous period, these Taoists considered too much craving and excitement as nefast for body, mind & spirit. Moral and societal values were also condemned. Rules and regulations were obstacles to the freedom of thought, the freedom of speech and a life in harmony with the Tao, the natural way. Wayfaring became a voice speaking out against hypocrisy. The hermit & recluse were models. All ideals to reform society for the better were left behind, and a throughout negative view on politics, culture & society became virulent (especially at the end of the Eastern Han Dynasty, ca. 219). This disillusioned, escapist pessimism stands in stark contrast to the optimism, idealism and engaged Taoism of the previous period. In the Chuang-tzŭ, the influence of the prevailing political chaos on the appreciation of the Tao must be noted. This negative stance towards society is absent in the teachings of Lao-tzŭ. The latter transforms the individual to change society for the better. The Tao is benevolent. This difference is more contextual than ideological. The fact Taoist religion returned to Lao-tzŭ proves the political dimensions of Taoism run deeper than the escapist & individualizing episode initiated by Chuang-tzŭ and his "neutral" interpretation of the Tao. His view is most probably a radical exception born out of traumatizing historical circumstances. In both periods, Taoism proclaimed the necessity to follow the natural way of the Tao. Only then can the individual change and achieve the ultimate state of enlightenment attained by the wise immortals. 2.2 The Emergence of Taoist Religion (220 BCE - 600 CE). "The first Taoist movement thus combined in its foundation the ancient worldview of the Taoist philosophers, the practices of the magico-technicians of the Former Han, and the messianic millenaristic dimensions of the popular cults of the Later Han." Kohn, L. : Op.cit., p.5. During the Han Dynasty (206 BCE - 219 CE), the practices of Shamanism were integrated in the religious and magical aspects of Taoism. Under the first rulers, the "Way of the Yellow Emperor and Lao-tzŭ" was introduced to the court. But the fundamental gulf between Confucianism and Taoism, between, on the one hand, the moral doctrine of imperial absolutism and central administration and, on the other hand, the real country with its local structures expressing a regional and unofficial form of religion, Taoism, was formed under Han Emperor Wu (140 - 86 BCE), who excluded all systems except Confucianism. The period between the beginning of the Eastern Han (25) and the end of the Six Dynasties (589), may be called the "Golden Age of Taoist Religion". Its emergence was stimulated by three factors : 1. The unification of China under the Ch'in Dynasty (221 - 207 BCE) made an end to the need for military and political advisors. The first Han emperors made sure the nobility could not became too powerful anymore and bring the renewed unity into danger. The formerly itinerant advisors recycled and focused on longevity, healing, divination, etc. A new class emerged, the "fang-shih" or "Masters of Formulae". During the early Han, the top layers of society were foremost interested in longevity and immortality, whereas the peasant population wanted their crops to be safeguarded from flooding and other natural disasters, and their families to be healthy to work the land. This prompted the advance of the use of magical talismans to protect and heal. 2. Another element was the emergence, at the end of the Period of the Warring States, of the Mohists, who developed the faith in a hierarchy of spirits and in the practice of honoring them through offerings. As a result, temples and local shrines were erected and people were trained to care for these sanctuaries. 3. Finally, state ceremonies performed by shamans were outlawed by the Han emperors. The shaman disappeared from the official scene and their role at court was taken over by the "fang-shih". In 150 CE, the Han emperor erected an altar for Lao-tzŭ and installed official ceremonies to honor him. He was transformed from a historical figure to a divinity or sacred power. To offer to these powers was a way to honor them and to thank them for their protection and assistance. After some time, Lao-tzŭ became the most important divinity of Taoism. The fact he is presented as "come again" ("hsin-ch'u") implied a continuity with the Classical Period and this popular organization of what would become the lineage of the "Heavenly Teachers" marked a turning point in the social history of China. These developments led, under the Eastern Han Dynasty (25 - 220 CE), to the revelation of Lao-tzŭ to Chang Tao-ling (ca. 34 - ca. 156 CE), who came from Southern China, an area renowned for its Shamanism and faith in magic. He was trained in Confucianism and got interested in Taoism in midlife. He lived in Shu, the Western part of China (present day Szechuan). The tribes living there had maintained their connection with shamanist practices. In ca. 142 CE, Chang Tao-ling claimed Lao-tzŭ had appeared to him to reveal the tenets of Taoism in terms of a religious practice, and to impart the ability of heal and repel evil spirits. He called Lao-tzŭ "T'ai-shang Lao-chün" or "the Great Lord up High". In his hands, Taoism truly became a religion, with a founder (Lao-tzŭ), a hierarchy of priests (the so-called "Heavenly Teachers"), acting as intermediaries between the believers and the divinities, and well-defined ceremonies. This new movement was called the "Way of Five Bushels of Rice". The descendants of Chang Tao-ling put in place a completely organised system, with a "papal" leader, a clergy, holy scriptures, liturgies, rituals, ceremonies and magical acts. Their central text, the T'ai-p'ing ching ("Book of Peace and Balance") was deemed to be written by divinities, the "keepers of the Tao". The book contained a theory on the creation of the universe, emphasized discipline & ceremony, had rewards and punishments and made an explicit connection between, on the one hand, adhering to religious ceremonies and, on the other hand, health & longevity. When the Han Dynasty ended, China was split in three warring kingdoms (220 - 280), the Wei (220 - 280), Shu (221 - 263) and Wu (222 - 280). When the Shu were defeated by the Wei, power resided between 220 and 265 in the hands of the Wei. During this time, the grandchild of Chang Tao-ling, Chang Lu, expanded the influence of the movement of the Heavenly Teachers, and it became the official, orthodox school of Taoism. The T'ai-shang ling-pao wu-fu ching (Book of the Highest Revelation of the Five Talismans of the Holy Spirit) emerged. It contained protective talismans, incantations, addresses to the divinities, a description of the heavenly hierarchy, meditation techniques, visualizations of the divinities and descriptions of herbs & minerals deemed to give immortality and the way to use them. Taoist religion became known as "t'ien-shih tao", or "Way of the Heavenly Teachers". It developed a Southern (Lu Hsiu-ching) and a Northern (K'ou Ch'ien-chih) branch. The latter emphasized ceremonies and liturgies instead of the traditional magic of talismans. The former, inspired by Buddhist scripture, began to collect and organize all available Taoist texts. In 417, the first Taoist Canon appeared, divided in seven parts. 2.3 The Emergence of Taoist Mysticism (300 - 600 CE). "... make your corporeal soul and your spiritual soul embrace the One and not be separated ..." Lao-tzŭ : Tao-te ching, chapter 10. "... Heaven obtained the One and is clear ; Earth obtained the One and is tranquil ... The Ten Thousand Things obtained the One and they have life." Lao-tzŭ : Tao-te ching, chapter 39. Mystical Taoism, "Shang-ch'ing" or "Mao-shan" Taoism was founded by Lady Wei Hua-ts'un, the daughter of a high ranking priest in the lineage of the Heavenly Teachers, who, herself a priestess, received revelations from the keepers of the Tao in the first period of the Chin Dynasty (265 - 420). In 228, she wrote these down in the Shang-ch'ing huang-t'ing nei-ching yü-ching (Book of the Yellow Court about Inner Images and the High Pure Realm). The founders of this mystical branch were members of the aristocracy. Two central concepts, already known during the Han Dynasty, were emphasized here : the importance of maintaining a direct relationship with "the One" (the so-called "preserving the One", "shou-I") and the notion five divine guardians resided in the body. The texts of mystical Taoism explain how Yang Hsi received a vision of Lady Wei Hua-ts'un, who had become an immortal, and wrote, under influence of Cannabis, the "shang-ch'ing" texts developing the mystical view on the Tao. In its early form, mystical Taoism had many practice in common with the Heavenly Teachers. During the Eastern Chin, nearly fifty texts are clearly part of this tradition. Summarizing the tenets : 1. in the inner universe, "the One" or the Tao-in-us, needs to be maintained. This primordial vapor, or secret embryo, keeps us alive. To embrace "the One" is to feed the secret embryo as a mother feeds its child. 2. the "san-yüan" or "Three-Ones" are emanations of the undifferentiated Tao, and called the "generative energy" (Realm of Water), the "vital energy or energy of life" (Earthly Realm) and the "spiritual energy" (Heavenly Realm). They also need to be preserved. 3. the Five : the heart (Fire), spleen (Earth), lungs (Metal), kidneys (Water) & liver (Wood) systems, associated with the five divine guardians of the body are also to be purified and filled with the primordial vapor. 4. in the outer universe, all things are manifestations of the Tao and its primordial vapor, in particular, Sun, Moon & stars. 5. unity with the Tao is realized by uniting the outer and inner universes, by uniting Heaven and Earth. This is done by "preserving the One", in particular by the "Three-Ones" and the deities dwelling in the cavities of the brain (the so-called "nine palaces"). 2.4 Taoist Alchemy (200 - 1200) "As the immortal Sang-feng put it, going along with the usual course of conditioning makes on an ordinary person, and going against it makes one an immortal ; it is all a matter of reversing the process." Chang Po-tuang : The Inner Teachings of Taoism (Cleary, T. : Op.cit., p.32). Taoist alchemy, often running parallel with Western examples, shows considerable similarity with Indian beliefs, for example the notion of a medicine able to prolong life, the so-called "Elixir of Immortality", appearing in India a millennium BCE. There is no proof of a common origin, although an exchange of thought is very likely. Ideas and symbols "migrate" from one country to another via trading and cultural contacts, and some rise spontaneously in different civilizations. However, in the West, the notion of an "Elixir of Life" did not appear as such until the twelfth century CE, introduced from China by the Arabs. But indeed similar ideas can be found in the Christian Eucharist (the Holy Host as "panacea" - cf. the Petition Before Receiving Communion - Matthew 8:8), as well as in Ancient Egyptian medicine & magic (giving water healing potency by pouring it over hieroglyphic spells). Chinese alchemy made a crucial distinction between external (inorganic, laboratory) alchemy and inner, philosophical alchemy. The former was concerned with making the Elixir or Pill of Immortality using plants & minerals, whereas the latter operated the own body of the alchemist, concerned with spiritual transformation and immortality (becoming a "hsien", an Immortal). The exoteric "outer elixir" ("wei tan") and the "inner elixir" ("nei tan") pointed to two radically different approaches. The aim of Taoist alchemy was to became a "True Man" ("chen jen"), in the sense of "purified" from all elements hindering the constant communication between Heaven (yang) and Earth (yin), the natural way of the Tao. This distinction also appears in Hermetism, namely in terms of the difference between "philosophical Hermetism" and its "technical" pole. Although outer & inner alchemy differ, they were not at first considered to be contradictory. The notion of an "inner pill" ("nei tan") only emerged in the T'ang Dynasty (618 - 906). Before that, Taoists were also always occupied with meditations, postures and sexual yoga. In fact, the first Taoist alchemists saw no need to make the distinction between outer & inner alchemy. The distinction rose when the goal of outer alchemy (finding the Elixir of Life) was deemed unattainable (namely at the end of the T'ang Dynasty). Alchemy is rooted in the quest for health & longevity found in the Classical Period of Taoism. Some "fang-shih" specialized in this effort and so pioneered Taoist alchemy. Some lived the life of a recluse, like Wei Po-yang at work during the Eastern Han (25 - 220). He tried to find the Pill of Immortality. When he found it, he gave it to his dogs and took it himself. They all collapsed and seemed to have died. Afterwards they regained life, and flew off as immortals ... He also composed the first Taoist text on alchemy, the Tsan-tung-chi ("The Triplex Unity"). In this book, in accordance with the tenets of Classical Taoism, the Tao is the origin of all things and the primordial energy of the Tao is the source of all life. As nature replenishes, so mortal beings can also renew and achieve immortality by living in accordance with the natural way of the Tao. The crucial concept advanced is the coupling of Heaven (yang) and Earth (yin). Alchemy then is the art and science to use these in such a way as to restore the original harmony. This happens when the impurities are driven out of the body. The alchemist Ko Hung, at work in the last period of the Chin Dynasty (265 - 420), composed an encyclopedic work (the P'ao-p'u-tzŭ), containing formulae, lists of ingredients, ways to prepare the Pill, methods to silence the mind, minimize craving, train the body, breathing techniques and ideas about "preserving the One". He combined outer and inner alchemy. But at the end of the Six Dynasties (ca. 589), Taoists began to doubt the methods of outer alchemy. Combinations of lead, mercury, cinnabar and sulphur were often lethal. The theoretical foundations of outer alchemy were reviewed. The effort itself was not yet abolished, but the use of dangerous substances was criticized. Nevertheless, with the rise of the T'ang Dynasty (618 - 906), outer alchemy received imperial backing. These alchemists thought there were two kinds of elixirs. The first has its origin in nature, and is composed of minerals & stones absorbing the primordial vapors of "yin" and "yang" over very long periods of time. The second is one produced in the laboratory, swiftly imitating the natural process. But after three hundred years of failed experiments, outer alchemy was discredited. Finding the Elixir of Immortality was deemed impossible. Under the influence of Ch'an Buddhism, the notion of "immortality" was equated with liberation from "samsâra" or identified with health & longevity. At the end of the T'ang Dynasty, outer alchemy was finally abandoned. "The science of spiritual alchemy is simply a matter of taking flexibility within strength and strength within flexibility, which are the two great medicines of true yin and true yang, and fusing them into one energy, thus forming the elixir." Liu I-Ming : The Inner Teachings of Taoism, in : Cleary, Th. : Op.cit., p.84. During the Sung Dynasty (960 - 1279), inner alchemy flourished. The patriarch of this approach was Lü Tung-pin, born at the end of the T'ang Dynasty. His pupil, Wang Ch'ung-yang (ca. 1113 - 1171), became the founder of the Northern School of Complete Reality, combining Taoism, Buddhism & Confucianism. But the most outstanding alchemist and founder of the Southern Complete Reality School was Chang Po-tuang (ca. 983 - 1082). His major work, Wu-jen p'ien (Understanding Reality), advances the thesis all ingredients & instruments of the alchemical process are to be found in the body, and the "outer" processes described in previous texts are deemed metaphors. This revolutionized alchemy. 2.5 The Great Synthesis : Complete Reality (1000 - today) "If You do not seek the Great Way to leave the path of delusion, even if You are intelligent and talented You are not great. A hundred years is like a spark. A lifetime is like a bubble. If You only crave material gain and prominence, without considering the deterioration of your body, I ask You, even if You accumulate a mountain of gold, can You buy off impermanence ?" Chang Po-tuang : Understanding Reality (Cleary, T. : Op.cit., p.27). Wang Ch'ung-yang (Wang Che, ca. 1113 - 1171) had a Confucian education, studied Buddhism, but at forty became a Taoist and pupil of Lü Tung-pin and Chung-li Ch'uan. For him, the integration of the experience of tranquility & emptiness (cf. Zen Buddhism), Confucian ethics and Taoists health and longevity techniques leads to a complete insight in the ultimate reality. His version of Taoism is therefore called the School of Complete Reality. From Confucianism he took the K'ao-ching (the Book of Child Duty), from Buddhism the Heart Sûtra and from Taoism the Tao-te ching and the Ch'ing-ching ching (Cultivating Silence). This is not an eclectical system, for Taoism is the foundation of the synthesis. The Tao is the formless & undifferentiated energy forming the background of reality. To unite with the Tao is to receive energy from this source, leading to longevity. As the highest reality, the Tao can only be experienced by the original spirit, free from thoughts, attachments and cravings. This original spirit is the immortal embryo ("yüan-shen"). All sentient beings have a seed of the Tao in them, but this can only develop if cravings and uncontrolled thoughts are eliminated. Eliminating these brings us back to the original spirit. Cultivating Tao begins by the experience of tranquility & emptiness as fostered in Ch'an Buddhism. The practice of virtue, benevolence and honor are deemed essential, for the original nature of goodness is equated with the original spirit. The latter is not only free from cravings, but also inclined towards goodness. Spiritual training leads to the transformation of body and spirit and this change is alchemical. This school had two branches. In the Southern school of Chang Po-tuang, who was not a pupil of Wang Che but of Liu Ts'ao (who in turn was a pupil of Chung-li Ch'uan & Lü Tung-pin and so a fellow student of Wang Che), one concentrated on the collecting of inner energy, purifying it to realize good health & longevity. Physical techniques were the condition for meditative work, and sexual yoga was part of the technology. In the Northern school of Chiu Ch'ang-ch'un (the most precious pupil of Wang che), meditative work came first and there was no room for sexual yoga. During the Ming Dynasty (1368 - 1644) a multitude of sects emerged. Differences in theory & practice and a profound interest in magic spurred the rise of various schools and subschools. In the Ch'ing Dynasty (1644 - 1911), a period of criticism followed. The preferences of the Ming were questioned, and the magical practices of Taoism were condemned, as well as all things deemed "irrational" (faith in spirits, divinities, magic and inner alchemy). A new synthesis of Confucianism, Buddhism and Taoism slowly emerged. Two outstanding figures are worth mentioning : • Liu I-ming (1734 - 1821), a Confucian who in midlife turned Taoist, considered inner alchemy as a psychological process, and so the whole transformation has the mind as object. Realizing the Tao is to rediscover the original nature of the mind goes hand in hand with developing true knowledge. "Yang" represents the innate goodness of the empty mind and "yin" the clear consciousness of the empty mind. The alchemical process refers to stabilizing firmness and flexibility, and not to the purification of the inner energies. Health & longevity are epiphenomena of a calm mind. The physical serves the mental. • For Liu Hua-yang (1736 - 1846 ?), a Buddhist who in midlife turned Taoist, the best of inner alchemy and Buddhism complement each other. Immortality and "Buddha-nature" refer to the same thing. Taoism is able to cultivate life, but not the original spirit. Buddhism is able to cultivate the original spirit, but cannot lead to health or longevity. Everybody has the essence of life, the energy of the Tao in his or her body. Craving, a negative mentality & emotional attachments cause this life-force to leave the body, leading to sickness and loss of immortality. When thoughts are silenced and cravings bridled, the life-force is able to circulate through the body, leading to the development of the spiritual embryo or the original spirit (Buddha-nature). This embryo is the consciousness of the original spirit and the energy feeding the body. When it grows, it forms a spiritual body travelling to other dimensions. When the body dies, it can become one with the energy of the universe. 3 Against Substantialism : Brother Buddhism & Sister Taoism. "But emptiness requires that emptiness reach the point when there is nothing to be emptied, only then is it called the ultimate of emptiness." Chang Po-tuang : The Inner Teachings of Taoism (Cleary, T. : Op.cit., p.6). 3.1 Buddhism in China : According to traditional Chinese sources, Buddhism was imported to China during the Eastern Han Dynasty (ca. 58 CE). They stressed still meditation ("śamatha") and ignored physical exercise. Western scholarship claims Buddhism penetrated China in the second century CE from Central Asia. In the centuries following the dissolution of the Han Dynasty, Buddhist texts were translated into Chinese and Buddhist monastic orders were established. Simultaneously, Taoist texts resembling Buddhist texts were composed and Taoist cloisters were set up on the Buddhist model. At first, Buddhism was deemed a "barbaric" form of Taoism. The oldest Chinese book on Buddhism was the Mou-tzŭ, an apologetic work dating from the second century. In this early period, Buddhism as divided in Hînayâna "Dhyâna" school, preoccupied with meditation, and the "Prajña" school, based on the Mahâyâna Prajñapâramitâ-Sûtras, promoted by Tao-an (312 - 385), who composed the first catalogue of Buddhist works translated into Chinese. Buddhist practice was identified with Calm Abiding ("śamatha"). It is interesting the note how Taoist scriptures contain elements only found in Tantric Buddhism. In contrast to what happened in Japan and Tibet, where the Vajrayâna took root and became predominant (in Tibet, Sutric training leads to Tantric practices), the latter never became popular in China. During the Sui and T'ang Dynasties (end of the 6th to beginning of the 10th century) Buddhism in China reached its high point. It was promoted by a series of emperors. During the T'ang Dynasty (618 - 907), Indian Buddhism had already started to decline, making China the world center of the Buddhadharma, from where it reached Japan and Tibet. In 845, emperor Wu-tsung, to acquire their wealth, closed thousands of monasteries. Although his successor tried to make amends, Buddhism never completely recovered. Except for Ch'an, the period of intellectual flourishing of Buddhism in China was over. Besides a few smaller schools (like the Mâdhyamaka San-lun & the Abhidharma Kośa), five great Chinese schools made their appearance : Ching-t'u, Fa-hsiang, T'ien-t'ai, Hua-yen and Ch'an. (a) "Ching-t'u" or Pure Land Buddhism : In Mahâyâna, Pure Lands are Buddha-realms presided over by a Buddha. There are as many Pure Lands as there are Buddhas, but the most important Pure Land is "Sukhâvatî", the Pure Land of Buddha Amitâbha, the Buddha of Infinite Light. These Lands are transcendent and the hope of believers who wish to be reborn in them. The decisive factor not being good "karma", but the aid of a given Buddha who took the vow to help all those who turn to him or her in loving faith. In popular belief, these paradises are places of bliss, while in fact they represent aspects of the awakened state of mind of a Buddha. These Pure Lands are not the final stage, but a stage before "nirvâna", realized in the ensuing rebirth. In a Pure Land, retrogression is no longer possible ! The Pure Land School ("Ching-t'u-tsung") was founded in 402 by the Chinese monk Hui-yuan. The goal was to be reborn in a Pure Land of Buddha Amitâbha. Faith in the power and active compassion of Buddha Amitâbha is all what counts, and the practice consists of the recitation of his name and the visualization of his paradise. These recitations give a vision of Amitâbha and foreknowledge of the time of one's death. These guarantee rebirth in "Sukhâvatî". (b) Fa-hsiang : The "Marks-of-Existence School", founded by Hsûan-tsang (600 - 664) and his pupil K'uei-chi (638 - 682), continues the teachings of the Yogâcâra (Mind Only), based on Vasubandhu and Asanga. Everything is only ideation. The "external world" is the product of consciousness and devoid of reality. Things exist insofar as they are contents of consciousness. These teachings have been discussed elsewhere when considering the Mahâyana Schools and emptiness. The consciousness or mind devoid of apprehended object and apprehending subject is a thoroughly established (perfect) nature, and thus truly established. This is also the definition of emptiness. Enlightenment is therefore identified with absence of duality. The Fa-hsiang denies all sentient beings possess Buddha-nature and can attain Buddhahood. Unbelievers cannot become Buddhas. (c) T'ien-tai Buddhism : T'ien-t'ai, or "School of the Celestial Platform", received its final form from Chih-i (538 - 597 CE). It is based on the Lotus Sûtra. All phenomena are an expression of the absolute or "suchness" ("tathatâ"). This idea gave rise to three truths : the truth of emptiness, the truth of temporal limitation and the truth of the middle. 1. the truth of emptiness : all "dharmas" lack independent reality ; 2. the truth of temporal limitation : a "dharma" has a functional, apparent existence perceived by the senses & grasped by the mind, i.e. they are not completely illusional or non-existent ; 3. the truth of the middle : includes both former truths and is equated with "suchness" ; the true state is not to be found elsewhere than in phenomena and so the absolute and phenomena are one. Emptiness (ultimate truth), phenomenality (conventional truth) and the middle (suchness) are aspects of a single existence. The practice of this school consists of meditations based on "chih-kuan". The first element ("chih" or collectedness) concentrates on the emptiness of all "dharmas". This prevents the arising of illusions. The second element ("kuan" or insight), causes us to recognize the apparent, functional, spatiotemporal existence of all "dharmas" despite their emptiness. (d) "Hua-yen" or "Flower Garland School": The "Flower Garland School" was founded by Fa-tsang (643 - 712), but began with the monks Tu-shun (557 - 640) and Chih-yen (602 - 668). Also called "Âvatamsaka School", it derives its name from the Chinese translation of the Buddhâvatamsaka-Sûtra, the largest text in the Buddhist Canon. Due to the refinement of its view, integrating Fa-hsiang and T'ien-t'ain, it is considered the intellectual culminating point of Chinese Buddhism, but due to the persecutions it rapidly declined. The school teaches the equality of all things and the interdependence of all things on one another. All things partake in a unity divided into many, allowing the manifold to be unified in this one (the teaching of totality). Everything in the universe arises simultaneously (the universal causality of the "dharmadhâtu", the uncaused, immutable totality in which all phenomena rise). Each "dharma" is either in a state of "suchness", the static aspect of which is emptiness, i.e. the realm of "principle" ("li") or the dynamic aspect of the realm of phenomena ("shih"). Interwoven, these two realms (principle & phenomena) are dependent on each other, and so the whole universe arises by interdependent conditioning. Nothing can subsist on its own (is essential, or possessing "svabhâva", "own-nature"). The teachings concentrate on the relationships between phenomena and not on that between the latter and the absolute. Fa-tsang explains the fundamental tenets of this school with the famous simile of the Golden Lion. The lion represents the phenomenal world and the gold the principle. The latter has no form of its own, but rather takes on any form according to conditions & circumstances (is empty). Every organ of the lion participates in the whole result, the lion made of gold. All phenomena (the organs & the lion) manifest one principle (emptiness) and each phenomenon encompasses all others. Gold and lion exist simultaneously and include each other mutually. Hence, each phenomenon (lion) represents the principle, emptiness or "li" (gold). These ideas bring about a division of the universe in four realms : 1. the realm of phenomena : the dynamic aspect of the "dharmas" ; 2. the realm of the absolute : the principle or static emptiness ; 3. the realm in which both mutually interpenetrate : the functional world of things ; 4. the realm in which every phenomenon exists in perfect harmony without obstructing each other : the ideal world. All "dharmas" possess six characteristics : 1. universality : the lion as a whole ; 2. specificity : the functional organs of the lion distinct from the lion as a whole ; 3. similarity : all functional organs are parts of the lion ; 4. distinctness : each organ has a distinct function ; 5. integration : all organs together make up the lion ; 6. differentiation : every organ takes its own particular place. All things are in complete harmony with one another, for manifestations of the same, one principle : emptiness. They are like individual waves of the same sea. Hence all phenomena are one with Buddha-mind, the "Dharmakâya". (e) Ch'an Buddhism : In the traditional account, Dhyâna Buddhism was introduced by Bodhidharma (ca. 470 - 543 CE) or Da Mo, the first patriarch of Ch'an Buddhism in China ("ch'an is an abbreviation of "ch'an-na", from the Sanskrit "dhyâna") and the twenty-eight patriarch of Dhyâni Buddhism in India. He is believed to be the second Indian priest to be invited to China (by Emperor Liang in 527 CE), Ba Tuo being the first Buddhist monk come to China to preach (called "Happy Buddha" or Mi Le Fo, ca. 495 CE). He placed particular emphasis on the harmony between the practice of meditation ("dhyâna") as a way to enlightenment ("bodhi") and physical exercises. He did not develop a philosophical view. Indeed, Da Mo is the author of the two classical texts on Ch'i Kung, namely the Yi Jin Jing (Muscle/Tendon Changing Classic) and the Xi Sui Jing (Marrow/Brain Washing Classic). He wrote these because he found the monks of the Shaolin Temple (on Shao Shi Mountain, Henan province) to be weak & sickly (for only practicing Nei Dan or "internal elixir"). These texts were fundamental in the further development of Ch'i Kung. The main teachings developed in the 6th & 7th centuries were a fruitful encounter of Dhyâna Buddhism with Taoism. Because it did not had large monasteries, it survived the persecutions at the end of the T'ang Dynasty, and so became, together with Pure Land Buddhism, the only form of Buddhism in China under the Sung, Yüan and Ming Dynasties. In the seventh century, Ch'an split in a Northern School (Shen-hsiu, 600 - 706), teaching (Indian) gradualism, and a Southern School (Hui-neng, 638 - 713), proposing (Chinese) suddenism. Ch'an stresses self-realization leading to complete enlightenment by way of intensive meditative self-discipline. Ritual practices and intellectual analysis of doctrine (analytical meditation) are deemed useless for the attainment of awakening. Ch'an Buddhism reached Japan in the 12th and at the beginning of the 13th century, were it was called "Zen". Sitting in meditative absorption ("zazen") is seen as the shortest & steepest way to complete enlightenment. It declined in China under the Sung and mixed with the Pure Land School of Buddhism during the Ming. It continued to exist until today. Ch'an can be summarized by these four statements : 1. special transmission : at Vulture Peak Mountain, Buddha is said to have held up a flower without speaking - his student Kâśyapa smiled and understood instantly on the spot what the Buddha meant. This was the first heart-mind to heart-mind transmission. Ch'an is therefore also called the "School of Buddha-Mind" or sudden enlightenment (suddenism is also found in Dzogchen) ; 2. nondependence on sacred writings : the experience of enlightenment is of primary concern, not the dry, thinglike reality of documents & dates ; 3. directly pointing to the heart : the pointing-out instruction is given by an enlightened master. The master identifies the nature of mind of the student and points it out to the student ; 4. realizing one's own nature : the essence of the whole discipline is the realization of Buddha-nature, the clear tranquil core of the mind. Although there are clearly parallels between, on the one hand, Tantric Buddhism and, on the other hand, Taoist methods of meditation and inner alchemy, Mantrayâna never took root in China. As a school, the Vajrayâna flourished briefly in the 8th century, but during the moralist Sung Dynasty (960 - 1278) most tantric texts disappeared. Emptiness and Dependent Arising. "All of these practices were taught By the Mighty One for the sake of wisdom. Therefore those who wish to pacify suffering Should generate this wisdom." Śântideva : A Guide to the Bodhisattva's Way of Life, IX:1. (a) Simultaneity in Wisdom-mind according to Tsongkhapa : "In order to be sure that a certain person is not present, you must know the absent person. Likewise, in order to be certain of the meaning of 'selflessness', or 'the lack of intrinsic existence', you must carefully identify the self, or intrinsic nature, that does not exist. For, if you do not have a clear concept of the object to be negated, you will also not have accurate knowledge of its negation." Tsongkhapa : Great Exposition of the Stages of the Path, vol.3, 2.10. Wisdom-mind is Buddha-mind, the enlightened body, speech, mind & activity of a Buddha, a former sentient being who entered Buddhahood. A Buddha experiences the Two Truths, conventional & ultimate truth, simultaneously, i.e. in the same cognitive act. To such an exalted & enlightened wisdom, every object is conventional and ultimate at the same time, in the same instance ; conventional insofar as it appears to sentient beings as interdependent (and so dependent on conditions & circumstances outside itself) and ultimate insofar as it lacks any kind of selfsubsistence (substance, essence, own-form or "svabhâva"). This great insight of Je Tsongkhapa (1357 - 1419) or the "Man from the Onion Valley" ended over thousand years of speculative investigations into the fundamental tenet of Buddhism : Selflessness of Persons ("anâtman" - Hînayâna) and Selflessness of Phenomena (Mahâyâna). Two extreme positions were thus avoided : eternalism & nihilism. In the former wrong view, substances (objects existing from their own side, self-powered) exist, whereas in the latter view, nothing truly exists (and so nothing really performs any function). For the Middle Way of Tsongkhapa, all objects (Buddhas included) are (a) ultimately empty of self-power (lack substance), but (b) conventionally exist logically & functionally, i.e. are valid names or labels and are operational, albeit appearing different as they truly are (i.e. presenting themselves as substances while they are not). Ultimate truth is valid and unmistaken, while conventional truth is valid but mistaken. "After I pass away, And my pure doctrine is absent, You will appear as an ordinary being, Performing the deeds of a Buddha, And establishing the Joyful Land, the Great Protector, In the Land of the Snows." Śâkyamuni's prediction of the coming of Tsongkhapa in the Root Tantra of Mañjuśrî. Tsongkhapa was a renowned Tibetan Buddhist spiritual reformer, yogi and scholar. Taking layman's vows at the age of three, he was ordained as "Lobsang Drakpa" ("Sumati Kirti" or "Perceptive Mind"), but simply called "Je Rinpoche". Founder of the doctrinal & influential Gelug school of Tibetan Buddhism, his direct inspiration came from the Kadam school, initiated by Atiśa (985 - 1054), as well as from the Sakya school. The results of his important systematic & complete organization of Buddhadharma (comparable to the Summa Theologica of Thomas Aquinas) were presented in the Lamrim Chenmo (Great Discourse on the Stages of the Path to Enlightenment) and the Ngagrim Chenmo (Great Discourse on Secret Mantra). As a Buddhist philosopher, Tsongkhapa attributed the proper logic to the system of the Middle Way founded by Nâgârjuna (ca. 2d/3d century), in particular the Prâsangika-Mâdhyamaka school, and was therefore a skillful teacher of "śûnyatâ", emptiness. His interpretation may be called "Critical Mâdhyamaka", for its central preoccupation is drawing the line between proper and improper objects of negation. Once we know what to negate when dealing with emptiness, namely self-powered, inherent existence, we can establish a valid foundation for Tantric practice. Negating too much (as in nihilism) results in eliminating conventional reality, bringing morality & compassion in jeopardy. Negating not enough (as in eternalism) creates permanent objects without good reason, substantializing or reifying what must be thought as lacking existence from its own side. The central texts of Mâdhyamaka are : • Nâgârjuna (2th CE) in Mûlamadhyamakakârikâ (A Fundamental Treatise on the Middle Way) & Shûnyatâsaptatikârikânâma (The Seventy Stanzas on Emptiness) ; • Chandrakîrti (ca. 600 – 650) in Mâdhyamakâvatâra (Entering the Middle Way) ; • Śântideva (8th CE) in his Bodhicharyâvatâra (A Guide to the Bodhisattva's Way of Life) & • Tsongkhapa (1357 - 1419) in The Great Treatise on the Stages of the Path to Enlightenment, The Ocean of Reasoning and The Essence of Eloquence. For Tsongkhapa, who refutes the definition of emptiness proposed by the Mind Only School (absence of duality between apprehended object and apprehending subject), duality itself is not a problem, only its reification is. The interaction between cognition and the cognitive field cannot be avoided, not even in the most evolved wisdom of Ârya Buddhas (cf. fully enlightened wisdom-minds directly apprehending, cognizing or perceiving emptiness). In his view, Buddhahood involves the simultaneous apprehension of the ultimate & the conventional of every phenomena in every cognitive act. For Tsongkhapa, Hearers, Solitary Buddhas & Superior Bodhisattvas of the Eighth to the Tenth Bhûmis are indeed totally free from even the subtlest latent (innate) reifying tendencies, but are nevertheless subject to nondeluded ignorance, the conditioned state of mind predisposed by the previously existent innate conception of inherent existence or essence. So they are not yet fully enlightened. They are predisposed to the assumption of dualities rather than their reification. Misconceptions of dualistic appearances remain. A Buddha no longer assumes duality, while the distinction between the cognitive act and its field is not gone. There is "merely" a witnessing, a sheer existential instantiation (cf. infra). The above Âryas are not yet enlightened because for them ultimate & conventional knowledge still come about sequentially, and so they have only alternating knowledge of the Two Truths. During meditation they known the ultimate. In postmeditation, they apprehend the conventional. But once they are capable of having direct knowledge of both truths simultaneously, able to cognize empty & dependently arisen phenomena concurrently, establishing the non-conceptual dual-union of the Two Truth (which is nondual but not a-dual), they become Buddhas, and the difference between meditation and post-meditation vanishes. Then, from their own perspective, only emptiness is apprehended, while all conventionality is explicitly known as it appears to sentient beings, i.e. as dependent arisings. (b) Six Instantiations explaining Emptiness : "Contemplating emptiness, it is also empty ; there is nothing for emptiness to empty." Wen-tzŭ, quoted in Clearly, Th. (transl), Practical Taoism, p. 18. In a general sense, "instantiation" means representing an idea in the form of an instance of it, i.e. as an item of information representative of the idea, clarifying it by giving an example of it. For Kant, a concept has "sense and meaning" ("Sinn und Bedeutung") when it is possible to experience an instantiation of this concept. For him, saying something "exists" merely points to its categorial instantiation, the fact it is an example or "instance" of a category of thought, and does not add anything substantial to the object (the fact it is deemed to exist as a "Ding an Sich" outside the subject of knowledge). For Kant, such substantial instantiation lies outside the possibilities of rational knowledge, bound to the categorial processing of appearances. "Existent" is not a determining predicate belonging to the set of predicates defining a concept. "Being" cannot be added to the concept of a thing, for it is not a property, nor a quality of anything. Neither does it report any details about it. At times, this verb and its variants behave as predicates, like in : "Unicorns don't exist.", and then seem to report something not done by unicorns, namely "existing". In fact, each time, the verb is only qualified as a grammatical or "logical" copula. In a logical sense, "Unicorns don't exist." is a short way to say : "Unicorn are never an instance of categorial processing." or "Unicorn cannot be posited." For Kant, "existence" only instantiates, designates, posits or imputes the concept. instantiates : the concept is an example of a category ; designates : the concept is assigned to a category ; posits : the concept is assumed to belong to a category ; imputes : the concept is attributed to a category. So when the "existence" of something or someone is posited, the totality of known predicates of a thing or an individual is indeed affirmed, adding nothing to it. When this existence is denied, the whole set of predicates vanishes and the referent with it. An object is what can be ascribed to it, nothing more. There is no "stable" core (or referent) as it were carrying the predicates or attributes without them. There is no fixed, substantail support or an Archimedean point providing something to hold on to. Ousiology (thinking "ousia" or "essence") is rejected. To affirm the set A "exists" is to instantiate (posit) its concept, but does not instantiate the richer concept "existing A". Every statement of existence ("there is", or "there are"), merely says about a concept it is instantiated, rather than it exists. Any legitimate statement of existence must be built out of propositions of the form : "There is an A.", where "A" stands for a determining predicate. This is strict nominalism ; the meaning of a concept is nothing but its name (or the category of which it is an example, an instantiation). The word "existence" can be grasped in terms of various instances, namely as specific sensate & mental objects said to "exist". The latter are identified as logical entities, functions, conventional empirico-formal propositions of science, substances, ultimate objects or mere existentials. Kant's criticism, as well as Tsongkhapa's analysis, shows how substantial instantiations are erroneous. • ЭLA logical instantiation : the existence of object A or Эx (x = A) is an instance of it being identifiable in classical logical terms LA according to the principles of identity (A = A) & non-contradiction (A ≠ ¬ A), and, classically, excluded third (A v ¬ A) or ЭLA ; (a) 0 = 0 ^ 1 = 1 (b) 0 ≠ 1 ^ 1 ≠ 0 (c) 0 v 1 ^ 1 v 0 This instantiation is not yet an empirico-formal object with synthetical content, but a mere formal or analytical object (as in logic & mathematics, attaching predicates to subjects by way of tautology). • ЭFA functional instantiation : the existence of object A is logically (LA) instantiated and identifiable in functional terms FA according to A = f(B) or B = f(A) or ЭFA ; This instantiation involves recognizing empirical functions and has all the properties of a direct empirico-formal object, i.e. one ostensively ascertained hic et nunc. This comes very close to mere existential instantiations, except for the fact the latter have purified all substantial connotations whatsoever, while logical & functional instantiations lead to or are suggestive of conventional instantiation. conventional instantiation : if the existence of object A is logically (LA) and functionally (FA) instantiated, then it is also substantially instantiated, or (ЭLA ^ ЭFA) »* (As = E!A) ; (*) the implication or "if A then B" This conventional instantiation is the way of conventional truth, valid to distinguish between conventionalities. It is a deceptive truth, for objects appear not as they truly are. Just as the Sun seems to rise & set, these objects seem to exist independently from the apprehending subject. In both cases this is a mere appearance, for cosmology teaches us the Earth rotates around its axis, and physics, neurology & observational psychology makes clear all observations depend on the observational frame adopted by the observer. This deception can however not be grasped & eliminated as long as one does not try to find, by way of ultimate analysis, this supposed "eidos" or enduring "essence", not realizing it cannot be found. Conventional instantiation is commonsense knowledge and insofar as it has been tested & discussed, triggering "correspondence" and "consensus", it is moreover scientific. As scientific knowledge does not probe into the deep to find whether there indeed is a substantial core, it is superficial and provisional, although logical, functional and synthetical (attaching predicates to subjects by way of sense objects). It presupposes analytic terms, always involves direct synthetic statements (statements of fact based on immediate sensing & thinking) and claims to articulate indirect synthetic propositions (holding a truth claim about the state of affair of the world no longer involving immediacy). • E!A substantial instantiation ("esse", being, true existence or inherent existence) : if object A has properties Z (or A(z)), then -by way of false ideation Cf- the essence of A, or As "having" these properties, necessarily inherently exist, or ЭA(z) ^ Cf » E! Эy (y = A) = As = E!A ; The substantial instantiation or false ideation Cf positing these attributes or accidents as inherent in the "real sense objects" is automatic. This automatism of grasping at an enduring "self" or self-grasping is innate & acquired. Infants, like animals, manifest it and in the course of our education humans are confirmed in attributing independent, self-powered reality to attended sense objects. To refute this instantiation is the job of ultimate analysis, probing into the object at hand, trying analytically to isolate the substantial, self-identical core. If, after exhausting all logical possibilities, no core can be found, then no rational ground is given to accept substance. This method leads to strict nominalism, always prompting its opponents to posit an enduring object ! ultimate instantiation : the existence of object A is logically LA and functionally FA instantiated without being -by way of true ideation Ct- substantially instantiated as inherently existing or (ЭLA ^ ЭFA) ^ Ct » {¬ (As = E!A)} ; ¬ (As = E!A) is a non-affirming negation, i.e. it negates substantial instantiation without positing anything else. So it is not empty of itself, for ЭLA ^ ЭFA conventionally endures. It only negates As by way of true ideation Ct, i.e. eliminates E!A. As under ultimate analysis no enduring "self" can be found or ¬ (As = E!A), the substantial instantiator E!A can be eliminated. When this is done, conventional objects appear together with their lack of inherent existence. This implies they appear as mere existential instantiations or dependent-arisings simultaneously with their lack of inherent existence. Whatever is a dependent arising does not inherently exist because inherent or independent existence is the opposite of dependent arising or E!A = ¬ {ЭLA ^ ЭFA}. mere existential instantiation ("existit" or mere existence) : the existence of object A is logically (LA) and functionally (FA) instantiated and absolutely nothing more : ЭA = ЭLA ^ ЭFA. Sentient, aware beings always conceive their objects as logical, functional and, by force of Cf, substantial. Because of their ignorant sentience, they, unlike computers, attribute selfhood to the objects they attend to. Because of this false attribution Cf, they possess the potential to consciously eliminate this and enter wisdom ! This potential to realize wisdom is what is meant by their Buddha-nature. Without the latter, beings, although merely existing, do so devoid of the possibility of enlightenment. Sentience preconditions Buddhahood. Buddhas perceive the absence of inherent existence, or ultimate instantiation, hand in hand with mere existential instantiation, seeing dependent arisings free of inherent existence. They know ultimate truth as ultimate, or space-like emptiness (without any obstruction), and simultaneously as merely existential, or illusion-like dependent arising. The former is ineffable, the latter a dependent-arising concealing its ultimate nature. Buddhas perceive all phenomena simultaneously as empty and as merely existing hic et nunc. This is merely seeing, merely hearing, merely touching, merely smelling, merely tasting sensate objects and merely consciously apprehending mental objects (of thought, affect & volition). (c) The Cognitive Activity of a Buddha : Technically, the ultimate nature of phenomena can be conceptualized as the absence of substantial instantiation, ending attributing own-form or existence to objects from their own side, or ¬ E!A. The mere apprehension of objects, exclusively instantiating their logical (name) & functional (operation) properties, i.e. the mere existential instantiation hic et nunc is all the enlightened wisdom-mind of a Buddha perceives. Wisdom-mind knows every phenomenon as one entity with two isolates, cognizing ultimate truth in two ways : 1. as space-like emptiness : This is the sphere where perception and sensation of objects fades. Where phenomena are no longer occupy the foreground. This is the non-differentiated experience, to be directly and personally experienced by the enlightened mind. It cannot however be conceptually known or linguistically described from the outside. Even a Buddha cannot offer any criterion to describe it. In this sphere, suffering, with its coming, going, stasis, passing away, arising, stance, foundations, support, etc. end. Consistent with the universals & the summit of the Via Negativa of mystical experience, nothing can be conceptualized or said about this "apex" or capstone of nondual cognition. While clearly cognitive, for the object of wisdom-mind is emptiness, it is ineffable. If something is actually uttered concerning this, science nor metaphysics are at hand, only sheer sublime poetry. 2. as illusion-like emptiness : In this mode of knowing ultimate truth, phenomena are apprehended as relational, interdependent and illusory. Relational because, as substantial instantiation has ceased, there are no independent objects and so all things are related. Interdependent because all objects are other-powered. Illusionary because they only appear as independent to conventional reason, while they are not. Although there is duality, this does not constitute a misconceived duality. When, with right discernment, one sees all phenomena as dependent co-arisings as they are actually present in this moment, one does not run after the past nor the future. The mere presence of duality, as mere existential instantiation is not problematic. Duality by itself causes no delusions, but the reification of its terms always does. Take this away, and the panacea against all suffering has been found ! (d) Dependent Arising : Functional co-relativity, correlational interdependence, universal interrelationality, conditioned co-production, interdependent co-arising, dependent origination, dependent arising ("pratîtya-samutpâda") are synonymous. A nuance can be observed. By saying objects are "dependent" we focus on the fact determining factors, conditions & circumstances outside them influence them. By saying objects are "interdependent", we affirm they are "dependent", but also add they all depend on one another. This is organicism, the idea the universe is a connected whole without "disjecta membra" or thoroughly isolated phenomena. All phenomena, "nirvâna" as well as Buddhahood, are dependent arisings, i.e. process-like instead of substance-like, interdependent instead of independent, without own-form instead of self-powered. When emptiness, the absence of inherent, substantial existence is realized, only dependent arisings remain. This is ¬ E!A, the negation of inherent existence ("svabhâva"). Emptiness makes process apparent ; process makes emptiness evident. Wisdom perfects method and method manifests wisdom. Compassion generates form, and wisdom truth. Form & truth are the bodies of a Buddha. Although on an absolute (deep, implicate, esoteric) level phenomena are devoid of substance (or empty), on a conventional (superficial, explicate, exoteric) level, functional, working & efficient interdependent relationships prevail. These conventional objects always appear cut-off as self-powered, independent mental or sensate objects. This aspect of their appearance is however false, for objects cannot be substantially initiated without absurdity. Although the notion of two "levels" or "Two Truth" is suggestive of a difference, this should not be viewed ontologically (as two levels of reality), but rather as two epistemic isolates of the same phenomena. A Platonic schism ("chorismos") is not implied, rather two perspectives on a single event. The event-continuum is all there is, for emptiness is not a subtle stratum of reality but a mere absence of inherent existence (cf. emptiness of emptiness). Ultimate reality exists conventionally ! Conventional reality is a process. This means change and impermanence are given to it. The Dharma refers to this cosmic law, ruled by the "king of logic" (Tsongkhapa), namely dependent arising ("pratîtya-samutpâda"). Phenomena arise as the result of determining conditions, abide for a certain time under influence of conditions and cease when the sustaining conditions vanish. This movement is universal and unchanging. While Buddhahood and "nirvâna" are often described as permanent, this only refers to their continuous dynamism, and the fact this dynamism has certain continuous features, like being totally emptied of any sense of substance or stasis. Compare this with a swimming style, simultaneous with the swimmer's movement and meaningless as a static notion. A swimming style is a dependent arising, for all phenomena are. Nevertheless, the conditions pertaining to Awakening are radically different from those ruling conventional reality or "samsâra". Awakened Ones are no longer under the spell of ignorance, but under the sway of wisdom. They acknowledge & apprehend the style of the movements while they are moving. Like a boat makes sense when it moves to cross the river, the characteristics of the dynamism are valid insofar as there is movement. A boat in a dock or wharf has lost its functionality and is only potentially useful. A swimmer outside the water no longer swims. The question at hand is whether a universal logic of dependent arising is possible ? The Buddha discussed this logic in terms of the twelve "nidânas" or Twelve Links of the causal chain ("nexus"). While all phenomena exist non-substantially (but not from their own side), they function in dependence on conditions & determination (like efficient causes). Because the Buddha was focused on awakening his disciples, this analysis is carried through from the side of the subject of experience and differs from a study of the conditions pertaining to the world (as in physics, cosmology, chemistry etc.). The latter comes into focus in Taoism (cf. the role of Chi-circulation in inner alchemy). The Twelve Links are : 1. ignorance ("avidyâ") : an old and sightless person with a stick : as the origin of the cycle, ignorance is the root-cause of all suffering, both mental & emotional. Innate ignorance is a state of distraction & confusion caused by being unaware of the true nature of phenomena. As a result of this ignorance, one "imputes", "imagines" or "hallucinates" a dual world (divided in a substantial subject & a substantial object), causing imaginary ignorance. The man is unable to see, yet believes he can use his stick. The small area covered by the stick is what the blind actually know, which is very limited. Likewise, the ignorant invent a dual world, locking themselves up within its narrow confines ; 2. volitional (karmic) formations ("samskârâ") : a potter : throwing all kinds of pots on his wheel, the potter represents the accumulation of conditioned, karma-bearing actions or impulses, manifesting in body, speech & mind as a result of ignorance. These can be virtuous (good karma), neutral or negative (bad karma). The form of the pot is the result of the activities of the potter. Too much or too little pressure makes an ugly pot. Likewise, because the ignorant exist in their make-up reality, the form of their experiences are co-relative with their own activities, whether physical, verbal (energetic) or mental ; 3. consciousness ("vijñâna") : a tree and a monkey jumping from branch to branch : the monkey seizes a fruit, plucks it and takes a bite while another fruit catches its eyes. It dashes off towards it, disregarding the fruit just plucked, swallowing it down in a hurry or dropping it. At the end of the day, there is a heap of half-chewed fruit left. Rebirthing consciousness is the result of past karma, arranging a new personality around this kernel. The jumping monkey represents the versatile, fluctuating, restless nature of deluded, karma-striken consciousness ; 4. name & form ("nâma-rûpa") : a boat with two people : as consciousness expands, it labels things. This name-giving is a form attributed to what appears, crystallizing phenomena into designated sensate & mental objects. The gross elements and the physical body are the result of this imputing activities of rebirth consciousness. So the two persons represent mind & body, the two major constituents of the individual ; 5. six sense bases ("śadâyatana") : a house with five windows & a door : the five senses (windows) and the door (mental sense) are the portals enabling consciousness to project outwards, allowing it to communicate with others, stepping outside itself to interact with the environment. The windows access the "lower" (visible) worlds, whereas the door of the mind offers an entry into the "higher" (invisible) worlds ; 6. contact ("sparśa") : a man & a woman embracing : the meeting of the senses with their object is made possible by the six sense bases, allowing physical interaction between beings ; 7. feeling/sensation ("vedanâ") : a man with an arrow in his right eye : because there is contact between beings, there are pleasant, neutral & painful sensations. The image conveys the strong vividness evoked by the sense organs ; The following two links tell us how we continue to create karma conditioning the future : 8. thirst/craving ("trisna") : a woman offering drink to a man slaking his thirst : the repetition of strong, afflictive emotions works addictive, and so conditioned by the experience of contact with an object, craving can be for (a) pleasure, (b) eternity, (c) existence & (d) annihilation (non-existence). These continue to produce negative effects ; 9. attachment/grasping/clinging ("upâdâna") : a woman grasping a fruit : craving itself begs for satisfaction and this leads to grasping or an exaggerated way to satisfy thirst. Once grasping is firmly established, we do anything to have our desires satisfied. Four kinds of clinging occur : (a) to sense pleasure, (b) to wrong views, (c) to rules & rituals & (d) to the notion of a soul or a self. These attachments cause an "automatic" form of rebirth, as by reflex ; The last three links point to issues related to this next life. They underline the notion of rebirth (in other words, the continuity of the continuum of consciousness), making it an integral part of Buddhist philosophy : 10. becoming/existence ("bhava") : a couple making love : conception occurs because during our previous life we constantly fed our karmic tendencies, which have now ripened. The conditions of our rebirth are thus determined by our karma, but conception (the actual, gross materialization of our rebirth consciousness) is determined by a couple making love ; 11. birth/rebirth ("jâti") : a woman in labour : the "newborn" is an "old born", carrying the karma of a previous existence. One is born in one of the six realms as a result of this old karma, and of all rebirths in "samsâra", being born as a human being with free choice offers the most opportunities for spiritual growth ; 12. old age & death ("jarâmarana") : a man carrying a corpse : it is in the nature of all transient things to end. Even gods die. When life-karma is exhausted, our gross body dies and the subtle elements are peeled away until the naked, empty & luminous nature of mind (the Clear Light of death) remains. (e) The View in the Heart-sûtra : The Four Profundities belong to the Heart Sûtra (Mahâprajñâpâramita-hridaya-sûtra), or "heartpiece of the perfection of wisdom sûtra", one of the shortest & most important sûtras of the Mahâyâna, belonging to the collection of forty sûtras constituting the Prajñâpâramitâ-sûtra. It formulates, in a very clear and concise way, the teachings on emptiness and was written in the first century CE. It is of major importance in Ch'an Buddhism, but is also widely discussed in the Vajrayâna. • The Profundity of the Ultimate : "Form is Empty." "Form" implies the five sense consciousnesses : Perception Sensation nose-consciousness of odors tongue-consciousness of tastes ions channels (?) body-consciousness of feels mechanical energy ear-consciousness of sounds eye-consciousness of lights All gross physical objects and a person's body are included. The aggregate of form is taken as the first basis for establishing emptiness. If form would be inherently existing or truly existing, i.e. substance-like, it would exist as it appears and be found from the side of the object itself without depending upon the apprehending consciousness. The body and its parts merely exist because they have a suitable basis to impute them, i.e. identify them and their dynamic functions. This is a merely nominalist designation, in no way establishing a static substance. Although a generic image of such a substance exists, it cannot be validated under ultimate analysis. While form appears to be static, it cannot be found to be so. The use of this false generic image is the false ideation to be removed. • The Profundity of the Conventional : "Emptiness is Form." Phenomena are seen as manifestations of emptiness. Ultimate truth and emptiness of inherent existence are synonyms. Emptiness is called a "sacred object truth" because its appearance to a non-conceptual direct perceiver is in accordance with its mode of existence. Unlike conventional truths, which do not appear as they ultimately are (they appear static but are in fact dynamic), emptiness does not conceal its true nature. To a wisdom-mind realizing emptiness directly, only emptiness appears and inherent existence does not appear (although conventional objects are known as they appear to deluded sentient beings, i.e. as inherently existing). Conventional truths are true with respect to the conventions of ordinary minds. Although they are deceptive regarding their mode of existence, they are not deceptive insofar as their logical identity & function go. If an object does not function as it appears, then a conventional falsehood is at hand (for example : a hallucination, a fata morgana, etc.). Such objects are "non-existent". Conventional objects are "truths for an obscurer" because self-grasping ignorantly conceives the apparent inherent existence, the substantial instantiation, to be true, which it is not. The profundity of the conventional aims to make clear the subtle nature of conventional objects. All conventional objects share the same fundamental, ultimate nature, emptiness. Each and every object is therefore not separate from its emptiness, but is an appearance arising out of its emptiness (cf. supra, the analysis of the Golden Lion). While objects do not inherently exist (First Profundity), we can establish the mere existence of form by pointing to its base of designation. This is a conventional appearance arising out of the ultimate nature of form, its subtle conventional nature (Second Profundity), just like the lion arises out of the gold ... • The Profundity of the Two Truths being the Same Entity : "Emptiness is not other than Form ..." If two phenomena are identical, they have the same generic image (logical identity & function). If they were not identical, they would have a different generic image. If two phenomena are not identical but are the same entity (like fire and its heat, or the body and its shape), this means they do not appear as separate to wisdom-mind, but appear as different to an ordinary conceptual mind. The same entity is at hand, but two different objects are known : the conventional nature or mode of existence is known by the deluded conceptual mind, the ultimate nature is known by enlightened wisdom-mind. • The Profundity of the Two Truths being Nominally Distinct : "... Form also is not other than Emptiness." Although the Two Truths are the same entity (Third Profundity), they are not identical. Being designated on the basis of the same form, they are two different epistemic isolates or two different objects of knowledge. The Two Truths can be distinguished on the basis of the difference between the conventional and ultimate nature of every object, not on the basis of two different objects (this would result in Platonism, positing a conventional world versus an ultimate world). The ultimate nature of an object is the object's emptiness of inherent existence established by wisdom-mind. The conventional nature of an object is the object's dependence on all other objects, i.e. it being other-powered. Hence, conventional objects are not independent substances, but interdependent, dependent-related phenomena. In order of increasing subtlety, this dependence of objects on other objects can be analyzed in five ways : 1. dependence on determinations : phenomena depend on laws determining their evolution from initial condition to outcome. These laws may be causal, interactive, teleological, statistical, etc. ; 2. dependence on parts : if phenomena were independent of parts, we would be able to remove the parts and find the phenomenon ; 3. dependence on names : phenomena can only be conceptualized by way of the names & labels given to them. Nameless phenomena cannot be objects of conventional reason ; 4. dependence on a basis of imputation : the names given to phenomena are given to them because some identity & some functions have been grasped. The latter serve as the basis of designation, allowing the conceptual mind to impute or posit the name ; 5. dependence on imputation by conceptualization : phenomena cannot be understood to depend on determinations, parts, names and a basis of imputations without the cognitive process itself allowing the conceptual mind to produce empirico-formal propositions about them. (f) The View in Hua-yen & T'ien-tai : In the Flower Garland School, the focus lies on the relationships between phenomena. The dynamic aspect of the "dharmas", featuring the dynamic interaction between whole (totality) and parts (specific), between singularity & multiplicity, brings in six characteristics shared by all possible phenomena : 1. universality : each phenomenon is a whole and should be considered as such ; 2. specificity : despite being a whole, phenomena have functional parts which can be posited distinct from the whole ; 3. similarity : these functional organs, although themselves wholes are nevertheless parts of the whole phenomenon ; 4. distinctness : each part of the whole has a distinct function, i.e. executes a specific, precise task ; 5. integration : all functional parts of the whole make up the whole ; 6. differentiation : every functional part has its particular place not shared by other parts. As these characteristics are shared by all phenomena, the universe is an organic totality interacting with its parts. Not a single phenomenon escapes this intrinsic dialectic between singular totality and multiple parts. In the School of the Celestial Platform, all phenomena are seen as an expression of the absolute of "suchness" ("tathatâ") or emptiness. Here, phenomena are not the focus, but their emergence from the ultimate. Temporal limitations are apparent existences emerging from emptiness, while the latter is not found "outside" phenomena. Each phenomenon shows how the absolute and the relative are the same reality. The Two Truths (ultimate and conventional) are in fact three truths : 1. the truth of emptiness : all "dharmas" lack independent reality ; nowhere is their a substance in existence, all things are process-like ; 2. the truth of temporal limitation : a "dharma" has a functional, apparent existence perceived by the senses & grasped by the mind ; the process-like nature of things falsely represents the state of affairs, for although things seem independent from an apprehending consciousness, they are not objective in that sense ; 3. the truth of the middle : the true state is not to be found elsewhere than in phenomena and so the absolute and phenomena are one ; suchness is not "another realm" or "another reality" above, beyond or next to phenomena, but coincides with them. 3.3 Absence of Essentialism in Classical Taoism. Let us now turn to the Taoism of Lao-tzŭ and Chuang-tzŭ and understand their take on the lack of inherent existence or the absence of essentialism, i.e. the rejection of the philosophical idea objects have a "support", essence ("ousia") or Archimedean point to hold on to. (a) The Nameless for Lao-tzŭ : Lao-tzŭ makes clear the Way, the Tao, is "nameless", "formless", "imageless", "invisible", "inaudible", etc. This comes down to saying the Tao is "Nothing" ("wu"), not to be understood as naught (zero), but as no-thing, undifferentiated. This Nothing, the One, is the beginning of the difference (the Two) between Heaven and Earth in potentia ("yu"). However, in no way is the Tao to be viewed as a substance, or a fixed, unchanging entity, quite on the contrary. The absolute Tao self-determines itself and is the Gateway of Myriad Wonders, or the foreboding of all things as sheer possibility. For Chuang-tzŭ, the Tao in its absoluteness defies all verbalization and language. At the level of language, the Way turns into a concept like "absolute" and then is exactly at the same rank as any other concept. To say the Tao is "non-differentiated", meaning there is no distinction between anything there, is no less a cognitive act as its opposition, "differentiated". The latter statement is typical for the empirical, common sense level of discourse, whereas the former points to the ontological indifferentiation characterizing the highest ontological level of the Tao. Unfortunately, although it points to this, it is not a well formed expression of this level, for it is nothing more than a contradiction of "differentiated". "So we posit Beginning. (But the moment we posit Beginning, our Reason cannot help going further back and) admit the idea of there having been no Beginning. (Thus the concept of No-Beginning is necessarily established. Not the moment we posit No-Beginning, our logical thinking goes further back by negating the very idea which it has just established, and) admits of there having been no 'there-having-been-no-Beginning'. (The concept of 'No No-Beginning' is thus established.)" Chuang-tzŭ : Chuang-tzŭ, section 2 (translation by Izutsu). Let us go through these steps : 1. the concept of "beginning" is the initial point of the world of "being". This is a relative concept, opposed to no-beginning ; 2. the concept of "no-beginning" is the negation of "beginning" and also a relative concept, and so we remain on the same logical level. To stop this circular process from "beginning" and "no-beginning" and back, arriving at an absolute "no-beginning", we have to transcend it by negating "no-beginning" ; 3. "no no-beginning" is disclosed in an intuitive way, indicating the grasp of logical reasoning has been exceeded. "In the same manner, (we begin by taking notice of the fact that) there is Being. (But the moment we recognize Being, our Reason goes further back and admits that) there is Non-Being (or Nothing). (But the moment we posit Non-Being we cannot but go further back and admit that) there has not been from the very beginning Non-Being. (The concept of No-(Non-Being) once established in this way, the Reason goes further back and admits that) there has been no 'there-having-been-no-Non-Being' (i.e. the negation of the negation of Non-Being, or No-No Non-Being)." The steps here are : 1. we posit "being", contradicted by "non-being" ; 2. we posit "non-being", contradicted by "no non-being" ; 3. we contradict "no non-being" and arrive at "no no-non-being", the absolute characterization beyond all possible further logic. The Tao in its original absoluteness, or absolute Tao, is conceptually the negation-of-negation-of-negation. The opposition of "being" and "non-being", i.e. this negation is itself negated. So the absolute Tao is not simply "nothing" or "non-being", but a transcendent, absolute Nothing lying beyond the relative opposition between "being" and "non-being". If we refuse to transcend the level of logic, the absolute characterization of the Tao will be naught, i.e. "no no-non-being" will equal zero. As such it can not do justice to the transcendent reality of the Tao in its absoluteness. The conceptual activity of the mind proves powerless in grasping this ultimate, "nameless" absolute Tao, i.e. the Way as it really is. (c) Classical Taoism and Śûnyatâ : The parallels with emptiness ("śûnyatâ"), "Dharmakâya" and "nirvâna" are clear. Ultimate reality cannot be conceptualized, and the best we can do is eliminate substantial concepts by ultimate analysis. When the mind is free from "Kaśyapa, it is like this. For example, two trees are dragged against each other by wind and from that a fire starts, burning the two trees. In the same way, Kaśyapa, if You have correct analytical discrimination, the power of a noble being's wisdom will emerge. With its emergence, correct analytical discrimination will itself be burned up." Śâkyamuni : Kaśyapa Chapter Sûtra. "Being" and "non-being" are relative concepts. Both belong to the level of conventional knowledge. Ultimate truth is not opposed to conventional truth, just as "nirvâna" is not opposed to "samsâra". If this were the case, ultimate truth would be relative to conventional truth and this would eclipse the true wisdom at hand. The point is to thoroughly transcended the oppositions prevalent on the common sense conventional level. Applying the logic of Nâgârjuna is accepting one cannot say emptiness is A, -A, not A and -A, nor not (A and -A). Directly seeing emptiness is ending Kamalaśîla's (ca. 700 - 750 CE) Path of Preparation and "burning up" substantial instantiation. This is a cognitive, but non-conceptual act. Taoism and Buddhism agree : the fundamental nature of phenomena is the absolutely Absolute, or emptiness ("nirvâna" or the "Dharmakâya"), the absolute Tao. After having reasoned his way up to the ultimate negation, Chuang-tzŭ typically asserts the futility of reasoning. He abandons all logical thinking concerning the Tao and immerses ecstatically in the non-conceptual, purely intuitive knowledge of the Way. Only in this way is a direct contact (what the Buddhists call "seeing") of the Tao possible. "The Great Way is not named ; Great Discriminations are not spoken ; Great Benevolence is not benevolent ; Great Modesty is not humble ; Great Daring does not attack. If the Way is made clear, it is not the Way. If discriminations are put into words, they do not suffice. If benevolence has a constant object, it cannot be universal. If modesty is fastidious, it cannot be trusted. If daring attacks, it cannot be complete. These five are all round, but they tend toward the square. Therefore understanding that rests in what it does not understand is the finest. Who can understand discriminations that are not spoken, the Way that is not a way ? If he can understand this, he may be called the Reservoir of Heaven. Pour into it and it is never full, dip from it and it never runs dry, and yet it does not know where the supply comes from. This is called the Shaded Light." Chuang-tzŭ : Chuang-tzŭ, section 2 (translation by Watson). Just like emptiness, the Tao in its ultimate reality transcends conceptual reasoning. This conclusion of Chuang-tzŭ forms the starting-point of Lao-tzŭ. Every name given to the Tao is manmade, and as one cannot refer to the Way without naming it, the designation "Tao" is not satisfactory. "The 'way' which can be designated by the word 'way' is not the real Way. The 'name' which can be designated by the word 'name' is not the real Name." Lao-tzŭ : Tao-te ching, chapter 1. The "Way" is not a human "way" or an ethical "way" as it was given by Confucius and his school. The "name" which is not the "real Name" refers to Confucian categories like "benevolence", "righteousness", "wisdom", etc. These cardinal virtues are not aimed at. The Way is not a principle of ethical conduct. Although for Confucius, the principle of ethical conduct was a reflection in human consciousness of the highest law of the universe, this "cosmic" conception is not the real Way, for the latter is essentially unknown and unknowable in a conceptual way. Lao-tzŭ goes even so far as to say "benevolence" and "righteousness" (the names of ethical conduct) arise when the great Way declines ! So, the only "real Name" ("ch'ang ming") is the absolute Name assumed by the Tao in its absoluteness, and this is, paradoxically, "nameless" i.e. beyond conceptual reason. Hence, the absolute Tao is the "Mystery of Mysteries", but also the "Gate of all Wonders" ! (d) the Non-Essentialist & Non-Conceptual Absolute Tao : We may conclude the likeness of the Tao in its absoluteness (i.e. the absolute Tao before any differentiation happened, i.e. before Heaven & Earth) is an absolute indifferentiation. This highest, ultimate stage of the absolute is non-essentialist and beyond conceptualization. non-essentialist : the absolute Tao has no "boundaries", no rigid, inflexible characteristics existing from their own side. Although essentialism is the view of the common man, the sage realizes this is not the ultimate view of things. There are no watertight compartments in existence becoming crystallized into fixed "things", given a "name" representing an "essential" fixity ensuring it from disintegration. All things ontologically interpenetrate one another, and so all things can be transformed in all other things, deemed impossible in essentialism (ontologizing or reifying the logical principle of identity). The original non-differentiated whole represented by the absolute Tao must not be divided up into fixed, unalterable substances or essences defining "this" or "that" once and for ever. Indeed, things are formed by their being designated by "this" or "that" particular name by virtue of relative social conventions. By fixing objects ontologically, all their other possibilities are nullified, and transformation (change) becomes impossible (a carved piece of wood has stopped being uncarved, i.e. receptive of all other forms). Hence, just as Nâgârjuna and the Mâdhyamaka with him underline, by confirming substance one negates change and by confirming interdependent change one negates substance (in other words, by establishing isolated substances, one nullifies any possible change) ; non-conceptual : the absolute Tao does not transcend cognition, for in the nondual mode of cognition the sage intuitively & ecstatically apprehends it. Any conceptual "name" can however not be applied and if so, error is the outcome. The same critical sounds are heard in the writings of Tsongkhapa and his refutation of idealist Mâdhyamaka, Mind Only ontology and other-emptiness Buddhism. To posit a name regarding emptiness is to conceptualize it and to do so is to create problems for the absolute cannot be conceptualized. This does not mean the absolute Tao cannot be an object of cognition. If this were the case, it could not be approached. But it can, in an intuitive, nondual (not a-dual), direct & ecstatic manner, namely by enlightened wisdom-mind ! These equations show how the "nameless" absolute Tao and Buddha's "śûnyatâ" (emptiness) refer to one and the same thing. The connection with "pratîtya-samutpâda" (dependent arising) is also firm, for to posit essentialism (affirm inherent existence or the lack of emptiness as in eternalism) is to fix fundamentally dynamical things. Once fixed, changed cannot be thought. But as all things are constantly changing (cf. the doctrine of the I ching), nothing static can be found. Moreover, the absolute Tao, the "Mystery of Mysteries", cannot be an object of the conceptual mind. Like emptiness, it is apprehended ecstatically by wisdom-mind transcending meditation & post-meditation ! 3.4 Brother Buddhism & Sister Taoism. What have we learned ? Buddhism focuses on wisdom-mind, bringing the non-substantial nature of phenomena to the fore. This absence of inherent existence is approached (a) by way of ultimate analysis (a reductio or argumentum ad absurdum proving positing substance leads to absurd conclusion) and (b) by exhausting the "king of logics", dependent arising. While the latter brings in the central conclusion substance cannot coexist with dependent arising, the Buddhadharma analyses the latter for the sake of realizing emptiness. The interdependent nature of phenomena is subjacent to the ultimate nature of phenomena. Even Tsongkhapa confirms this, for according to him Buddha-mind witnesses emptiness only (while simultaneously knowing how interdependent phenomena appear to ever-deluded sentient beings). Taoism aims to understand the interdependent and changing nature of phenomena emerging from the empty, process-like absolute Tao. Interdependence is assessed by (a) apprehending how phenomena rise out of the absolute Tao (the object of the Flower Garland School) and (b) grasping how the ever-changing, ongoingness of the transformations of the "Ten Thousand Things" can be used to realize permanent health & longevity by way of Chi-circulation for the sake of fusing with the One (cf. infra) and attaining the state of the immortals ("hsien"). Identical fundamental categories (emptiness/absolute Tao and dependent arising/Tai Chi) are stressed differently. In Buddhism, realizing wisdom-mind stands out, in Taoism the 64 stages of the law of change & transformation. This reminds of Liu Hua-yang (1736 - 1846 ?) and his view on the complementarity between Buddhism & Taoism, seeing immortality and "Buddha-nature" as the same thing and stating Taoism is able to cultivate life, but not Buddha-nature, while Buddhism is able to cultivate the original spirit, but cannot lead to health or longevity. Hence, in a metaphorical language, and grosso modo, we may say Buddhism and Taoism are like brother and sister. The masculine (Solar) approach of Buddhism aims at the direct realization of wisdom-mind (discussing interdependence in a secondary way, namely insofar as the analysis of the eleven effects of ignorance is at hand), while the feminine (Lunar) approach of Taoism is focuses on the five-phase elemental cycle (Wood-Fire-Earth-Metal-Water) and the 64 stages of interdependence to realize immortality, another name for awakening. The ultimate nature of reality is realized by probing the harmony between Heaven & Earth, not by surging into Heaven while leaving Earth behind (cf. Buddhist renunciation & Tantra). Buddhism aims at Heaven, Taoism at Earth ... Buddhism wants to escape Earth, Taoism wants to harmonize Earth with Heaven to attain spiritual immortality. 4 The Tao : the Way in Absolute & Relative Terms. {Ø} > 1 > 2 > 3 > ... The Tao has an one absolute (non-differentiated) and various relative (differentiated) stages. These stages represent the absolute, self-existent Tao in various moments of self-determination. Each of them is the absolute Tao in a secondary, derivative and limited sense. The stage next to the absolute Tao, namely non-being or the One differs only slightly from the absolute Tao and so is almost the same. (a) {Ø} The Absolute Tao - Uncreated and Creating : The Absolute Tao or Mystery of Mysteries The Nameless as Beyond All the Great Mystery (black) & Gateway of Myriad Wonders (red) So far we discussed the absolute Tao, non-local, non-temporal, non-differentiated, nameless, and empty of substance or inherent existence, without permanent and unalterable distinctions. This absolute Tao is beyond conceptualization and object of ecstatic, nondual apprehension. "Even if we try to see it, it cannot be seen. In this respect it is called 'figureless'. Even if we try to hear it, it cannot be heard. In this respect it is called 'inaudibly faint'. Even if we try to grasp it, it cannot be touched. In this respect it is called 'extremely minute'. In these three aspects, it is totally unfathomable. They merge into One." Lao-tzŭ : Tao-te ching, chapter 14. The absolute Tao is not turned towards phenomena, nor is it wholly self-referential. This "abstract of abstractions" cannot be conceptualized and named. It is Nameless. To reach the ultimate and absolute stage of the Way, we have to negate the opposition between being and non-being, positing "no no-non-being". This level can only be apprehended ecstatically, and this absolutely ineffable Lao-tzŭ symbolically calls the "Mystery of Mysteries". Mystery ("hsüan") originally means black with a mixture of redness. The absolute, unfathomable Mystery or "black" does reveal itself, at a certain stage, as being "pregnant" of the "Ten Thousand Things" or "red" in their stage of potentiality. In the Mystery of Mysteries being and non-being are not yet differentiated, and in this state "these two are one and the same thing". Although the absolute Tao cannot be said to be turned towards the phenomena, in this utter darkness of the Great Mystery ("black"), a faint foreboding of the appearance of phenomena lurks ("red"). So the Mystery of Mysteries is also the "Gateway of Myriad Wonders". Hence, the "Ten Thousand Things" stream forth out of this Gateway ! So the absolute Tao ({Ø}) has two components : 1. a black component : the Great Mystery or ineffable utter darkness, absolutely invisible transcending being and non-being, the ultimate metaphysical state lacking even a shadow of possibility ; 2. a red component : the Gateway of Myriad Wonders, or the foreboding of all things as sheer possibility, pregnant with all things in potentia. This has again two components : the potential of non-being ("wu") and the potential of being ("yu"). (b) "1" WU : the One - Created Potential Non-Being : The One : The Nameless as Potentiality of Non-Being When Lao-tzŭ introduces the Way as "the Granary of the Ten Thousand Things" (chapter 62), he aims at a stage slightly lower than the Mystery of Mysteries, the absolute Tao. At this stage, the Tao begins to manifest its creativity. The image of a "granary" conveys the sense all things are contained therein, not actually but in a state of potentiality. He refers to this aspect of the absolute Tao as "the eternal non-being", or "wu". At this stage, the absolute Tao is potentially already Heaven and Earth, i.e. being. Hence, the non-being referred to is not a passive Nothing, pure negative absence of being or existence (naught or zero), but a "something" in the sense of an "act", the act of existence itself or Actus Purus. It exists as the very act of existing and making things exist. This is called "the One". "The Way does have a reality and its evidence. But (this does not imply that it) does something intentionally. Nor does it possess any (tangible) form. (...) It is the thing that makes the Heavenly Emperor divine. It produces Heaven. It produces Earth." Chuang-tzŭ : Chuang-tzŭ, section 6 (translation by Izutsu). This Actus Purus does not exist as a substance. In order not to reify it by way of concepts, the One can only be ecstatically intuited by "sitting in oblivion" (Chuang-tzŭ). The One is darkness not because it is deprived of light, but because it is too full of light, too luminous, i.e. Light Itself. "A 'way' which is (too) bright seems dark." Lao-tzŭ : Tao-te ching, chapter 41. From the point of view of the One itself, the One is bright. From the point of view of man, it is dark or Nothing. The One is the Great Singularity, a homogeneous & single plane not externally articulated, a unity ready to diversify, the absolute Tao as the principle of eternal and endless creativity. From the absolute Tao the One emerges as the unity of all things, the primordial unity in which all things lie hidden in a state of "chaos" without being as yet actualized as the Ten Thousand Things. The One is the Unbounded Wholeness because it embraces in itself "the Ten Thousand Things under Heaven" (Tao-te ching, chapter 40) in the state of pure possibility or potency. The One is the "Urgund" of being. If the absolute Tao is called "Nameless" because it is beyond all possible names, the One is called "Nameless" because for human consciousness it is as Nothing. "The Way begets 'one' ; 'one' begets 'two' ; 'two' begets 'three' ; and 'three' begets the Ten Thousand Things. The Ten Thousand Things carry on their backs the Yin energy, and embrace in their arms the Yang energy and these two are kept in harmonious unity by the (third) energy emerging out of (the blending and interaction of) them." Lao-tzŭ : Tao-te ching, chapter 1 (translation by Izutsu). (c) "2" YU : The Two - Created Potential Being : The Two : The Named as the Mother of the Ten Thousand Things The Potentiality of Being "The Nameless is the beginning of Heaven and Earth. The Named is the Mother of the Ten Thousand Things." Lao-tzŭ : Tao-te ching, chapter 1. When it enters its first stage of "pure" self-manifestation or mere self-determination, Lao-tzŭ admits the One or active non-being assumes a positive "name". This name is "existence" or "being" ("yu"). The latter is also called "Heaven and Earth" ("t'ien ti"). The Way at this stage is not yet the actual order of Heaven and Earth, but only all possible things as "pure" being, i.e. again in potentia. The One begets the Two : Heaven (Yang) and Earth (Yin), the cosmic duality. They are the self-evolvement of the absolute Tao, the Way itself. The One is the initial virtual point of self-determination of the Way, the Two brings about (as a mother) the possibility or probability of actuality and carries this over into actual reality. In this way, the One is the ontological ground of all things, acting as its ontological energy, while the Two develops this activity into a particular ontological structure, Yin and Yang and the Three, i.e. the blending & interaction between these ("Tai Chi"). Hence Heaven is limpid and clear, and Earth is solid and settled ... The driving force giving to all things birth, growth, flourishing and return to its origin, allowing each and every thing to possess its own characteristics or nature, is nothing else than the absolute Tao as it actualizes Itself in a limited way in every thing or "Tai Chi", the universe of distance, dimensions, time, space and the world of interchangeable extremes where nothing is absolute. "The Way is permanently inactive, yet it leaves nothing undone." Lao-tzŭ : Tao-te ching, chapter 37. This happens naturally, without the Way "forcing" anything. Non-doing ("wu wei") is precisely letting each of the Ten Thousand Things be what they are of themselves. As the Way is not conscious of its own creative activity, it is unconscious of its results either. Infinitely gracious to all things, its activity is beneficial to all without counting the benefits and favors it never ceases to confer upon all things. "It works, yet does not boast of it. It makes (things) grow, and yet exercises no authority upon them. This is what I would call the Mysterious Virtue." Lao-tzŭ : Tao-te ching, chapter 10. 5 Taoist Metaphysics : Objective & Subjective Considerations. Classical Taoism approached the absolute Tao from two directions : Lao-tzŭ articulated the creative activity of the Way (cosmology), while Chuang-tzŭ was more interested in the epistemological stages involved in the step-by-step ecstatic absorption into the Tao. (a) The Cosmological Approach of Lao-tzŭ : The objective side was the object of the previous paragraph. Lao-tzŭ : the Cosmological Approach Great Limitless emptiness Mystery of Mysteries the absolute Tao The One potential non-being or WU The Two potential being or YU actuality dependent Tai Chi Great Ultimate The Five Forces (b) The Epistemological Approach of Chuang-tzŭ : "When the discriminating spirit does not arise, aberrant fire goes out ; when aberrant fire goes out, true fire arises. When true fire arises, the harmonious energy is fertile and the mechanism of life does not cease ; so there is hope of attaining the universal Tao." Chuang-tzŭ is interested in the epistemological process preceding the final stage of illumination and tries to describe the experiential content at hand symbolically. The first point he considers is the centrifugal activity of the mind establishing boundary, fixed structure & limitation. "The Way has absolutely no 'boundaries'. Nor has language absolutely any permanency. But (when the correspondence becomes established between the two) there arises real (essential) 'boundaries'". Chuang-Tzŭ : Chuang-Tzŭ, II. Futile verbalizations are caused by thinking one is a self-subsistent entity endowed with ontological independence (i.e. existing from its own side). This ego, the point of co-ordination of the disparate physical & mental elements of personality will cause the mind ("hsin") of an ordinary person to constantly move, going this way and then another way etc., and this in response to myriad impressions coming from the outside, attracting attention and arousing curiosity unceasingly. This centrifugal movement of the mind is like "sitting-galloping" ("tso ch'ih"), for while the body is sitting still, the mind is running around. This basic situation of the deluded mind is "shin hsin", or "making the mind one's own teacher", a disastrous situation. On an intellectual level, such a turbulent, dispersed mind has taken on a fixed, coagulated form, it is a "finished mind" ("ch'êng hsin"). Discriminating and passing judgments, such a mind falls deeper & deeper into the limitless swamp of ridicule and absurdity. "Everybody follows his own 'finished mind' and venerates it as his own teacher. In this respect we might say no one lacks a teacher." Lao-tzŭ shares this view. He writes of a "constant or unchangeable mind" ("ch'ang hsin") loosing its natural "softness". Unnatural rigidity goes hand in hand with distinguishing and discriminating, perceiving right and wrong, good and bad etc. "Thus the 'sacred man', while he lives in this world, keeps his mind wide open and 'chaotifies" his own mind toward all." Lao-tzŭ : Tao-te ching, chapter 49. When the cognitive act, usually tending toward the outside, is curbed and brought back toward the inside, "illumination" ("ming") is the outcome. The centrifugal tendency must be turned into the centripetal direction. "He who know others is a 'clever' man, but he who knows himself is an 'illumined' man." Lao-tzŭ : Tao-te ching, chapter 33. The Tao is present in the "inside" of very human being. All humans are able to intuit the palpitating life of the Tao working there. The further one moves "outside", the less one is in touch with the Tao. "Without going out of the door, one can know everything under Heaven. Even without peeping out of the window, one can see the working of Heaven. The further one goes out, the less one knows." Lao-tzŭ : Tao-te ching, chapter 47. Chuang-tzŭ is interested in the process by which the phenomenal "returns" to the original state of absolute Unity, to the One. In order to do this, one has to totally "forget" the mental activity of the ego, resulting in "the void" ("hsü"). In this subjective spiritual state or attitude nothing obstructs the all-pervading activity of the Tao, and the activity of the mind corresponds with the structure of the Way itself. The void has no positive sense, it is totally negative (not virtual or potential), but identical with naught (mathematical zero or the total absence of nothingness). Then, "sitting-galloping" becomes "sitting-forgetting" or "sitting in oblivion". "It means that all the members of the body become dissolved, and the activities of the ears and eyes become abolished, so that the man makes himself free from both form and mind, and becomes united and unified with the All-Pervader." The All-Pervader ("ta t'ung") is "ta Tao", the Great Way (cf. Ch'êng Hsüan Ying), for the Way pervades all things and enlivens them. One who lost ego rediscovers a "cosmic ego", freely transforming into all things transforming themselves into each other. "Being unified, You have no liking. Being transmuted, You have no fixity". Thus transformed, the mind is like a clear, polished mirror, a firmly closed empty room mysteriously & calmly illuminating itself with a while light of its own. "Look into that closed room and see how its empty 'interior' produces bright whiteness. All blessings of the world come in to reside in that stillness." Chuang-tzŭ : Chuang-tzŭ, section 4 (translation by Izutsu). Such a man cannot be intruded by things replacing one another before his eyes, for he maintains his "innermost treasure" "in a peaceful harmony with (all these changes) so that he becomes one with them without obstruction, and never loses his spiritual delight. (...) Such a state I would call the perfection of the human potentiality." (Chuang-tzŭ, section 5). Chuang-tzŭ : the Epistemological Approach 5. no death and no life 4. perceiving the Oneness experiencing the Tao 3. to put life outside the mind opening the "inner eye" 2. putting the things outside the mind putting the world outside the mind Taking the ascending course, Chuang-tzŭ describes the stages by way of a conversation between the old Nü Yü and Nan Po Tzŭ K'uei, astonished at the young complexion of the old man. These stages are as follows : 1. "putting the world outside the mind" & 2. "putting the things outside the mind" represent the external aspects of the world. Forgetting the world is the first stage of renunciation. Here, the "world" implies impersonal objects far from the mind. Next come the things needed in daily life, close to the ego. These are more difficult to forget, for they serve us daily and are very familiar to us. These first two return in Buddhism as the Eight Worldly Concerns, all characterized by clinging (affirming, craving) & aversion (negating, rejecting) : 1. Attachment to getting & keeping material things. 2. Aversion to not getting material things or being separated from them. 3. Attachment to praise, hearing nice words, and feeling encouraged. 4. Aversion to getting blamed, ridiculed, and criticized. 5. Attachment to having a good reputation. 6. Aversion to having a bad reputation. 7. Attachment to sense pleasures. 8. Aversion to unpleasant experiences. 3. "to put life outside the mind" : At this stage, the common ego is dropped and disappears from consciousness. When this happens, illumination immediately follows, for one's "inner eye" is opened and the "first light of dawn breaks through". The next two "stages" happen simultaneously and so occur all together after this inner eye has been opened : 4. "perceiving the absolute Oneness" : When all and things become absolutely one, the opposition between object & subject is gone, for the seer and the seen are completely unified. Distinctions between "this" and "that" vanish, and the original unity of the One is restored in consciousness, timeless & abiding in the Eternal Now. 5. "no death and no life" : As time has been transcended, all sense of sequence (past, present, future) is nullified and consciousness is in the midst of the Way, which is beyond life and death. Epistemological multiplicity is brought back to the absolute unity of the One. This is not a static state, but a dynamic non-movement, concealing within itself endless possibilities of action. This unity is itself a potential multiplicity and a stillness concealing a possible unrestrained expression. "That which kills life does not die. That which brings to life everything that lives does not live. By its very nature it sends off everything, and welcomes everything. There is nothing that it does not destroy. There is nothing that is does not perfect. It is, in this aspect, called 'Commotion-Tranquility' ('ying ning'). The name Commotion-Tranquility refers to the fact that it sets in turmoil and agitation and then leads them to tranquility." A certain parallel between these stages and the Five Paths of Kamalaśîla (ca. 700 - 750 CE) are apparent. The first two cover issues dealt with on the Path of Accumulation. The Path of Preparation preludes the opening of the inner eye, an event happening on the Path of Seeing. Entering this Path immediately initiates the work of the Ten Stages of the Superior Bodhisattva, finished in nine steps on the Path of Meditation (perceiving the One) and the Path of No More Learning (no death, no life). 6 Ontological Tradition of the West. "Chinese religion and philosophy did not have the other-worldly outlook of the Mesopotamian-Mediterranean beliefs, since in Chinese thought spirit and matter were not sharply divided ; both were held to operate together in the world of Nature, so when the body had been sufficiently purified and etherealized it could continue to exist in this world, or in the heavens, or both." Cooper, J.C. : Op.cit., 1984, p.35. Broadly speaking, the metaphysical tradition of Mediterranean thought, ranging from the start of the Pharaonic Period (ca. 3000 BCE) to the publication of Process and Reality (1929), can be characterized as substantialist, designating objects & subjects by attributing a fixed "core" or "essence" to them, an unchanging support as it were carrying their accidents, attributes or predicates. After nearly five millennia, this grand project can be said to have failed ! No reliable substance could be isolated. Although this substantialist tradition evidences a vast complexity, it can be divided in four phases : Ancient Egyptian Heliopolitanism, Hellenism, Abrahamism & Modernism. Let me first summarize these grosso modo : Heliopolitanism : before creation, in the vast, dark & undifferentiated Nun, the primordial ocean, the primordial Atum or "becoming totality" was "afloat" in a preexistent fashion. Heaven ("pet") and Earth ("ta") were not yet divided. At some point, Atum self-generated as the beginning of light (Re) and simultaneously divided in a company of nine primordial forces of nature ("paut" or "Ennead"), eventually actualizing as "Horus" ("heru"), "he upon high", the origin of the world-order presided by the divine king ("nesu"). The sky of Re was the world of self-subsistent lights ("akhu"), the ontological roots of all possible being. While this scheme is proto-substantial, it betrays a shamanistic intent bringing it close to what we found in Taoism ; Hellenism : with the advent of formal thought, the pre-existent order is an Olympic world of being, and although Heraclitus evidences a non-substantial exception -even more apparent in the Orphic & Dionysian mysteries- the overall Hellenic concept is substantial, finally identifying an "idea of ideas" (Plato's "agathon") or an "Unmoved Mover" (Aristotle) at the origin of existence. In the philosophy of Plotinus, Greek substantial thinking reached its climax, positing a substance of substances (the Plotinian One) beyond the world of ideas. With this philosophy, the Greek inability to think emptiness, affirming relation as of lesser importance than independence, got epitomized ; Abrahamism : the three religions "of the book" (Judaism, Christianity & Islam), inspired -in various meandering courses- by Heliopolitanism and the Ancient Egyptian heritage, worked out an onto-theology, an ontology of an objective, self-subsisting, substantial Supreme Being, conceptualizing it (a) in terms of the (neo)Platonic tradition, i.e. as a "summum bonum" (cf. Philo of Alexandria, Al-Kindi, Augustine) or (b) in tune with the Peripatetic emphasis on empirical reality (cf. Maimonides, Averroes, Thomas Aquinas). This ultimate God-as-substance created the world "ex nihilo", and is believed to be the ontological "imperial" root of all possible existence. Only in the more mystical traditions of these faiths do we find another, less positive affirmation of this substance-God's necessary supremacy : the negative veils "Ain", "Ain Soph" and "Ain Soph Aur" in Qabalah (Luria), the ineffable hyper-existence of God in negative theology (ps.-Dionysius the Areopagite, Marguerite Porete) and the unknowability of the Divine essence in Sufism (Ibn Arabi). But these refined mystical "apophatic" speculations were muted by the overall "katapathic" noise produced by the theologians, as always preoccupied by apologetic concerns and manipulative, power-based mass-indoctrination ; Modernism : from the Renaissance onwards, empirism and rationalism become the two main organs of scientific thought, discarding fideism and the "revealed truths" of scriptures (Torah, New Testament, Koran), found to be man-made literary compilations of small scientific interest. Descartes designates three substances ("res extensa", "res cognitans" and God), whereas the empirists (Locke, Berkeley, Hume) try to erect the foundation of true knowledge on "sense data" (impressions derived from the five senses). Although intuitive knowledge is still part of the equation (cf. Cusanus, Spinoza), its role become fainter and then disappears. Finally, with Kant, substantialism comes under severe critical attack and neo-Kantianism validly argued why the possibility of knowledge cannot find a subjective (ideal) or objective (real) "sufficient ground" in anything outside the cognitive apparatus itself (cf. Criticosynthesis, 2008). Insofar as postmodernism does not leap into irrationalism (as protest philosophers like Schopenhauer, Kierkegaard, Nietzsche & Bergson had done in the 19th century), a new kind of modular modernism or hyper-modernism may see the light. Ending substantialism, process philosophy emerges as an alternative embracing the conclusions of relativity & quantum mechanics, heralding a radical paradigm shift. And although as yet the West has not really come to terms with process (still clinging to quasi-substantial forms of cognizing), there can be no doubt the process paradigm, integrating the core teachings of the Eastern "dharmic" view, as it appears in Buddhism & Taoism, is the paradigm of the future, integrating physical & social sciences, as well as the emerging green revolution of ecology. (a) Ancient Egyptian Heliopolitan Cosmo-Metaphysics : Nun, Atum-Re, the Ennead & Horus. Before rational thought rose as the result of the "Greek miracle", ante-rationalism (featuring mythical, pre-rational & proto-rational strands of cognition) dominated Antiquity. The oldest, most outstanding and longest example of this way of cognizing is given with Ancient Egyptian civilization. For more information : www.maat.sofiatopia.org. Contrary to other cultures of the time, the Egyptians had a very pronounced interest in sapience (given formal thought was absent, the word "philosophy" is avoided), a fact recently acknowledged : Hornung, 1992, p.13. Following characteristics of Egyptian thought played a prominent role in the constitution of Greek philosophy : • the words of god and the love of writing : in Ancient Egypt, it should be emphasized, both spoken and written words were very important : hieroglyphs were "divine words", endowed with magical properties, "set apart" and distinguished from everyday language and writing (in hieratic and later demotic). Pharaoh Unis (ca. 2378 - 2348 BCE) was the first to decorate his tomb with hieroglyphs to assure his ascension and subsequent arrival in heaven. Even if the offerings to his Ka would end, the hieroglyphs -hidden in the total obscurity of the tomb- contained enough "inner" power ("sekhem") to assure Wenis' felicity ad perpetuam ... While producing a vast literary corpus, Egyptian thought never reached the rational mode of cognition. Egypt's attachment to the contextual and the local, as well as the special pictorial nature of the "sacred script", all point to an ante-rational mentality, rooted in the mythical, pre-rational (pre-concepts) and proto-rational (concrete concepts) layers of early African cognition ; • accomplished discourse : the fundamental categories of Memphite philosophy were "heart/tongue/heart" insofar as theo-cosmology, logoism and magic were at hand and "hearing/listening/hearing" in moral, anthropological, didactical and political matters. The first category reflected the excellence of the active and outer (the father), the second the perfection of the passive and inner (the son). The active polarity was linked with Pharaoh's "Great Speech", which was an "authoritative utterance" ("Hu") and a "creative command" no counter-force could stop ("heka"). The passive polarity was nursed by the intimacy of the teacher/pupil relationship, based on the subtle and far-reaching encounters of excellent discourse with a perfected hearing, i.e. true listening. The "locus" of Egyptian wisdom was this intimacy ; • truth and the plummet of the balance : in Middle Egyptian, the word "maat" ("mAat") is used for "truth" and "justice" (in Arabic, "al-haq", is both "truth" and "real"). Truth is linked with a measurable state of affairs as given by the balance : "Pay attention to the decision of truth Papyrus of Ani, Plate 3 - XXVIIIth Dynasty - British Museum This exhortation summarizes the practice of wisdom and its pursuit of truth found in Ancient Egypt. It also points to their philosophy of well-being and art of living happily & light-heartedly (for the outcome of the weighing is determined by the condition of the heart or mind alone). In this short sentence, the "practical method of truth" of the Ancient Egyptians springs to the fore : concentration, observation, quantification (analysis, spatiotemporal flow, measurements) & recording (fixating) with the sole purpose of rebalancing, reequilibrating & correcting concrete states of affairs, using the plumb-line of the various equilibria in which these actual aggregates of events are dynamically -scale-wise- involved, causing Maat (truth and justice personified as the daughter of Re, equivalent with the Greek Themis, daughter of Zeus - cf. "maâti" as the Greek "dike") to be done for them and their environments and the proper Ka, at peace with itself, to flow between all vital parts of creation. The "logic" behind this operation involves four rules : 1. inversion : when a concept is introduced, its opposite is also invoked (the two scale of the balance) ; 2. asymmetry : flow is the outcome of inequality (the feather-scale of the balance is a priori correct) ; 3. reciprocity : the two sides of everything interact and are interdependent (the beam of the balance) ; Although Egypt had five schools of divinity (Memphis, Heliopolis, Hermopolis, Abydos & Thebes), the Pharaonic cult, with the divine king as "son of Re", was intimately connected with Heliopolis ("Iunnu"), the city of the supreme deity of the Pantheon, Re. The Heliopolitan cosmogony developed there dominated Egyptian thought for three millennia and indeed the whole Mediterranean basin, in particular its monotheisms (Judaism, Christianity, Islam) and metaphysics. The oldest text available to evidence this connection is the Pyramid Text of Unas. Plan of the Valley temple and Pyramid-complex of Unas (after Lehner, 1997, p.154) King Unas, Unis or Wenis (ca. 2378 - 2348 BCE) was the last Pharaoh of the Vth Dynasty (ca. 2487 - 2348 BCE). His pyramid at Saqqara, called "Perfect are the Placed of Unas", is at the South-western corner of Djoser's enclosure and the smallest of all known Old Kingdom pyramids. King Unas was also the first to include hieroglyphic inscriptions in his royal tomb, namely in its corridor, antechamber, passage-way & burial-chamber. The area around the sarcophagus and the serdab are left uninscribed. This coincides with a general increase of writing in general in the later Vth Dynasty. The Unas text, carved and filled with blue pigment, contains, in 228 of the 759 (Faulkner, 1969) known "utterances", the first historical account of the (Heliopolitan) religion of the Old Kingdom, in particular its royal cult. It precedes the textualization of the Vedas, reckoned at ca. 1900 BCE (Unas died ca. 2348 BCE). "The Pyramid Texts reflect not only an Egyptian vision of the afterlife but also the entire background of Old Kingdom religious and social structures, and they incorporate an ancient worldview much different from that of more familiar cultures." Allen, 2005, p.13. Technically, the Pyramid Texts are a corpus consisting of "utterances" or "spells", so called because the expression "Dd mdw" ("Dd" = "word" ; "mdw" = "speech"), "to say" or "to say the words", i.e. the sacred words to be recited is, as a rule, atop most texts, allowing for a classification. The one introduced by Sethe (1910, with 714 utterances), is an inventory of all texts, irrespective of the kind of text or its placement in the tombs. Discovered by Maspero in 1881, the Unas text had been buried and left undisturbed for ca. 4200 years. An untainted primary religious source ! Together with the texts found in the tombs of King Unas' successors, Pharaohs Teti, Pepi I, Merenre & Pepi II (ca. 2270 - 2205 BCE) of the VIth Dynasty, these compositions form the first known religious corpus in world literature, as well as the earliest example of extended writing worldwide (including a rich pallet of various styles, forms & intentions) "... the Unas texts were evidently regarded as an integral work in their own right, and seem to have acquired 'canonical' status ..." Naydler, 2005, p.149. Maspero (1884, p.3) assumed these texts were exclusively funerary and divided them in ritual texts, prayers and magical spells. In the previous century, authors realized they include drama, hymns, litanies, glorifications, magical texts, offering rituals, prayers, charms, divine offerings, the ascension of Pharaoh, his arrival & settling in heaven, etc. They offer a glimpse of an African, ante-rational perspective on death, rebirth & illumination. According to Allen (2005), the Pyramid Texts : "are largely concerned with the deceased's relationship to two gods, Osiris and the Sun. Egyptologists once considered these two themes as independent views of the afterlife that had become fused in the Pyramid Texts, but more recent research has shown that both belong to a single concept of the deceased's eternal existence after death - a view of the afterlife that remained remarkably consistent throughout ancient Egyptian history." Allen, 2005, p.7. • the Duat (burial-chamber) : though a part of the world (Earth), but neither Nun or sky, the Netherworld is inaccessible to the living and outside normal human experience. It is separate from the sky and reached prior to it. The Field of Reeds is the realm of the deceased and the deities and the mystery of Osiris. The Horus-king has perpetuated offerings, and stands at the door of the horizon to emerge from the Duat and start his spiritualization ; • the Imperishable sky (northern corridor) : the process of transfiguration (ultimate spiritualization) being completed, the Akh-spirit leaves the tomb and ascends to the northern stars, becoming an Imperishable One. Eyre (2002) suggests the training and initiation of the funerary priests points to this-life rituals. Perhaps the king rehearsed his forthcoming burial during life ? Eyre, 2002, p.72. Recently, Naydler (2005), by suspending the funerary interpretation, evidenced that the Pyramid Texts in general and the Unas texts in particular, reveal an experiential dimension, and so also represent this-life initiatic experiences consciously sought by the divine king (cf. Egyptian initiation). These may be classified in two categories : Lunar Osirian rejuvenation (cf. the texts of the burial-chamber), already at work in the Sed festival, and Solar Heliopolitan ascension (cf. the texts in the antechamber). Apparently the former was celebrated regularly, whereas the latter is foremost funerary. Egyptian spirituality was two-tiered : 1. VIA THE MOON : the (lower) sky of Osiris : the ultimate state of human blessedness is to live the life of an "Osiris NN", with a court, humbling servants and a kingdom situated in the vast darkness of the Duat (like creation is a bubble of moist air suspended in chaos). Even the smallest offer made with a sincere heart during earthly life might be enough to be helped by Isis or Osiris, and so the commoners made sure the holy family would notice them. This economy is inclusive of everyman, but conditional, except for Pharaoh - the Eye of Horus ; 2. ENDING IN THE SUN : the (upper) sky of Re : the sky of Osiris and the sky of Re are proximate, and after the highest spirituality of servitude has been fulfilled, the "Ba" of the deceased is transformed, in the horizon, into an "Akh" of Re, sailing, among the other pure beings of light, on the Bark of Re, illuminating the beings of day and night, including the deities and the justified blessed dead of Osiris (who otherwise sleep). The sacred knowledge regarding this spiritual evolution was for the very few and, when first written down, portrayed in the tomb of kings only. This economy is exclusive of everyman, reserved to the deities (as the king and his high priests) and unconditional - the Eye of Re. Summarizing the scholarly finding regarding these texts : • date of inception : the beginning of the IIIth Dynasty (ca. 2670 BCE) ; • aim of the texts : to assist the divine king in his royal cult, both during his life on Earth (namely through Lunar regeneration), and in the afterlife (to ascend to Re) ; • spatial semantics : there is a spatial symbolism at work in the actual placement of the texts in the chambers, passage-way & corridor : Lunar Duat (sarcophagus room) and Solar Akhet (antechamber) are at work in four directions : West (Duat, sarcophagus, false door, dusk), North (Imperishables, the sky of Re), South (cyclic stars, the inundation) & East (Eastern Horizon, rise of Re). The texts circumambulate the theme of the king's glorious being, both as a living Horus (a reigning monarch), a living Osiris (rejuvenated by the Sed festival) and, finally, a divine ancestor, a "power of powers" and "image of images", a god one with Atum ; • composition : the texts form a literary unity insofar as they represent a careful and conscious selection out of the available body of ritual utterances (cf. those found in the tomb of his successors plus very likely others). They are not narrative and do not represent the actual funerary ritual, nor the pyramid complex. As a ritual and magical anthology, they bring together all what is needed to bring about for the divine king his regeneration (in the Lunar Duat) and ascension (via the Solar Akhet) to the stellar Imperishables. The composition is not available as a linear narrative. There is matter of choice guided by spatial semantic, although an overall story-line is discernable ; • cognitive limitations : to back the unstable concepts of pre-rationality, a regression into myth is a common strategy, as are conservatism, contextualism and multiple approaches. As a lot of these myths are meaningless today, some connotations may seem pointless to a contemporary reader. Careful study of the images and the actual hieroglyphs used is often rewarding but seldom conclusive ; • hermeneutical typology : the Unas text contain short pieces of drama, hymns, litanies, glorifications, magical texts, offering rituals, prayers, protective charms and divine offerings. They invoke the regeneration of Osiris King Unas, the ascension of King Unas, his arrival in heaven, settling in heaven, eating the deities, etc. Predynastic, Heliopolitan, Hermopolitan, Osirian, royal, funerary, ecstatic, magical, occult & funerary registers can be isolated, making its unity and integration (in one tomb) even more remarkable. The regeneration of the king happens against a specific cosmogonic background, given isolated attention in the Coffin Texts, composed later. This cosmogony, influencing Greek cosmogony and the Abrahamic notion of a Creator-God creating ex nihilo, had several stages : • Nun : the unmanifested, chaotic sameness of everything ; • Atum : unmanifested light diffused in Nun ; • Atum-Kheprer : the unmanifested, self-created first occurrence of eternally recurrent light, splitting into a company of natural forces (the Ennead of deities) ; • Re : the manifest presence of Atum as light on the primordial "hill", the stable foundation escaping Nun. "I was born in Nun before the Sky existed, before the Earth existed, before that which was to be made form existed, before turmoil existed, before that fear which arose on account of the Eye of Horus existed." Pyramid Texts, utterance 486. 1. before creation : Nun : the container or milieu of the "Lord of Life" : The issue of autogenous activity is another important concept. Light and life are spontaneous. Precreation is the conjunction of Nun and the sheer possibility of something preexisting as a nonexistent, virtual singularity. Precreation is the dual-union of Nun and Atum, of infinite energy-field and primordial atom. Hornung, 1986, p.169, my italics. 2. during creation : Atum : he who is a virtual completeness : Hornung, 1986, pp.157-158. Coffin Texts, utterance 587 - § 1587 3. the First Occurrence : This difficult notion is touched upon in this remarkable text : Ancient Egyptian Temporality Phenomenal Time "of men on Earth" Eternal time : the repetition and duration "of the gods" Allen, 1988, p.8. With the emergence of "ta-Tenen", the "first land" rising out of precreation (cf. the islands emerging after the inundation), i.e. the primordial Earth (cf. the hypostyle hall in the Egyptian temple), and with the first Sun-ray (of Horus-Re in the sky) touching it (cf. the Benben, the prototype for later obelisks, as a petrified beam), the first occurrence is over. not creating & Ennead eternal first time Horus, Re & Pharaoh not creating phenomenal time & Osiris eschatological time How to identify substantialism here ? Nun acts as the undifferentiated, primordial "stuff" of creation. This chaos remains at the background even after creation is initiated. In many ways, Nun represents the mythical deep-structure or matrix of ante-rational cognition. But with Atum, who self-generates (causa sui), the self-subsisting nature of existents is underlined. Atum is not produced by Nun, remaining passive, but by "putting his own seed in his own mouth", i.e. Atum generates Atum (cf. logical identity). The deities are the forces of nature rooted in this recurrent self-creative act of the fugal Atum. But given the "split" of Atum in Shu (heaven) & Tefnut (moist), occurring simultaneously with Atum's self-creation, we may say substantialist fixation is minimal. Indeed, Egyptian ante-rational thought has a fugacity defying the permanence first given with Greek concept-realism. Nevertheless, creation cannot exist without the quasi-permanent, eternally recurrent Ennead and so in Heliopolitan thought, the forces of nature (starting with Atum creating Atum) and their harmonious concert (represented by Maat and the balance) represent the first stirring of the substantialist intention to fixate objects from their own side. The deities are projected "outside" and represent the luminous constants of creation. To return to these Polar "Imperishables" is the goal of Pharaoh's transformation, who tries to escape the Lunar vicissitudes of the Osirian realm, the Duat. Although truly African, and rooted in Shamanism and its awareness of the ongoing processes of nature, Egyptian spirituality tries to isolate and exalt the "fixed stars" in the various constellations of nature, while remaining aware of the constant unpredictable change undergone by the latter (cf. the strange attractor ruling the flood of the Nile). Heliopolitanism represents substantialism in its ante-rational stage, still steeped in the dynamics of the natural world, but trying to escape it by establishing the first solid foundations (the primordial hill) upon which to erect a lasting Pharaonic model, transcending changing opposites in a higher, more enduring order (cf. the plummet of the scales of Maat). Just as Pharaoh assimilated the magical powers of the pre-historical great sorceress without eliminating her (cf. the Wadjet on the brow of the royal crown) and represents the quest for a stablility encompassing all opposites, Heliopolitanism integrates elements of Shamanism while introducing the need to find a solid, enduring (cyclical) order of proto-substances, of the deities rooted in the self-creating, fugal Atum-Re, and of the divine king who is the sole deity incarnating his spirit on Earth. (b) Hellenism : Formal Reason and Concept-Realism. From a philosophical point of view, the fact the Greek word "nous" (mind, thinking, perceiving) may be derived from the Egyptian "nw", "to see, look, perceive, observe", is noteworthy. The "logoic" nature of Greek philosophy, as well as its preoccupation with "aletheia" or "truth", are thus possibly linearizations of the Memphite philosophy to be found in both the work of Ptahhotep, the sapiental authors, and the theology of the priests of Ptah. In their ante-rational discourse, the pre-Socratics sought the foundation or "archē" of the world. It explained existence as well as the moral order. For Anaximander of Miletus (ca. 611 - 547 BCE), the cosmos developed out of the "apeiron", the boundless, infinite and indefinite (without distinguishable qualities). Later, Aristotle would add : immortal, Divine and imperishable. The Archaic, pre-Socratic stratum of the "Greek Miracle" was itself layered : • Milesian "archē", "phusis" & "apeiron" : the elemental laws of the cosmos are rooted in substance, which is all ; • Pythagorian "tetraktys" : the elemental cosmos is rooted in numbers which form man, gods & demons ; • Heraclitian "psuche" & "logos" : a quasi-reflective self-consciousness, symbolical & psychological ; • Parmenidian "aletheia" : the moment of truth is a decision away from opinion ("doxa") entering "being" ; • Protagorian "anthropos" : man is the measure of all things and the relative reigns. The Eleatic effort (cf. Parmenides of Elea (ca. 515 - 440 BCE), inspired by Pythagoras of Samos (ca. 580 BCE - 500)), to posit the necessity of logic & unity was turned into rhetoric by the wandering Sophists. By so introducing the relativity of thought (skepticism and humanism), they prompted a new quest for a comprehensive system. In it, the various facets developed since Thales of Miletus (ca. 652 - 545 BCE) would have to be brought together in such a way that true knowledge would remain certain and eternal (and not circumstantial and probable). "Nothing exists. If anything existed, it could not be known. If anything did exit, and could be known, it could not be communicated." Gorgias of Leontini : On What is Not, or On Nature, 66 - 86. The systems of Plato (428 - 347 BCE) & Aristotle (384 - 322 BCE) are also a reply to the Sophists. Protagorian relativism is wrong. To refute this skepticism, i.e. the unwillingness to accept there is only "doxa", opinion, not "aletheia", truth, Classical philosophy opts for substantialism, the idea some permanence exists in the things that change. This core or essence is subjective or objective. In the former, it is a subject modified by change while remaining "the same", acting as the common support of its successive inner states. In the latter, it is the real stuff out of which everything consists, allowing the manifestation of the real world "out there". Both Plato & Aristotle are concept-realists, and their systems are examples of foundational thinking. Truth is eternalized and static. Concept-realism will always ground our concepts in a reality outside knowledge. Plato cuts reality in two qualitatively different worlds. True knowledge is remembering the world of ideas. Aristotle divides the mind in two functionally different intellects. To draw out & abstract the common element, an intellectus agens is needed. The first substance is "eidos", i.e. the form, or Platonic idea realized in matter (cf. hylemorphism). The foundationalism inherent in concept-realism seeks permanence and cannot find it. It therefore ends the infinite regress ad hoc and posits something to be possessed by the subject. This is either an object of the mind (a permanent soul) or an object of the world (the permanent stuff of reality). Greek concept-realism seeks substance ("ousia") and substrate ("hypokeimenon"). This core is permanent, unchanging and existing from its own side. In a further reification of this foundationalism, subtle substance is introduced, and the eternalizing tendency gives rise to "universalia", eternal ideas (in the mind of God). Substance is the eternal, permanent, unchanging core or essence of every possible thing, existing from its own side, and never an attribute of or in relation with any other thing. So Greek concept-realism, in tune with the tendency of thought to fossilize and substantialize, developed these two radical answers and two major epistemologies : the Platonic and the Peripatetic. These were foremost intended to serve ontology, the study of "real" beings and being, as does the logic that underpins them. Indeed, neither Plato or Aristotle developed the quantitative view of the world as proposed by Democritus of Abdera (ca. 460 - 380/370 BCE). Their systems are devoid of mathematical physics. In Greek concept-realism, concepts must refer to something "real". Our thoughts are always about some thing. The "real" is a sufficient ground guaranteeing the identity of every thing. For these Greeks, the "real" had to be universal ("ta katholou", or applicable everywhere and all the time). Either these universals exist by themselves outside the sensoric world (the real is ideal) or they only exist as the form of things in each individual thing (the ideal is real). In the former, a cleavage occurs and dualism emerges (between being and becoming), in the latter, a monism ensues. For Plato , strongly influenced by Pythagoras and the Eleatics, there is a real, Divine world of ideas "out there" or, as in neo-Platonism, "in here", a transcendent realm of Being, in which the things of this fluctuating world participate. Ideas are those aspects of a thing which do not change. Obviously then, truth is the remembrance (anamnesis) of (or return to) this eternally good state of affairs, conceived as the limit of limits of Being or even beyond that. These Platonic ideas, like particularia of a higher order, are no longer the truth of this world of becoming but of another, better world of Being, leaving us with the cleaving impasse of idealism : Where is the object ? The Platonic ideas exist objectively in a reality outside the thinker. Hence, the empirical has a derivative status. The world of forms is outside the permanent flux characteristic of the former, and also external to the thinking mind and its passing whims. A trans-empirical, Platonic idea is a paradigm for the singular things which participate in it ("methexis"). Becoming participates in Being, and only Being, as Parmenides taught, has reality. The physical world is not substantial (without sufficient ground) and posited as a mere reflection. If so, it has no true existence of its own (for its essence is trans-empirical). Plato projects the world of ideas outside the human mind. He therefore represents the transcendent pole of Greek concept-realism, for the "real" moves beyond our senses as well as our minds. To eternalize truth, nothing less will do. Aristotle (384 - 322 BCE) rejects the separate, Platonic world of real proto-types, but not the "ta katholou", the generalities ("les généralités", "die Allgemeinen"), conceived, as concept-realism demands, in terms of the "real", essential and sufficient ground of knowledge, the foundation of thought. So general, universal ideas do exist, but they are always immanent in the singular things of this world. There is no world of ideas "out there". There is no cleavage in what "is" and there is only one world, namely the actual world present here and now. The indwelling formal and final causes of things are known by abstracting what is gathered by the passive intellect, fed by the senses, witnessing material and efficient causes. The actual process of abstraction is performed by the intellectus agens, a kind of Peripatetic "Deus ex machina", reflective of the impasse of realism : Where is the subject ? "The faculty of thinking then thinks the forms in the images, and as what is to be pursued or avoided is already marked out for it in these forms, the faculty can, by being engaged upon the images, be moved, and this also in a way independent from perception." Aristotle : De Anima, III.7. How is this first intellect able to derive by abstraction the universal on the basis of the particular ? How does it recognize the forms in the images without (Platonic) proto-types ? Even a very large number of particulars does not logically justify a universal proposition, as Aristotle knew. Induction has no final clause, for all past causes can never be known. How does this active intellect then recognize the similarities between properties offered by the passive intellect, if not by virtue of a measure which is independent from perception (and so again introducing a world of ideas) ? Aristotle posits the objective forms in the actual world. In the latter, both being and becoming operate. This was a major step forward, for ontological dualism is explicitly avoided, although implicitly reintroduced within psychology. The forms are realized in singulars, but known by accident of a universal intellect he does not study. For him, the "real" is known through the senses and the curious abstracting abilities of the mind. The workings of the intellectus agens remain dark. This concept-realism is immanent. All things are explained in terms of four causes : causa materialis, causa efficiens, causa formalis and causa finalis. Experience of the first two causes, triggers the process of cognition and knowledge of material bodies. Abstracting the last two causes, allows one to understand the "form" or essence of things. In Platonic concept-realism, one cannot avoid asking the question : How can another world be the truth of this world ? The ontological cleavage is unacceptable. Peripatetic thought summons a psychological critique, for how can the human soul possibly know anything if not by virtue of this remarkable active intellect ? Both reductions are problematic. Because they try to escape, in vain, the Factum Rationis, and so represent the two extreme poles of the concordia discors of thought, they form an apory. Plato, being an idealist, lost grip on reality. Aristotle, the realist, did not fully probe his own mind. Composite forms of both systems do not avoid the conflict, although they may conceal it better. The crucial tension of thought was not solved by Greek concept-realism. (c) Abrahamic Traditions : God as Caesar. "The notion of God as the 'unmoved mover' is derived from Aristotle, at least so far as Western thought is concerned. The notion of God as 'eminently real' is a favourite doctrine of Christian theology. The combination of the two into the doctrine of an aboriginal, eminently real, transcendent creator, at whose fiat the world came into being, and whose imposed will it obeys, is the fallacy which has infused tragedy into the histories of Christianity and of Mahometanism. (...) The Church gave unto God the attributes which belonged exclusively to Caesar." Whitehead, A.N. : PR, §§ 519 - 520. The monotheisms introduce theo-ontology : existence is created by the revealed God. This singular God is the sole Supreme Being, the substantial absolute of absoluteness creating a plural creation ex nihilo. As the "summum bonum", God does not tolerate evil, considered as the mere absence of goodness ("privatio boni"). In these religions, the focus is not on truth & ontology, but on salvation, the restoration of the link with God. But in the process of erecting the salvic model, a theology was invented build upon Greek concept-realism. This superstructuring of religious experience using "heathen" intellectual constructs would prove to be detrimental to the survival of fundamental theology. These religious philosophies tried to bring faith and reason together, but failed. By identifying the mind of God with Plato's world of ideas, the Platonists had to exchange Divine grace for intuitive reason. The Peripatetics introduced perception as a valid source of knowledge and so prepared the end of Christian theology, the rational explanation of the "facts" of revelation. There seemed to be no facts after all ! When Peripatetic metaphysics got integrated in monotheist theology, the end of fundamental theology could not be far off. Indeed, how to assimilate the more empirical approach of Aristotle without harming the God of revelation ? As soon as the natural world became focus of attention, the "facts" of revelation could no longer be believed at their face value. Moreover, Aristotle's concept of the "Unmoved Mover" reaffirmed the general Greek prejudice against relationality, identifying objects entertaining relationships with other objects as of "lower rank" compared to objects removed from empirical actuality, looking down at the world from their unmoved Olympic heights. Indeed, for Thomas Aquinas (1225 - 1274), the relation between God and the world is a "relatio rationis", not a real or mutual bond. This scholastic notion can be explained by taking the example of a subject apprehending an object. From the side of the object only a logical, rational relationship persists. The object is not affected by the subject apprehending it. From the side of the subject however, a real relationship is at hand, for the subject is really affected by the perception of the object. According to Thomism, God is not affected by the world, and so God is like an object, not a subject ! The world however is affected by this object-God, clearly not "Emmanuel", God-with-us. Hence, the relationship between God and the world is deemed not to be reciprocal. If so, the world only contributes to the glory of God ("gloria externa Dei"). The finite is nothing more than a necessary "explicatio Dei". This is the only way the world can contribute to God. In the line of this reasoning, the monotheist God, like a Caesar of sorts, is omnipotent and omniscient. This means God knows what is possible as possible, what is presently real as real and also the future of what is real. Moreover, God can do what He likes and so is directly responsible for all events. These views make it however impossible not to attribute all possible evil, like the slaying of the innocent, to God ! Such a theology turns the good God into a brutal monster or proves the point He cannot exist (cf. Sartre). Finally, free will cannot be combined with this view of God as the sufficient condition of all things, for freedom only harmonizes with a view of God as the necessary condition. In a philosophical discourse on the Divine influenced by the data of science, no longer a priori -as a handmaiden- forced to take sides with the dogma's of revelation, these inconsistencies in monotheist theology could no longer be maintained. Fundamental theology was finally shipwrecked, and the distinction between the discourse of faith and the reasons of metaphysics became more pertinent (cf. deism). The Age of Enlightenment would eliminate the more "scientific" pretensions of the revelations (like the story of creation, geocentrism, the position of woman, slavery and other contra-factual & immoral views), and by the beginning of the XXth century, relativity & quantum mechanics introduced a new, post-Newtonian view on spatio-temporality and the physical categories of determination (replacing efficient causality with neo-causality, interaction, statistical probabilism, teleological determination, etc.). The Judeo-Christian socio-political grip on humanity was incapacitated. In Islam, the revolution of "an age of enlightened reason" is still on its way and can today be felt in the so-called "European Islam". Clearly, a new philosophical view on God is needed. (d) The Renaissance and Modern Scientific Thought. Influenced by the "Orientale Lumen" and Arabic scholarship, the cultural movement known as "the Renaissance", born in Florence as early as the 14th century and spreading over Europe in the following three centuries, placed the human phenomenon center stage, rediscovered Late Hellenism and tried to end Catholic supremacy on knowledge, learning and the arts. The "via antiqua" was over. Times of religious turmoil were at hand. The Renaissance and its humanism sparked the Reformation and other debates & conflicts. With the French Revolution (1789) the political translation of modernist thinking was on its way. Renaissance thinking is still foundational. It still clings to substance in terms of the Platonic world of ideas being the mind of God, or posits a Peripatetic active intellect able to abstract the essential core of sense objects. Saturated with centuries of Christian idealism, substance itself was not (yet) rejected, only its fixation in terms of the Judeo-Christian & Catholic monopoly. Renaissance thinkers are self-conscious. With the birth of reflection as a cultural phenomenon, European thought was liberated from the chains of authority and magisterial dogmas. As reflection was immature, only the intellectual freedom to do so was demanded, so the fundamental substances could be scrutinized by facts & arguments, unassuaged by oppressing clerical influence. The ontological system of René Descartes (1596 - 1650) foresaw three fundamental substances : "res cogitans" or the thinking substance (consciousness), "res extensa" or the extended substance (matter) and God. The ontologies after him return to this division and either introduce reductions (of mind to matter, or matter to mind) or rename the Cartesian triad, this summary of all previous ontologies. Descartes himself was not a reductionist. The three substances have their own kind of (interacting) existence. Mind points to consciousness and its freedom. Matter is limited and bound to cause & effect. God is the ultimate guarantee things happen as they happen. Leibniz (1646 - 1716) occupies a central place in both philosophy and science. He invented the infinitesimal calculus independently of Newton, with a notation in general use since then, as well as the binary system, making him the founding father of all modern computer architectures ... He also made contributions to physics, technology, anticipated notions surfacing later in biology, medicine, geology, probability theory, psychology, linguistics, information science, politics, law, ethics, theology, history & philology ! In philosophy, Leibniz was an optimist. In his theodicy, he explains the world as the best possible combination available to God. Ontologically, Leibniz was not a triadist or a monist but a pluralist, focusing on how a plurality of substances can form a unity. In his view, there are infinitely many simple substances (monads). In his The Monadology, Leibniz explains how monads are metaphysical points, animate points or metaphysical atoms. In contrast to atomism, they are not extended (not bodies). Neither are they immaterial ! Monads consist of two principles inseparable from each other, but together constituting a complete substance. The innermost center of a monad, i.e. the mathematical point, where the entelechy, soul or spirit is located, is its inner form. This has no existence in itself, but is incarnated in a physical point or an infinitesimally small sphere, which is the "vehicle of the soul". This hull consists of a special matter, called primary matter ("materia prima"). Monads have "no windows" or portals. So nothing can enter them from the outside or could escape from the inside. Despite this, the monad, in a spontaneous act, represents the surrounding world with an individual perspective, constituted by its punctual inner structure of centre, radius and circumference. In his Ethica, Spinoza (1632 - 1677) rethinking Descartes and Leibniz, tries to prove his monist version of rationalism "de more geometrico". With the Spinozist definition of "substance" (nature or God), the rational definition of substance matured. The stuff of existence is an infinite, closed, solitary, singular, unchanging, eternal & everlasting monad from its own side, the only free Supreme Being, an abstract "God" (also called "Nature") or "Godhead", the root of theo-ontology, involved in the permanent direct experience "Face-to-Face" of God with God. "By God, I mean the absolutely infinite Being - that is, a substance consisting in infinite attributes, of which each expresses for itself an eternal and infinite essentiality." Spinoza : Ethics, Part I, definition VI. "That thing is called 'free', which exists solely by the necessity of its own nature, and of which the action is determined by itself alone. That thing is inevitable, compelled, necessary, or rather constrained, which is determined by something external to itself to a fixed and definite method of existence or action." Spinoza : Ethics, Part I, definition VII. At the end of the XVIIIth century, a variety of ontological systems had been proposed and substantialism had come under severe attack by empirism. What if only direct experience is valid ? Is there a permanent, fixed Archimedean "support" or sufficient ground outside or inside the subject of experience ? Perhaps science, in the sense of eternalizing statements about the world, is impossible, as Hume (1711 - 1776) conjectured ? Moreover, how can two contradictory answers to the same question, seeming equally reasonable, be both true (cf. "antinomies") ? Kant (1724 - 1804) deemed the situation, giving him sleepless nights, scandalous, and his Kritik der reinen Vernunft (KRV) initiated the "Copernican Revolution" of philosophy. This major revolution in Western thought and its strong influence on the critical tenets of contemporary epistemology have been studied elsewhere. 7 A New Theology. Nietzsche, F. : Die fröhliche Wissenschaft (section 125 : The Madman). (a) Reasons to Resuscitate God ... When Nietzsche's madman cried "God is dead", he was pointing to the "God" of scholasticism, in particular the Christian God construed in terms of the Apollinic (Platonic) model. So he is not lamenting the physical death of an "imaginary being" called "God", but the end of an external, absolute basis for morality, leaving humanity with the responsibility of coming up with our own morality. This burden may be too great for ordinary mortals, and so only an "Übermensch" has the strength to live in a Godless world without falling into nihilism. So for Nietzsche, the death of this God implies we are no longer able to believe in any such cosmic order. We no longer recognize it. The death of this God will lead, so Nietzsche thought, to the rejection of a belief in a cosmic order but also to a rejection of absolute values themselves and the adherence to an objective and universal moral law, binding upon all individuals. And for ordinary men (the Nazi "Untermenschen"), this conjectured loss of an absolute basis for morality leads to nihilism. Besides the fact Nietzsche's proposed Dionysian Will at the core of his irrational protest philosophy remains dependent on the Platonic (Apollinic) "summum bonum" he rejected, being its mere reversal, there are valid reasons to doubt whether the dead of the monotheist "Deo revelatio" indeed leads to the rejection of a cosmic order or nihilism. Firstly, because the Christian God is not the best model of God available and secondly, because there are good reasons to resuscitate the idea of God. Let us first discuss these reasons. 1. Reasons from Logic : 1.1 The Argument of the First Conserver from Conservation : "All the conserving causes simultaneously concur for the conservation of an effect ; if, therefore, in the order of conserving causes we go on ad infinitum, then an infinite number of things would be actually existing at the same time. This, however, is impossible ..." Ockham : Questionis in lib. I Physicorum, Q.cxxxvi. For William of Ockham (1290 - 1350), who took the equipment to develop his terminist logic from his predecessors, empirical data were primordial and exclusive to establish the existence of a thing. So the only way to prove God's existence would be as efficient cause of all things, remaining within the finite order. Indeed, Ockham stops at the first efficient cause. The reasons for this move also explain his rejection of the arguments of God from necessity and from perfections. Infinite transcendence is thus avoided. But to identify this cause with God is not possible, for this cause could be a heavenly body (Quodlibet). It cannot be proved this supposed heavenly body is caused by God, for we have only immediate and mediate sense data of corruptible things, not of any transcending concept. In the traditional argument from efficient causes, it is assumed an infinite regress in causes of the same kind is not possible. The world was deemed finite and the world of ideas infinite. For the scholastics, to say the world is infinite is sheer blasphemy, for it ruins the strict line drawn by these theists between a finite creation and an infinite Creator. In such a context, free natural inquiry is repressed. For Ockham, the finitude of the world cannot be strictly demonstrated. Maybe an infinite series exists, maybe not. All previous proofs presupposed the truth of the proposition "The world is not infinite.", but this is not necessarily so. Nevertheless, probabilities may be assessed and calculated. To avoid the question of the infinite ingress in time, i.e. as a horizontal sequence of interacting and interdependent efficient causes, Ockham's argument ingeniously jumps to the actual, vertical order of events "here and now", i.e. as they are happening in every moment. By doing so, it avoids an infinite regress, for it is a solid logical premiss to affirm the world is not infinite in each actual moment ! Ockham's argument of the First Conserver from conservation runs as follow. As a contingent thing coming into being and is conserved in being as long as it exists, its conserver is dependent, for its own conservation, on another conserver or not. To suppose a thing is not conserved is absurd, for its actuality proves it is conserved in the vertical order of things here and now. So things are conserved in each actual moment. As only necessary beings conserve themselves and the world only contains contingent things, it follows every conserver must depend on another conserver, etc. As there cannot be an infinite number of actual conservers "hic et nunc", i.e. in each actual moment (cf. supra), there must be a first Conserver. An infinite regress in the case of things existing one after the other (like horizontal causes of the same kind) is indeed conceivable (and that's why all these arguments fail). But an infinite regress in the actual, empirical world here and now would give an actual infinity, which is absurd. Indeed, to avoid the first Conserver, actual reality would become infinite ! Ergo, the first Conserver probably exists. This is a terministic (probabilistic) proof because it is based on reasonable assumptions, namely (a) things are conserved as long as they exist, (b) the world is not infinite in each actual moment and (c) the world contains no actual infinity. This elegant proof of the first Conserver is completely a posteriori. It avoids the order of infinity, and considers the world finite. No limit-concept is invoked, no transcendent being deduced. The "essence" of God cannot be known, lies outside reason. The existence of God cannot be demonstrated by necessity, but argued by probability, for the finite order of contingent beings cannot be conserved without a first Conserver. So, according to Ockham, in the order of rational, empirical knowledge, natural necessity and a first Conserver is all philosophy can infer as proven, probable knowledge. Nothing which is really God can be known by us without something other than God being involved as object. There is no simple concept proper to God mirroring the essence of God adequately. We are left with the first Conserver, and reason cannot advance further. So far William of Ockham. 1.2 The Argument of the Architect of the World : Although Kant is associated with rejecting the proofs of God, it is often forgotten he too favored the proof of the "architect of the world". Kant reclassified the proofs of the existence of God as follows : 1. ontological : whatever our concept of an object may contain (for example, the idea of the "ens realissimum" as the idea of an absolutely necessary being), we must always step outside it in order to attribute existence to it. Existence is not a predicate and adds nothing to an object, not even in the unique case of the most perfect being. To say something "exists" is to posit the subject with all its predicates. To say "God does not exist." is to annihilate all the predicates, not just "existence". Hence, the ontological argument fails ; 2. cosmological : this proof will always complete the series of phenomena in the unconditioned unity of a necessary Being, and by doing so, overstep the boundaries of reason, for the categorial principle "everything contingent has a cause" is only valid in the realm of sense-experience (the world) and it is only there it has meaning, never outside it (cf. the arguments from motion, efficient causes, perfections & necessity). Again the argument fails ; 3. physico-theological : this proof of finality, aim or design is based on an analogy from human adaptation of means to ends. We can move from the idea of design to the idea of a Designer, but not from the latter to the transcendent Creator of the world. This would again involve a misuse of the transcendental ideas of reason, a crossing over of the ring-pass-not of pure reason. The argument fails. Kant retained a real respect for the argument from design, being the oldest, clearest and most in conformity with reason. It can prepare the mind for practical theological knowledge and give it "a right and natural direction" (KRV, B665). Moreover, it gives life to the study of nature, "deriving its own existence from it, and thus constantly acquiring new vigour" (KRV, B649). To posit a necessary & all-sufficient Being (the monotheist God of scholasticism) means it is so overwhelming and so high above everything empirical and conditioned, we never would find enough material in experience to fill such a concept. If it is part of the chain of conditions, it would require further investigation with regard to its own still higher cause, but if it stands by itself, it is outside the chain and thus a purely intelligible Being. But then, "what bridge is then open for reason to reach it, considering that all rules determining the transition from effect to cause, nay, all synthesis and extension of our knowledge in general, refer to nothing but possible experience, and therefore to the objects of the world of sense only, and are valid nowhere else ?" (KRV, B649). With regard to causality, we cannot do without a last and highest Being, but such a transcendental idea, although agreeing with the demands of reason, would only give a faint outline of an abstract concept (emerging when we represent all possible perfections united in one substance). It would favour the extension of the employment of reason in the midst of experience, guiding it towards order and system, and would not oppose any experience. But this is not the same as proving the existence of a necessary and self-sufficient God and Creator à la monotheism. "The transcendental idea of a necessary and all-sufficient original Being is so overwhelming, so high above everything empirical, which is always conditioned, that we can never find in experience enough material to fill such a concept, and can only grope about among things conditioned, looking in vain for the unconditioned, of which no rule of any empirical synthesis can ever give us an example, or ever show the way towards it." Kant, I. : Critique of Pure Reason, B646. The inference, proceeding from the order and design observed in the world as a contingent arrangement (one with a possibility of happening) to the concept of a cause proportionate to it, teaches us something quite definite about this first cause, namely that it is a very great being of an astounding and immeasurable might and virtue, but not what the thing is by itself. Or, in other words, the harmony existing in nature proves the contingency of the form, but not of the matter or the substance in the world (we grasp the form, but do not observe the matter). To prove the contingency of matter itself would require us to show that in the substance of the things of the world, the product of a supreme wisdom exists. But the latter is not part of the world and thus no object of the senses. The conclusion is clear : "The utmost, therefore, that could be established by such a proof would be an architect of the world, always very much hampered by the quality of the material with which he has to work, not a creator, to whose idea everything is subject. This would by no means suffice for the purposed aim of proving an all-sufficient original Being. If we wished to prove the contingency of matter itself, we must have recourse to a transcendental argument, and this is the very thing which was to be avoided." Kant, I. : Critique of Pure Reason, B653. This argument, although using a variant terminology (rooted in the transcendental method) is in tune with Ockham's first Conserver (of each entity hic et nunc). In the vertical order of simultaneity, the a posteriori series (of conservers) has to be stopped before exiting the order of the world. Hence, the apex reached is well within the world and at the top of the chain. The first Conserver too is a cause proportional to the arrangements within the world, and does not step outside the world. This first Conserver is the "anima mundi", but not the transcendent, omnipotent & omniscient God of the monotheisms. 2. Reasons from Science or the Argument from Design : The Platonic strategy of the ontological argument a priori favored by traditional theism fails. Its aim was to prove a necessary, absolute Being beyond nature, not a principle existing inside nature. This peculiar immanence is not the ultimate, absolute cause, which is transcendent, but exists within nature, as it were coinciding with her. The degree of perfection of this cause lies within what is possible in experience, and so could be called the first immanent cause. It explains the over-arching unity, order and harmony of the world without advancing further, without stepping from this likelihood of immanent excellence to its determining concept as an all-embracing Divine transcendence, as it were bridging the broad abyss between immanent existence of actual entities and the necessary transcendent Being. The cause advanced in the argument from design is not the absolute unity of a transcendent Being beyond reason, but the peculiar unity explaining the skilful edifice, a cause proportionate to the order and design everywhere to be observed in the world. "This present world presents to us so immeasurable a stage of variety, order, fitness and beauty, whether we follow it up in the infinity of space or in its unlimited division, that even with the little knowledge which our poor understanding has been able to gather, all language, with regard to so many and inconceivable wonders, loses its vigour, all numbers their power of measuring, and all our thoughts their necessary determinations ; so that our judgment of the whole is lost in a speechless, but all the more eloquent astonishment." Kant, I. : Critique of Pure Reason, B649. The logical core of the argument from design is a procession from the observed contingent order to the existence of a very great cosmic might, one making the peculiar unity of the world possible, i.e. the first immanent cause. As no cause outside the world can ever be definite, no rational principle of transcendent theology (the theist concept of a necessary Being), forming the base of religion, can be given. But, if we can infer an immanent cause of the world, then an immanent metaphysics can be used to construct a natural religious philosophy, the pantheist ideal of a necessary being inside the world. Although such a concept merely suggests a still higher cause, one explaining Ultimate Authorship, no transgression is allowed and so, from this natural vantage point, in strict rational terms the concept of the Author of the World must remain empty (in the sense of zero). Summarize the logical steps of the traditional argument from design as follows : 1. Major Premiss 1 : the world is an organized, contingent whole, evidencing variety, order, fitness and beauty ; 2. Major Premiss 2 : it is impossible for this arrangement to be inherent in the things existing in the world, i.e. the different entities could never spontaneously co-operate towards such obvious definite aims ; 3. Minor Premiss : definite aims need a selecting and arranging purposeful rational disposing principle ; 4. Conclusion 1 : ergo, there exists a sublime and intelligent cause (or many) which is the cause of the world, not only in terms of natural necessity (blind and all-powerful), but as an intelligence, by freedom ; 5. Conclusion 2 : the unity of this cause (or these causes) may be inferred with certainty from the unity of the reciprocal relation of the parts of the world as portions of a skilful edifice so far as our experience reaches. Ergo, the intelligent cause or causes of the world forms or form a unity of design ; 6. Lemma : if this cause is projected outside the world to explain its activity, then the domain of reason is left and the argument from design becomes the refuted argument from necessity (cf. the cosmological argument). Ergo, the argument from design does not prove an ultimate, but a proximate cause. For Kant, the argument from design led to the "stage of admiration" of the greatness, the intelligence and the power of the Architect of the World, who, unlike a Creator or Author, who is self-sufficient, necessary and transcendent, is very much hampered by the quality of the material with which to work. This argument from design works well together with Ockham's revised a posteriori argument from efficient causes : 1. Major Premiss : in the contingent order of the world nothing can be the cause of itself or it would exist before itself ; 2. Minor Premiss 1 : an infinite series is conceivable in the case of efficient causes (existing horizontally one after the other), but impossible in the actual (vertical) order of conservation "hic et nunc" ; 3. Minor Premiss 2 : an infinite regress in the actual, empirical world here and now would give an actual infinity, which is absurd ; 4. Minor Premiss 3 : a contingent thing coming into being is conserved in being as long as it exists ; 5. Minor Premiss 4 : as only necessary beings conserve themselves and the world contains contingent things only, every conserver depends on another conserver, etc. ; 6. Conclusion 1 : ergo, as there is no infinite number of actual conservers, there is a first Conserver ; 7. Lemma : if we suppose an infinite regress in the actual, empirical world here and now, then an actual infinity would exist, which is absurd, ergo, the first Conserver exists. The conclusions of both arguments, given the terministic nature of logic, are not certain but probable. This is in tune with our non-foundational epistemology. They support a conserving cause of the world, intelligently pre-planning the universe in a design, like an architect or demiurge, with a freedom limited by the own-forms of the actual entities "at hand", working on the "tick" of the cosmic clock to conserve and maintain the universe. Clearly such a very great being, possessing the highest natural wisdom, is not a final concept. But immanent metaphysics cannot advance further. The Intelligent Conserving Cause itself cannot be explained by ante-rationality, reason or the creativity of immanence. A "desperate leap" across the "broad abyss" between the unity of the world and the Author of the world may be attempted, but without any valid reason. For it is all together a different thing to be creative thanks to casual intellectual flashes in an airy, shaded room, than to be constantly a witness of the full blaze of the Sun and its brightest light. As Ionescu (1909 - 1994), the founder of Absurd Theater, one may choose to walk away from it ... To posit transcendence is impossible. This truth is the major obstacle in any serious apology of the traditional theist God. Absolute totality can only be suggested by sublime poetry. Religions are poetical constructs of a certain quality. Transcendent meta-rationality (nondual intuition) is non-conceptual, like an intuition without image, a merging without seed, a union without means, an experience of silent namelessness. The meaning of grand poetry is the object of metaphysics. Arguments can be presented. But in a transcendent metaphysics, these poetical forms become revealed cosmogonies explaining the creation of the universe. In the deepest sense they try to fathom the unconditional, and have, like koans, an exemplaric relevance. But to those who adhere to them, they are windows to the transcendent God. To solidify the argument from design even more, its pivotal second major premiss needs to be studied and backed in more detail : • Major Premiss 2 : the different entities composing the world could never spontaneously co-operate towards definite aims. Indeed, central to the debate (cf. Dembski & Behe (1998) and Hamilton (2002)), is the question whether the organization of the universe and the emergence of life are accidental ? Hoyle (1986) concluded random events and change occurrences are insufficient to account for the complexity of living organisms. Hoyle compared the likelihood of the random emergence of higher forms of life with the probability of a tornado sweeping through a junk-yard ending up assembling a Boeing 747 ! A highly unlikely event. He also seriously tried to show why Darwin's theory is not supported by the mathematics of evolution. Perhaps the "grand story" of (neo-) Darwinism is over too ... Since Prigogine (1917 - 2003) wrote La Nouvelle Alliance (1979), a weak form of finality is gaining ground in science. He suggested the return of finality in open, dissipative (physical, biological and social) systems. Four analogies provide a strong backing for the case presenting the non-spontaneous becoming of the actual world process. How to detect non-spontaneous "design" ? 1. design by analogy of human products : the proximate cause proportional to the order, harmony, fitness & freedom observed in the world can be identified (named) by following the analogy of products of human design. In doing so, only the "form" aspect of the world is observed to identify design. In this way, the "matter", or substance of the world, is not targeted, and it is no longer necessary to prove in addition, that the things of the world, given the laws of nature, were in themselves incapable of such order and harmony. Hence, to avoid backing the premiss, it is accepted no supreme intelligence exists in the material substance of the things of the world. In the traditional Peripatetic account, four causes are at work in the world : material, efficient, formal & final. By analogy of human products, the design involves the formal and final causes only ; 2. design by analogy of outcomes in living organisms : all living things seem tailor-made for their function and appear to interact purpose-fully with their environments : animals use camouflage, most parts of our bodies, down to our DNA helix, are very delicately engineered, and large numbers of apparent coincidences exist between various living organisms, etc. These highly ordered biological schemata seem places of reference to back the premiss, for how could such a complexity rise out of simplicity without a pattern of intelligent choices ? The chances are small enough, given what science demands in other areas, to dismiss spontaneous, random activity. Nevertheless, this study of outcomes was seriously affected by the discovery of the Darwinian principle organisms evolve by natural selection, adaptations and (random) mutations. If all biological events can be explained by this principle (turned into a paradigm), then indeed there is no "purpose" behind the grand natural symphony. Darwin (1809 - 1882) and neo-Darwinism were able to explain much of the data of his time and the first half of the previous century. Even societies could be studied in terms of the survival of the fittest (Monod, 1970). But, recent studies show how the theory has been unable to account for certain more subtle phenomena uncovered by the biochemistry of the last 50 years, mostly related to complex events such as protein transport, blood clotting, closed circular DNA, electron transport, photosynthesis etc. Progressive metamorphosis, with the emergence of increasingly complex and intelligent species in a step-wise, sequential pattern was recently proposed (Joseph, 2002). Large-scale protein innovation (Aravind, 2001), "silent genes" (Henikoff, 1986, Watson, 1992), the precise regulatory control of genome novelty (Courseaux & Nahon, 2001) and the overall genetically predetermined "molecular clockwise" fashion of the unfoldment of the human being (Denton, 1998), underline the evolutionary metamorphosis theory of life and intelligent design. So, beyond the grip of Darwin's macroscopic view, on those more subtle levels of biology and biochemistry, design may be detected and purposeful arrangement of parts suspected. A revised analogy of subtle outcomes becomes thus again possible, leading to a more comprehensive backing of the premiss ; 3. design by analogy of the forms of the laws of nature : Maxwell (1831 - 1879) pointed to molecules as entities not subject to selection, adaptation & mutation. The contrast between the evolution of species, featuring biological changeability, and the existence of identical building blocks for all observed actual physical entities is crucial. Given the effectiveness of Newton's laws on the mesolevel (the inverse-square law of gravity being optimal for the becoming of the Solar system), our knowledge of what happens in stars (in particular the production of carbon and oxygen) and the cosmology of the Big Bang, then calculate the odds of spontaneous emergence. A choice has to be made between either an intelligent design (which does not offend intelligence by leaping into silly forms of creationism) or a monstrous random and blind sequence of accidents producing a gigantic complexity, in other words either a natural higher intelligence or the ongoing mathematical miracles of a blind nature morte. Indeed, ad contrario, the form of the laws of nature underlines the presence of a deep-laid scheme, representing an accurate mathematical descriptions of the natural order (both in genesis as in effect). Although no "consensus omnium" has been reached, the laws of nature likely accommodate biology ; 4. design by analogy of fundamental constants : the actual irreducible mathematical presence of immutable natural building blocks such as the natural constants, seems to give a palpable proof of the existence of something independent of every human measurement (and its biological constitutive). These constants define the fabric of physical reality and determine the nature of light, electricity and gravity. They make particles come into existence and fundamental forces work. They actualize the laws of physics by giving equations numerical quantity and are necessary in the logic of physics. What can be said about the particular values takes by these constants ? The conditions for order and eventually life to develop have been found to heavily depend upon these constants. Indeed, although mathematically, the equations of physics, representing the fundamental architecture of the order of the world, also produce outcomes when other quantities of the same constants are introduced, the world would be lifeless and barren (instead of a haven for incredible complexity) if even a small amount of these values would be changed. Ergo, the various values of the constants of nature were designed, and pre-planned. An infinite number of different worlds are possible, but only in one are order, fitness, beauty and life actual. Only our universe has observers witnessing it. The chances of all of this happening at random are so small that someone versed in the simple basics of probability theory would frown at the idea. Leibniz was right in assuming it to be impossible for this precise combination to arise without Divine intervention. And this backed his optimism ! 3. Reasons from Metaphysics : "... speculative philosophy (= metaphysics) is the endeavour to frame a coherent, logical, necessary system of general ideas in terms of which every element of experience can be interpreted." Whitehead : Process & Reality, Op.cit., § 4. The foregoing arguments make clear the manifold we call "world" can only be understood if we reintroduce (a) a principle of order and (b) a level of all-comprehensive synthesis. 3.1 The Necessity of Synthesis : The world does not manifest itself to us and our sciences as a single substance (as Spinoza claimed), but as a multiplicity under unity, as Leibniz conjectured. A multitude of experiential happenings constantly occur, often in complex networks with several hubs. Reality is what works, what happens and a lot is constantly going on. Leibniz' ontological pluralism is confirmed by science. Hence, the problem of unity (both in the micro-, meso- and macro- dimensions of the universe) becomes more acute : How is this incredible variety of singular events brought under unity. How is a pluriverse avoided and the universe possible ? Unity does not result by virtue of some overarching ontological super force or "vis a tergo", but is constantly regained based on previous agents, and so this begs the question of a higher-order level of synthesizing, the act of bringing the manifold under unity by way a principle beyond the nominal order of events characterizing the manifold. To be a universe, multiplicity must be connected, related and so interdependent. Such a higher-order level must take the dynamical features of the universe into account and so avoid (a) the idea of a "ruling Caesar" (a Higher Principle deciding over everything) and (b) a "Dieu hologer" (regulating events from the outside). To see this Higher Principle, philosophically (not religiously) called "God" or "Godhead" as all-encompassing, does not necessitate pantheism, for Godhead does not necessarily exhaust Itself in these finite agents. Instead of pantheism, pan-en-theism is a hand, positing both an immanent and a transcendent side of the Divine. The former coincides with the notion of the Architect of the World, whereas the latter points to the dimension of the Author of the World. The Architect remains within the boundaries of (conceptual) immanent metaphysics, whereas the Author, insofar as a "Supreme Being" is envisaged, refers to the nondual, transcendent side. Avoiding the substance-theology of a "Supreme Being", and thus Kant's decisive criticism, is introducing a new way to approach this transcendent side of the Higher Principle. 3.2 The Principle of Order : There can be no order in the universe without an ordering principle. This is the teleological argument. This can be conceived in two ways. Either this principle acts from the "outside" of the universe it created (as in traditional substance-theology), and in this case it is either (a) a cruel and oppressive Caesar or despot, or (b) a kind of watchmaker or silly puppeteer, regulating all events reduced to marionettes. As such a transcendent principle moves outside the limitations of reason, it cannot function in a valid metaphysics. Hence, although the principle of order functionally differs from all other events in the universe, it cannot be another static ontological level, splitting the world in a Platonic dyad, with a "perfect" world of being posited against an "imperfect" world of becoming. This catastrophic move was made by all monotheisms and lead to various irrationalities covered by the "mysterium fidei". "Undoubtedly, the intuitions of Greek, Hebrew, and Christian thought have alike embodied the notions of a static God condescending to the world, and of a world either thoroughly fluent, or accidentally static, but finally fluent - 'heaven and earth shall pass away'." Whithead, A.N. : Process and Reality, 1929, § 526. Order is not a random property of events, but the conditions of their existence. The world cannot produce its own order, but needs it to actually exist. However, this necessary Principle of Order invoked to explain this order must, to allow for panoramic overview, somehow transcend these events, and this without belonging to another order of reality (as was the case in traditional Greek-based theology). To succeed in successfully positing it, it must altogether embody a different functional operation while remaining fully an ontological part of the Real, i.e. not a static stratum outside the world (creating the world ex nihilo). Such a Divine possibility must therefore encompass (a) the actual dynamism of God being near all events (concrete) and (b) the abstract Godhead carefully considering and thus weighing the probabilities of all possible events in terms of unity, harmony & beauty (cf. infra). In Process Theology, the former is called the "consequent nature" and the latter the "primordial nature" of God (cf. infra). This division is not an ontological split. 4. Reasons from Personal Experience or Religious Existentialism : Is the production of the Divine fact possible ? Can empirico-formal propositions objectify the Divine ? Is there an experimental methodology, itinerary or protocol leading towards spiritual experience ("cognitio Dei experimentalis") ? If so, then an experimental argument a posteriori may be inferred. Finally, if the mystics give an exemplaric account of a bi-polar, i.e. pan-en-theist Divinity (transcendent as well as immanent), then can we allow transcendent metaphysics to merely poetically suggest the conceptually improvable existence of the absolute totality, entirely impossible on rational (conceptual) grounds ? Can the religions, as institutions of poetry of a certain quality, be given new meaning and momentum ? Ockham's & Kant's general arguments in favour of the intelligent design of the world, the fitness and harmony existing in the works of nature point to an Architect of the World. Although intelligent, this being is always hampered by the quality of the materials used, but nevertheless shows us the "right and natural" direction. For Ockham, contingent beings are unable to conserve themselves and if we take the complete vertical chain of conservers hic et nunc, we must conclude, hand in hand with natural necessity, the first Conserver exists. Both positions are strong. To make clear what an immanent perspective means, let us take the example of the rejected a posteriori argument from necessity. If it is legitimate to ask how the world composed of contingent objects was caused, then the totality of objects must have a reason external to itself. Why ? This reason cannot be part of the contingent world (rise and perish), for then it could not be a satisfactory explanation of the reality of the world (it would also rise and perish). Hence, and here the category-mistake creeps in, a transcendent necessary being exists, for an infinite series is deemed impossible. Moreover, the question remain whether it is indeed impossible to explain the reality of the world in terms of rising & perishing ? Perhaps one can posit a holomovement, a continuous series of symmetry-transformations, a "style" manifesting itself solely in and through movement (as is the case of a swimming style) ? The arguments of motion, efficient causes and perfections also stop this infinite regress as hoc by "filling the gap" and jumping outside the order of the world. Only the argument from design avoids this problem. However, if Bertrand Russell is right, and the world is "just there and that's all" or "actual process", as Whitehead thought, and together with Kant we reject any illegitimate transgression in the use of the ideas of reason, then the "optimum" our reason seems to arrive at, is a strong form of pantheism, positing the concept of a necessary, first conserving, most perfect, intelligent immanent Conserver of the world. Is it possible to say more ? How to defend pan-en-theism, introduce transcendence without an ontological division between the "world" and "Godhead" ? The valid argument a posteriori calls forth the following witnesses : 1. the fact of design : the world is not the work of a blind watchmaker, but of an intelligent Designer ; 2. the fact of spiritual experience : the experience of the Divine can be (re)produced and its protocol transmitted ; 3. the possible entelechy of the world : the order and beauty of the world point to a final end : to actualize all possibilities (which is an ongoing, endless process). The fact of design can be demonstrated without the fact of spiritual experience. But, by fulfilling the conditions to experience Divine immanence, one furthermore acquires the necessary "form" or "spiritual attitude", a key to open the "doors of perception" (cf. Huxley). Indeed, the direct, immediate observation of the Divine is not self-evident, nor necessary. Self-realization is only triggered by a free intention. There is no "natural" necessity to seek out, see and meet the soul of the world or the beyond. By a strong focus on orthopraxis, the problem of the production of the spiritual fact comes into perspective. A direct plug-in or access to the supposed "soul" of the world and beyond must, ex hypothesi, be given. Otherwise, the concept of an immanent Designer would imply remoteness and inaccessibility, which is in contradiction with the relatedness shown in the design. The Architect is not in one place, but in all places all the time. Moreover, if a plug-in (a software) is postulated, then a material manager (a hardware) must be identified to compute & process (execute) this own-form of human spirituality. This line of argument boils down to the presentation of a spiritual protocol with minimal orthodoxy, one which is all about doing, practice, discipline and constant devotion (a userware). This spiritual methodology is then a series of actions, affects and thoughts producing at least a direct experience of the immanent totality conserving the world-process, if not more. (b) Desubstantializing Western Theology. "So long as the temporal world is conceived as a self-sufficient completion of the creative act, explicable by its derivation from an ultimate principle which is at once eminently real and the umoved mover, from this conclusion there is no escape : the best that we can say of the turmoil is, 'For so he giveth his beloved - sleep.' This is the message of religions of the Buddhistic type, and in some sense it is true." Whithead, A.N. : Process and Reality, 1929, § 519. In the traditional onto-theological scheme, initiated in Heliopolitan thought, finding its rationality in Greek philosophy and developed by Abrahamic theology into the concept of a static, substantial, essentialist & self-powered God, revealing Himself in likewise unchanging holy scriptures, there is no room for a relational Deity, for God is a Supreme Object, unstirred by His creation. To maintain this view, posing problems in terms of soteriology, creation has no good reason. Why would an impassible, ineffable, self-powered, omnipotent & omniscient Supreme Being (who's essence is only known by Himself) want to create anything ? If God is totally self-sufficiently perfect, without being touched by anything except God, then creation can have no other reason than being His free gift to His creatures. Creation is willed by God because God wills it so ! Further than this circular logic one cannot move. In this line of thought, before creation actually happens, there cannot be anything besides God. In monotheism, the Greek solution of primordial matter (chaos) fashioned by a "demiurgos" is rejected. God is the sole, unique, singular Supreme Being, and by positing this remote God, His involvement in the world becomes paradoxical and therefore the object of mystifications called "mysteries". How can such a hidden God show interest and participate ? Can He be more than a "Deus absconditus", an absent God ? Only Christianity posits a human factor in God (Christ) and so is able to remedy this theology, but not without causing new problems & schisms (cf. Trinitarism, Christology, Christocentrism & the status of the Holy Ghost). Before creation, concepts such as the "outside" or "inside" of God have no meaning. In the three "religions of the book", they are posited by the Will of God creating creation, separating "before" & "after", "inside" & "outside". Nothingness has no existence of its own, nor are potentiality, virtuality or possibility considered, for the essence of God is deemed a self-subsistent super-substance. The "nihil" in "creatio ex nihilo" merely indicates nothing but God's Will rose creation. Hence, God's creative Will is not bound by any necessity, but lawless (not random) and absolutely indeterminate (but not disorganized). How this has to be conceptualized is unclear. God is beyond the created order, and in no way in need of creation or bound by any necessity to create - creation is "ex nihilo", i.e. with the absence of any possible necessity "ex parte Dei", in other words, the result of a Divine contingency in the act of the creative Will of God. So the whole of creation exists by the grace of the Will of God, who is not a "deus ex machina", nor an impersonal Power of powers or Principle of principles. Clearly, this substantialist view, not really backed by revelation itself, was the result of what Ockham considered to be unnecessary infiltrations of Greek metaphysics into (Christian) theology. Indeed, for Ockham, the metaphysics of essences was introduced into (Christian) theology and philosophy from Greek sources. His strict nominalism did not incorporate them. There are no universal subsistent forms, for otherwise God would be limited in His creative act by these eternal ideas or self-subsisting substances ! This non-Christian invention had no place in Christian thought. Universals are only "termini concepti", terms signifying individual things standing for them in propositions. Unfortunately, like many other formidable intuitions of Ockham, his views were ahead of his time and so sidetracked. To be able to think Godhead more consistently, reducing the part of paradox, irrationalism, fideism & mystification and introducing the "God of the philosophers" instead of a Hellenized "Deo revelatio", this traditional ontological approach must be relinquished. Strict nominalism implies the Divine, as all other things in existence, is empty of inherent existence, and so does not "exist" from its own side, but as a result of determinations, circumstances & conditions, i.e. is other-powered. Then God can be understood as passible and influenced by the world, making sense of the notion of "covenant" and the alliances of God with humanity. And this hand in hand with a more abstract appreciation of God, i.e. "Godhead". If we accept there is only one reality, namely the order of abstractions & events or what is just there, then the Divine is never "outside" the world, for the world is all there is. So the notion of "creation" is also dubious, for suggestive of a period the world was not present and only Godhead was. This same idea is given with the event seemingly "starting up" the universe, the "Big Bang". But the question : "What was there before the Big Bang ?" is nonsensical, for the advent of the spatiotemporal order, as general relativity claims, coincides with this event. Suppose we introduce, in the physical order of things, a "fourth time", escaping the order of past, present & future (a kind of Eternal Now), then when the "Big Bang" was "not yet", mere virtual particles & forces (a potential, folded or implicate space-time configuration) existed, while after this (lesser) singularity particles & forces unfolded to become manifest. But both virtual & manifest particles belong to the world, or the Real. And if the actual world of events is deemed a dynamic network, then the manifest God -or Architect of the World- is its "hub of hubs", being near all events (all-encompassing) and, for all of eternity, as abstract Godhead, or Author of the World, weighing in favour for the possibility of Beauty (eternal). The conditions of these two aspects are not identical, for to be near all events God must encompass all what happens, and to lure events towards their greatest possible harmony, Godhead must be an abstract principle "next" to or "with" all events, but not beyond any occurrence, or instance of existence. 8 The God of Process Theology. "... God is to be relied upon to do for the world all that ought to be done for it, and with as much survey of the future as there ought to be or as is ideally desirable, leaving for the members of the world community to do for themselves and each other all that they ought to be left to do. We cannot assume that what ought to be done for the world by deity is everything that ought to be done at all, leaving the creatures with nothing to do for themselves and for each other." Hartshorne, Ch. : Op.cit., 1964, p.24. (a) The Fundamental Categories of Process Philosophy. "... how an actual entity becomes constitutes what that actual entity is ; so that the two descriptions of an actual entity are not independent. Its 'being' is constituted by its 'becoming'. This is the 'principle of process'". Whitehead, A.N. : PR, §§ 34 - 35. § 1 Alfred North Whitehead (1861 - 1947), the mathematician who, together with his ex-pupil Bertrand Russell (1872 - 1970), wrote Principia Mathematica and accepted to teach philosophy at Harvard at 63, developed a system of thought no one will ever succeed in writing a short account about. His work evidences shifts of opinion and in the course of his long life, he developed many loose and at times obscure expressions, producing desperation in anyone trying to be his chronicler. Hence, Religion in the Making (1926) and Process and Reality (1929) are fundamental, and while dispensing, as much as possible, with the technicalities, we shall focus on the latter. He is an important figure because he integrated mathematics, biology, relativity and quantum physics into his thought (cf. his The Principle of Relativity, 1922). He is often associated with Charles Hartshorne (1987 - 2003), who, during one semester, was his assistant, and who focused on the status of God in process philosophy. In his The Concept of Nature (1920), we learn about his view on the philosophical ideal in general, and metaphysics in particular, as the attainment of "some unifying concept" able to unify science. The metaphysician has a descriptive role to play. He seeks to understand the general characteristics of reality, setting these up tentatively as categories. This description of the most general features of experience is not argumentative, but rather in accord with the "I'm telling You !" method. The word "process philosophy" was probably coined by Bernard Loomer, and in a general sense the idea of the interconnectedness between all events in the universe as well as the importance of becoming, was preluded in the work of Schelling, Hegel, Peirce, James, Bergson & de Chardin. § 2 "The notion of 'substance' is transformed into that of 'actual entity'. (...) The ontological principle can be summarized as : no actual entity, then no reason." Whitehead, A.N. : PR, § 28. The basic intuitions of this system are : • we live in a universe, not a pluriverse : it is a philosophy of organicism, thinking the unity of all what happens ; • part of this unity evidenced by the universe can be grasped by reason, allowing for science. Not a single generalization would be possible if the universe were totally random & chaotic ; • the universe appears to be a dynamic whole, and so growth and becoming are fundamental to it ; • the displayed dynamism implies novelty and this means an event is never completely determined by what happened before it, for otherwise nothing would truly "happen". The universe is always an incomplete abiding synthesis and must be "remade" every time. This is "creative synthesis" or "creative advance" ; • this creative becoming is from the inside aimed at the realization of esthetic value or harmony. This beauty is the result of multiple adaptations of multiple elements to each other. Harmony is the result of this multiplicity brought under unity. § 3 For Whitehead, actual entities are the basic category of his system. Events are a nexus of actual entities. Everything that exists is an actual entity. When something is real, it is a happening, and occasion. Hence, there is a plurality of nodes of activity. Actual entities are like Leibniz' monads, with the exception they do have "doors & windows", i.e. they enter each other's selfbecoming or "concrescence". Besides real spatiotemporal actual entities, i.e. compounds or societies of actual occasions, Nature also encompasses three abstract formative elements escaping space & time : creativity, eternal objects & God. Creativity is formless and eternal objects are pure possibilities. These two formative elements are not actual, merely potential. God however, is actual but nevertheless escapes the spatio-temporal order. Basic Categories of Process Ontology the Real temporal actual world real actual non-temporal God abstract actual eternal objects pure possibilities This scheme makes clear God is a non-temporal actual entity giving relevance to the realm of pure possibility in the becoming of the actual world, encompassing non-temporal everlastingness & temporal (recurrent) eternity. God, both potential & actual, is the meeting ground of the actual world & pure possibilities. Together, the realm of abstract possibilities and the actual world form reality or the Real. § 4 Whitehead seeks to introduce a new "ontological principle" able to think becoming and change. The "ousiology" of past thinkers was unable to do this, for it was based on the changeless, permanent nature of the essence and its identity (cf. the Platonic "eidos"). In this traditional view, only accidents change and the "ousia" remains identical with itself. This creates a difference between a "supposed but unknown support" (Locke) and the subjective accidents of predication, returning in Cartesian thought as the polarization between "res extensa" & "res cogitans". Whitehead disagrees with this distinction and seeks to integrate it on a higher level. The Cartesian "ego", which is ontological (as Kant also stressed), is also rejected. To distance oneself from substantialist thinking means to deobjectify all elements of metaphysics. Being more radical than Kant, Whitehead underlines the subjective nature of reality. He does not need the "fuel" of "objective" sensations to turn on the "engine" of the categories to guarantee the possibility of synthetic propositions a priori. On the contrary, all is subject. Hence, the actual world is a subject. So the actual whole is an organic unity of those elements disclosed in the analysis of the experience of subjects. We cannot go further. We cannot pull ourselves outside ourselves. Knowledge is subjective, for nobody escapes his or her own form of definiteness. This "subjectivist principle" is another way to state the principle of relativity. All things are qualifications of actual occasions and there is nothing else. The Platonic world is unmasked as the root of all ousiological constructs. The world is a unity of actual entities and without the latter there is nothing. There is no transcendent world, no ontological stratum "above" the world we observe. The exercise of metaphysics is immanent, not transcendent. In this "self model", the "cogito" is thus the definition of actuality. Only "actual occasions" of "actual entities" are the building-blocks of the universe. Only actual ntities exist. An event is then a "nexus" of actual entities. Causality is also implied. If there are no events, then there can be no causality. But events happen. If event A exerts its influence on event B (or "causal efficacy"), then B cannot be totally explained by A. This because the "novelty" of event B cannot be explained in terms of past initial events only. So, besides efficient causality, he conjectures a "formal causality", which is the cause of the becoming of the "novelty" incorporated in B. This formal causality aims at self-realization and self-creation. "... nexus is a set of actual entities in the unity of the relatedness constituted by their prehensions of each other, or -what is the same thing conversely expressed- constituted by their objectification in each other." Whitehead, A.N. : PR, § 35. This self-creation of the actual entities is the self-constitution of an experience. In the process of the non-I exerting an influence, something is experienced (this is the causal efficacy). Besides, there is the "subjective immediacy" of the self-experience, accomplishing a new synthesis between the multiplicity of the many influences and the own form of definiteness. Hence, the actual entities are not solipsist (like monads), but continuously enter in each other's self-creation. "Being" is hence always to be in another. Being (events) & becoming (self-creation) imply the capacity to enter in another, new actual entity. The universe is hyper-social. "... it belongs to the nature of a 'being' that it is a potential for every 'becoming'. Thus all things are to be conceived as qualifications of actual occasions." Whitehead, A.N. : PR, § 252. Whitehead understands being from the vantage point of becoming. He does not eliminate the eternal, for not only does he wish to replace a teaching on substance with a teaching on events, but he virulently reacts against the "vicious separation" between "flux" & "permanence". This distinction introduced the bi-polarity between temporality (becoming) and eternity (being) and the adjacent aporic pendulum-movement between the two (the same dyad returns in all areas of Greek, scholastic and pre-Kantian thought and influenced most religions). Traditional metaphysics conceptualized being and identity and so construed a static God, an "aboriginal, eminently real, transcendent creator". Instead, metaphysics thinks "permanency in fluency, fluency in permanence". Although becoming is the sole point of view, one cannot grasp the ultimate nature of the universe without simultaneously thinking both the changing world of events and the eternal realm of pure potency. The dyad remains, but devoid of possible substantialist antagonism. The universe is dual, for it is both transient (conventional or actual) and eternal (ultimate or potential). There is nothing "outside" reality, constituted by both formative elements and actual entities. § 5 Although nothing except actual entities exist, the world of actual events is not the Real as a whole. Although there is no world "behind" the world of events, and this changing, phenomenal reality is all there is, one is able to think (conceptualize) the eternal and the permanent. This is not an ontological realm, source of being, transcendent sufficient ground, "prima materia" or pre-creation initiating creation "ex nihilo", for actual entities are the only existing things, i.e. the ultimate exists conventionally. In separation from actual entities, there is nothing, merely nonentity. But a "category of the ultimate" can and should be thought. In Religion in the Making, the three "formative elements" called in to guarantee order & novelty in the actual world are explained thus : 1. creativity realized in actual entities : "'Creativity' is the universal of universals characterizing ultimate matter of fact. It is that ultimate principle by which the many, which are the universe disjunctively, become the one actual occasion, which is the universe conjunctively. It lies in the nature of things that the many enter into complex unity. 'Creativity' is the principle of novelty. An actual occasion is a novel entity diverse from any entity in the 'many' which it unifies." Whitehead, A.N. : PR, § 31. Thanks to creativity, the real actual world lapses into a new world order. The dynamism of the world of actual entities, grasped by the senses, implies novelty, for the unity of experience here and now is an original concrescence of previous experiences and my own form of definiteness and determination. The creativity of the actual universe demands everything influences everything, bringing multiplicity to unity. The actual course of events is thus not self-evident. The sheer ongoingness of the universe speaks of permanent creativity, from the smallest subatomic particle to God's eternal valuation of possibilities. Creativity is the "natural matrix of all things" and real when realized in an actual entity. The self-creativity of entities is an instance of this creativity, which itself is not a substance, nor an entity, nor a reality. It is a "category" qualifying (determining, limiting) all actual entities ; 2. potential eternal objects forming actual entities : the "perpetual perishing" of actual entities cannot be "saved" by something which is itself an entity, for all entities are "on the move", all actual, concrete things change (impermanence). Next to (not behind, nor underneath or above) the world of actual entities, Whitehead postulates a world of pure potency and possibility. This abstract world is the domain of "pure potential for the specific determination of fact". These eternal objects are implied by the fact no two actual entities are completely identical although similarities can be determined. The latter point to a "form of definiteness". These forms participate in the becoming of actual entities, but are themselves not actual or concrete. Neither are they unreal, but potential, i.e. indicative of possibility. Because they remain identical with themselves, these objects are called "eternal". They escape the permanent change of the real world, and because they are in no way "subject", i.e. an actual, real entity, they are "objective" and "grasped" by mental "prehension". The "objective" is not "the concrete" (for only actual, subjective entities are so), nor "unreal" (as nonentity or fiction). The objective is sheer potentiality ; 3. God harmonizing endless potentiality : the domain of pure potentiality is per definition limitless. The eternal objects give form to actual entities but are themselves without borders. By giving "graded relevance" to these various endless possibilities, God harmonizes the different possibilities and so orders the becoming of the actual entities from within, receiving form & structure. The "key" used by God is called "harmony" and "beauty". God embraces all possibilities but offers them as the esthetic possibility of self-creation. God rules all possibilities and is also the principle of definiteness. God grasps all possibilities and harmonizes them. God limits the limitless domain of pure potentiality so something may enter actuality. Every valuation is contingent, and without God no possibility can become actual. Because of God's "vision of beauty", continuous pressure is put on all events, giving them their "initial aim". As God is not creativity itself, God is not responsible for all what happens ! (b) The Primordial Nature of God. "Viewed as primordial, he is the unlimited conceptual realization of the absolute wealth of potentiality. In this aspect, he is not before all creation, but with all creation. But, as primordial, so far is he from 'eminent reality', that in this abstraction he is 'deficiently actual' - and this in two ways. His feelings are only conceptual and so lack the fullness of actuality. Secondly, conceptual feelings, apart from complex integration with physical feelings, are devoid of consciousness in their subjective forms." Whitehead, A.N. : PR, §§ 521. Among the formative elements, God is an actual entity, while the eternal objects are not. God is the anterior ground guaranteeing a fraction of all possibilities may enter into the factual becoming of the spatiotemporal world. Without God, nothing of what is possible, can become some thing, change and create. The universe, its order and creativity are the result of a certain valuation of possibilities. However, God is not the universe, nor its order (derived from eternal objects) or the creativity at work in actual entities. The latter are concrete actual entities, while God is an abstract actual entity, while creativity & eternal objects are non-actual formative elements. 1. concrete actual entities (the actual world) : all what exists in the world of facts and events ; 2. abstract actual entity (the abstract) : God "the organ of novelty, aiming at intensification" is the Artist who makes a beautiful world more likely ; 3. potential eternal objects (the potential Realm of Possibilities) : selfsame, "pure" forms outside the stream of actual entities, organizing them ; 4. creativity : the formless "matrix" of all things, the principle of the continuous becoming of novel unity and creative advance out of multiplicity. God is the instance grounding the permanence and continuous novelty characterizing the universe. This primordial nature of God is completely separated from the actual world. For although an actual entity, God's activity is "abstract", namely in the esthetic (artistic) process of valuating possibilities, which are no fictions. But God is engaged in the factual becoming of the actual entities, but cannot be conceived as a concrete actual entity, a fact among the facts. God is the only "abstract" actual entity possible. Besides being an abstract Godhead, God is also a Divine consciousness prehending all events. This is his consequent nature. In these two ways, God is related to the realm of actualities. God's primordial nature is transcendent and does not touch the universe, the actual world. This aspect of Deity is God as the "Lord of All Possibilities". It offers all events the possibility to constitute themselves. If not, nothing would happen. Possibilities, although highly abstract, are no fictions, and enter concrete entities (cf. Popper's propensity-interpretation of the Schrödinger equation). Although there is no imaginary heavenly (Platonic) museum displaying the statue of David before Michelangelo fashioned it, the latter did not invent the material, the possibility allowing him to do so. So the fact formless creativity received definite form is attributed to God as Principle of Definiteness. By way of conceptual valuation, God imposes harmony on all possibilities, for actuality implies choice & limitation. But as all order is contingent, lots of things always remain possible. Whitehead never speaks of God as the "Creator of the Universe" (too suggestive of the total dependence of the world). The "ideal harmony" is only realized as an abstract virtually, and God is the actual entity bringing this beauty into actuality, turning potential harmony into actual esthetic value. Taking into account everything given in the field of existence of all actual events, God's highest purpose for each is for it to contribute to the realization of the purpose of the whole, namely the unity of harmony in diversity. God does not decide, but lures, i.e. makes beauty more likely. There is no efficient causality at work here, but a teleological pull inviting creative advance. Given the circumstances, a tender pressure is present to achieve the highest possible harmony. God is the necessary condition, but not the sufficient condition for events. Classical omnipotence & omniscience are thus eliminated. God knows all actual events as actual and all possible (future) events as possible. He does not know all future events as actual. This is a category mistake. He cannot hamper creativity. Giving metaphysical complements to God is relinquished. God's purpose for each and every event, given all conditions determining it, is that it may contribute to the realization of the purpose of the whole universe, the harmony in diversity. God is the unique abstract actual entity making it possible for the multiplicity of events to end up in harmony. This aspect of God is permanent, eternal and not linked to time & space. It is a permanent property of reality, resulting in a uni-verse. Call this aspect of Deity "Godhead". (c) The Consequent Nature of God. "Love neither rules, nor is it unmoved ; also it is a little oblivious as to morals. It does not look to the future, for it finds its own reward in the immediate present." Whitehead, A.N. : PR, §§ 520 - 521. God's consequent nature is God's concrete, super-conscious presence in the universe, actually being near all possible events and valorizing them to bring out harmony and the purpose of the whole. God, with infinite care, is a tenderness loosing nothing that can and wants to be saved. Hence, God's experience of the world changes. It always grows and can never be given as a whole. God is loyal and will never forsake any event. The two natures of God are not two parts or elements, but two ways of dealing with the world. Primordially, God is always offering possibilities and realizing unity and order, and this in all possible worlds. Consequentially, God takes the self-creation of all actual events in this concrete universe into account, considering what they realize of what is made possible. These two ways, initiating & responding, permanent & alternating are God's bi-polar, pan-en-theist approaches of the actual world. 9 Towards a Synthesis. The Tao, both transcendent & immanent, is the one reality, the Real encompassing both the world of pure potency and the realm of actual entities. In an absolute sense, the Tao is the ultimate truth or most profound, implicate nature of all phenomena, but non-differentiated, nameless and empty of fixed substantial essence. In a relative sense, the Tao is the relative truth or explicate nature of these same phenomena, differentiated, named and interdependent with all other phenomena. Just like the Nun (the undifferentiated ocean) and Atum (the principle of unity & differentiation), the absolute Tao and the One are pre-existent, potential, virtual. They are the aspect of the Real harboring all possibilities and the principle of limitations bringing out, albeit still in potentia, their forms of definiteness. Caught by the limitations of classical formal logic, emptiness (ultimate truth) and interdependent arising (conventional truth), i.e., on the one hand, the Tao, the One, the Two & the Three, and on the other hand, the Ten Thousand Things, seem disjunctive. Indeed, conceptual thought is unable to cognize both in conjunction. The ultimate experience of the Tao exceeds all possible conceptual categories and so only direct spiritual experience is the only final arbiter. Even if we succeed to come near to a conceptual realization of emptiness, we fail to grasp is exhaustively. The conceptual mind may relinquish classical formal logic and embrace non-classical approaches (seeing emptiness in terms of symmetry transformations & interdependent arisings as symmetry breaks), and understand the totality of the Real as a movement of totality (holomovement), no conceptual approach will satisfy and silence the mind. Only the direct apprehension of the nature of mind, its "Clear light" ("rigpa") realizes such a feat. But, this is the way of silence, ending all possible conceptualization. Despite the fact reason must try to move to the outer frontiers of possibilities offered by concepts, and so must not leap into mystification and paradox before doing its utmost most to first achieve conceptual clarity, distinctness and argumentative excellence, one can never replace such endeavors with the datum of direct experience. As mystical experience, bathing in the ecstatic, moves beyond the conceptual, no scholasticism is able to catch it. The direct experience of the Tao remains hidden for discursive thought, even in its most abstract, metaphysical countenance, given way to the ultimate stretch of intellectualism. Three fundamental thoughts persist : 1. all phenomena lack inherent existence, i.e. are process-like instead of substance-like. Realizing this truth is apprehending the ultimate nature of all phenomena ; 2. simultaneously all phenomena are other-powered, i.e. depend on something other than themselves. Even the absolute Tao is not a "substance of substances", but a phenomenon depending on conditions. The Way follows Nature ; 3. all dependent phenomena rise out of emptiness, i.e. are definite actualizations of an indefinite potential. (a) Rationality & Experience of Emptiness : "Ontological 'essentialism' is dangerous because as soon as we take up such an attitude, we are doomed to lose our natural flexibility of mind and consequently lose sight of the absolute 'undifferentiation' which is the real source and basis of all existent things." Izutsu, T. : Op.cit., 1983, p.359. Dharmic spirituality (in its Buddhist & Taoist variants) and Process Theology both reject substantialism and emphasize change. On the most fundamental, implicate or ultimate level, phenomena are devoid of self-subsisting, inherently existing essence, but considered as "empty", i.e. lacking static, thing-like, self-powered own-nature. Emptiness itself is also "empty", implying the concept of emptiness has to be eliminated too. Emptiness can be approached by way of concepts or by way of experience. The right meditation first eliminates concepts by way of concepts (the rational method), allowing this clearing to bring in the light of the nature of mind. Wrong meditation eliminates concepts and then stops, producing the passive void of nihilism. • In a strict rational, conceptual path (as in Critical Mâdhyamaka), emptiness is nothing "in itself", but merely points to an absence, a lack or a non-affirming negation. Here, emptiness is introduced to eliminate concepts by means of concepts, stopping the grasping, deluded, samsaric mind to bring about the luminous, enlightened mind. Insofar as emptiness is conceptualized without this intent to realize the nirvanic, shining mind, emptiness becomes an obscuration to the mind, making it addicted to the medicine, producing the disease of nihilism, equating emptiness with sheer nothingness, zero or naught. This is missing the point. The rational view precludes direct experience. • In the experiential view of the Tantras, Dzogchen yogis and Wayfarers, emptiness refers to having no fixations at all, allowing the basic nature of mind to spontaneously appear, i.e. extinguishing the stirring mind without extinguishing the shining mind. The shining mind, nature of mind, Clear Light of mind, original spirit or Buddha-nature point to a non-conceptual, nondual cognition, the experience or apprehension of emptiness as a limitless field of all possibilities out of which all objects emerge. This idea, of all conventional dependent arisings emerging out of emptiness (as the golden lion out of gold) is the pivotal contribution of Chinese Buddhism, in particular Hua-yen & T'ien-tai. The rational view does not preclude direct experience, but facilitates it ... Both in Buddhism, Taoism and Process Theology, emptiness is more than mere absence of inherent existence. While classical logic can do not more than identify emptiness with the non-affirming negation, these systems posit a limitless field or energy of possibilities, virtuality & potentialities. Emptiness appears as the absolute absoluteness of a virtual realm of all possibilities. The latter is an endless ocean of non-differentiated energy giving rise to all actual, conventional events. (b) Dependent Arising : "The very driving force by which a thing is born, grows up, flourishes, and then goes back to its own origin - this existential force which everything possesses as its own 'nature' - is in reality nothing other than the Way as it actualizes itself in a limited way in everything." Izutsu, T. : Op.cit., 1983, p.403. On the implicate, ultimate level, phenomena are empty of inherent existence, existing as Actus purus in a state of sheer potency. Buddhahood is the direct experience or ecstatic apprehension of this level of reality, known as the ultimate aspect of every (conventional) phenomenon (the ultimate therefore exists conventionally, not "split off" from the actual world). On the explicate, conventional level, phenomena are interdependent & interconnected, constantly changing & creative, i.e. ongoingly entering the self-becoming of other actual entities. On this level, phenomena appear as if they exist from their own side, independent from subjects or other objects. This mistaken appearance does not hinder their validity as conventional, functional occurrences. Although illusionary, they do allow the mind to logically & functionally distinguish them and know them as valid in conventional terms. These two aspects of reality as a whole, namely the ultimate, abstract reality of emptiness and the conventional, concrete reality do not exist as two realities, but as two simultaneous sides of every single event. There is no Platonic split in being, for the division does not refer to two ontic realities (one Supreme and another not), but to one ontic reality (the selfsame, unique, singular reality) simultaneously harboring two polarities, the one concrete, the other potential. The one reality is therefore bipolar. "The final facts are, all alike, actual entities ; and these actual entities are drops of experience, complex and interdependent." Whitehead, A.N. : PR, § 28. (c) The One : "... most of the Taoist texts depict the One as residing within the body in the form of the three Primordial Breaths - namely the Three-One (san-i) or Three Originals (san-yüan). These are the deities that must be 'preserved' or maintained within the body by the means of meditative thought." Robinet, I., Op.cit., 1993, p.123. The One is not an ontological entity above and beyond all possible entities (as in Plotinus), but a principle of definiteness bringing, in potentia, limitations to the limitless creativity of the absolute Tao of limitless creativity. Without the One, the infinite would remain infinite and no possibility would be able to receive the potential form of being an identity (A = A). This potential form is of course not yet actual form, for the One is merely a non-differentiated, abstract principle of unity and harmony. The eternal objects (Heaven and Earth) add a principle of differentiation, allowing, in potentia, the limitations under unity advanced by the One to differentiate into multiplicities. With both potential unity (harmony) and differentiation, the actual world can manifest as a series of actual events constantly lured into contributing to the manifest unity & esthetic value of the universe. "The One is present in everything as its ontological ground. It acts in everything as its ontological energy. It develops its activity in everything in accordance with the latter's particular ontological structure (...) If it were not for this activity of the One, nothing in the world would keep its existence as it should." Izutsu, T. : Op.cit., 1983, p.402. Insofar as the One, as Author of the World, is merely a principle of unity, It is an impersonal, eternal super force, energy or power establishing potential identities within the limitless. But as all actual entities exist by the grace of this principle, the One, as the Architect of the World, is also a super-consciousness or super-mind in which all entities endure and last forever. The One is thus simultaneously the eternal principle of unity (Godhead) and the everlasting, all-encompassing conscious actual entity who is near all events (God). As Godhead, the One is impersonal and unconscious, but as God, the One is personal and conscious, aware of every event as it has happened yesterday, as it happens now and as it possibly may happen tomorrow. "When the One is attained, all problems are solved." Chao Pi Ch'en, in Luk, Ch. : Op.cit., 1973, p.5. Given the self-becoming of all entities, playing together in the alleatoric symphony of the actual world, God cannot know every event that actually will happen. Insofar as God does not know the actual future beforehand, the world influences God, co-determines the Divine Comedy, making God no longer the impassible, solipsist Caesar above & beyond the world, but the vulnerable fellow-sufferer with everlasting patience, one who cares without succumbing to this suffering ! (d) Towards a Synthetic Ontological Scheme : "... the formless spontaneously produces form and the immaterial produces substance." Reality, the Tao, is more than just the temporal actual world of concrete actual entities. It is the unity of the concrete (the Ten Thousand Things) and the abstract (the absolute Tao, the One, the Two, the Three), of actuality & potentiality. Synthetic Ontological Scheme temporal concrete actual the Ten Thousand Things dependent valid but mistaken abstract actual potential differentiation Heaven & Earth emptiness valid and potential wholeness The One formless matrix of All Reality is the unity of the actual world and the realm of all abstract possibilities. The former is spatiotemporal, the latter abides, as Actus Purus, in the "fourth time" of the Eternal Present. This realm of virtuality itself is organized in degrees of determination : 1. the non-differentiated absolute in its absoluteness (the ultimate nature of all phenomena beyond conceptualization), the absolute Tao or limitless & formless creativity is the "matrix" or "receptacle" of all possibilities itself, boundless ongoing symmetry-transformations. This absolute Tao, the "Dharmakâya" of Buddhahood, is the fundamental absolute reality, the Mystery of Mysteries ; "Ultimate nonbeing, it contains ultimate being ; ultimate emptiness, it contains ultimate fulfillment." 2. this Mystery of Mysteries is also the Gateway of Myriad Wonders, or the One (Wu-Chi), bringing these formless possibilities under the principle of unity & order of harmony in potentia, allowing formlessness to become potential form, accommodating potential wholeness (impling unity-under-variety). Here we find the Sambhogakâya of Buddhahood, the absolute Tao bridging formlessness and actuality, linked with great compassion and great benevolence ; 3. the One determines Heaven & Earth, i.e. differentiation in potential. Yang and Yin interact (the Three) and manifest as actual entities, the Ten Thousand things, the Nirmânakâya of Buddhahood, the body of manifestation (Tai Chi). This body first becomes actual as a singular super-mind, the conserving Architect of the world, encompassing the temporal world of actual entities. This super-mind (God) is the One (Godhead) insofar as it with all actual events. It is the One conscious of the actual world, luring it towards the greatest possible unity & esthetic value. This progression in degrees of determination of the absolute Tao is not temporal, but logical (abstract). The realm of potentiality and the realm of actuality, of ultimate & conventional reality are simultaneous, a condition only to be grasped by non-dual, non-conceptual ecstacy or enlightened wisdom-mind. These degrees represent three aspects of the absolute Tao, namely insofar as it is the absolute absoluteness of creativity (Mystery of Mysteries), the absolute principle of potential definiteness (the One) and the absolute principle of potential being or manifestation (Heaven & Earth). To clear the possessive, deluded mind from substantial concepts is a necessary condition for Wayfaring and the realization of the wisdom-mind taught by the Buddhas and the Immortals. Then and only then can phenomena be apprehended as interdependent and no longer as substances, the mode of cognizing of deluded beings. To the enlightened, immortal mind, all is empty, and this ultimate truth is directly experienced in an ineffable, ecstatic way, apprehending it in terms of a nondual cognition beyond all possible affirmation & denial. To realize this luminous mind of "Clear Light", concepts have to be silenced and this can be done using various methods. The contemplative mind mostly does so by merely thoroughly silencing the mind, entering the vast space of its natural state, while the more intellectual mind eliminates fixed concepts by ultimate concepts, raising the Sword of Wisdom to cut through the correct object of negation : inherent existence. The most profound mind witnesses how both approaches combine : Calm Abiding and its meditative equipoise stimulates Insight Meditation and insight increases tranquility. Such minds quickly enter "nirvâna", attaining the immortal spirit or the indestructible, very subtle mind, the core of sentience. Given the presence of intelligence in contemplation and the presence of contemplation in intelligence, one should not confront tranquility with insight or seek the latter without the former. But, although conceptual analysis is necessary, it is far from sufficient. Indeed, spiritual practice is the only way to realization, and conceptual elucidation is merely a way to prepare the mind with the correct view. Wrong views fix the mind, rendering it incapable to approach the ultimate truth. As a shield, such views blind one from suchness, things as they truly are. As an anchor, they stop progress, and so no harbor can be found. But playing with correct views without the direct experience of suchness addict the mind to the medicine, leading to nihilism, the view reality is not functional, operational and so not the mother of the concrete. In both Wayfaring and the Buddhadharma, spiritual practice aims at the immortal & enlightened wisdom-mind. But while the Buddhas develop methods, by meditating on emptiness, to generate this mind, Wayfarers seek to "circulate the breaths" and feed the "elixir fields" to harmonize the polarities in ever-dynamic ways, bringing about immortality by the "king of logics", dependence & interdependence. Both approaches complement each other, for Wayfaring without emptiness is folly and emptiness without dependent arising lacks compassion. Hence, to integrate these spiritual practices in one spiritual exercise (in fact a series of such exercises) is the correct concentration sought after. So both Buddhist wisdom and Taoist philosophy are cripple without spiritual practice. Clearing concepts to arrive at the correct view serves the purpose of the correct spiritual practice, combining meditation & Chi-circulation. Divorced from the latter, the intellectual pursuit is vain, unnecessary and dangerous. Vain because nothing lasting is achieved. Unnecessary because it is a mere waste of good time. Dangerous because by overabusing the medicine, emptiness becomes a poison. And with the sole medicine gone, how can one be healed ? The personal experience of non-conceptual wisdom is the sole defense against the ignorance of eternalism & nihilism. Philosophy without the actual practice of wisdom is a heap of dead bones, like trying to make a skeleton compete with a living whole. The integration of Taoism & Confucianism (being a Confucian during the day and a Taoist at night) belongs to the intention to "cull yin to augment yang", never negating Earth to reach for Heaven. In Mâhayâna Buddhism, the "pâramitâs" are cultivated to accumulate merit in the light of generating Bodhicitta, so the realm of "samsâra" can be more quickly exited for the benefit of all sentient beings (for as Buddhas helping others becomes flawless). Moreover, the monastic way is preferred, thus separating the spiritual intent from the world (renunciation), making monks beg for their subsistence. In a general way, Indian spirituality reaches for Heaven by negating Earth, while in the Chinese mentality, the concrete is spiritualized and the subtle materialized. This is also reflected in the yogas & tantras ; Indians focus on the subtle channel near the spine (raising the Kundalinî-Śakti), while the Chinese make the Chi circulate, connecting back (yang) & front (yin) channels of the body. So by adding the Confucian intent of harmonizing the individual with society, the overall combination becomes more potent, turning spirituality, by transforming individuals, into a force to change society for the better. • Confucianism : living in harmony with society ; • Taoism : transforming the individual in harmony with nature ; • Buddhism : realizing wisdom-mind. Finally, by "preserving the One", experiencing the "God of process" next to meditation and Chi-circulation, prayer & mystic experience enter as the third pole of spiritual life. Then, the Tao can also be addressed as a benevolent super-consciousness, not merely as life-force "driven" by intent as an objective series of (generative, vital & spiritual) powers. The latter are impersonal, while a personal super-mind intimately knows me as a person, establishing an intersubjective dialogue affecting both parties involved. Such a practice must avoid the age-old tendency of the mind to equate Deity with the concept of a Supreme Being, i.e. avoid the reification or eternalization of God. The God of process is not a substance, nor does the One supersede the absolute absoluteness of the Tao, nor dominate the limitless field of creative possibilities. By keeping this "existential" view as the correct view on God, a new non-imperialist onto-theology is possible, and intelligent persons may once again with confidence pray : "Kyrie eleison !", "Lord, have mercy !" For the Bibliography on Ancient Egypt click here. For the Bibliography on General Philosophy click here. Beinfield, H. & Korngold, E. : Between Heaven and Earth, Random House - London, 1991. Chang, G.C.C. : The Buddhist Teaching of Totality, Pennsylvania State University Press - Pennsylvania, 1994. Chia, M. : Awaken Healing Energy Through the Tao, Aurora - Santa Fe, 1983. Chia, M. : Tan Tien Chi Kung, Destiny Books - Rochester, 2004. 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Yudelove, E.S. : 100 Days to Better Health, Good Sex & Long Life, Llewellyn Publications - Minnesota, 1997. © Wim van den Dungen, Antwerp - 2015 philo@sofiatopia.org l Acknowledgments l SiteMap l Bibliography May all who encounter the Dharma accumulate compassion & wisdom. initiated : 25 III 2009 - last update : 24 IX 2014 - version n°2
87189aef75a8a5b2	Trend: Is a room-temperature, solid-state quantum computer mere fantasy? • Marshall Stoneham, London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK Physics 2, 34 Creating a practical solid-state quantum computer is seriously hard. Getting such a computer to operate at room temperature is even more challenging. Is such a quantum computer possible at all? If so, which schemes might have a chance of success? Illustration: Alan Stonebraker Figure 1: In optically controlled spintronics [11], the active region is a thin (a few nm) silicon layer on a silica substrate. A thin diamond film may ultimately prove better. This thin film is randomly doped with qubit donors (red circles) and control donors (green circles).In optically controlled spintronics [11], the active region is a thin (a few nm) silicon layer on a silica substrate. A thin diamond film may ultimately prove better. This thin film is randomly doped with qubit donors (red circles) and control donors... Show more Illustration: Alan Stonebraker Figure 2: The entanglement of the qubit electron spins is controlled by exciting a chosen control donor into a spatially more extended state. Dopant wave functions fix the scale of the separations over which entanglement is effective to a mere 1020nm. But light can only be focused down to 10002000nm. However, different controls (hence different gates) can be selected by exploiting the variations in excitation energies from one place to another, from whatever cause, even from surface steps. So here the red laser excites one control, entangling two qubits. Next, the green laser excites another control, in the case shown entangling a different pair of qubits without affecting those controlled by the red laser. Spectral selectivity is combined with spatial selectivity. In this way, a sequence of optical pulses of chosen wavelengths and durations can control quantum operations in a “patch” of about one optical wavelength across. Each patch might contain perhaps 20 gates.The entanglement of the qubit electron spins is controlled by exciting a chosen control donor into a spatially more extended state. Dopant wave functions fix the scale of the separations over which entanglement is effective to a mere 1020nm. But lig... Show more Illustration: Alan Stonebraker Figure 3: Optically controlled spintronic patches might be linked by flying qubits to form a larger processor. Even 20 qubits linked within a patch would provide only a very modest quantum computer. Linking 10 or 12 patches would be much more impressive. This figure shows schematically such a linkage to form a larger processor. If each patch is to be accessed by separate optical inputs, the spacings must be more than optical wavelengths, so of order 1–2 micronsOptically controlled spintronic patches might be linked by flying qubits to form a larger processor. Even 20 qubits linked within a patch would provide only a very modest quantum computer. Linking 10 or 12 patches would be much more impressive. This ... Show more Illustration: Alan Stonebraker Figure 4: In each cavity is a diamond with an NV center in a specific state, the two tuned to have identical excitation energies. Both systems are exposed to a weak laser pulse that, on average, will achieve one excitation. The single NV center excited will emit a photon that, after passing though beam splitters and an interferometer, is detected without giving information as to which system was excited. This leaves the two NV centers entangled.In each cavity is a diamond with an NV center in a specific state, the two tuned to have identical excitation energies. Both systems are exposed to a weak laser pulse that, on average, will achieve one excitation. The single NV center excited will ... Show more Illustration: Alan Stonebraker Figure 5: Larger arrays of diamond center qubits could be linked together for scale-up to a quantum computer. The many pairwise entanglements (Fig. 4) can be linked via a fast-switched optical multiplexer, in readiness for the final measurement step. In his 2008 Newton Medal talk, Anton Zeilinger of the University of Vienna said: “We have to find ways to build quantum computers in the solid state at room temperature—that’s the challenge.” [1] This challenge spawns further challenges: Why do we need a quantum computer anyway? What would constitute a quantum computer? Why does the solid state seem essential? And would a cooled system, perhaps with thermoelectric cooling, be good enough? Some will say the answer is obvious. But these answers vary from “It’s been done already” to “It can’t be done at all.” Some of the “not at all” group believe high temperatures just don’t agree with quantum mechanics. Others recognize that their favored systems cannot work at room temperature. Some scientists doubt that serious quantum computing is possible anyway. Are there methods that might just be able to meet Zeilinger’s challenge? The questions that challenge What is a computer? Standard classical computers use bits for encoding numbers, and the bits are manipulated by the classical gates that can execute AND and OR operations, for example. A classical bit has a value of 0 or 1, according to whether some small subunit is electrically charged or uncharged. Other forms are possible: the bits for a classical spintronic computer might be spins along or opposite to a magnetic field. Even the most modest computers on sale today incorporate complex networks of a few types of gates to control huge numbers of bits. If there are so few bits that you can count them on your fingers, it can’t seriously be considered a computer. What do we mean by quantum? Being sure a phenomenon is “quantum” isn’t simple. Quantum ideas aren’t intuitive yet. Could you convince your banker that quantum physics could improve her bank’s security? Perhaps three questions identify the issues. First, how do you describe the state of a system? The usual descriptors, wave functions and density matrices, underlie wavelike interference and entanglement. Entanglement describes the correlations between local measurements on two particles, which I call their “quantum dance.” Entanglement is the resource that could make quantum computing worthwhile. The enemy of entanglement is decoherence, just as friction is the enemy of mechanical computers. Second, how does this quantum state change if it is not observed? It evolves deterministically, described by the Schrödinger equation. The probabilistic results of measurements emerge when one asks the third question: how to describe observations and their effects. Measurement modifies entanglement, often destroying it, as it singles out a specific state. This is one way that you can tell if an eavesdropper intercepted your message in a quantum communications system. Proposed quantum computers have qubits manipulated by a few types of quantum gates, in a complex network. But the parallels are not complete [2]. Each classical bit has a definite value, it can only be 0 or 1, it can be copied without changing its value, it can be read without changing its value and, when left alone, its value will not change significantly. Reading one classical bit does not affect other (unread) bits. You must run the computer to compute the result of a computation. Every one of those statements is false for qubits, even that last statement. There is a further difference. For a classical computer, the process is Load Run Read, whereas for a quantum computer, the steps are Prepare Evolve Measure, or, as in one case discussed later, merely Prepare Measure. Why do we need a quantum computer? The major reasons stem from challenges to mainstream silicon technology. Markets demand enhanced power efficiency, miniaturization, and speed. These enhancements have their limits. Future technology scenarios developed for the semiconductor industry’s own roadmap [3] imply that the number of electrons needed to switch a transistor should fall to just 1 (one single electron) before 2020. Should we follow this innovative yet incremental roadmap, and trust to new tricks, or should we seek a radical technology, with wholly novel quantum components operating alongside existing silicon and photonic technologies? Any device with nanoscale features inevitably displays some types of quantum behavior, so why not make a virtue of necessity and exploit quantum ideas? Quantum-based ideas may offer a major opportunity, just as the atom gave the chemical industry in the 19th century, and the electron gave microelectronics in the 20th century. Quantum sciences could transform 21st century technologies. Why choose the solid state for quantum computing? Quantum devices nearly always mean nanoscale devices, ultimately because useful electronic wave functions are fairly compact [4]. Complex devices with controlled features at this scale need the incredible know-how we have acquired with silicon technology. Moreover, quantum computers will be operated by familiar silicon technology. Operation will be easier if classical controls can be integrated with the quantum device, and easiest if the quantum device is silicon compatible. And scaling up, the linking of many basic and extremely small units is a routine demand for silicon devices. With silicon technologies, there are also good ways to link electronics and photonics. So an ideal quantum device would not just meet quantum performance criteria, but would be based on silicon; it would use off-the-shelf techniques (even sophisticated ones) suitable for a near-future generation fabrication plant. A cloud on the horizon concerns decoherence: can entanglement be sustained long enough in a large enough system for a useful quantum calculation? All the objections It has been done already? Some beautiful work demonstrating critical steps, including initializing a spin system and transfer of quantum information, has been done at room temperature with nitrogen-vacancy ( NV-) centers in diamond [5]. Very few qubits were involved, and scaling up to a useful computer seems unlikely without new ideas. But the combination of photons—intrinsically insensitive to temperature—with defects or dopants with long decoherence times leaves hope. It can’t be done: serious quantum computing simply isn’t possible anyway. Could any quantum computer work at all? Is it credible that we can build a system big enough to be useful, yet one that isn’t defeated by loss of entanglement or degraded quantum coherence? Certainly there are doubters, who note how friction defeated 19th century mechanical computers. Others have given believable arguments that computing based on entanglement is possible [6]. Of course, it may prove that some hybrid, a sort of quantum-assisted classical computing, will prove the crucial step. It can’t be done: quantum behavior disappears at higher temperatures. Confusion can arise because quantum phenomena show up in two ways. In quantum statistics, the quantal ħ appears as ħω/kT. When statistics matter most, near equilibrium, high temperatures T oppose the quantum effects of ħ. However, in quantum dynamics, ħ can appear unassociated with T, opening new channels of behavior. Quantum information processing relies on staying away from equilibrium, so the rates of many individual processes compete in complex ways: dynamics dominate. Whatever the practical problems, there is no intrinsic problem with quantum computing at high temperatures. It can’t be done: the right qubits don’t exist. True, some qubits are not available at room temperature. These include superconducting qubits and those based on Bose-Einstein condensates. In Kane’s seminal approach [7], the high polarizability needed for phosphorus-doped silicon ( Si:P) corresponds to a low ionization donor energy, so the qubits disappear (or decohere) at room temperature. In what follows, I shall look at methods without such problems. What needs to be done: Implementing quantum computing David DiVincenzo at IBM Research Labs devised a checklist [8] that conveniently defines minimal (but seriously challenging) needs for a credible quantum computer. There must be a well-defined set of quantum states, such as electron spin states, to use as qubits. One needs scalability, so that enough qubits (let’s say 20, though 200 would be better) linked by entanglement are available to make a serious quantum computer. Operation demands a means to initialize and prepare suitable pure quantum states, a means to manipulate qubits to carry out a desired quantum evolution, and means to read out the results. Decoherence must be slow enough to allow these operations. What does this checklist imply for solid-state quantum computing? Are there solid-state systems with decoherence mechanisms, key energies, and qubit control systems that might work at useful temperatures, ideally room temperature? Solid-state technologies have good prospects for scalability. There is a good chance that there are ingenious ways to link the many qubits and quantum gates needed for almost any serious application. However, decoherence might be fast. This may be less of a problem than imagined, for fast operating speeds go hand in hand with fast decoherence. Fast processing needs strong interactions, and such strong interactions will usually cause decoherence [9]. For spin-based solid-state quantum computing, most routes to initialization group into four categories. First, there are optical methods (including microwaves), based on selection rules, such as those used for NV- experiments. Then there are spintronic approaches, using a source (perhaps a ferromagnet) of spin-polarized electrons or excitons. (Note that spins have been transferred over distances of nearly a micron at room temperature [10].) Then there are brute force methods aiming for thermal equilibrium in a very large magnetic field, where the ratio of Zeeman splitting to thermal energy kBT is large. And finally there are tricks involving extra qubits that are not used in calculations. Of these methods, the optical and spintronic concepts seem most promising for room-temperature operation. For readout, there are two broad strategies. Most ideas for spin-based quantum information processing aim at the sequential readout of individual spins. However, there are other less-developed ideas in which the ensemble of relevant spins is looked at together, as in some neutron scattering studies of antiferromagnetic crystals. What methods are there for probing single spins, if the sequential strategy is chosen? First, there is direct frequency discrimination, including the use of Zeeman splitting, of hyperfine structure, and so on. Ideas from atom trap experiments suggest that one can continue to interrogate a spin with a sequence of photons that do not change the qubit [11]. Such methods might work at room temperature, at least if the relevant spectral lines remain sharp enough. Second, there are many ways to exploit spin-dependent rates of carrier scatter or trapping. One might examine how mobile polarized spins are scattered by a fixed spin that is to be measured. Or the spin of a mobile carrier might be measured by its propensity for capture or scatter by fixed spin, or by some combination of polarized mobile spins and interferometry. At room temperature, the problem is practice rather than principle, and acceptable methods seem possible. A third way is to use relative tunnel rates, where one spin state can be blocked. Tunneling-based methods can become very hard at higher temperatures. There are then various ideas, all of which seem to be both tricky and relatively slow, but I may be being pessimistic. These include the use of circularly polarized light and magneto-optics, the direct detection of spin resonance with a scanning tunneling microscope, the exploitation of large spin-orbit coupling, or the direct measurement of a force with a scanning probe having a magnetic tip. For the manipulations during operation, probably the most important ways use electromagnetic radiation, whether optical, microwave or radio frequency. Other controls, such as ultrasonics or surface acoustic waves, are less flexible. Electromagnetic methods might well operate at room temperature. Other suggestions invoke nanoscale electrodes. I do not know of any that look both credible and scalable. Hopes for higher temperature operation In what follows, I shall concentrate on two proposals as examples, with apologies to those whose suggestions I am omitting. Both of the proposals use optical methods to control spins, but do so in wholly different ways. The first is a scheme for optically controlled spintronics that I, Andrew Fisher, and Thornton Greenland proposed [11,12]. The second route exploits entanglement of states of distant atoms by interference [13] in the context of measurement-based quantum computing [14]. A broader discussion of the materials needed is given in Ref. [15]. Optically controlled spintronics [11,12]. Think of a thin film of silicon, perhaps 10nm thick, isotopically pure to avoid nuclear spins, on top of an oxide substrate (Fig. 1). The simple architecture described is essentially two dimensional. Now imagine the film randomly doped with two species of deep donor—one species as qubits, the other to control the qubits. In their ground states, these species should have negligible interactions. When a control donor is excited, the electron’s wave function spreads out more, and its overlap with two of the qubit donors will create an entangling interaction between those two qubits (Fig. 2). Shaped pulses of optical excitation of chosen control donors guide the quantum dance (entanglement) of chosen qubit donors [16]. For controlling entanglement in this way, typical donor spacings in silicon must be of the order of tens of nanometers. Optically, one can only address regions of the order of a wavelength across, say 1000nm. The limit of optical spatial resolution is a factor 100 larger than donor spacings needed for entanglement. How can one address chosen pairs of qubits? The smallest area on which we can focus light contains many spins. The answer is to exploit the randomness inevitable in standard fabrication and doping. Within a given patch of the film a wavelength across, the optical absorptions will be inhomogeneously broadened from dopant randomness. Even the steps at the silicon interfaces are helpful because the film thickness variations shift transition energies from one dopant site to another. Light of different wavelengths will excite different control donors in this patch, and so manipulate the entanglements of different qubits. Reasonable assumptions suggest one might make use of perhaps 20 gates or so per patch. Controlled links among 20 qubits would be very good by present standards, though further scale up—the linking of patches—would be needed for a serious computer (Fig. 3). The optically controlled spintronics strategy [11,12] separates the two roles: qubit spins store quantum information, and controls manipulate quantum information. These roles require different figures of merit. To operate at room temperature, qubits must stay in their ground states, and their decoherence—loss of quantum information—must be slow enough. Shallow donors like Si:P or Si:Bi thermally ionize too readily for room-temperature operations, though one could demonstrate principles at low temperatures with these materials. Double donors like Si:Mg+ or Si:Se+ have ionization energies of about half the silicon band gap and might be deep enough. Most defects in diamond are stable at room temperature, including substitutional N in diamond and the NV- center on which so many experiments have been done. What about decoherence? First, whatever enables entanglement also causes decoherence. This is why fast switching means fast decoherence, and slow decoherence implies slow switching. Optical control involves manipulation of the qubits by stimulated absorption and emission in controlled optical excitation sequences, so spontaneous emission will cause decoherence. For shallow donors, like Si:P, the excitation energy is less than the maximum silicon phonon energy; even at low temperatures, one-phonon emission causes rapid decoherence. Second, spin-lattice relaxation in qubit ground states destroys quantum information. Large spin-orbit coupling is bad news, so avoiding high atomic number species helps. Spin lattice relaxation data at room temperature are not yet available for those Si donors (like Si:Se+) where one-phonon processes are eliminated because their first excited state lies more than the maximum phonon energy above the ground state. In diamond at room temperature, the spin-lattice relaxation time for substitutional nitrogen is very good ( 1ms) and a number of other centers have times 0.1ms. Third, excited state processes can be problems, and two-photon ionization puts constraints on wavelengths and optical intensities. Fourth, the qubits could lose quantum information to the control atoms. This can be sorted out by choosing the right form of excitation pulses [16]. Fifth, interactions with other spins, including nuclear spins, set limits, but there are helpful strategies, like using isotopically pure silicon [17]. The control dopants require different criteria. The wave functions of electronically excited controls overlap and interact with two or more qubits to manipulate entanglements between these qubits. The transiently excited state wave function of the control must have the right spatial extent and lifetime. While centers like Si:As could be used to show the ideas, for room-temperature operation one would choose perhaps a double donor in silicon, or substitutional phosphorus in diamond. The control dopant must have sharp optical absorption lines, since what determines the number of independent gates available in a patch is the ratio of the spread of excitation energies, inhomogeneously broadened, to the (homogeneous) linewidth. The spread of excitation energies—inhomogeneous broadening is beneficial in this optical spintronics approach [11,12]—has several causes, some controllable. Randomness of relative control-qubit positions and orientations is important, and it seems possible to improve the distribution by using self-organization to eliminate unusable close encounters. Steps on the silicon interfaces are also helpful, provided there are no unpaired spins. Overall, various experimental data and theoretical analyses indicate likely inhomogeneous widths are a few percent of the excitation energy. A checklist of interesting systems as qubits or controls shows some significant gaps in knowledge of defects in solids. Surprisingly little is known about electronic excited states in diamond or silicon, apart from energies and (sometimes) symmetries. Little is known about spin lattice relaxation and excited state kinetics at temperatures above liquid nitrogen, except for the shallow donors that are unlikely to be good choices for a serious quantum computer. There are few studies of stabilities of several species present at one time. Can we be sure to have isolated P in diamond? Would it lose an electron to substitutional N to yield the useless species P+ and N- ? Will most P be found as the irrelevant (spin S=0) PV- center? What limits the number of gates in a patch is the number of control atoms that can be resolved spectroscopically one from another. As the temperature rises, the lines get broader, so this number falls and scaling becomes harder. Note the zero phonon linewidth need not be simply related to the fraction of the intensity in the sidebands. Above liquid nitrogen temperatures, these homogeneous optical widths increase fast. Thus we have two clear limits to room-temperature operation. The first is qubit decoherence, especially from spin lattice relaxation. The second is control linewidths becoming too large, reducing scalability, which may prove a more powerful limit. Entangled states of distant atoms or solid-state defects created by interference. A wholly different approach generates quantum entanglement between remote systems by performing measurements on them in a certain way [13]. The systems might be two diamonds, each containing a single NV- center prepared in specific electron spin states, the two centers tuned to have exactly the same optical energies (Fig. 4). The measurement involves “single shot” optical excitation. Both systems are exposed to a weak laser pulse that, on average, will achieve one excitation. The single system excited will emit a photon that, after passing though beam splitters and an interferometer, is detected without giving information as to which system was excited (Fig. 5). “Remote entanglement” is achieved, subject to some strong conditions. The electronic quantum information can be swapped to more robust nuclear states (a so-called brokering process). This brokered information can then be recovered when needed to implement a strategy of measurement-based quantum information processing [14]. The materials and equipment needs, while different from those of optically controlled spintronics, have features in common. For remote entanglement, a random distribution of centers is used, with one from each zone chosen because of their match to each other. The excitation energies of the two distant centers must stay equal very accurately, and this equality must be stable over time, but can be monitored. There are some challenges here, since there will be energy shifts when other defect species in any one of the systems change charge or spin state (the difficulty is present but less severe for the optical control approach). As for optically controlled spintronics [11,12], scale-up requires narrow lines, and becomes harder at higher temperatures, though there are ways to reduce the problem. Remote entanglement needs interferometric stability, avoiding problems when there are different temperature fluctuations for the paths from the separate systems. Again, there are credible strategies to reduce the effects. So is room-temperature quantum computing feasible? Spectroscopy is a generic need for both optically controlled spintronics and remote entanglement approaches. Both need qubits (the electron qubit for the measurement-based approach) with slow decoherence, a significant multiple of switching times. Both need sharp optical transitions with weak phonon sidebands to avoid loss of quantum information. A few zero phonon lines do indeed remain sharp at room temperature. The sharp lines should have frequencies stable over extended times. This mix of properties is hard to meet, but by no means impossible. Perhaps the hardest conditions have yet to be mentioned. A quantum gate is no more a quantum computer than a transistor is a classical computer. Putting all the components of a quantum computer together could prove really hard. System integration may be the ultimate challenge. Quantum information processing (QIP) will need to exploit standard silicon technology to run the quantum system; and QIP must work alongside a feasible laser optics system. The optical systems are seriously complicated, though each feature seems manageable. It may be necessary to go to architectures even more complicated than those I have described. It might even prove useful to combine elements of remote entanglement and optical spin control, whether this is regarded as using remote entanglement to link spin patches, or as having spin patches instead of NV- centers as nodes for remote entanglements. A short article like this has to miss out many features of importance, not least questions of error correction, but a major message is that, even in the most rudimentary approaches, we have to think through all of the system when talking of a possible computer. And what would you do with a quantum computer if you had one? Proposals that do not demand room temperature range from probable, like decryption or directory searching, to the possible, like modeling quantum systems, and even to the difficult yet perhaps conceivable, like modeling turbulence. More frivolous applications, like the computer games that drive many of today’s developments, make much more sense if they work at ambient temperatures. And available quantum processing at room temperature would surely stimulate inventive new ideas, just as solid-state lasers led to compact disc technology. Summing up, where do we stand? At liquid nitrogen temperatures, say 77K, quantum computing is surely possible, if quantum computing is possible at all. At dry ice temperatures, say 195K, quantum computing seems reasonably possible. At temperatures that can be reached by thermoelectric or thermomagnetic cooling, say 260K, things are harder, but there is hope. Yet we know that small (say 2–3 qubit) quantum devices operate at room temperature. It seems likely, to me at least, that a quantum computer of say 20 qubits will operate at room temperature. I do not say it will be easy. Will such a QIP device be as portable as a laptop? I won’t rule that out, but the answer is not obvious on present designs. This work was supported in part by EPSRC through its Basic Technologies program. I am especially grateful for input from Gabriel Aeppli, Polina Bayvell, Simon Benjamin, Ian Boyd, Andrea Del Duce, Andrew Fisher, Tony Harker, Andy Kerridge, Brendon Lovett, Stephen Lynch, Gavin Morley, Seb Savory, and Jason Smith. I am particularly grateful to Simon Benjamin and Stephen Lynch for preparing the initial versions of the figures. 2. C. P. Williams and S. H. Clearwater, Ultimate Zero and One: Computing at the Quantum Frontier (Copernicus, New York, 2000)[Amazon][WorldCat] 3. International Technology Roadmap for Semiconductors, 4. General discussions relevant here: R. W. Keyes, J. Phys. Condens. Matter 17, V9 (2005); R. W. Keyes, 18, S703 (2006); T. P. Spiller and W. J. Munro, 18, V1 (2006); R. Tsu, Int. J. High Speed Electronics and Systems 9, 145 (1998); R. W. Keyes, Appl. Phys. A 76, 737 (2003); M. I. Dyakonov, Future Trends in Microelectronics: Up the Nano Creek, edited by S. Luryi, J. Xu, and A. Zaslavsky (Wiley, Hoboken, NJ, 2007)[Amazon][WorldCat] 5. Examples include: E. van Oort, N. B. Manson, and M. Glasbeek, J. Phys. C 21, 4385 (1988); F. T. Charnock and T. A. Kennedy, Phys. Rev. B 64, 041201 (2001); J. Wrachtrup et al., Opt. Spectrosc. 91, 429 (2001); J. Wrachtrup and F. Jelezko, J. Phys. Condens. Matter 18, S807 (2006); R. Hanson, F. M. Mendoza, R. J. Epstein, and D. D. Awschalom, Phys. Rev. Lett. 97, 087601 (2006); A. D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J. R. Rabeau, P. Reichart, B. C. Gibson, S. Rubanov, S. T. Huntington, D. N. Jamieson, and S. Prawer, J. Phys. Condens. Matter 18, S825 (2006) 6. M. B. Plenio and P. L. Knight, Philos. Trans. R. Soc. London A 453, 2017 (1997) 7. B. E. Kane, Nature 393, 133 (1998) 8. D. P. DiVincenzo and D. Loss, Superlattices Microstruct. 23, 419 (1998) 9. A. J. Fisher, Philos. Trans. R. Soc. London A 361, 1441 (2003); 10. V. Dediu, M. Murgia, F. C. Matacotta, C. Taliani, and S. Barbanera, Solid State Commun. 122, 181 (2002) 11. A. M. Stoneham, A. J. Fisher, and P. T. Greenland, J. Phys Condens. Matter 15, L447 (2003) 12. R. Rodriquez, A .J. Fisher, P. T. Greenland, and A. M. Stoneham, J. Phys. Condens. Matter 16, 2757 (2004) 13. C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, Phys. Rev. A 58, 1025 (1999) 14. S. C. Benjamin,B. W. Lovett and J. M. Smith, Laser Photonics Rev. (to be published) 15. A. M. Stoneham, Materials Today 11, 32 (2008) 16. A. Kerridge, A. H. Harker, and A. M. Stoneham, J. Phy. Condens. Matter 19, 282201 (2007); E. M. Gauger et al., New J. Phys. 10, 073027 (2008) 17. A. M. Tyryshkin, J. J. L. Morton, S. C. Benjamin, A. Ardavan, G. A. D. Briggs, J. W. Ager, and S. A. Lyon, J. Phys. Condens. Matter 18, S783 (2006) About the Author Image of Marshall Stoneham Marshall Stoneham is Emeritus Massey Professor of Physics at University College London. He is a Fellow of the Royal Society, and also of the American Physical Society and of the Institute of Physics. Before joining UCL in 1995, he was the Chief Scientist of the UK Atomic Energy Authority, which involved him in many areas of science and technology, from quantum diffusion to nuclear safety. He was awarded the Guthrie gold medal of the Institute of Physics in 2006, and the Royal Society’s Zeneca Prize in 1995. He is the author of over 500 papers, and of a number of books, including Theory of Defects in Solids, now an Oxford Classic, and The Wind Ensemble Sourcebook that won the 1997 Oldman Prize. Marshall Stoneham is based in the London Centre for Nanotechnology, where he finds the scope for new ideas especially stimulating. His scientific interests range from new routes to solid-state quantum computing through materials modeling to biological physics, where his work on the interaction of small scent molecules with receptors has attracted much attention. He is the co-founder of two physics-based firms. Subject Areas Quantum InformationSpintronics Related Articles Synopsis: Flip-Flopping the Bands Synopsis: Flip-Flopping the Bands Viewpoint: Photon Qubit is Made of Two Colors Viewpoint: Photon Qubit is Made of Two Colors Single particles of light can be prepared in a quantum superposition of two different colors, an achievement that could prove useful for quantum information processing. Read More » Synopsis: Ten Photons in a Tangle Quantum Information Synopsis: Ten Photons in a Tangle An entangled polarization state of ten photons sets a new record for multiphoton entanglement. Read More » More Articles
660765963df9d649	Stanford Encyclopedia of Philosophy The SEP will be down intermittently from 8am–9am PST December 11. We apologize for any inconvenience. During that time, please use our Australian or Dutch mirror site. Causal Determinism First published Thu Jan 23, 2003; substantive revision Thu Jan 21, 2010 Causal determinism is, roughly speaking, the idea that every event is necessitated by antecedent events and conditions together with the laws of nature. The idea is ancient, but first became subject to clarification and mathematical analysis in the eighteenth century. Determinism is deeply connected with our understanding of the physical sciences and their explanatory ambitions, on the one hand, and with our views about human free action on the other. In both of these general areas there is no agreement over whether determinism is true (or even whether it can be known true or false), and what the import for human agency would be in either case. 1. Introduction In most of what follows, I will speak simply of determinism, rather than of causal determinism. This follows recent philosophical practice of sharply distinguishing views and theories of what causation is from any conclusions about the success or failure of determinism (cf. Earman, 1986; an exception is Mellor 1994). For the most part this disengagement of the two concepts is appropriate. But as we will see later, the notion of cause/effect is not so easily disengaged from much of what matters to us about determinism. Traditionally determinism has been given various, usually imprecise definitions. This is only problematic if one is investigating determinism in a specific, well-defined theoretical context; but it is important to avoid certain major errors of definition. In order to get started we can begin with a loose and (nearly) all-encompassing definition as follows: Determinism: The world is governed by (or is under the sway of) determinism if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law. The italicized phrases are elements that require further explanation and investigation, in order for us to gain a clear understanding of the concept of determinism. The roots of the notion of determinism surely lie in a very common philosophical idea: the idea that everything can, in principle, be explained, or that everything that is, has a sufficient reason for being and being as it is, and not otherwise. In other words, the roots of determinism lie in what Leibniz named the Principle of Sufficient Reason. But since precise physical theories began to be formulated with apparently deterministic character, the notion has become separable from these roots. Philosophers of science are frequently interested in the determinism or indeterminism of various theories, without necessarily starting from a view about Leibniz' Principle. Since the first clear articulations of the concept, there has been a tendency among philosophers to believe in the truth of some sort of determinist doctrine. There has also been a tendency, however, to confuse determinism proper with two related notions: predictability and fate. Fatalism is easily disentangled from determinism, to the extent that one can disentangle mystical forces and gods' wills and foreknowledge (about specific matters) from the notion of natural/causal law. Not every metaphysical picture makes this disentanglement possible, of course. As a general matter, we can imagine that certain things are fated to happen, without this being the result of deterministic natural laws alone; and we can imagine the world being governed by deterministic laws, without anything at all being fated to occur (perhaps because there are no gods, nor mystical forces deserving the titles fate or destiny, and in particular no intentional determination of the “initial conditions” of the world). In a looser sense, however, it is true that under the assumption of determinism, one might say that given the way things have gone in the past, all future events that will in fact happen are already destined to occur. Prediction and determinism are also easy to disentangle, barring certain strong theological commitments. As the following famous expression of determinism by Laplace shows, however, the two are also easy to commingle: In this century, Karl Popper defined determinism in terms of predictability also. Laplace probably had God in mind as the powerful intelligence to whose gaze the whole future is open. If not, he should have: 19th and 20th century mathematical studies have shown convincingly that neither a finite, nor an infinite but embedded-in-the-world intelligence can have the computing power necessary to predict the actual future, in any world remotely like ours. “Predictability” is therefore a façon de parler that at best makes vivid what is at stake in determinism; in rigorous discussions it should be eschewed. The world could be highly predictable, in some senses, and yet not deterministic; and it could be deterministic yet highly unpredictable, as many studies of chaos (sensitive dependence on initial conditions) show. Predictability does however make vivid what is at stake in determinism: our fears about our own status as free agents in the world. In Laplace's story, a sufficiently bright demon who knew how things stood in the world 100 years before my birth could predict every action, every emotion, every belief in the course of my life. Were she then to watch me live through it, she might smile condescendingly, as one who watches a marionette dance to the tugs of strings that it knows nothing about. We can't stand the thought that we are (in some sense) marionettes. Nor does it matter whether any demon (or even God) can, or cares to, actually predict what we will do: the existence of the strings of physical necessity, linked to far-past states of the world and determining our current every move, is what alarms us. Whether such alarm is actually warranted is a question well outside the scope of this article (see the entries on free will and incompatibilist theories of freedom). But a clear understanding of what determinism is, and how we might be able to decide its truth or falsity, is surely a useful starting point for any attempt to grapple with this issue. We return to the issue of freedom in Determinism and Human Action below. 2. Conceptual Issues in Determinism Recall that we loosely defined causal determinism as follows, with terms in need of clarification italicized: 2.1 The World Why should we start so globally, speaking of the world, with all its myriad events, as deterministic? One might have thought that a focus on individual events is more appropriate: an event E is causally determined if and only if there exists a set of prior events {A, B, C …} that constitute a (jointly) sufficient cause of E. Then if all—or even just most—events E that are our human actions are causally determined, the problem that matters to us, namely the challenge to free will, is in force. Nothing so global as states of the whole world need be invoked, nor even a complete determinism that claims all events to be causally determined. For a variety of reasons this approach is fraught with problems, and the reasons explain why philosophers of science mostly prefer to drop the word “causal” from their discussions of determinism. Generally, as John Earman quipped (1986), to go this route is to “… seek to explain a vague concept—determinism—in terms of a truly obscure one—causation.” More specifically, neither philosophers' nor laymen's conceptions of events have any correlate in any modern physical theory.[1] The same goes for the notions of cause and sufficient cause. A further problem is posed by the fact that, as is now widely recognized, a set of events {A, B, C …} can only be genuinely sufficient to produce an effect-event if the set includes an open-ended ceteris paribus clause excluding the presence of potential disruptors that could intervene to prevent E. For example, the start of a football game on TV on a normal Saturday afternoon may be sufficient ceteris paribus to launch Ted toward the fridge to grab a beer; but not if a million-ton asteroid is approaching his house at .75c from a few thousand miles away, nor if the phone is about to ring with news of a tragic nature, …, and so on. Bertrand Russell famously argued against the notion of cause along these lines (and others) in 1912, and the situation has not changed. By trying to define causal determination in terms of a set of prior sufficient conditions, we inevitably fall into the mess of an open-ended list of negative conditions required to achieve the desired sufficiency. Moreover, thinking about how such determination relates to free action, a further problem arises. If the ceteris paribus clause is open-ended, who is to say that it should not include the negation of a potential disruptor corresponding to my freely deciding not to go get the beer? If it does, then we are left saying “When A, B, C, … Ted will then go to the fridge for a beer, unless D or E or F or … or Ted decides not to do so.” The marionette strings of a “sufficient cause” begin to look rather tenuous. They are also too short. For the typical set of prior events that can (intuitively, plausibly) be thought to be a sufficient cause of a human action may be so close in time and space to the agent, as to not look like a threat to freedom so much as like enabling conditions. If Ted is propelled to the fridge by {seeing the game's on; desiring to repeat the satisfactory experience of other Saturdays; feeling a bit thirsty; etc}, such things look more like good reasons to have decided to get a beer, not like external physical events far beyond Ted's control. Compare this with the claim that {state of the world in 1900; laws of nature} entail Ted's going to get the beer: the difference is dramatic. So we have a number of good reasons for sticking to the formulations of determinism that arise most naturally out of physics. And this means that we are not looking at how a specific event of ordinary talk is determined by previous events; we are looking at how everything that happens is determined by what has gone before. The state of the world in 1900 only entails that Ted grabs a beer from the fridge by way of entailing the entire physical state of affairs at the later time. 2.2 The way things are at a time t The typical explication of determinism fastens on the state of the (whole) world at a particular time (or instant), for a variety of reasons. We will briefly explain some of them. Why take the state of the whole world, rather than some (perhaps very large) region, as our starting point? One might, intuitively, think that it would be enough to give the complete state of things on Earth, say, or perhaps in the whole solar system, at t, to fix what happens thereafter (for a time at least). But notice that all sorts of influences from outside the solar system come in at the speed of light, and they may have important effects. Suppose Mary looks up at the sky on a clear night, and a particularly bright blue star catches her eye; she thinks “What a lovely star; I think I'll stay outside a bit longer and enjoy the view.” The state of the solar system one month ago did not fix that that blue light from Sirius would arrive and strike Mary's retina; it arrived into the solar system only a day ago, let's say. So evidently, for Mary's actions (and hence, all physical events generally) to be fixed by the state of things a month ago, that state will have to be fixed over a much larger spatial region than just the solar system. (If no physical influences can go faster than light, then the state of things must be given from a spherical volume of space 1 light-month in radius.) But in making vivid the “threat” of determinism, we often want to fasten on the idea of the entire future of the world as being determined. No matter what the “speed limit” on physical influences is, if we want the entire future of the world to be determined, then we will have to fix the state of things over all of space, so as not to miss out something that could later come in “from outside” to spoil things. In the time of Laplace, of course, there was no known speed limit to the propagation of physical things such as light-rays. In principle light could travel at any arbitrarily high speed, and some thinkers did suppose that it was transmitted “instantaneously.” The same went for the force of gravity. In such a world, evidently, one has to fix the state of things over the whole of the world at a time t, in order for events to be strictly determined, by the laws of nature, for any amount of time thereafter. In all this, we have been presupposing the common-sense Newtonian framework of space and time, in which the world-at-a-time is an objective and meaningful notion. Below when we discuss determinism in relativistic theories we will revisit this assumption. 2.3 Thereafter For a wide class of physical theories (i.e., proposed sets of laws of nature), if they can be viewed as deterministic at all, they can be viewed as bi-directionally deterministic. That is, a specification of the state of the world at a time t, along with the laws, determines not only how things go after t, but also how things go before t. Philosophers, while not exactly unaware of this symmetry, tend to ignore it when thinking of the bearing of determinism on the free will issue. The reason for this is that we tend to think of the past (and hence, states of the world in the past) as done, over, fixed and beyond our control. Forward-looking determinism then entails that these past states—beyond our control, perhaps occurring long before humans even existed—determine everything we do in our lives. It then seems a mere curious fact that it is equally true that the state of the world now determines everything that happened in the past. We have an ingrained habit of taking the direction of both causation and explanation as being past—-present, even when discussing physical theories free of any such asymmetry. We will return to this point shortly. Another point to notice here is that the notion of things being determined thereafter is usually taken in an unlimited sense—i.e., determination of all future events, no matter how remote in time. But conceptually speaking, the world could be only imperfectly deterministic: things could be determined only, say, for a thousand years or so from any given starting state of the world. For example, suppose that near-perfect determinism were regularly (but infrequently) interrupted by spontaneous particle creation events, which occur on average only once every thousand years in a thousand-light-year-radius volume of space. This unrealistic example shows how determinism could be strictly false, and yet the world be deterministic enough for our concerns about free action to be unchanged. 2.4 Laws of nature In the loose statement of determinism we are working from, metaphors such as “govern” and “under the sway of” are used to indicate the strong force being attributed to the laws of nature. Part of understanding determinism—and especially, whether and why it is metaphysically important—is getting clear about the status of the presumed laws of nature. In the physical sciences, the assumption that there are fundamental, exceptionless laws of nature, and that they have some strong sort of modal force, usually goes unquestioned. Indeed, talk of laws “governing” and so on is so commonplace that it takes an effort of will to see it as metaphorical. We can characterize the usual assumptions about laws in this way: the laws of nature are assumed to be pushy explainers. They make things happen in certain ways , and by having this power, their existence lets us explain why things happen in certain ways. (For a recent defense of this perspective on laws, see Maudlin (2007)). Laws, we might say, are implicitly thought of as the cause of everything that happens. If the laws governing our world are deterministic, then in principle everything that happens can be explained as following from states of the world at earlier times. (Again, we note that even though the entailment typically works in the future past direction also, we have trouble thinking of this as a legitimate explanatory entailment. In this respect also, we see that laws of nature are being implicitly treated as the causes of what happens: causation, intuitively, can only go past future.) It is a remarkable fact that philosophers tend to acknowledge the apparent threat determinism poses to free will, even when they explicitly reject the view that laws are pushy explainers. Earman (1986), for example, explicitly adopts a theory of laws of nature that takes them to be simply the best system of regularities that systematizes all the events in universal history. This is the Best Systems Analysis (BSA), with roots in the work of Hume, Mill and Ramsey, and most recently refined and defended by David Lewis (1973, 1994) and by Earman (1984, 1986). (cf. entry on laws of nature). Yet he ends his comprehensive Primer on Determinism with a discussion of the free will problem, taking it as a still-important and unresolved issue. Prima facie at least, this is quite puzzling, for the BSA is founded on the idea that the laws of nature are ontologically derivative, not primary; it is the events of universal history, as brute facts, that make the laws be what they are, and not vice-versa. Taking this idea seriously, the actions of every human agent in history are simply a part of the universe-wide pattern of events that determines what the laws are for this world. It is then hard to see how the most elegant summary of this pattern, the BSA laws, can be thought of as determiners of human actions. The determination or constraint relations, it would seem, can go one way or the other, not both! On second thought however it is not so surprising that broadly Humean philosophers such as Ayer, Earman, Lewis and others still see a potential problem for freedom posed by determinism. For even if human actions are part of what makes the laws be what they are, this does not mean that we automatically have freedom of the kind we think we have, particularly freedom to have done otherwise given certain past states of affairs. It is one thing to say that everything occurring in and around my body, and everything everywhere else, conforms to Maxwell's equations and thus the Maxwell equations are genuine exceptionless regularities, and that because they in addition are simple and strong, they turn out to be laws. It is quite another thing to add: thus, I might have chosen to do otherwise at certain points in my life, and if I had, then Maxwell's equations would not have been laws. One might try to defend this claim—unpalatable as it seems intuitively, to ascribe ourselves law-breaking power—but it does not follow directly from a Humean approach to laws of nature. Instead, on such views that deny laws most of their pushiness and explanatory force, questions about determinism and human freedom simply need to be approached afresh. A second important genre of theories of laws of nature holds that the laws are in some sense necessary. For any such approach, laws are just the sort of pushy explainers that are assumed in the traditional language of physical scientists and free will theorists. But a third and growing class of philosophers holds that (universal, exceptionless, true) laws of nature simply do not exist. Among those who hold this are influential philosophers such as Nancy Cartwright, Bas van Fraassen, and John Dupré. For these philosophers, there is a simple consequence: determinism is a false doctrine. As with the Humeans, this does not mean that concerns about human free action are automatically resolved; instead, they must be addressed afresh in the light of whatever account of physical nature without laws is put forward. See Dupré (2001) for one such discussion. 2.5 Fixed We can now put our—still vague—pieces together. Determinism requires a world that (a) has a well-defined state or description, at any given time, and (b) laws of nature that are true at all places and times. If we have all these, then if (a) and (b) together logically entail the state of the world at all other times (or, at least, all times later than that given in (b)), the world is deterministic. Logical entailment, in a sense broad enough to encompass mathematical consequence, is the modality behind the determination in “determinism.” 3. The Epistemology of Determinism How could we ever decide whether our world is deterministic or not? Given that some philosophers and some physicists have held firm views—with many prominent examples on each side—one would think that it should be at least a clearly decidable question. Unfortunately, even this much is not clear, and the epistemology of determinism turns out to be a thorny and multi-faceted issue. 3.1 Laws again As we saw above, for determinism to be true there have to be some laws of nature. Most philosophers and scientists since the 17th century have indeed thought that there are. But in the face of more recent skepticism, how can it be proven that there are? And if this hurdle can be overcome, don't we have to know, with certainty, precisely what the laws of our world are, in order to tackle the question of determinism's truth or falsity? The first hurdle can perhaps be overcome by a combination of metaphysical argument and appeal to knowledge we already have of the physical world. Philosophers are currently pursuing this issue actively, in large part due to the efforts of the anti-laws minority. The debate has been most recently framed by Cartwright in The Dappled World (Cartwright 1999) in terms psychologically advantageous to her anti-laws cause. Those who believe in the existence of traditional, universal laws of nature are fundamentalists; those who disbelieve are pluralists. This terminology seems to be becoming standard (see Belot 2001), so the first task in the epistemology of determinism is for fundamentalists to establish the reality of laws of nature (see Hoefer 2002b). Even if the first hurdle can be overcome, the second, namely establishing precisely what the actual laws are, may seem daunting indeed. In a sense, what we are asking for is precisely what 19th and 20th century physicists sometimes set as their goal: the Final Theory of Everything. But perhaps, as Newton said of establishing the solar system's absolute motion, “the thing is not altogether desperate.” Many physicists in the past 60 years or so have been convinced of determinism's falsity, because they were convinced that (a) whatever the Final Theory is, it will be some recognizable variant of the family of quantum mechanical theories; and (b) all quantum mechanical theories are non-deterministic. Both (a) and (b) are highly debatable, but the point is that one can see how arguments in favor of these positions might be mounted. The same was true in the 19th century, when theorists might have argued that (a) whatever the Final Theory is, it will involve only continuous fluids and solids governed by partial differential equations; and (b) all such theories are deterministic. (Here, (b) is almost certainly false; see Earman (1986),ch. XI). Even if we now are not, we may in future be in a position to mount a credible argument for or against determinism on the grounds of features we think we know the Final Theory must have. 3.2 Experience Determinism could perhaps also receive direct support—confirmation in the sense of probability-raising, not proof—from experience and experiment. For theories (i.e., potential laws of nature) of the sort we are used to in physics, it is typically the case that if they are deterministic, then to the extent that one can perfectly isolate a system and repeatedly impose identical starting conditions, the subsequent behavior of the systems should also be identical. And in broad terms, this is the case in many domains we are familiar with. Your computer starts up every time you turn it on, and (if you have not changed any files, have no anti-virus software, re-set the date to the same time before shutting down, and so on …) always in exactly the same way, with the same speed and resulting state (until the hard drive fails). The light comes on exactly 32 µsec after the switch closes (until the day the bulb fails). These cases of repeated, reliable behavior obviously require some serious ceteris paribus clauses, are never perfectly identical, and always subject to catastrophic failure at some point. But we tend to think that for the small deviations, probably there are explanations for them in terms of different starting conditions or failed isolation, and for the catastrophic failures, definitely there are explanations in terms of different conditions. There have even been studies of paradigmatically “chancy” phenomena such as coin-flipping, which show that if starting conditions can be precisely controlled and outside interferences excluded, identical behavior results (see Diaconis, Holmes & Montgomery 2004). Most of these bits of evidence for determinism no longer seem to cut much ice, however, because of faith in quantum mechanics and its indeterminism. Indeterminist physicists and philosophers are ready to acknowledge that macroscopic repeatability is usually obtainable, where phenomena are so large-scale that quantum stochasticity gets washed out. But they would maintain that this repeatability is not to be found in experiments at the microscopic level, and also that at least some failures of repeatability (in your hard drive, or coin-flipping experiments) are genuinely due to quantum indeterminism, not just failures to isolate properly or establish identical initial conditions. If quantum theories were unquestionably indeterministic, and deterministic theories guaranteed repeatability of a strong form, there could conceivably be further experimental input on the question of determinism's truth or falsity. Unfortunately, the existence of Bohmian quantum theories casts strong doubt on the former point, while chaos theory casts strong doubt on the latter. More will be said about each of these complications below. 3.3 Determinism and Chaos If the world were governed by strictly deterministic laws, might it still look as though indeterminism reigns? This is one of the difficult questions that chaos theory raises for the epistemology of determinism. A deterministic chaotic system has, roughly speaking, two salient features: (i) the evolution of the system over a long time period effectively mimics a random or stochastic process—it lacks predictability or computability in some appropriate sense; (ii) two systems with nearly identical initial states will have radically divergent future developments, within a finite (and typically, short) timespan. We will use “randomness” to denote the first feature, and “sensitive dependence on initial conditions” (SDIC) for the latter. Definitions of chaos may focus on either or both of these properties; Batterman (1993) argues that only (ii) provides an appropriate basis for defining chaotic systems. A simple and very important example of a chaotic system in both randomness and SDIC terms is the Newtonian dynamics of a pool table with a convex obstacle (or obstacles) (Sinai 1970 and others). See Figure 1: Billiard table with convex obstacle Figure 1: Billiard table with convex obstacle The usual idealizing assumptions are made: no friction, perfectly elastic collisions, no outside influences. The ball's trajectory is determined by its initial position and direction of motion. If we imagine a slightly different initial direction, the trajectory will at first be only slightly different. And collisions with the straight walls will not tend to increase very rapidly the difference between trajectories. But collisions with the convex object will have the effect of amplifying the differences. After several collisions with the convex body or bodies, trajectories that started out very close to one another will have become wildly different—SDIC. In the example of the billiard table, we know that we are starting out with a Newtonian deterministic system—that is how the idealized example is defined. But chaotic dynamical systems come in a great variety of types: discrete and continuous, 2-dimensional, 3-dimensional and higher, particle-based and fluid-flow-based, and so on. Mathematically, we may suppose all of these systems share SDIC. But generally they will also display properties such as unpredictability, non-computability, Kolmogorov-random behavior, and so on—at least when looked at in the right way, or at the right level of detail. This leads to the following epistemic difficulty: if, in nature, we find a type of system that displays some or all of these latter properties, how can we decide which of the following two hypotheses is true? 1. The system is governed by genuinely stochastic, indeterministic laws (or by no laws at all), i.e., its apparent randomness is in fact real randomness. 2. The system is governed by underlying deterministic laws, but is chaotic. In other words, once one appreciates the varieties of chaotic dynamical systems that exist, mathematically speaking, it starts to look difficult—maybe impossible—for us to ever decide whether apparently random behavior in nature arises from genuine stochasticity, or rather from deterministic chaos. Patrick Suppes (1993, 1996) argues, on the basis of theorems proven by Ornstein (1974 and later) that “There are processes which can equally well be analyzed as deterministic systems of classical mechanics or as indeterministic semi-Markov processes, no matter how many observations are made.” And he concludes that “Deterministic metaphysicians can comfortably hold to their view knowing they cannot be empirically refuted, but so can indeterministic ones as well.” (Suppes (1993), p. 254) There is certainly an interesting problem area here for the epistemology of determinism, but it must be handled with care. It may well be true that there are some deterministic dynamical systems that, when viewed properly, display behavior indistinguishable from that of a genuinely stochastic process. For example, using the billiard table above, if one divides its surface into quadrants and looks at which quadrant the ball is in at 30-second intervals, the resulting sequence is no doubt highly random. But this does not mean that the same system, when viewed in a different way (perhaps at a higher degree of precision) does not cease to look random and instead betray its deterministic nature. If we partition our billiard table into squares 2 centimeters a side and look at which quadrant the ball is in at .1 second intervals, the resulting sequence will be far from random. And finally, of course, if we simply look at the billiard table with our eyes, and see it as a billiard table, there is no obvious way at all to maintain that it may be a truly random process rather than a deterministic dynamical system. (See Winnie (1996) for a nice technical and philosophical discussion of these issues. Winnie explicates Ornstein's and others' results in some detail, and disputes Suppes' philosophical conclusions.) The dynamical systems usually studied under the label of “chaos” are usually either purely abstract, mathematical systems, or classical Newtonian systems. It is natural to wonder whether chaotic behavior carries over into the realm of systems governed by quantum mechanics as well. Interestingly, it is much harder to find natural correlates of classical chaotic behavior in true quantum systems. (See Gutzwiller (1990)). Some, at least, of the interpretive difficulties of quantum mechanics would have to be resolved before a meaningful assessment of chaos in quantum mechanics could be achieved. For example, SDIC is hard to find in the Schrödinger evolution of a wavefunction for a system with finite degrees of freedom; but in Bohmian quantum mechanics it is handled quite easily on the basis of particle trajectories. (See Dürr, Goldstein and Zhangì (1992)). The popularization of chaos theory in the past decade and a half has perhaps made it seem self-evident that nature is full of genuinely chaotic systems. In fact, it is far from self-evident that such systems exist, other than in an approximate sense. Nevertheless, the mathematical exploration of chaos in dynamical systems helps us to understand some of the pitfalls that may attend our efforts to know whether our world is genuinely deterministic or not. 3.4 Metaphysical arguments Let us suppose that we shall never have the Final Theory of Everything before us—at least in our lifetime—and that we also remain unclear (on physical/experimental grounds) as to whether that Final Theory will be of a type that can or cannot be deterministic. Is there nothing left that could sway our belief toward or against determinism? There is, of course: metaphysical argument. Metaphysical arguments on this issue are not currently very popular. But philosophical fashions change at least twice a century, and grand systemic metaphysics of the Leibnizian sort might one day come back into favor. Conversely, the anti-systemic, anti-fundamentalist metaphysics propounded by Cartwright (1999) might also come to predominate. As likely as not, for the foreseeable future metaphysical argument may be just as good a basis on which to discuss determinism's prospects as any arguments from mathematics or physics. 4. The Status of Determinism in Physical Theories John Earman's Primer on Determinism (1986) remains the richest storehouse of information on the truth or falsity of determinism in various physical theories, from classical mechanics to quantum mechanics and general relativity. (See also his recent update on the subject, “Aspects of Determinism in Modern Physics” (2007)). Here I will give only a brief discussion of some key issues, referring the reader to Earman (1986) and other resources for more detail. Figuring out whether well-established theories are deterministic or not (or to what extent, if they fall only a bit short) does not do much to help us know whether our world is really governed by deterministic laws; all our current best theories, including General Relativity and the Standard Model of particle physics, are too flawed and ill-understood to be mistaken for anything close to a Final Theory. Nevertheless, as Earman (1986) stressed, the exploration is very valuable because of the way it enriches our understanding of the richness and complexity of determinism. 4.1 Classical mechanics Despite the common belief that classical mechanics (the theory that inspired Laplace in his articulation of determinism) is perfectly deterministic, in fact the theory is rife with possibilities for determinism to break down. One class of problems arises due to the absence of an upper bound on the velocities of moving objects. Below we see the trajectory of an object that is accelerated unboundedly, its velocity becoming in effect infinite in a finite time. See Figure 2: object accelerates to reach infinity Figure 2: An object accelerates so as to reach spatial infinity in a finite time By the time t = t, the object has literally disappeared from the world—its world-line never reaches the t = t surface. (Never mind how the object gets accelerated in this way; there are mechanisms that are perfectly consistent with classical mechanics that can do the job. In fact, Xia (1992) showed that such acceleration can be accomplished by gravitational forces from only 5 finite objects, without collisions. No mechanism is shown in these diagrams.) This “escape to infinity,” while disturbing, does not yet look like a violation of determinism. But now recall that classical mechanics is time-symmetric: any model has a time-inverse, which is also a consistent model of the theory. The time-inverse of our escaping body is playfully called a “space invader.” space invader comes from infinity Figure 3: A ‘space invader’ comes in from spatial infinity Clearly, a world with a space invader does fail to be deterministic. Before t = 0, there was nothing in the state of things to enable the prediction of the appearance of the invader at t = 0+.[2] One might think that the infinity of space is to blame for this strange behavior, but this is not obviously correct. In finite, “rolled-up” or cylindrical versions of Newtonian space-time space-invader trajectories can be constructed, though whether a “reasonable” mechanism to power them exists is not clear.[3] A second class of determinism-breaking models can be constructed on the basis of collision phenomena. The first problem is that of multiple-particle collisions for which Newtonian particle mechanics simply does not have a prescription for what happens. (Consider three identical point-particles approaching each other at 120 degree angles and colliding simultaneously. That they bounce back along their approach trajectories is possible; but it is equally possible for them to bounce in other directions (again with 120 degree angles between their paths), so long as momentum conservation is respected.) Moreover, there is a burgeoning literature of physical or quasi-physical systems, usually set in the context of classical physics, that carry out supertasks (see Earman and Norton (1998) and the entry on supertasks for a review). Frequently, the puzzle presented is to decide, on the basis of the well-defined behavior before time t = a, what state the system will be in at t = a itself. A failure of CM to dictate a well-defined result can then be seen as a failure of determinism. In supertasks, one frequently encounters infinite numbers of particles, infinite (or unbounded) mass densities, and other dubious infinitary phenomena. Coupled with some of the other breakdowns of determinism in CM, one begins to get a sense that most, if not all, breakdowns of determinism rely on some combination of the following set of (physically) dubious mathematical notions: {infinite space; unbounded velocity; continuity; point-particles; singular fields}. The trouble is, it is difficult to imagine any recognizable physics (much less CM) that eschews everything in the set. Finally, an elegant example of apparent violation of determinism in classical physics has been created by John Norton (2003). As illustrated in Figure 4, imagine a ball sitting at the apex of a frictionless dome whose equation is specified as a function of radial distance from the apex point. This rest-state is our initial condition for the system; what should its future behavior be? Clearly one solution is for the ball to remain at rest at the apex indefinitely. Norton's dome Figure 4: A ball may spontaneously start sliding down this dome, with no violation of Newton's laws. (Reproduced courtesy of John D. Norton and Philosopher's Imprint) But curiously, this is not the only solution under standard Newtonian laws. The ball may also start into motion sliding down the dome—at any moment in time, and in any radial direction. This example displays “uncaused motion” without, Norton argues, any violation of Newton's laws, including the First Law. And it does not, unlike some supertask examples, require an infinity of particles. Still, many philosophers are uncomfortable with the moral Norton draws from his dome example, and point out reasons for questioning the dome's status as a Newtonian system (see e.g. Malament (2007)). 4.2 Special Relativistic physics Two features of special relativistic physics make it perhaps the most hospitable environment for determinism of any major theoretical context: the fact that no process or signal can travel faster than the speed of light, and the static, unchanging spacetime structure. The former feature, including a prohibition against tachyons (hypothetical particles travelling faster than light)[4]), rules out space invaders and other unbounded-velocity systems. The latter feature makes the space-time itself nice and stable and non-singular—unlike the dynamic space-time of General Relativity, as we shall see below. For source-free electromagnetic fields in special-relativistic space-time, a nice form of Laplacean determinism is provable. Unfortunately, interesting physics needs more than source-free electromagnetic fields. Earman (1986) ch. IV surveys in depth the pitfalls for determinism that arise once things are allowed to get more interesting (e.g. by the addition of particles interacting gravitationally). 4.3 General Relativity (GTR) Defining an appropriate form of determinism for the context of general relativistic physics is extremely difficult, due to both foundational interpretive issues and the plethora of weirdly-shaped space-time models allowed by the theory's field equations. The simplest way of treating the issue of determinism in GTR would be to state flatly: determinsim fails, frequently, and in some of the most interesting models. To leave it at that would however be to miss an important opportunity to use determinism to probe physical and philosophical issues of great importance (a use of determinism stressed frequently by Earman). Here we will briefly describe some of the most important challenges that arise for determinism, directing the reader yet again to Earman (1986), and also Earman (1995) for more depth. 4.3.1 Determinism and manifold points In GTR, we specify a model of the universe by giving a triple of three mathematical objects, <M, g,T>. M represents a continuous “manifold”: that means a sort of unstructured space (-time), made up of individual points and having smoothness or continuity, and dimensionality (usually, 4-dimensional), but no further structure. What is the further structure a space-time needs? Typically, at least, we expect the time-direction to be distinguished from space-directions; and we expect there to be well-defined distances between distinct points; and also a determinate geometry (making certain continuous paths in M be straight lines, etc.). All of this extra structure is coded into g. So M and g together represent space-time. T represents the matter and energy content distributed around in space-time (if any, of course). For mathematical reasons not relevant here, it turns out to be possible to take a given model spacetime and perform a mathematical operation called a “hole diffeomorphism” h* on it; the diffeomorphism's effect is to shift around the matter content T and the metric g relative to the continuous manifold M.[5] If the diffeomorphism is chosen appropriately, it can move around T and g after a certain time t = 0, but leave everything alone before that time. Thus, the new model represents the matter content (now h* T) and the metric (hg) as differently located relative to the points of M making up space-time. Yet, the new model is also a perfectly valid model of the theory. This looks on the face of it like a form of indeterminism: GTR's equations do not specify how things will be distributed in space-time in the future, even when the past before a given time t is held fixed. See Figure 5: Hole diffeomorphism shifts contents of spacetime Figure 5: “Hole” diffeomorphism shifts contents of spacetime Usually the shift is confined to a finite region called the hole (for historical reasons). Then it is easy to see that the state of the world at time t = 0 (and all the history that came before) does not suffice to fix whether the future will be that of our first model, or its shifted counterpart in which events inside the hole are different. This is a form of indeterminism first highlighted by Earman and Norton (1987) as an interpretive philosophical difficulty for realism about GTR's description of the world, especially the point manifold M. They showed that realism about the manifold as a part of the furniture of the universe (which they called “manifold substantivalism”) commits us to a radical, automatic indeterminism in GTR, and they argued that this is unacceptable. (See the hole argument and Hoefer (1996) for one response on behalf of the space-time realist, and discussion of other responses.) For now, we will simply note that this indeterminism, unlike most others we are discussing in this section, is empirically vacuous: our two models <M, g, T> and the shifted model <M, hg, h*T> are empirically indistinguishable. 4.3.2 Singularities The separation of space-time structures into manifold and metric (or connection) facilitates mathematical clarity in many ways, but also opens up Pandora's box when it comes to determinism. The indeterminism of the Earman and Norton hole argument is only the tip of the iceberg; singularities make up much of the rest of the berg. In general terms, a singularity can be thought of as a “place where things go bad” in one way or another in the space-time model. For example, near the center of a Schwarzschild black hole, curvature increases without bound, and at the center itself it is undefined, which means that Einstein's equations cannot be said to hold, which means (arguably) that this point does not exist as a part of the space-time at all! Some specific examples are clear, but giving a general definition of a singularity, like defining determinism itself in GTR, is a vexed issue (see Earman (1995) for an extended treatment; Callender and Hoefer (2001) gives a brief overview). We will not attempt here to catalog the various definitions and types of singularity. Different types of singularity bring different types of threat to determinism. In the case of ordinary black holes, mentioned above, all is well outside the so- called “event horizon”, which is the spherical surface defining the black hole: once a body or light signal passes through the event horizon to the interior region of the black hole, it can never escape again. Generally, no violation of determinism looms outside the event horizon; but what about inside? Some black hole models have so-called “Cauchy horizons” inside the event horizon, i.e., surfaces beyond which determinism breaks down. Another way for a model spacetime to be singular is to have points or regions go missing, in some cases by simple excision. Perhaps the most dramatic form of this involves taking a nice model with a space-like surface t = E (i.e., a well-defined part of the space-time that can be considered “the state state of the world at time E”), and cutting out and throwing away this surface and all points temporally later. The resulting spacetime satisfies Einstein's equations; but, unfortunately for any inhabitants, the universe comes to a sudden and unpredictable end at time E. This is too trivial a move to be considered a real threat to determinism in GTR; we can impose a reasonable requirement that space-time not “run out” in this way without some physical reason (the spacetime should be “maximally extended”). For discussion of precise versions of such a requirement, and whether they succeed in eliminating unwanted singularities, see Earman (1995, chapter 2). The most problematic kinds of singularities, in terms of determinism, are naked singularities (singularities not hidden behind an event horizon). When a singularity forms from gravitational collapse, the usual model of such a process involves the formation of an event horizon (i.e. a black hole). A universe with an ordinary black hole has a singularity, but as noted above, (outside the event horizon at least) nothing unpredictable happens as a result. A naked singularity, by contrast, has no such protective barrier. In much the way that anything can disappear by falling into an excised-region singularity, or appear out of a white hole (white holes themselves are, in fact, technically naked singularities), there is the worry that anything at all could pop out of a naked singularity, without warning (hence, violating determinism en passant). While most white hole models have Cauchy surfaces and are thus arguably deterministic, other naked singularity models lack this property. Physicists disturbed by the unpredictable potentialities of such singularities have worked to try to prove various cosmic censorship hypotheses that show—under (hopefully) plausible physical assumptions—that such things do not arise by stellar collapse in GTR (and hence are not liable to come into existence in our world). To date no very general and convincing forms of the hypothesis have been proven, so the prospects for determinism in GTR as a mathematical theory do not look terribly good. 4.4 Quantum mechanics As indicated above, QM is widely thought to be a strongly non-deterministic theory. Popular belief (even among most physicists) holds that phenomena such as radioactive decay, photon emission and absorption, and many others are such that only a probabilistic description of them can be given. The theory does not say what happens in a given case, but only says what the probabilities of various results are. So, for example, according to QM the fullest description possible of a radium atom (or a chunk of radium, for that matter), does not suffice to determine when a given atom will decay, nor how many atoms in the chunk will have decayed at any given time. The theory gives only the probabilities for a decay (or a number of decays) to happen within a given span of time. Einstein and others perhaps thought that this was a defect of the theory that should eventually be removed, by a supplemental hidden variable theory[6] that restores determinism; but subsequent work showed that no such hidden variables account could exist. At the microscopic level the world is ultimately mysterious and chancy. So goes the story; but like much popular wisdom, it is partly mistaken and/or misleading. Ironically, quantum mechanics is one of the best prospects for a genuinely deterministic theory in modern times! Even more than in the case of GTR and the hole argument, everything hinges on what interpretational and philosophical decisions one adopts. The fundamental law at the heart of non-relativistic QM is the Schrödinger equation. The evolution of a wavefunction describing a physical system under this equation is normally taken to be perfectly deterministic.[7] If one adopts an interpretation of QM according to which that's it—i.e., nothing ever interrupts Schrödinger evolution, and the wavefunctions governed by the equation tell the complete physical story—then quantum mechanics is a perfectly deterministic theory. There are several interpretations that physicists and philosophers have given of QM which go this way. (See the entry on quantum mechanics.) More commonly—and this is part of the basis for the popular wisdom—physicists have resolved the quantum measurement problem by postulating that some process of “collapse of the wavefunction” occurs from time to time (particularly during measurements and observations) that interrupts Schrödinger evolution. The collapse process is usually postulated to be indeterministic, with probabilities for various outcomes, via Born's rule, calculable on the basis of a system's wavefunction. The once-standard, Copenhagen interpretation of QM posits such a collapse. It has the virtue of solving certain paradoxes such as the infamous Schrödinger's cat paradox, but few philosophers or physicists can take it very seriously unless they are either idealists or instrumentalists. The reason is simple: the collapse process is not physically well-defined, and feels too ad hoc to be a fundamental part of nature's laws.[8] In 1952 David Bohm created an alternative interpretation of QM—perhaps better thought of as an alternative theory—that realizes Einstein's dream of a hidden variable theory, restoring determinism and definiteness to micro-reality. In Bohmian quantum mechanics, unlike other interpretations, it is postulated that all particles have, at all times, a definite position and velocity. In addition to the Schrödinger equation, Bohm posited a guidance equation that determines, on the basis of the system's wavefunction and particles' initial positions and velocities, what their future positions and velocities should be. As much as any classical theory of point particles moving under force fields, then, Bohm's theory is deterministic. Amazingly, he was also able to show that, as long as the statistical distribution of initial positions and velocities of particles are chosen so as to meet a “quantum equilibrium” condition, his theory is empirically equivalent to standard Copenhagen QM. In one sense this is a philosopher's nightmare: with genuine empirical equivalence as strong as Bohm obtained, it seems experimental evidence can never tell us which description of reality is correct. (Fortunately, we can safely assume that neither is perfectly correct, and hope that our Final Theory has no such empirically equivalent rivals.) In other senses, the Bohm theory is a philosopher's dream come true, eliminating much (but not all) of the weirdness of standard QM and restoring determinism to the physics of atoms and photons. The interested reader can find out more from the link above, and references therein. This small survey of determinism's status in some prominent physical theories, as indicated above, does not really tell us anything about whether determinism is true of our world. Instead, it raises a couple of further disturbing possibilities for the time when we do have the Final Theory before us (if such time ever comes): first, we may have difficulty establishing whether the Final Theory is deterministic or not—depending on whether the theory comes loaded with unsolved interpretational or mathematical puzzles. Second, we may have reason to worry that the Final Theory, if indeterministic, has an empirically equivalent yet deterministic rival (as illustrated by Bohmian quantum mechanics.) 5. Chance and Determinism Some philosophers maintain that if determinism holds in our world, then there are no objective chances in our world. And often the word ‘chance’ here is taken to be synonymous with 'probability', so these philosophers maintain that there are no non-trivial objective probabilities for events in our world. (The caveat “non-trivial” is added here because on some accounts all future events that actually happen have probability, conditional on past history, equal to 1, and future events that do not happen have probability equal to zero. Non-trivial probabilities are probabilities strictly between zero and one.) Conversely, it is often held, if there are laws of nature that are irreducibly probabilistic, determinism must be false. (Some philosophers would go on to add that such irreducibly probabilistic laws are the basis of whatever genuine objective chances obtain in our world.) The discussion of quantum mechanics in section 4 shows that it may be difficult to know whether a physical theory postulates genuinely irreducible probabilistic laws or not. If a Bohmian version of QM is correct, then the probabilities dictated by the Born rule are not irreducible. If that is the case, should we say that the probabilities dictated by quantum mechanics are not objective? Or should we say that we need to distinguish ‘chance’ and ‘probabillity’ after all—and hold that not all objective probabilities should be thought of as objective chances? The first option may seem hard to swallow, given the many-decimal-place accuracy with which such probability-based quantities as half-lives and cross-sections can be reliably predicted and verified experimentally with QM. Whether objective chance and determinism are really incompatible or not may depend on what view of the nature of laws is adopted. On a “pushy explainers” view of laws such as that defended by Maudlin (2007), probabilistic laws are interpreted as irreducible dynamical transition-chances between allowed physical states, and the incompatibility of such laws with determinism is immediate. But what should a defender of a Humean view of laws, such as the BSA theory (section 2.4 above), say about probabilistic laws? The first thing that needs to be done is explain how probabilistic laws can fit into the BSA account at all, and this requires modification or expansion of the view, since as first presented the only candidates for laws of nature are true universal generalizations. If ‘probability’ were a univocal, clearly understood notion then this might be simple: We allow universal generalizations whose logical form is something like: “Whenever conditions Y obtain, Pr(A) = x”. But it is not at all clear how the meaning of ‘Pr’ should be understood in such a generalization; and it is even less clear what features the Humean pattern of actual events must have, for such a generalization to be held true. (See the entry on interpretations of probability and Lewis (1994).) Humeans about laws believe that what laws there are is a matter of what patterns are there to be discerned in the overall mosaic of events that happen in the history of the world. It seems plausible enough that the patterns to be discerned may include not only strict associations (whenever X, Y), but also stable statistical associations. If the laws of nature can include either sort of association, a natural question to ask seems to be: why can't there be non-probabilistic laws strong enough to ensure determinism, and on top of them, probabilistic laws as well? If a Humean wanted to capture the laws not only of fundamental theories, but also non-fundamental branches of physics such as (classical) statistical mechanics, such a peaceful coexistence of deterministic laws plus further probabilistic laws would seem to be desirable. Loewer (2004) argues that this peaceful coexistence can be achieved within Lewis' version of the BSA account of laws. 6. Determinism and Human Action In the introduction, we noted the threat that determinism seems to pose to human free agency. It is hard to see how, if the state of the world 1000 years ago fixes everything I do during my life, I can meaningfully say that I am a free agent, the author of my own actions, which I could have freely chosen to perform differently. After all, I have neither the power to change the laws of nature, nor to change the past! So in what sense can I attribute freedom of choice to myself? Philosophers have not lacked ingenuity in devising answers to this question. There is a long tradition of compatibilists arguing that freedom is fully compatible with physical determinism. Hume went so far as to argue that determinism is a necessary condition for freedom—or at least, he argued that some causality principle along the lines of “same cause, same effect” is required. There have been equally numerous and vigorous responses by those who are not convinced. Can a clear understanding of what determinism is, and how it tends to succeed or fail in real physical theories, shed any light on the controversy? Physics, particularly 20th century physics, does have one lesson to impart to the free will debate; a lesson about the relationship between time and determinism. Recall that we noticed that the fundamental theories we are familiar with, if they are deterministic at all, are time-symmetrically deterministic. That is, earlier states of the world can be seen as fixing all later states; but equally, later states can be seen as fixing all earlier states. We tend to focus only on the former relationship, but we are not led to do so by the theories themselves. Nor does 20th (21st) -century physics countenance the idea that there is anything ontologically special about the past, as opposed to the present and the future. In fact, it fails to use these categories in any respect, and teaches that in some senses they are probably illusory.[9] So there is no support in physics for the idea that the past is “fixed” in some way that the present and future are not, or that it has some ontological power to constrain our actions that the present and future do not have. It is not hard to uncover the reasons why we naturally do tend to think of the past as special, and assume that both physical causation and physical explanation work only in the past present/future direction (see the entry on thermodynamic asymmetry in time). But these pragmatic matters have nothing to do with fundamental determinism. If we shake loose from the tendency to see the past as special, when it comes to the relationships of determinism, it may prove possible to think of a deterministic world as one in which each part bears a determining—or partial-determining—relation to other parts, but in which no particular part (i.e., region of space-time) has a special, stronger determining role than any other. Hoefer (2002) uses these considerations to argue in a novel way for the compatiblity of determinism with human free agency. Other Internet Resources Related Entries compatibilism \| free will \| Hume, David \| incompatibilism: (nondeterministic) theories of free will \| laws of nature \| Popper, Karl \| probability, interpretations of \| quantum mechanics \| quantum mechanics: Bohmian mechanics \| Russell, Bertrand \| space and time: supertasks \| space and time: the hole argument \| time: thermodynamic asymmetry in The author would like to acknowledge the invaluable help of John Norton in the preparation of this entry. Thanks also to A. Ilhamy Amiry for bringing to my attention some errors in an earlier version of this entry.
c10a70f8c283cd23	Saturday, January 21, 2012 Some parallels between classical and quantum mechanics This isn't really a blog post. More of something I wanted to interject in a discussion on Google plus but wouldn't fit in the text box. I've always had trouble with the way the Legendre transform is introduced in classical mechanics. I know I'm not the only one. Many mathematicians and physicists have recognised that it seems to be plucked out of a hat like a rabbit and have even written papers to address this issue. But however much an author attempts to make it seem natural, it still looks like a rabbit to me. So I have to ask myself, what would make me feel comfortable with the Legendre transform? The Legendre transform is an analogue of the Fourier transform that uses a different semiring to the usual. I wrote briefly about this many years ago. So if we could write classical mechanics in a form that is analogous to another problem where I'd use a Fourier transform, I'd be happier. This is my attempt to do that. When I wrote about Fourier transforms a little while back the intention was to immediately follow it with an analogous article about Legendre transforms. Unfortunately that's been postponed so I'm going to just assume you know that Legendre transforms can be used to compute inf-convolutions. I'll state clearly what that means below, but I won't show any detail on the analogy with Fourier transforms. Free classical particles Let's work in one dimension with a particle of mass whose position at time is . The kinetic energy of this particle is given by . Its Lagrangian is therefore . The action of our particle for the time from to is therefore The particle motion is that which minimises the action. Suppose the position of the particle at time is and the position at time is . Then write for the action minimising path from to . So where we're minimising over all paths such that . Now suppose our system evolves from time to . We can consider this to be two stages, one from to followed by one from to . Let be the minimised action analogous to for the period to . The action from to is the sum of the actions for the two subperiods. So the minimum total action for the period to is given by Let me simply that a little. I'll use where I previously used and for . So that last equation becomes: Now suppose is translation-independent in the sense that . So we can write . Then the minimum total action is given by Infimal convolution is defined by so the minimum we seek is So now it's natural to use the Legendre transform. We have the inf-convolution theorem: where is the Legendre transform of given by and so (where we use to represent Legendre transform with respect to the spatial variable). Let's consider the case where from onwards the particle motion is free, so . In this case we clearly have translation-invariance and so the time evolution is given by repeated inf-convolution with and in the "Legendre domain" this is nothing other than repeated addition of . Let's take a look at . We know that if a particle travels freely from to over the period from to then it must have followed the minimum action path and we know, from basic mechanics, this is the path with constant velocity. So and hence the action is given by So the time evolution of is given by repeated inf-convolution with a quadratic function. The time evolution of is therefore given by repeated addition of the Legendre transform of a quadratic function. It's not hard to prove that the Legendre transform of a quadratic function is also quadratic. In fact: Addition is easier to work with than inf-convolution so if we wish to understand the time evolution of the action function it's natural to work with this Legendre transformed function. So that's it for classical mechanics in this post. I've tried to look at the evolution of a classical system in a way that makes the Legendre transform natural. Free quantum particles Now I want to take a look at the evolution of a free quantum particle to show how similar it is to what I wrote above. In this case we have the Schrödinger equation Let's suppose that from time onwards the particle is free so . Then we have Now let's take the Fourier transform in the spatial variable. We get: We can write this as So the time evolution of the free quantum particle is given by repeated convolution with a Gaussian function which in the Fourier domain is repeated multiplication by a Gaussian. The classical section above is nothing but a tropical version of this section. I doubt I've said anything original here. Classical mechanics is well known to be the limit of quantum mechanics as and it's well known that in this limit we find that occurrences of the semiring are replaced by the semiring . But I've never seen an article that attempts to describe classical mechanics in terms of repeated inf-convolution even though this is close to Hamilton's formulation and I've never seen an article that shows the parallel with the Schrödinger equation in this way. I'm hoping someone will now be able to say to me "I've seen that before" and post a relevant link below. I'm not sure how the above applies for a non-trivial potential . I wrote this little Schrödinger equation solver a while back. As might be expected, it's inconvenient to use the Fourier domain to deal with the part of the evolution due to . In order to simulate a time step of the code simulates in the Fourier domain assuming the particle is free and then spends solving for the -dependent part in the spatial domain. So even in the presence of non-trivial it can still be useful to work with a Fourier transform. Almost the same iteration could be used to numerically compute the action for the classical case. 1 comment: John Baez said... Great blog post! I feel pretty sure this material is known, since there's a long tradition of 'idempotent analysis' in Russia which seeks to treat classical mechanics using linear algebra over the tropical semirig. I've provided a short list of references here. I'm not sure they contain what you want, but they should give a reasonably good picture of the state of the art.
33ff9ec107931845	This is a good article. Click here for more information. Condensed matter physics From Wikipedia, the free encyclopedia (Redirected from Condensed Matter) Jump to: navigation, search Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter,[1] where particles adhere to each other. Condensed matter physicists seek to understand the behavior of these phases by using physical laws. In particular, they include the laws of quantum mechanics, electromagnetism and statistical mechanics. The most familiar condensed phases are solids and liquids while more exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. The study of condensed matter physics involves measuring various material properties via experimental probes along with using methods of theoretical physics to develop mathematical models that help in understanding physical behavior. The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists,[2] and the Division of Condensed Matter Physics is the largest division at the American Physical Society.[3] The field overlaps with chemistry, materials science, and nanotechnology, and relates closely to atomic physics and biophysics. The theoretical physics of condensed matter shares important concepts and methods with that of particle physics and nuclear physics.[4] A variety of topics in physics such as crystallography, metallurgy, elasticity, magnetism, etc., were treated as distinct areas until the 1940s, when they were grouped together as solid state physics. Around the 1960s, the study of physical properties of liquids was added to this list, forming the basis for the new, related specialty of condensed matter physics.[5] According to physicist Philip Warren Anderson, the term was coined by him and Volker Heine, when they changed the name of their group at the Cavendish Laboratories, Cambridge from Solid state theory to Theory of Condensed Matter in 1967,[6] as they felt it did not exclude their interests in the study of liquids, nuclear matter, and so on.[7] Although Anderson and Heine helped popularize the name "condensed matter", it had been present in Europe for some years, most prominently in the form of a journal published in English, French, and German by Springer-Verlag titled Physics of Condensed Matter, which was launched in 1963.[8] The funding environment and Cold War politics of the 1960s and 1970s were also factors that lead some physicists to prefer the name "condensed matter physics", which emphasized the commonality of scientific problems encountered by physicists working on solids, liquids, plasmas, and other complex matter, over "solid state physics", which was often associated with the industrial applications of metals and semiconductors.[9] The Bell Telephone Laboratories was one of the first institutes to conduct a research program in condensed matter physics.[5] References to "condensed" state can be traced to earlier sources. For example, in the introduction to his 1947 book Kinetic Theory of Liquids,[10] Yakov Frenkel proposed that "The kinetic theory of liquids must accordingly be developed as a generalization and extension of the kinetic theory of solid bodies. As a matter of fact, it would be more correct to unify them under the title of 'condensed bodies'". Classical physics[edit] Heike Kamerlingh Onnes and Johannes van der Waals with the helium liquefactor in Leiden (1908) One of the first studies of condensed states of matter was by English chemist Humphry Davy, in the first decades of the nineteenth century. Davy observed that of the forty chemical elements known at the time, twenty-six had metallic properties such as lustre, ductility and high electrical and thermal conductivity.[11] This indicated that the atoms in John Dalton's atomic theory were not indivisible as Dalton claimed, but had inner structure. Davy further claimed that elements that were then believed to be gases, such as nitrogen and hydrogen could be liquefied under the right conditions and would then behave as metals.[12][notes 1] In 1823, Michael Faraday, then an assistant in Davy's lab, successfully liquefied chlorine and went on to liquefy all known gaseous elements, except for nitrogen, hydrogen, and oxygen.[11] Shortly after, in 1869, Irish chemist Thomas Andrews studied the phase transition from a liquid to a gas and coined the term critical point to describe the condition where a gas and a liquid were indistinguishable as phases,[14] and Dutch physicist Johannes van der Waals supplied the theoretical framework which allowed the prediction of critical behavior based on measurements at much higher temperatures.[15]:35–38 By 1908, James Dewar and Heike Kamerlingh Onnes were successfully able to liquefy hydrogen and then newly discovered helium, respectively.[11] Paul Drude in 1900 proposed the first theoretical model for a classical electron moving through a metallic solid.[4] Drude's model described properties of metals in terms of a gas of free electrons, and was the first microscopic model to explain empirical observations such as the Wiedemann–Franz law.[16][17]:27–29 However, despite the success of Drude's free electron model, it had one notable problem: it was unable to correctly explain the electronic contribution to the specific heat and magnetic properties of metals, and the temperature dependence of resistivity at low temperatures.[18]:366–368 In 1911, three years after helium was first liquefied, Onnes working at University of Leiden discovered superconductivity in mercury, when he observed the electrical resistivity of mercury to vanish at temperatures below a certain value.[19] The phenomenon completely surprised the best theoretical physicists of the time, and it remained unexplained for several decades.[20] Albert Einstein, in 1922, said regarding contemporary theories of superconductivity that "with our far-reaching ignorance of the quantum mechanics of composite systems we are very far from being able to compose a theory out of these vague ideas".[21] Advent of quantum mechanics[edit] Drude's classical model was augmented by Wolfgang Pauli, Arnold Sommerfeld, Felix Bloch and other physicists. Pauli realized that the free electrons in metal must obey the Fermi–Dirac statistics. Using this idea, he developed the theory of paramagnetism in 1926. Shortly after, Sommerfeld incorporated the Fermi–Dirac statistics into the free electron model and made it better able to explain the heat capacity. Two years later, Bloch used quantum mechanics to describe the motion of a quantum electron in a periodic lattice.[18]:366–368 The mathematics of crystal structures developed by Auguste Bravais, Yevgraf Fyodorov and others was used to classify crystals by their symmetry group, and tables of crystal structures were the basis for the series International Tables of Crystallography, first published in 1935.[22] Band structure calculations was first used in 1930 to predict the properties of new materials, and in 1947 John Bardeen, Walter Brattain and William Shockley developed the first semiconductor-based transistor, heralding a revolution in electronics.[4] A replica of the first point-contact transistor in Bell labs In 1879, Edwin Herbert Hall working at the Johns Hopkins University discovered a voltage developing across conductors transverse to an electric current in the conductor and magnetic field perpendicular to the current.[23] This phenomenon arising due to the nature of charge carriers in the conductor came to be termed the Hall effect, but it was not properly explained at the time, since the electron was experimentally discovered 18 years later. After the advent of quantum mechanics, Lev Landau in 1930 developed the theory of landau quantization and laid the foundation for the theoretical explanation for the quantum Hall effect discovered half a century later.[24]:458–460[25] Magnetism as a property of matter has been known in China since 4000 BC.[26]:1–2 However, the first modern studies of magnetism only started with the development of electrodynamics by Faraday, Maxwell and others in the nineteenth century, which included classifying materials as ferromagnetic, paramagnetic and diamagnetic based on their response to magnetization.[27] Pierre Curie studied the dependence of magnetization on temperature and discovered the Curie point phase transition in ferromagnetic materials.[26] In 1906, Pierre Weiss introduced the concept of magnetic domains to explain the main properties of ferromagnets.[28]:9 The first attempt at a microscopic description of magnetism was by Wilhelm Lenz and Ernst Ising through the Ising model that described magnetic materials as consisting of a periodic lattice of spins that collectively acquired magnetization.[26] The Ising model was solved exactly to show that spontaneous magnetization cannot occur in one dimension but is possible in higher-dimensional lattices. Further research such as by Bloch on spin waves and Néel on antiferromagnetism led to developing new magnetic materials with applications to magnetic storage devices.[26]:36–38,48 Modern many-body physics[edit] A magnet levitating over a superconducting material. The Sommerfeld model and spin models for ferromagnetism illustrated the successful application of quantum mechanics to condensed matter problems in the 1930s. However, there still were several unsolved problems, most notably the description of superconductivity and the Kondo effect.[30] After World War II, several ideas from quantum field theory were applied to condensed matter problems. These included recognition of collective excitation modes of solids and the important notion of a quasiparticle. Russian physicist Lev Landau used the idea for the Fermi liquid theory wherein low energy properties of interacting fermion systems were given in terms of what are now termed Landau-quasiparticles.[30] Landau also developed a mean field theory for continuous phase transitions, which described ordered phases as spontaneous breakdown of symmetry. The theory also introduced the notion of an order parameter to distinguish between ordered phases.[31] Eventually in 1965, John Bardeen, Leon Cooper and John Schrieffer developed the so-called BCS theory of superconductivity, based on the discovery that arbitrarily small attraction between two electrons of opposite spin mediated by phonons in the lattice can give rise to a bound state called a Cooper pair.[32] The quantum Hall effect: Components of the Hall resistivity as a function of the external magnetic field[33]:fig. 14 The study of phase transition and the critical behavior of observables, termed critical phenomena, was a major field of interest in the 1960s.[34] Leo Kadanoff, Benjamin Widom and Michael Fisher developed the ideas of critical exponents and widom scaling. These ideas were unified by Kenneth G. Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory.[34] The quantum Hall effect was discovered by Klaus von Klitzing in 1980 when he observed the Hall conductance to be integer multiples of a fundamental constant .(see figure) The effect was observed to be independent of parameters such as system size and impurities.[33] In 1981, theorist Robert Laughlin proposed a theory explaining the unanticipated precision of the integral plateau. It also implied that the Hall conductance can be characterized in terms of a topological invariable called Chern number.[35][36]:69, 74 Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now a rational multiple of a constant. Laughlin, in 1983, realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational method solution, named the Laughlin wavefunction.[37] The study of topological properties of the fractional Hall effect remains an active field of research. In 1986, Karl Müller and Johannes Bednorz discovered the first high temperature superconductor, a material which was superconducting at temperatures as high as 50 Kelvin. It was realized that the high temperature superconductors are examples of strongly correlated materials where the electron–electron interactions play an important role.[38] A satisfactory theoretical description of high-temperature superconductors is still not known and the field of strongly correlated materials continues to be an active research topic. In 2009, David Field and researchers at Aarhus University discovered spontaneous electric fields when creating prosaic films[clarification needed] of various gases. This has more recently expanded to form the research area of spontelectrics.[39] In 2012 several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator [40] in accord with the earlier theoretical predictions.[41] Since samarium hexaboride is an established Kondo insulator, i.e. a strongly correlated electron material, the existence of a topological surface state in this material would lead to a topological insulator with strong electronic correlations. Theoretical condensed matter physics involves the use of theoretical models to understand properties of states of matter. These include models to study the electronic properties of solids, such as the Drude model, the Band structure and the density functional theory. Theoretical models have also been developed to study the physics of phase transitions, such as the Ginzburg–Landau theory, critical exponents and the use of mathematical methods of quantum field theory and the renormalization group. Modern theoretical studies involve the use of numerical computation of electronic structure and mathematical tools to understand phenomena such as high-temperature superconductivity, topological phases, and gauge symmetries. Main article: Emergence Theoretical understanding of condensed matter physics is closely related to the notion of emergence, wherein complex assemblies of particles behave in ways dramatically different from their individual constituents.[32] For example, a range of phenomena related to high temperature superconductivity are understood poorly, although the microscopic physics of individual electrons and lattices is well known.[42] Similarly, models of condensed matter systems have been studied where collective excitations behave like photons and electrons, thereby describing electromagnetism as an emergent phenomenon.[43] Emergent properties can also occur at the interface between materials: one example is the lanthanum aluminate-strontium titanate interface, where two non-magnetic insulators are joined to create conductivity, superconductivity, and ferromagnetism. Electronic theory of solids[edit] The metallic state has historically been an important building block for studying properties of solids.[44] The first theoretical description of metals was given by Paul Drude in 1900 with the Drude model, which explained electrical and thermal properties by describing a metal as an ideal gas of then-newly discovered electrons. He was able to derive the empirical Wiedemann-Franz law and get results in close agreement with the experiments.[17]:90–91 This classical model was then improved by Arnold Sommerfeld who incorporated the Fermi–Dirac statistics of electrons and was able to explain the anomalous behavior of the specific heat of metals in the Wiedemann–Franz law.[17]:101–103 In 1912, The structure of crystalline solids was studied by Max von Laue and Paul Knipping, when they observed the X-ray diffraction pattern of crystals, and concluded that crystals get their structure from periodic lattices of atoms.[17]:48[45] In 1928, Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, called the Bloch wave.[46] Calculating electronic properties of metals by solving the many-body wavefunction is often computationally hard, and hence, approximation methods are needed to obtain meaningful predictions.[47] The Thomas–Fermi theory, developed in the 1920s, was used to estimate system energy and electronic density by treating the local electron density as a variational parameter. Later in the 1930s, Douglas Hartree, Vladimir Fock and John Slater developed the so-called Hartree–Fock wavefunction as an improvement over the Thomas–Fermi model. The Hartree–Fock method accounted for exchange statistics of single particle electron wavefunctions. In general, it's very difficult to solve the Hartree–Fock equation. Only the free electron gas case can be solved exactly.[44]:330–337 Finally in 1964–65, Walter Kohn, Pierre Hohenberg and Lu Jeu Sham proposed the density functional theory which gave realistic descriptions for bulk and surface properties of metals. The density functional theory (DFT) has been widely used since the 1970s for band structure calculations of variety of solids.[47] Symmetry breaking[edit] Main article: Symmetry breaking Some states of matter exhibit symmetry breaking, where the relevant laws of physics possess some symmetry that is broken. A common example is crystalline solids, which break continuous translational symmetry. Other examples include magnetized ferromagnets, which break rotational symmetry, and more exotic states such as the ground state of a BCS superconductor, that breaks U(1) phase rotational symmetry.[48][49] Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons. For example, in crystalline solids, these correspond to phonons, which are quantized versions of lattice vibrations.[50] Phase transition[edit] Main article: Phase transition Phase transition refers to the change of phase of a system, which is brought about by change in an external parameter such as temperature. Classical phase transition occurs at finite temperature when the order of the system was destroyed. For example, when ice melts and becomes water, the ordered crystal structure is destroyed. In quantum phase transitions, the temperature is set to absolute zero, and the non-thermal control parameter, such as pressure or magnetic field, causes the phase transitions when order is destroyed by quantum fluctuations originating from the Heisenberg uncertainty principle. Here, the different quantum phases of the system refer to distinct ground states of the Hamiltonian. Understanding the behavior of quantum phase transition is important in the difficult tasks of explaining the properties of rare-earth magnetic insulators, high-temperature superconductors, and other substances.[51] Two classes of phase transitions occur: first-order transitions and continuous transitions. For the later, the two phases involved do not co-exist at the transition temperature, also called critical point. Near the critical point, systems undergo critical behavior, wherein several of their properties such as correlation length, specific heat, and magnetic susceptibility diverge exponentially.[51] These critical phenomena poses serious challenges to physicists because normal macroscopic laws are no longer valid in the region and novel ideas and methods must be invented to find the new laws that can describe the system.[52]:75ff The simplest theory that can describe continuous phase transitions is the Ginzburg–Landau theory, which works in the so-called mean field approximation. However, it can only roughly explain continuous phase transition for ferroelectrics and type I superconductors which involves long range microscopic interactions. For other types of systems that involves short range interactions near the critical point, a better theory is needed.[53]:8–11 Near the critical point, the fluctuations happen over broad range of size scales while the feature of the whole system is scale invariant. Renormalization group methods successively average out the shortest wavelength fluctuations in stages while retaining their effects into the next stage. Thus, the changes of a physical system as viewed at different size scales can be investigated systematically. The methods, together with powerful computer simulation, contribute greatly to the explanation of the critical phenomena associated with continuous phase transition.[52]:11 Experimental condensed matter physics involves the use of experimental probes to try to discover new properties of materials. Such probes include effects of electric and magnetic fields, measuring response functions, transport properties and thermometry.[54] Commonly used experimental methods include spectroscopy, with probes such as X-rays, infrared light and inelastic neutron scattering; study of thermal response, such as specific heat and measuring transport via thermal and heat conduction. Image of X-ray diffraction pattern from a protein crystal. Further information: Scattering Several condensed matter experiments involve scattering of an experimental probe, such as X-ray, optical photons, neutrons, etc., on constituents of a material. The choice of scattering probe depends on the observation energy scale of interest. Visible light has energy on the scale of 1 electron volt (eV) and is used as a scattering probe to measure variations in material properties such as dielectric constant and refractive index. X-rays have energies of the order of 10 keV and hence are able to probe atomic length scales, and are used to measure variations in electron charge density.[55]:33–34 Neutrons can also probe atomic length scales and are used to study scattering off nuclei and electron spins and magnetization (as neutrons have spin but no charge). Coulomb and Mott scattering measurements can be made by using electron beams as scattering probes.[55]:33–34[56]:39–43 Similarly, positron annihilation can be used as an indirect measurement of local electron density.[57] Laser spectroscopy is an excellent tool for studying the microscopic properties of a medium, for example, to study forbidden transitions in media with nonlinear optical spectroscopy.[52] :258–259 External magnetic fields[edit] In experimental condensed matter physics, external magnetic fields act as thermodynamic variables that control the state, phase transitions and properties of material systems.[58] Nuclear magnetic resonance (NMR) is a method by which external magnetic fields are used to find resonance modes of individual electrons, thus giving information about the atomic, molecular, and bond structure of their neighborhood. NMR experiments can be made in magnetic fields with strengths up to 60 Tesla. Higher magnetic fields can improve the quality of NMR measurement data.[59]:69[60]:185 Quantum oscillations is another experimental method where high magnetic fields are used to study material properties such as the geometry of the Fermi surface.[61] High magnetic fields will be useful in experimentally testing of the various theoretical predictions such as the quantized magnetoelectric effect, image magnetic monopole, and the half-integer quantum Hall effect.[59]:57 Cold atomic gases[edit] The first Bose–Einstein condensate observed in a gas of ultracold rubidium atoms. The blue and white areas represent higher density. Main article: Optical lattice Ultracold atom trapping in optical lattices is an experimental tool commonly used in condensed matter physics, and in atomic, molecular, and optical physics. The method involves using optical lasers to form an interference pattern, which acts as a lattice, in which ions or atoms can be placed at very low temperatures. Cold atoms in optical lattices are used as quantum simulators, that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets.[62] In particular, they are used to engineer one-, two- and three-dimensional lattices for a Hubbard model with pre-specified parameters, and to study phase transitions for antiferromagnetic and spin liquid ordering.[63][64] In 1995, a gas of rubidium atoms cooled down to a temperature of 170 nK was used to experimentally realize the Bose–Einstein condensate, a novel state of matter originally predicted by S. N. Bose and Albert Einstein, wherein a large number of atoms occupy one quantum state.[65] Computer simulation of nanogears made of fullerene molecules. It is hoped that advances in nanoscience will lead to machines working on the molecular scale. Research in condensed matter physics has given rise to several device applications, such as the development of the semiconductor transistor,[4] laser technology,[52] and several phenomena studied in the context of nanotechnology.[66]:111ff methods such as scanning-tunneling microscopy can be used to control processes at the nanometer scale, and have given rise to the study of nanofabrication.[67] In quantum computation, information is represented by quantum bits, or qubits. The qubits may decohere quickly before useful computation is completed. This serious problem must be solved before quantum computing may be realized. To solve this problem, several promising approaches are proposed in condensed matter physics, including superconducting Josephson junction qubits, spintronic qubits using the spin orientation of magnetic materials, or the topological non-Abelian anyons from fractional quantum Hall effect states.[67] Condensed matter physics also has important uses for biophysics, for example, the experimental mehtod of magnetic resonance imaging, which is widely used in medical diagnosis.[67] See also[edit] 1. ^ Both hydrogen and nitrogen have since been liquified, however ordinary liquid nitrogen and hydrogen do not possess metallic properties. Physicists Eugene Wigner and Hillard Bell Huntington predicted in 1935[13] that a state metallic hydrogen exists at sufficiently high pressures (over 25 GPa), however this has not yet been observed. 1. ^ Taylor, Philip L. (2002). A Quantum Approach to Condensed Matter Physics. Cambridge University Press. 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ece55164d997770d	Critical Review of "A New Kind of Science" 12 July 2002 With extreme hubris, Wolfram has titled his new book on cellular automata "A New Kind of Science". But it's not new. And it's not science. Solo Discovery? The main text of the book gives the strong impression that everything pictured and described in the book is Wolfram's own invention. Moreover, he often implies that various areas of science have been much more limited in their scope than they actually are. Consequently, it is vital to read the notes at the back of the book in combination with the main text, in order to restore a little of the balance. The notes are much better at giving proper credit to the vast reams of work that Wolfram is building on. Part of the problem is Wolfram's insistance on using his own terminology for concepts and ideas that have perfectly good names in regular mathematics and science. Examples: he always referring to fractals as "nested" (and never makes clear whether the term includes less structured fractals or not), he doesn't refer to well-known pictures by their common name in the main text (such as the Sierpinski gasket or the Koch curve), he calls refers to lossy compression as "irreversible", he insists on using Mathematica notation rather than standard mathematical notation (there may indeed be a million Mathematica users, but there are considerably more who understand normal notation and don't have access to this expensive tool). I suspect Wolfram's reasons for this are pedagogical, so as not to put off his less widely educated readers and to ensure a consistency of style throughout the book. However, this approach is likely to put off the serious scientific reader—particularly in combination with his other individual ways of expressing his ideas (such as starting many paragraphs with a conjunction, or avoiding the use of color for figures). More seriously, even allowing for the notes, the book is seriously misleading in its description of some areas of science, and plain wrong in a few others. One misleading example is the section on partial differential equations in chapter 4. Wolfram states that "in fact almost all the work—at least in one dimension—has concentrated on just the three specific equations" (p.162) which are the diffusion equation, the wave equation and the sine-Gordon equation. The notes to the chapter themselves bely this, as they give (p.925) pictures of another three one-dimensional equations which have been studied (Burger's equation, a nonlinear Schrödinger equation and the Kuramoto-Shivshinsky equation), and of course there are many more—for example the Ginsburg-Landau equation (which his nonlinear Schödinger equation is a special case of) or the Korteweg-de Vries equation. A more serious example where Wolfram is simply wrong is in his treatment of chaos theory. Throughout the book, he equates chaos theory with the phenomenon of sensitive dependence on initial conditions (SDIC). This allows him to claim that any randomness that occurs in a chaotic system is just a consequence of the inherent randomness in the least significant digits of the initial condition. In turn, this sets the stage for what he claims is one of his own major discoveries: that simple programs can inherently generate complex behavior and randomness. However, SDIC is just one of the attributes of chaotic behavior. Another important attribute of chaos theory—and indeed the reason why the field is called "chaos" in the first place—is the observation that complicated, apparently random behavior can arise from simple systems (when they are nonlinear). When I made a quick survey of five books on chaos theory on my bookshelf, only three of them mentioned SDIC as part of their definition of chaos, and in those cases it was only part of the definition. So why is Wolfram so comprehensively ignoring the normal understanding in this field? A cynical part of me suggests that it would be too inconvenient for him to completely give credit to a field whose key observation is that complicated, apparently random behavior can arise from simple systems. This is exactly the key observation that he himself is trying to lay sole claim to—in over 30 places he mentions that this observation is one of the "main discoveries" of the book. (To be fair, Wolfram's own earlier work was indeed one of the many strands that helped propel this observation to the forefront of chaos theory.) This key point undermines the whole of the first half of the book, and makes much of the second half sound familiar—when chaos theory appeared on the scene, its proponents also made the case for it being a key ingredient of little understood complex behavior in a range of fields from fluid dynamics to population biology to cardiology to to economics. Another area where Wolfram misleads his audience is in his presentation of the Principle of Computational Equivalence, the centrepoint of his final chapter. He only tangentially makes clear that the main content of this Principle is mostly just a restatement of the universality of computation, a result known since the 1960s; over and above this, the Principle boils down to the observation that he suspects that such universal systems are far more ubiquitous than people have previously realised. A final example where Wolfram neglects to mention in the main text that he's not the only bold explorer in a new intellectual landscape is regarding the potential use of cellular automata and other rule-based systems to model fundamental physics. The work of Zuse, Fredkin, Toffoli et al is mentioned in the notes, but how many readers are going to wade through that 8 point text? I'll admit that some part of this section arise from a traditional scientist's horror at seeing the normal forms for crediting predecessors skimped on. Wolfram is careful never to actually claim credit for something he hasn't produced; however, he's good at wording the main text so that it implies he has discovered things. But there is a much more serious point than mere pique. When all of the so-called discoveries of the book are added up, the net total is not an intellectual equivalent of a new "Principia Mathematica", but instead an intellectual equivalent of a Scientific American survey article—albeit one with a unusual breadth of scope and a slightly unusual point of view. My claim in the prologue of this review that Wolfram's book does not qualify as science is perhaps a little overstated. However, there is a key point: serious science should be predictive, not just descriptive. To qualify as science that applies to the real world, I would have expected to see some kind of claim in the book which could be verified against the behavior of the real world. Note that I'm not expecting him to have actually performed the verification yet (the book has only just come out, after all), but that there should be some indication of a path that would lead to verifiable, falsifiable predictions. This is not a new accusation. The fields of chaos theory and complexity theory (which Wolfram is essentially summarizing) have had similar accusations levelled at them, with some justification. However, in those fields there are genuine concrete results that can be pointed at, examined and potentially disproved—for example, the universal behavior of any iterated unimodal map (Feigenbaum[1979], Lanford [1982]), the route to chaos in a homoclinic system (Shil'nikov [1970]), the evolution of a more optimal sorting network (Hillis [1990]). Even within the boundaries of descriptive science, Wolfram leaves himself escape routes in the event of serious criticism—the book is littered with weasel words like "seems", "almost always", "appears". Let's be clear here: my complaint here is not that Wolfram hasn't verified his predictions, but that he hasn't made any predictions that admit verification. When discussing his Principle of Computational Equivalence and the phenomenon of computational irreducibility, Wolfram starts to make clear the point that his approach to science and modelling may never be able to produce model behavior in a shorter time than the system itself produces the actual behavior. Again, this phenomenon of computational irreducibility is not Wolfram's discovery, but the key point here is that you can begin to see why "traditional" science has not devoted much attention to this kind of modelling. For if you cannot get conceptual understanding or useful predictions from a model, what use is it? The Good Stuff To try to bring a little balance to this review, I should point out that there are definite areas of the book that I enjoyed reading. Much of the background material presented in the notes is well-presented and thorough, and presentation of the operations of various rule based systems is clear and easily understood. There are also some genuine nuggets of real science in the book, such as the models for pigmentation and branching in chapter 8. The rough outline of a rule-based approach to fundamental physics presented in chapter 9 is the most tantalizing chapter, however. This chapter does actually show signs of promise for generating an interesting (and verifiable) scientific model. Personally, I think that Wolfram should have concentrated on fleshing out this material for the last ten years, rather than exploring a zillion different cellular automata models—if he had, there's just a chance that he might have been able to live up to the title of his book. I wouldn't be quite as negative as Freeman Dyson (whose alleged one word review was "worthless"), but this book definitely does not live up to its own hype. The book actually reminds me of Wolfram's other huge creation, Mathematica. For Mathematica Wolfram built on a lot of well-known techniques, a few of which he himself had actually invented, and tied it together into one coherent whole—but which was nowhere near as unique and spectacular as its own publicity labelled it. With "A New Kind of Science", Wolfram has brought together existing observations from a range of disciplines, combined them with his own particular worldview to attempt to produce a coherent whole. It's an interesting tour of modern science (particularly in the notes)—but that's not what the book presents itself as. The book presents itself as a earth-shaking change to the fundamental paradigms of science, and it just plain isn't. This review is driven from my notes on reading the book, together with the references that are relevant to ideas in the book and which Wolfram refuses to give (with a reasonable justification that the book is already too large). Some days I suspect that I may be one of the few people outside of Wolfram's crack team of hagiographers who has actually read the entire book. My main concern with the book is that a reader who is not already aware of the work done in related areas is going to come away from the book with a very misleading impression of Wolfram's contribution to knowledge. Hopefully, I can challenge that impression. Copyright (c) 2002-2003 David Drysdale Back to Home Page Contact me
710cff213bc4a94d	tisdag 21 juni 2016 New Quantum Mechanics 2: Computational Results I have now tested the atomic model for an atom with $N$ electrons of the previous post formulated as a classical free boundary problem in $N$ single-electron charge densities with non-overlapping supports filling 3d space with joint charge density as a sum of electron densities being continuously differentiable across inter-electron boundaries. I have computed in spherical symmetry on an increasing sequence of radii dividing 3d space into a sequence of shells filled by collections of electrons smeared into spherically symmetric shell charge distribution. The electron-electron repulsive energy is computed with a reduction factor of $\frac{n-1}{n}$ for the electrons in a shell with $n$ electrons to account for lack of self repulsion. Below is a typical result for Xenon with 54 electrons organised in shells with 2, 8, 18, 18 and 8 electrons with ground state energy -7413 to be compared with -7232 measured and with the energy distribution in the 5 shells displayed in the order of total energy, kinetic energy, kernel potential energy and inter-electron energy. Here the blue curve represents electron charge density, green is kernel potential and red is inter-electron potential. The inter-shell boundaries are adaptively computed so as to represent a preset 2-8-18-18-8 configuration in iterative relaxation towards a ground state of minimal energy. In general computed ground state energies agree with measured energies within a few percent for all atoms up to Radon with 86 electrons. The computations indicate that it may well be possible to build an atomic model based on non-overlapping electronic charge densities as a classical continuum mechanical model with electrons keeping individuality by occupying different regions of space, which agrees reasonably well with observations. The model is an $N$-species free boundary problem in three space dimensions and as such is readily computable for any number of $N$ for both ground states, excited states and dynamic transitions between states. We recall the the standard model in the form of Schrödinger's equation for a wave function depending on $3N$ space dimensions, is computationally demanding already for $N=2$ and completely beyond reach for larger $N$. As a result the full $3N$-dimensional Schrödinger equation is always replaced by some radically reduced model such as Hartree-Fock with optimization over a "clever choice" of a few "atomic orbitals", or Thomas-Fermi and Density Functional Theory with different forms of electron densities. The present model is an electron density model, which as a free boundary problem with electric individuality is different from Thomas-Fermi and DFT. We further recall that the standard Schrödinger equation is an ad hoc model with only formal justification as a physical model, in particular concerning the kinetic energy and the time dependence, and as such should perhaps better not be taken as a given ready-made model which is perfect and as such canonical (as is the standard view). Since this standard model is uncomputable, it is impossible to show that the results from the model agree with observations, and thus claims of perfection made in books on quantum mechanics rather represent an ad hoc preconceived idea of unquestionable ultimate perfection than true experience. 2 kommentarer: 1. Atomic energies are not that interesting, do you have any result for chemical quantities. Like bond lengths and angles for molecules or lattice sizes for compounds? 2. I agree, but it is a necessary starting point.
f7ec8a9b64abbaba	Math for Everyone shares glimpses of soliton theory Author: Shadia Ajam Alex Kasmen, Math for Everyone speaker This past Thursday (Nov. 7) at the Math for Everyone lecture series, Alex Kasman from the Department of Mathematics at the College of Charleston gave a presentation about soliton theory, which combines algebra and geometry with the study of waves and elementary particles. Kasman opened his presentation by explaining the origins of the theory. Solitons, self-reinforcing solitary waves, were first discovered by Scottish ship designer and civil engineer John Scott Russell in 1834. One day Russell observed what he thought was a solitary wave on a Scottish canal and decided to study this phenomenon further. However, the response from the scientific community was extremely negative. Recently, however, the importance of the theory has resurfaced. Russell was ahead of his time and the math needed to fully understand his discovery did not exist until the 20th century. Soliton theory is still an active area of research today and provides a better look at a side of both math and theoretical physics and has found applications in diverse fields such as molecular biology and telecommunications. Mathematicians and scientists are now recognizing and creating soliton equations and applying them to math, the sciences, and engineering. Some equations that revolve around this theory include the non-linear Schrödinger equation which involves applications to optics and water waves, the sine-Gordon equation that models positrons/electrons in one dimension or frictionless coupled pendula, and the bilinear KP equation which focuses on ocean waves. “Russell made an important observation; you can notice something that you might think is important, even though no one believes it,” Kasman said. Kasman’s book, Glimpses of Soliton Theory, aims to introduce the algebro-geometric structure of soliton equations to undergraduate math majors. Math for Everyone is a series of math-related lectures, especially for undergraduates. The next Math for Everyone lecture, "All Tied Up in Knots" will be held on December 5, 2013 in 101 Jordan Hall of Science. Everyone, from Fields medalists to math phobes, is welcome to attend
52307b3945092006	General · Language · Site Eigen what? In reading about vibrations in fields and wave functions and wave packets, some words that start with ‘e’ occurred a lot: eigenvalue, eigenvector, eigenfunction, eigenvalue equation, eigenequation [1]. I needed to refresh my understanding of the mathematical models for macroscopic and microscopic systems [2]. Everyday we are surrounded by things that vibrate when excited. Some of these we sense and some not. Some are pleasant like harmonic tones from musical instruments. Some are unpleasant like when we drive on a rough road. Many things vibrate with natural or characteristic frequencies. We see and hear due to such frequencies. Colors and tunes enrich our lives. Science and math help us understand how all this works and empower us to shape vibrations in practical ways. In particular, to analyze and make systems where vibrations are sinusoidal oscillations. At the macroscopic level, classical mechanics allows us to model oscillating systems. One of the simplest being the simple harmonic oscillator, which exhibits regular sinusoidal motion with a frequency dependent on some properties of such a system. Classically, a model of a single mass (object) oscillating with one degree of freedom (up-down). When we get to more interesting systems, ones with more than one degree of freedom and multiple masses, their analysis rapidly gets harder — their models are more complex. More powerful mathematical techniques are required. Many of these systems still involve sinusoidal motion. What we discover is that we can determine certain properties of these systems because they exhibit natural or “inherent” or “characteristic” frequencies. The mathematical equations can be solved for those specific frequencies. The German word “eigen” is used to describe such values. For the physical models, the solutions to eigenequations are eigenvalues and eigenvectors (typically written in matrix format), which describe the vibration modes of the system. Understanding eigenfunctions and eigenstates tells us more about the systems being analyzed. We can make predictions. [3] When we get to microscopic systems, much of that classical mathematical framework applies to quantum mechanics. In new ways. Objects are no longer tiny balls in an exactly characterized (certainly valued) state of space and time — in definite eigenstates. The math is even more daunting, but we’re still looking at an eigen landscape because it’s all about vibrations (in fields). These days we know that the electrons don’t really “orbit” at all, because they don’t really have a “position” or “velocity.” Quantum mechanics says that the electrons persist in clouds of probability known as “wave functions,” which tell us where we might find the particle if we were to look for it. — Carroll, Sean (2012-11-13). The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World (Kindle Locations 646-648). Penguin Publishing Group. Kindle Edition. [] When an object can definitely be “pinned down” in some respect, it is said to possess an eigenstate. … when the wavefunction collapses because the position of an electron has been determined, the electron’s state becomes an “eigenstate of position”, meaning that its position has a known value, an eigenvalue of the eigenstate of position. 1. Eigenvalue aka characteristic value or characteristic root associated with the eigenvector The prefix eigen- is adopted from the German word eigen for “proper”, “inherent”; “own”, “individual”, “special”; “specific”, “peculiar”, or “characteristic”. … In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation T(v) = λv An example of an eigenvalue equation where the transformation T is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics: E = EψE where H, the Hamiltonian, is a second-order differential operator and ψE, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy. 2. This Khan Academy video is a helpful overview of eigenvectors and eigenvalues (eigenvalues are associated with a corresponding eigenvector), providing you are somewhat familiar with vector spaces (fields) and matrix mathematics (as a way to encode linear maps between vector spaces). In particular the Hilbert vector space. Fourier analysis (which involves the superposition of sine waves) usually uses the Hilbert space. 3. Quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known. 4. Degrees of freedom and mode shapes The simple mass–spring damper model is the foundation of vibration analysis, but what about more complex systems? The mass–spring–damper model described above is called a single degree of freedom (SDOF) model since the mass is assumed to only move up and down. In more complex systems, the system must be discretized into more masses that move in more than one direction, adding degrees of freedom. … This is referred to an eigenvalue problem in mathematics … 5. “The simple harmonic oscillator … is an excellent model for a wide range of systems in nature.” 6. Using eigenvalues and eigenvectors to study vibrations … a brief introduction to the use of eigenvalues and eigenvectors to study vibrating systems for systems with no inputs. MatLab code is also included on the “Vibrating Systems” page. Analyzing a system in terms of its eigenvalues and eigenvectors greatly simplifies system analysis, and gives important insight into system behavior. For example, once the eigenvalues and eigenvectors of the system above have been determined, its motion can be completely determined simply by knowing the initial conditions and solving one set of algebraic equations. 7. Wolfram MathWorld Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector; there is no analogous distinction between left and right for eigenvalues). The decomposition of a square matrix A into eigenvalues and eigenvectors is known in this work as eigen decomposition, and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of A is square is known as the eigen decomposition theorem. 8. “If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.” 9. “The Schrödinger equation is the basic equation for obtaining the constant energy states of atoms, molecules, etc. … Boundary conditions [as in particle in a box] give additional equations, on top of Schrödinger’s equation, and this “narrows down” the number of acceptable wavefunctions. … One of the postulates … of quantum mechanics … is that to each physical property (energy, momentum, position, kinectic energy, number of particles, etc.) there is an associated operator. … [and] that the result of a measurement of that property must give one of the eigenvalues associated with that operator.” — re eigenvalue equation [Useful overview in 6 pages] The wavefunction for a given physical system contains the measurable information about the system. To obtain specific values for physical parameters, for example energy, you operate on the wavefunction with the quantum mechanical operator associated with that parameter. The operator associated with energy is the Hamiltonian, and the operation on the wavefunction is the Schrodinger equation. Solutions exist for the time independent Schrodinger equation only for certain values of energy, and these values are called “eigenvalues*” of energy. 11. The Schrödinger Equation is an Eigenvalue Problem It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate. [1] Dictionary definitions (comb. form) proper; characteristic: eigenfunction. ORIGIN from the German adjective eigen ‘own.’ 1. each of a set of values of a parameter for which a differential equation has a nonzero solution (an eigenfunction) under given conditions. 2. any number such that a given matrix minus that number times the identity matrix has a zero determinant. eigenvector: a vector that when operated on by a given operator gives a scalar multiple of itself. eigenfunction: each of a set of independent functions that are the solutions to a given differential equation. [2] As an undergrad, I studied linear (matrix) algebra. But I “hit the wall” in math while taking a class about the calculus of complex variables in n-space. An applied mathematics class dealing with complex analysis or the theory of functions of a complex variable. Perhaps my struggle was due to the way the course was taught. The TA merely came into the classroom and started writing his notes on a roll-up section of blackboards, moving on to the next section when each filled up and erasing earlier ones (the blackboards wrapped around three sides of the room). Probably was an early morning class as well. I remember barely keeping up taking down those notes. No Q&A. When I met with the TA and asked why he just didn’t handout copies of his notes for discussion, he said something like “what would be the point of the class?” Sigh. [3] At the macroscopic level, every object (rather than a dynamic system of objects) may be viewed as in its own “eigenstate.” A baseball, for example, is in a state of composite quantum decoherence. All its (emergent) properties appear definite. No quantum behavior is observable — there are no coherent aggregates (as in lasers and superconductors). And the de Broglie wavelength of a baseball is too small to ever measure (~10^–34 m). Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects, with an enormous number of degrees of freedom, in their natural environment. The process explains why we tend not to observe quantum behavior in everyday macroscopic objects. It also explains why we do see classical fields emerge from the properties of the interaction between matter and radiation for large amounts of matter.
f71484f922ee4335	Exploring "equivalence" of Biology and Boolean Logic What can we learn by attempting to implement biologies systems using symbolic logic, and what improvements can be made in our implementations of symbolic logic through the study of biology? That is the purpose of this exploration. I've been considering the dichotomy of "symbolic" systems, such as pure math (which exists in the theoretical domain, not subject to the effects of physics), and "physical" systems (which are subject to hidden variables and uncertainty, e.g. Heisenberg uncertainty). I've attempted to document these raw thoughts in the preface of the following essay. This comparison (which can be oversimplified as physics v. theory) and the role of "computers" as a medium for translation between these two domains is ancillary to the essay's main objective: to explore the implementations and properties of Biological computation (how symbolic computation emerges "naturally" from biological, physical systems) and Boolean Logic (Von Neumann architecture and how present day computers implement symbolic logic over physical matter). Ideally, this comparison will evolve over time to examine the most basic biological components and the most basic electronic implementations of boolean logic gates from both directions (creating boolean gates using these axiomatic biological components and created the axiomatic biological components using electronics). The outcome would be the ability to manipulate physical systems (using techniques and components like schmitt trigger triggers in electronics to enable circuits to be robust against the uncertainty of physics) to construct playgrounds where we can make promises about deduction about the equivalence of biology and computer programs (like the Curry-Howard correspondence, but including isomorphism to biological systems) The reason pure math "works" (i.e. has the property of being proveable and internally consistent) is because it is abstract and symbolic. That is, it exists in a theoretical domain which is not subject to the physics of the universe and its complex variables and their uncertainties over time. Within the static, unchanging context of the symbolic domain, we can freely test axioms and their consequences, control the construction and scope of a problem, examine properties of expressions using different approaches, and prove the consistency, correctness, and properties of theorems. While these properties of the symbolic domain convenience us to make deep promises about the integrity, inferences, and comparison of purely symbolic mathematical systems, it doesn't eliminate the reality that many of the systems we care about are subject to the chaotic laws of our physical universe. In such physical domains, we have less luxury of controlling our axioms. We may eliminate some variables by controlling our environment, even manipulate the powers of physics within reason of our technical capabilities (e.g. Bose-Einstein Conensate or the world's biggest vacuum chamber), but we cannot change that the universe is made of quarks -- a confounding constraint pure mathematics can ignore. The need for symbolic systems is necessitated by the complexity of our physical world. It seems there is always something more we do not know, in which every other computation depends: "For centuries [people] lived in the belief that truth was slim and elusive and that once [it was] found [...], the troubles of [our]kind would be over. [One] of knowledge in our time is bowed down under a burden [one] never imagined [one] would ever have: The overproduction of truth that cannot be consumed." -- The Denial of Death (page xviii, paragraph 1) by Ernest Becker To fully understand the physical world in its entirety requires that it be exhaustively, deterministically computable, and fully observable. At the quantum level, this "unfolding of the layers of the onion" requires creativity and finese as objects become so small that even observing them changes their behaviour. Instead of hedging all of our bets on speculatation on whether we will someday be able to compute what happened before the big bang, or the exact location of an electron with full confidence without relying on heisenberg uncertainty principle, for instance, there's reason to create incrementally better (more accurate) tools and systems for approximating these answers (or function). Fortunately, our universe is sufficiently complex as to support physical systems (made of matter and subject to the laws of physics) capable of faithfully emulating isolatable symbolic domains at high (albeit finite) accuracy. A computer is one example of a "sub-universe" we can create whose axioms are those of symbolic logic, a subset of symbolic mathematics. In later sections, it will be explored how the ingenuity of electronic design and engineering has enabled computers to uphold such a promise; to achieve resiliance and robustness to the seemingly untamable chaos and stochasticism of physics. The importance of the computer is that it provides a medium for integrating symbolic systems into our physical lives; as a frictionless translation layer between the physical and the symbolic. Why is important to be able to engineer faithfully symbolic systems from within our physical reality? First, it affords us interoperability; the ability for observations to be taken directly from the physical word and then analyzed within a controllable, symbolic environment. Second, we benefit from one the standard features of symbolic systems, which is that they are fully observable; they allow us to simulate and test complex interactions while maintaining full control over the environment's variables and axioms, giving us insights into the workings of the physical world. Most importantly, as "implementations" of symbolic logic, they provide us with a extensible, evolvable framework for performing repeatable computation (executing algorithms). How much can we trust a computer proof? See: Curry-Howard Correspondence. Is the computer the only viable approach of implementing symbolic systems? The dichotomy of pure symbolic mathematics and the physical universe means that we can make progress in either domain/direction. Towards a perfect understanding of the physical universe, or in advancing discrete tools and systems for solving certain problems in isolation. But so too can we discuss the dichotomy of different systems. Boolean logic is not the only implementable system for computation. We also have biology, an evolutionary system which evolved independently and from different axioms than boolean logic. Because biology is a physical system and boolean logic a symbolic one, perhaps the same dichotomy exists as does at the higher level comparison of the physical universe and symbolic math (in their totality). What can we learn by attempting to implement biology's systems using symbolic logic, and what improvements can be made in our implementations of symbolic logic through the study of biology? How and where will the brain and the transistor "meet in the middle"? And can it be demonstrated that computationally they are equivalent? That is the purpose of this exploration. A mess of unintelligable notes: Electrical engineering, avoiding inaccuracies in electrical current capacitors (integrate input of noisy buttons, for instance -- debouncer) or Schmitt trigger as detector * Schmitt triggers "When the necessity rises to determine which of the two signals is stronger or to determine which of the two signals reaches a specified value a comparitor is used." * Inspiration schrodener - "what is life" (recommended by Drew Winget) Acting on 1st principles, determining boundaries between the supervening fields deductions based on 1st principles - size constraints - electromagnetic bonds / type * 2 directions: 1. boolean logic using/from biological components 2. biological components using boolean logic (electronics) * What are the axiomatic components of biology (the simplest units made only of the elements) * What are the axiomatic components of boolean logic and electronics? NAND gates, etc. * Do computers make "errors"? Error Prevention, Safeguards, Guardbands MPE (Manchester Encoding) Types of errors 1. Floating point errors; deterministic inaccuracies in calculation due to hardware limitation 2. Design mistakes (e.g. CPU); formal methods can be used to verify correctness 3. Environmental Corruption and Hardware error: radiation from cosmic rays, electrical noise or spikes (e.g. electrostatic discharge) - https://en.wikipedia.org/wiki/Engineering_tolerance#Electrical_component_tolerance - https://en.wikipedia.org/wiki/Allowance_(engineering) - https://en.wikipedia.org/wiki/Allowance_(engineering)#Confounding_of_the_engineering_concepts_of_allowance_and_tolerance ** Error correction https://www.youtube.com/watch?v=5sskbSvha9M error correction * How do the brain's mistakes differ from those of Von Neumann computers? The difficulties of executing simple algorithms: why brains make mistakes computers don't. * Heisenberg's uncertainty principle & Schrödinger equation * Synthetic Biology (applying engineering principles to nature) Designing Input module, Response module, and Output module Creating Life - The Ultimate Engineering Challenge. (Synthetic Biology documentary) https://www.youtube.com/watch?v=VhuiMRIn6GM Labster - Synthetic Biology Virtual Lab Simulation Designing an Apoptotic Biological Circuit * electroporation * pasmid isolation (http://vlab.amrita.edu/?sub=3&brch=77&sim=314&cnt=1) * gel electrophoresis ! systems biology Previous works http://www.irisa.fr/dyliss/public/asiegel/Articles/SchaubSiegelVidela.pdf https://books.google.com/books?id=qGREBAAAQBAJ&pg=PT60&lpg=PT60&dq=equivalence+of+boolean+logic+and+biology&source=bl&ots=A3erNXBCxB&sig=MICDGNmyE-Tcgp631qOYBODYhYk&hl=en&sa=X&ei=sn6MVfPzI4r6sAWliJmYDQ&ved=0CB4Q6AEwAA#v=onepage&q=equivalence%20of%20boolean%20logic%20and%20biology&f=false Tangent (Essay) Conceptually, "Labster" (MIT virtual science laboratory) is an amazing technology and resource for anyone interested in practicing Synthetic Biology Provenance Trail The past few days I've gone down a rabbit hole involving biology and computational theory. I became particularly interested in the relationship between electric circuits (physically engineered implementations of boolean logic) and biological systems. What might be, or has been, learned about their comparative computational abilities or properties, and how might understanding one inform advances within the other? While my journey has just begun, I've recorded breadcrumbs of my findings[1] to organize myself and others interested in the topic. This whole spiral started in a very "Mark P Xu Neyerian" kind of way, by thinking about Curry-Howard Correspondence (Isomorphism), which demonstrates computational equivalence between mathematics and computer programs (i.e. boolean logic). I wondered if a similar correspondence might exist between biological systems and boolean logic. I admit this question is more than slightly naive (it is in-concrete what "equivalence" means in this context), given biology is built of the chemical elements and on top physics, both of which are subject to measurement uncertainty. If a loose correspondence can be demonstrated and we are able to determine both (a) mappings between biological systems and boolean logic, as well as (b) determine the chemical and thus mathematical thresholds under which these biological systems predictably operate equivalently (in the same way a Schmitt trigger is used in electronics to make electrical circuits robust against uncertainty and variability in electrical flow), this may expand the way we can use computers to cheaply and accurately test, evolve, or even prove biological and medical results. I also must admit, if this question of Curry-Howard-Biology correspondence is new, it's only because it hasn't been phrased "exactly" as such and because it's implications are mostly philosophical and don't contribute to the underlying sciences and experiments -- both which are anything but new and have been around for many years. Systems Biology, for instance, attempts to understand and emulate biological components, and vice versa through computational models[2]. The field of Biocomputing attempts to achieve some of the same learnings from the opposite direction, by creating physical boolean logic gates and other computational systems from biological material[3]. There is yet another field, Synthetic Biology, which constitutes the interaction of engineering practices and biology to design and synthesize biological components for specific applications. In order to connect the dots and make headway in any of these directions towards my goal, I needed a better understanding of what the simplest units of biology are. I had to consider, what is below the "cell"? What are the simplest biological constructs (molecules, systems) made only of the elements (which rise only from chemistry)? This question too has two directions. One, we can deduce the structure and components of the "cell" (from the cell down to its lowest level constituent parts) or two, we can inductively determine the arrangement of chemicals and circumstances required for biological, organic matter and simple biological systems to emerge from pure chemistry (abiogenesis/biopoiesis -- see Miller-Urey experiment)[4]. Synthetic biology may be the answer to "Schmitt trigger" analogy I previously entertained, as a medium for controlling the uncertainty effects of biological variables, determine attribution/consequences of variables, and in retroactively verifying results. Which made me wonder, how does synthetic biology work? I first watched this video[5] which followed a team of new biology researchers in a project to create bacterial capable of changing color within parasite infested water. The video is very practical and also follows environment consequences and implications, design considerations, and the reality of experimentation (regression testing), and team/lab dynamics. While the video was quite understandable, what the video didn't outline was any sort of terminology for contextualizing or reproducing the different experimental steps. I found myself wanting to know what processes the scientists were running and how. That's when I saw Labster[6] and was sufficiently impressed by the idea. Even in the video, the steps of Synthetic Biological are demystified. The best part is, the program seems like an amazingly safe, fast, scalable and accessible alternative to laboratories and hazardous material. In the interest in time-boxing this tangent, my next step is to continue exploring the simplest systems within biology, as well as how they may be functionally emulated through boolean logic. Updates soon! green chemistry artificial evolution Stanford karl deisseroth [1] Exploring "equivalence" of Biology and Boolean Logic [2] An Introduction to Systems Biology: Design Principles of Biological Circuits [3] Synthesizing Biomolecule-based Boolean Logic Gates http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3603578/ [4] Miller–Urey Experiment https://en.wikipedia.org/wiki/Miller%E2%80%93Urey_experiment [5] Creating Life - The Ultimate Engineering Challenge. (Synthetic Biology documentary) [6] Labster - Synthetic Biology Virtual Lab Simulation https://www.youtube.com/watch?v=VhuiMRIn6GM [7] http://www.nature.com/nbt/journal/v32/n6/full/nbt.2891.html [8] https://www.quantamagazine.org/20160128-ecorithm-computers-and-life/ "ecorithms"
88f17f0c9b6e4b2c	Superposition principle Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation. The superposition principle,[1] also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). A function that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity for scalar a. This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour. The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. Relation to Fourier analysis and similar methods By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific, simple form, often the response becomes easier to compute. For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses. Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves. Wave superposition Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave. Waves are usually described by variations in some parameter through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point. In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at top.) Wave diffraction vs. wave interference With regard to wave superposition, Richard Feynman wrote:[2] Other authors elaborate:[3] The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically they are two limiting cases of superposition effects. Yet another source concurs:[4] Inasmuch as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is therefore a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront. Wave interference The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, such as in a line array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference. green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Interference of two waves.svg wave 1 wave 2 Two waves in phase Two waves 180° out of phase Departures from linearity In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics. Quantum superposition In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type—stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[5] The projective nature of quantum-mechanical-state space makes an important difference: it does not permit superposition of the kind that is the topic of the present article. A quantum mechanical state is a ray in projective Hilbert space, not a vector. The sum of two rays is undefined. To obtain the relative phase, we must decompose or split the ray into components where the and the belongs to an orthonormal basis set. The equivalence class of allows a well-defined meaning to be given to the relative phases of the .[6] There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics." According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."[7] Boundary value problems A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation with some boundary specification For example, in Laplace's equation with Dirichlet boundary conditions, F would be the Laplacian operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R. In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: while the boundary values superpose: Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary value problems. Additive state decomposition Consider a simple linear system : By superposition principle, the system can be decomposed into Superposition principle is only available for linear systems. However, the Additive state decomposition can be applied not only to linear systems but also nonlinear systems. Next, consider a nonlinear system where is a nonlinear function. By the additive state decomposition, the system can be ‘additively’ decomposed into This decomposition can help to simplify controller design. Other example applications • In electrical engineering, in a linear circuit, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. The use of Fourier analysis on this basis is particularly common. For another, related technique in circuit analysis, see Superposition theorem. • In physics, Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields which arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation. • In mechanical engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[8] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[9] • In hydrogeology, the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer. • In process control, the superposition principle is used in model predictive control. • The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization. • In music, theorist Joseph Schillinger used a form of the superposition principle as one basis of his Theory of Rhythm in his Schillinger System of Musical Composition. According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.[10] See also 1. ^ The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London 2. ^ Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1] 3. ^ N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [2] 4. ^ Tim Freegard, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [3] 5. ^ Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ISBN 978-0-486-66772-0 6. ^ Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623. 7. ^ Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14. 8. ^ Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0-07-252036-1 9. ^ Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 ISBN 0-13-301458-4 10. ^ Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2. Further reading External links
5830bab8eef65405	Implications of a deeper level explanation of the deBroglie–Bohm version of quantum mechanics • G. Grössing • S. Fussy • J. Mesa PascasioEmail author • H. Schwabl Regular Paper Elements of a “deeper level” explanation of the deBroglie–Bohm (dBB) version of quantum mechanics are presented. Our explanation is based on an analogy of quantum wave-particle duality with bouncing droplets in an oscillating medium, the latter being identified as the vacuum’s zero-point field. A hydrodynamic analogy of a similar type has recently come under criticism by Richardson et al. (On the analogy of quantum wave-partile duality with bouncing droplets, 2014), because despite striking similarities at a phenomenological level the governing equations related to the force on the particle are evidently different for the hydrodynamic and the quantum descriptions, respectively. However, said differences are not relevant if a radically different use of said analogy is being made, thereby essentially referring to emergent processes in our model. If the latter are taken into account, one can show that the forces on the particles are identical in both the dBB and our model. In particular, this identity results from an exact matching of our emergent velocity field with the Bohmian “guiding equation”. One thus arrives at an explanation involving a deeper, i.e., subquantum, level of the dBB version of quantum mechanics. We show in particular how the classically local approach of the usual hydrodynamical modeling can be overcome and how, as a consequence, the configuration-space version of dBB theory for N particles can be completely substituted by a “superclassical” emergent dynamics of N particles in real three-dimensional space. Quantum mechanics Hydrodynamics DeBroglie–Bohm theory Guiding equation Configuration space Zero-point field Mathematics Subject Classification 1 Introduction The Schrödinger equation for $N>1$ particles does not describe a wave function in ordinary three-dimensional space, but instead in an abstract $3N$-dimensional space. For quantum realists, including Schrödinger and Einstein, for example, this has always been considered as “indigestible”. This holds even more so for a realist, causal approach to quantum phenomena such as the deBroglie–Bohm (dBB) version of quantum mechanics. David Bohm himself has admitted this, calling it a “serious problem”: “While our theory can be extended formally in a logically consistent way by introducing the concept of a wave in a $3N$-dimensional space, it is evident that this procedure is not really acceptable in a physical theory, and should at least be regarded as an artifice that one uses provisionally until one obtains a better theory in which everything is expressed once more in ordinary three-dimensional space” [1]. (For more detailed accounts of this discussion already in the early years of quantum mechanics, see [17, 18].) In the present paper, we shall refer to our attempt towards such a “better theory” in terms of a deeper level, i.e., subquantum, approach to the dBB theory, and thus to quantum theory in general. In fact, with our model, we have in a series of papers already obtained several essential elements of nonrelativistic quantum theory [8, 9, 13, 14]. They derive from the assumption that a particle of energy $E=\hbar \omega $ is actually an oscillator of angular frequency $\omega $ phase-locked with the zero-point oscillations of the surrounding environment, the latter containing both regular and fluctuating components and being constrained by the boundary conditions of the experimental setup via the buildup and maintenance of standing waves. The particle in this approach is an off-equilibrium steady-state oscillation maintained by a constant throughput of energy provided by the (“classical”) zero-point energy field. We have, for example, applied the model to the case of interference at a double slit, thereby obtaining the exact quantum mechanical probability density distributions on a screen behind the double slit, the average trajectories (which because of the averaging are shown to be identical to the Bohmian ones), and the involved probability density currents. Our whole model is constructed in close analogy to the bouncing/walking droplets above the surface of a vibrated liquid in the experiments first performed by Couder and Fort [4, 5], Fort and co-workers [6], which in many respects can serve as a classical prototype guiding our intuition for the modeling of quantum systems. However, there are also obvious differences between the mentioned physics of classical bouncers/walkers on the one hand, and the hydrodynamic-like models for quantum systems like our own model or the dBB on the other hand. In a recent paper, Richardson et al. [20] have probed more thoroughly into the hydrodynamic analogy of dBB-type quantum wave-particle duality with that of the classical bouncing droplets. Apart from the obvious difference in that Bohmian theory is distinctly nonlocal, whereas droplet–surface interactions are rooted in classical hydrodynamics and thus in a manifestly local theory, Richardson et al. focus on the following observation: the evidently different nature of the Bohmian force upon a quantum particle as compared to the force that a surface wave exerts upon a droplet. In fact, wherever the probability density in the dBB picture is close to zero, the quantum force becomes singular and will very quickly push any particle away from that area. Conversely, the hydrodynamic force directs the droplet into the trough of the wave! So, the probability of finding a droplet in the minima never reaches zero as it does for a quantum particle. The authors conclude that these discrepancies between the two models highlight “a major difference between the hydrodynamic force and the quantum force” [20]. Although these authors generally recover in numerical hydrodynamic simulations the results of the Paris group (later confirmed also by the group of Bush [3] at MIT) on single-slit diffraction and double-slit interference, they also point out the (already known) striking contrast between the trajectory behaviors for the bouncing droplet systems and dBB-type quantum mechanics, respectively. Whereas the latter exhibits the well-known no-crossing property, the trajectories of the former do to a large extent cross each other. So, again, the physics in the two models is apparently fundamentally different, despite some striking similarities on a phenomenological level. As to the differences, one may very well expect that they will even become more severe when moving from one-particle to $N$-particle systems. So, all in all, the paper by Richardson et al. [20] cautions against the assumption of too close a resemblance of bouncer/walker systems and the hydrodynamic-like modeling of quantum systems like the dBB, with their main argument being that the hydrodynamic force on a droplet strikingly contrasts with the quantum force on a particle in the dBB theory. However, we shall here argue against the possible conclusion that one has thus reached the limits of applicability of the hydrodynamic bouncer analogy for quantum modeling. On the contrary, as we have already pointed out in previous papers, it is a more detailed model inspired by the bouncer/walker experiments that can show the fertility of said analogy. It enables us to show that our model, being of the type of an “emergent quantum mechanics” [10, 11], can provide a deeper level explanation of the dBB version of quantum mechanics (Sect. 2). Moreover, as we shall also show, it turns out to provide an identity of an emergent force on the bouncer in our hydrodynamic-like model with the quantum force in Bohmian theory (Sect. 3). Finally, in Sect. 4, we shall discuss the “price” to be paid to arrive at our explanation of dBB theory in that some kind of nonlocality, or a certain “systemic nonlocality”, has to be admitted in the model from the start. However, the simplicity and elegance of our derived formalism, combined with arguments about the reasonableness of a corresponding hydrodynamic-like modeling, will show that our approach may be a viable one w.r.t. understanding the emergence of quantum phenomena from the interactions and contextualities provided by the combined levels of classical boundary conditions and those of a subquantum domain. 2 Identity of the emergent kinematics of $N$ bouncers in real three-dimensional space with the configuration-space version of deBroglie–Bohm theory for $N$ particles Consider one particle in an $n$-slit system. In quantum mechanics, as well as in our quantum-like modeling via an emergent quantum mechanics approach, one can write down a formula for the total intensity distribution $P$ which is very similar to the classical formula. For the general case of $n$ slits, it holds with phase differences $\varphi _{ij}=\varphi _{i}-\varphi _{j}$ that $$\begin{aligned} P=\sum _{i=1}^{n}\left( P_{i}+\sum _{j=i+1}^{n}2R_{i}R_{j}\cos \varphi _{ij}\right) , \end{aligned}$$ where the phase differences are defined over the whole domain of the experimental setup. Apart from the role of the relative phase with important implications for the discussions on nonlocality [14], there is one additional ingredient that distinguishes (1) from its classical counterpart, namely the “dispersion of the wavepacket”. As in our model the “particle” is actually a bouncer in a fluctuating wave-like environment, i.e., analogously to the bouncers of Couder and Fort’s group, one does have some (e.g., Gaussian) distribution, with its center following the Ehrenfest trajectory in the free case, but one also has a diffusion to the right and to the left of the mean path which is just due to that stochastic bouncing. Thus the total velocity field of our bouncer in its fluctuating environment is given by the sum of the forward velocity $\mathbf {v}$ and the respective diffusive velocities $\mathbf {u}_{\mathrm {L}}$ and $\mathbf {u}_{\mathrm {R}}$ to the left and the right. As for any direction $i$ the diffusion velocity $\mathbf {u}_{{i}}=D\frac{\nabla _{i}P}{P}$ does not necessarily fall off with the distance, one has long effective tails of the distributions which contribute to the nonlocal nature of the interference phenomena [14]. In sum, one has three, distinct velocity (or current) channels per slit in an $n$-slit system. We have previously shown [7, 15] how one can derive the Bohmian guidance formula from our bouncer/walker model. To recapitulate, we recall the basics of that derivation here. Introducing classical wave amplitudes $R(\mathbf {w}_{i})$ and generalized velocity field vectors $\mathbf {w}_{i}$, which stand for either a forward velocity $\mathbf {v}_{i}$ or a diffusive velocity $\mathbf {u}_{i}$ in the direction transversal to $\mathbf {v}_{i}$, we account for the phase-dependent amplitude contributions of the total system’s wave field projected on one channel’s amplitude $R(\mathbf {w}_{i})$ at the point $(\mathbf {x},t)$ in the following way: We define a conditional probability density $P(\mathbf {w}_{i})$ as the local wave intensity $P(\mathbf {w}_{i})$ in one channel (i.e., $\mathbf {w}_{i}$) upon the condition that the totality of the superposing waves is given by the “rest” of the $3n-1$ channels (recalling that there are three velocity channels per slit). The expression for $P(\mathbf {w}_{i})$ represents what we have termed “relational causality”: any change in the local intensity affects the total field, and vice versa, any change in the total field affects the local one. In an $n$-slit system, we thus obtain for the conditional probability densities and the corresponding currents, respectively, i.e., for each channel component $ i $, $$\begin{aligned} P(\mathbf {w}_{i})&= R(\mathbf {w}_{i})\hat{\mathbf {w}}_{i}\cdot {\displaystyle \sum _{j=1}^{3n}}\hat{\mathbf {w}}_{j}R(\mathbf {w}_{j})\end{aligned}$$ $$\begin{aligned} \mathbf {J}\mathrm {(}\mathbf {w}_{i}\mathrm {)}&= \mathbf {w}_{i}P(\mathbf {w}_{i}),\qquad i=1,\ldots ,3n, \end{aligned}$$ $$\begin{aligned} \cos \varphi _{i,j}:=\hat{\mathbf {w}}_{i}\cdot \hat{\mathbf {w}}_{j}. \end{aligned}$$ Consequently, the total intensity and current of our field read as $$\begin{aligned} P_{\mathrm {tot}}=&{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})=\left( {\displaystyle \sum _{i=1}^{3n}}\hat{\mathbf {w}}_{i}R(\mathbf {w}_{i})\right) ^{2}\end{aligned}$$ $$\begin{aligned} \mathbf {J}_{\mathrm {tot}}=&\sum _{i=1}^{3n}\mathbf {J}(\mathbf {w}_{i})={\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i}), \end{aligned}$$ leading to the emergent total velocity $$\begin{aligned} \mathbf {v}_{\mathrm {tot}}=\frac{\mathbf {J}_{\mathrm {tot}}}{P_{\mathrm {tot}}}=\frac{{\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})}{{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})}\,. \end{aligned}$$ In [7, 15], we have shown with the example of $n=2,$ i.e., a double-slit system, that Eq. (7) can equivalently be written in the form $$\begin{aligned} \mathbf {v}_{\mathrm {tot}}=\frac{R_{1}^{2}\mathbf {v}_{\mathrm {1}}+R_{2}^{2}\mathbf {v}_{\mathrm {2}}+R_{1}R_{2}\left( \mathbf {v}_{\mathrm {1}}+\mathbf {v}_{2}\right) \cos \varphi +R_{1}R_{2}\left( \mathbf {u}_{1}-\mathbf {u}_{2}\right) \sin \varphi }{R_{1}^{2}+R_{2}^{2}+2R_{1}R_{2}\cos \varphi }\,. \end{aligned}$$ The trajectories or streamlines, respectively, are obtained according to $\dot{{\mathbf {x}}}=\mathbf {v}_{\mathrm {tot}}$ in the usual way by integration. As first shown in [13], by re-inserting the expressions for convective and diffusive velocities, respectively, i.e., $\mathbf {v}_{i}=\frac{\nabla S_{i}}{m}$, $\mathbf {u}_{i}=-\frac{\hbar }{m}$ $\frac{\nabla R_{i}}{R_{i}}$, one immediately identifies Eq. (8) with the Bohmian guidance formula. Naturally, employing the Madelung transformation for each path $j$ ($j=1$ or $2$), $$\begin{aligned} \psi _{j}=R_{j}\mathrm{e}^{{i}S_{j}/\hbar }, \end{aligned}$$ and thus $P_{j}=R_{j}^{2}=\|\psi _{j}\|^{2}=\psi _{j}^{}\psi _{j}$, with $\varphi =(S_{1}-S_{2})/\hbar $, and recalling the usual trigonometric identities such as $\cos \varphi =\frac{1}{2}\left( \mathrm{e}^{{i}\varphi }+\mathrm{e}^{-{i}\varphi }\right) $, one can rewrite the total average current immediately in the usual quantum mechanical form as $$\begin{aligned} \begin{array}{ll} {\displaystyle \mathbf {J}_{\mathrm{tot}}} &{} =P_{\mathrm{tot}}\mathbf {v}_{\mathrm{tot}}\\ &{} ={\displaystyle (\psi _{1}+\psi _{2})^{}(\psi _{1}+\psi _{2})\frac{1}{2}\left[ \frac{1}{m}\left( -{i}\hbar \frac{\nabla (\psi _{1}+\psi _{2})}{(\psi _{1}+\psi _{2})}\right) +\frac{1}{m}\left( {i}\hbar \frac{\nabla (\psi _{1}+\psi _{2})^{}}{(\psi _{1}+\psi _{2})^{}}\right) \right] }\\ &{} ={\displaystyle -\frac{{i}\hbar }{2m}\left[ \Psi ^{}\nabla \Psi -\Psi \nabla \Psi ^{}\right] ={\displaystyle \frac{1}{m}{Re}\left\{ \Psi ^{}(-{i}\hbar \nabla )\Psi \right\} ,}} \end{array} \end{aligned}$$ where $P_{\mathrm{tot}}=\|\psi _{1}+\psi _{2}\|^{2}=:\|\Psi \|^{2}$. Equation (7) has been derived for one particle in an $n$-slit system. However, it is straightforward to extend this derivation to the many-particle case. Due to the purely additive terms in the expressions for the total current and total probability density, respectively, also for N particles, the only difference now is that the currents’ nabla operators have to be applied at all of the locations of the respective N particles, thus providing the quantum mechanical formula $$\begin{aligned} {\displaystyle \mathbf {J}_{\mathrm{tot}}}\left( N\right) ={\displaystyle \sum _{i=1}^{N}}\frac{1}{m_{i}}{Re}\left\{ \Psi \left( t\right) ^{}(-{i}\hbar \nabla _{i})\Psi \left( t\right) \right\} , \end{aligned}$$ where $\Psi \left( t\right) $ now is the total $N$-particle wave function, whereas the total velocity fields are given by $$\begin{aligned} \mathbf {v}_{i}\left( t\right) =\frac{\hbar }{m_{i}}\mathrm {Im}\frac{\nabla _{i}\Psi \left( t\right) }{\Psi \left( t\right) }\;\forall i=1,\ldots ,N. \end{aligned}$$ Note that this result is similar in spirit to that of Norsen et al. [17, 18] who with the introduction of a conditional wave function $\tilde{\psi }_{i}$, as opposed to the configuration-space wave function $\Psi $, rewrite the guidance formula, for each particle, in terms of the $\tilde{\psi }_{i}$: $$\begin{aligned} \frac{\,\mathrm {d}X_{i}\left( t\right) }{\,\mathrm {d}t}=\frac{\hbar }{m_{i}}\mathrm {Im}\left. \frac{\nabla \Psi }{\Psi }\right\| _{\mathbf {\varvec{x}=\varvec{X}\left( t\right) }}\equiv \frac{\hbar }{m_{i}}\mathrm {Im}\left. \frac{\nabla \tilde{\psi }_{i}}{\tilde{\psi }_{i}}\right\| _{x=X_{i}\left( t\right) }, \end{aligned}$$ where the $X_{i}$ denote the location of one specific particle and $\mathbf {X}\left( t\right) =\left\{ X_{1}\left( t\right) ,\ldots ,X_{N}\left( t\right) \right\} $ the actual configuration point. Thus, in this approach, each $\tilde{\psi }_{i}$ can be regarded as a wave propagating in physical three-dimensional space. In sum, with our introduction of a conditional probability $P(\mathbf {w}_{i})$ for channels $\mathbf {w}_{i}$, which include subquantum velocity fields, we obtain the guidance formula also for $N$-particle systems. Therefore, what looks like the necessity in the dBB theory to superpose wave functions in configuration space in order to provide an “indigestible” guiding wave, can equally be obtained by superpositions of all relational amplitude configurations of waves in real three-dimensional space. The central ingredient for this to be possible is to consider the emergence of the velocity field from the interplay of the totality of all of the system’s velocity channels. We have termed the framework of our approach a “superclassical” one, because in it are combined classical levels at vastly different scales, i.e., at the subquantum and the macroscopic levels, respectively. 3 Identity of the emergent force on a particle modeled by a bouncer system and the quantum force of the deBroglie–Bohm theory With the results of the foregoing Chapter, we can now return to and resolve the problem discussed in Sect. 1 of the apparent incompatibility between the Bohmian force upon a quantum particle and the force exerted on a bouncing droplet as formulated by Richardson et al. [20]. In fact, already a first look at the bouncer/walker model of our group provides a clear difference as compared to the hydrodynamical force studied by Richardson et al. Whereas the latter investigates the effects of essentially a single bounce on the fluid surface and the acceleration of the bouncer as a consequence of this interaction, our bouncer/walker model for quantum particles involves a much more complex dynamical scenario: we consider the effects of a huge number of bounces, i.e., typically of the order of $1/{\omega }$, like approximately $10^{21}$ bounces per second of an electron, which constitute effectively a “heating up” of the bouncer’s surrounding, i.e., the subquantum medium related to the zero-point energy field. Note that as soon as a microdynamics is assumed, the development of heat fluxes is a logical necessity if the microdynamics is constrained by some macroscopic boundaries like that of a slit system, for example. As we have shown in some detail [12], the thermal field created by such a huge number of bounces in a slit system leads to an emergent average behavior of particle trajectories which is identified as anomalous, and more specifically as ballistic, diffusion. As such, the particle trajectories exiting from, say, a Gaussian slit behave exactly as if they were subject to a Bohmian quantum force. We were also able to show that this applies also to $n$-slit systems, such that one arrives at a subquantum modeling of the emergent interference effects at $n$ slits whose predicted average behavior is identical to that provided by the dBB theory. It is then easily shown that the average force acting on a particle in our model is the same as the Bohmian quantum force. Due to the identity of our emerging velocity field with the guidance formula, and because they essentially differ only via the notations due to different forms of bookkeeping, their respective time derivatives must also be identical. Thus, from Eq. (7), one obtains the particle acceleration field (using a one-particle scenario for simplicity) in an $n$-slit system as $$\begin{aligned} a_{\mathrm {tot}}\left( t\right)&= \frac{\,\mathrm {d}\mathbf {v}_{\mathrm{tot}}}{\,\mathrm {d}t}=\frac{\,\mathrm {d}}{\,\mathrm {d}t}\left( \frac{{\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})}{{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})}\right) \nonumber \\&= \frac{1}{\left( {\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})\right) ^{2}}\left\{ \sum _{i=1}^{3n}\left[ P(\mathbf {w}_{i})\frac{\,\mathrm {d}\mathbf {w}_{i}}{\,\mathrm {d}t}+\mathbf {w}_{i}\frac{\,\mathrm {d}P(\mathbf {w}_{i})}{\,\mathrm {d}t}\right] \left( {\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})\right) \,-\left( {\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})\right) \left( {\displaystyle \sum _{i=1}^{3n}\frac{\,\mathrm {d}P(\mathbf {w}_{i})}{\,\mathrm {d}t}}\right) \right\} .\nonumber \\ \end{aligned}$$ Note in particular that (14) typically becomes infinite for regions $\left( \mathbf {x},t\right) $ where $P_{\mathrm {tot}}={\sum _{i=1}^{3n}}P(\mathbf {w}_{i})\rightarrow 0$, in accordance with the Bohmian picture. From (14), we see that even the acceleration of one particle in an $n$-slit system is a highly complex affair, as it nonlocally depends on all other accelerations and temporal changes in the probability densities across the whole experimental setup! In other words, this force is truly emergent, resulting from a huge amount of bouncer–medium interactions, both locally and nonlocally. This is of course radically different from the scenario studied by Richardson et al. where the effect of only a single local bounce is compared with the quantum force. From our new perspective, it is then hardly a surprise that the comparison of the two respective forces provides distinctive differences. However, as we just showed, with the emergent scenario proposed in our model, complete agreement with the Bohmian quantum force is established. 4 Choose your poison: how to introduce nonlocality in a hydrodynamic-like model for quantum systems? As already mentioned in the introduction of this paper, purely classical hydrodynamical models are manifestly local and thus inadequate tools to explain quantum mechanical nonlocality. Although nonlocal correlations may also be obtainable within hydrodynamical modeling [2], there is no way to also account for dynamical nonlocality [21] in this manner. So, as correctly observed by Richardson et al. [20], droplet–surface wave interaction scenarios are not enough to serve as a full-fledged analogy of the distinctly nonlocal dBB theory, for example. The question thus arises how in our much more complex, but still “hydrodynamic-like” bouncer/walker model nonlocal, or nonlocal-like, effects can come about. To answer this, one needs to consider in more detail how the elements of our model are constructed, which finally provide an elegant formula, Eq. (7), identical with the guidance formula in a (for simplicity: one-particle) system with $n$ slits. (As shown above, the extension to $N$ particles is straightforward.) As we consider, without restriction of generality, the typical example of Gaussian slits, we introduce the Gaussians in the usual way, with $\sigma $ related to the slit width, for the probability density distributions (which in our model coincide with “heat” distributions due to the bouncers’ stirring up of the vacuum) just behind the slit. The important feature of these Gaussians is that we do not implement any cutoff for the distributions, but maintain the long tails which actually then extend across the whole experimental setup, even if these are only very small and practically often negligible amplitudes in the regions far away from the slit proper. As the emerging probability density current is given by the denominator of Eq. (8), we see that in fact the product $R_{1}R_{2}$ may be negligibly small for regions where only a long tail of one Gaussian overlaps with another Gaussian, nevertheless the last term in (8) can be very large despite the smallness of $R_{1}$ or $R_{2}$. It is this latter part which is responsible for the genuinely quantum-like nature of the average momentum, i.e., for its nonlocal nature. This is similar in the Bohmian picture, but here given a more direct physical meaning in that this last term refers to a difference in diffusive currents as explicitly formulated in the last term of (8). Because of the mixing of diffusion currents from both channels, we call this decisive term in $\mathbf {J_{\mathrm { \mathrm {tot} }}=P_{\mathrm{tot}}\mathbf {v}_{\mathrm{tot}}}$ the “entangling current” [16]. Thus, one sees that formally one obtains genuine quantum mechanical nonlocality in a hydrodynamic-like model with one particular “unusual” characteristic: the extremely feeble but long tails of (Gaussian or other) distribution functions for probability densities exiting from a slit extend nonlocally across the whole experimental setup. So, we have nonlocality by explicitly putting it into our model. After all, if the world is nonlocal, it would not make much sense to attempt its reconstruction with purely local means. Still, so far we have just stated a formal reason for how nonlocality may come about. Somewhere in any theory, so it seems, one has to “choose one’s poison” that would provide nonlocality in the end. But what would be a truly “digestible” physical explanation? Here is where at present only some speculative clues can be given. For one thing, strict nonlocality in the sense of absolute simultaneity of space-like separated events can never be proven in any realistic experiment, because infinite precision is not attainable. This means, however, that very short time lapses must be admitted in any operational scenario, with two basic options remaining: (1) either there is a time lapse due to the finitely fast “switching” of experimental arrangements in combination with instantaneous information transfer [but not signaling; see Walleczek and Grössing (forthcoming)], or (2) the information transfer itself is not instantaneous, but happens at very high speeds $v\ggg c$. How, then, can the implementation of nonlocal or nonlocal-like processes with speeds $v\ggg c$ be argued for in the context of a hydrodynamic-like bouncer/walker model? We briefly mention two options here. Firstly, one can imagine that the “medium” we employ in our model is characterized by oscillations of the zero-point energy throughout space, i.e., between any two or more macroscopic boundaries as given by experimental setups. Between these boundaries standing wave configurations may emerge (similar to the Paris group’s experiments, but now explicitly extending synchronously over nonlocal distances). Here it might be helpful to remind ourselves that we deal with solutions of the diffusion (heat conduction) equation. At least (but perhaps only) formally, any change of the boundary conditions is effective “instantaneously” across the whole setup. Alternatively, if the experimental setup is changed such that old boundary conditions are substituted by new ones, due to the all-space-pervading zero-point energy oscillations, one “immediately” (i.e., after a very short time of the order $t\sim \frac{1}{\omega }$) obtains a new standing wave configuration that now effectively implies an almost instantaneous change of probability density distributions, or relative phase changes, for example. The latter would then become “immediately” effective in that changed phase information is available across the whole domain of the extended probability density distribution. We have referred to this state of affairs as “systemic nonlocality” [14]. So, one may speculate that it is something like “eigenvalues” of the universe’s network of zero-point fluctuations that may be responsible for quantum mechanical nonlocality–eigenvalues which (almost?) instantaneously change whenever the boundary conditions are changed. A second option even more explicitly refers to the universe as a whole, or, more particularly, to spacetime itself. If spacetime is an emergent phenomenon as some recent work suggests [19], this would very likely have strong implications for the modeling and understanding of quantum phenomena. Just as in our model of an emergent quantum mechanics we consider quantum theory as a possible limiting case of a deeper level theory, present-day relativity and concepts of spacetime may be approximations of, and emergent from a superclassical, deeper level theory of gravity and/or spacetime. It is thus a potentially fruitful task to bring both attempts together in the near future. We thank Jan Walleczek for many enlightening discussions, and the Fetzer Franklin Fund for partial support of the current work. 1. 1. Bohm, D.: Causality and Chance in Modern Physics. Routledge, London (1997)Google Scholar 2. 2. 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61daf397b45a0bd6	Author Archives: Admin Solving the Constitutional Paradox The Constitutional Paradox and superposition, in this essay is seen as a kind of legal purgatory. However, the two row wampum allows for us to solve the Superposition and break free from the Constitutional Paradox, the two row wampum itself is a tool for quantum positioning, to discover the true position of the Kanienkahaka, solving the superposition the box is open and the constitution of the Kanienkehaka is realized. Like the Cat in Schrödinger’s equation, the ‘Indian’ is seeing life or death(from inside the box), and inside with him is the Great Law, the observer (outside the box) can only guess the status of the Indian; death or life, assimilation or independant. The situation for the Kanienkhaka is similar, but the assumption is that without opening the box, the Canadian courts and all concerned have plausible deniability, so the dilemma is how to get the Canadian people to open the box themselves to reveal what constitution the Haudnanshuanee live by, are we alive or dead, and whether the Wampum laws have been broken. How does the cat get the observer to open the box, or Let the cat out of the box. The Host and guest-friend relationship that was once cherished by all who enjoyed the bounty of the lands and resources, but until the Box is re-opened or the wampum are polished, and the truth is then set free about The Great Law of Peace and Wampum laws, the original Hosts will remain hostage to the guests who have become an enemy to the peace. Constitutional Paradox: Schrödinger’s Quantum Theory on Superposition A constitution is a set of fundamental principles or established precedents according to which a state or other organization is governed, written or unwritten, that establishes the character of a government by defining the basic principles to which a society must conform; by describing the organization of the government and regulation, distribution, and limitations on the functions of different government departments; and by prescribing the extent and manner of the exercise of its sovereign powers. A Quantum superposition is a fundamental principle of quantum mechanics. It refers to a property of pure state solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it. Canadian Constitution or Wampum Law Our Guests: Homer of Brothers in Christ In ancient Greece and Rome, the concept of hospitality was considered to be a guest’s divine right and the host’s divine duty. Other cultures also followed such hospitality relationships though they referred to these relationships by other names. During the time of Homer, strangers (without any exception) were protected by Zeus Xenios who was the god of strangers and suppliants. Strangers had the right to be treated with respect and honor. As soon as a guest entered the home of a Greek host, he or she would be clothed as well as entertained and no questions would be asked of them regarding their antecedents and name. Only after all hospitality duties were completed would the guest be questioned. When the guest was ready to leave, they would be given a parting gift. This helped to establish a family connection and the gift (normally, a die) would serve as recognition of the fact that the host would protect the guest if the latter ever required protection. Those who violated these hospitality relationships would have to suffer the wrath of the gods. In ancient Rome, private hospitality was well defined in both legal and other terms. The relationship between guest and host was almost the same as that of a client and patron. When a guest and host clasped hands, a strong relationship was established between them and a written agreement would also be exchanged by them. Xenia is the Greek concept of hospitality in which guests who were far from their homes were to be treated with generosity as well as courtesy by their hosts. The hospitality relationship created between the two existed at two levels. The first level involved material benefits and the second level involved non-material benefits. At the material level the host gave gifts to the guest while at the non-material level, the host would provide protection and shower favors as well as give shelter to the guest. In Greek, the word Xenos implies a stranger though this term can be interpreted in different ways. In 1215 King John ceded England and Ireland to the Roman Pope and continues to pay tribute, Canada also uses the papal bulls as a source of authority. This however makes the Kings and Queen of Britain agents for the Pope, carrying the papal offer to be our father, and we the creatures or sons of the Pope. The Two Row Wampum treaty is an agreement between the Iroquois’s Five Nations and representatives of the government of Holland. This treaty was signed in 1613 and the agreement was recorded in a wampum belt called the Two Row Wampum. The meaning of the belt is, “You say that you are our Father and I am your Son. We say ‘We will not be like Father and Son, but like Brothers”. Papal Bulls of the fifteenth century gave license to Spanish and Portuguese kings to usurp lands and enslave non-Christian populations. The possessions and resources of people who were enslaved according to the papal bulls would then be expropriated by the kings of Spain and Portugal. Lately, however, there is a move underfoot to revoke as well as denounce these documents. If we trust the Two Row Wampum treaty, then we find that both parties to the agreement are to be treated as Brothers and not as Father and Son. Hence, this treaty may be treated as a treaty that voids the papal bulls. Additional Info: Ξ, ξ (XI) - The Greek letter XI (pronounced iksee) sounds like the x in axe or box. xenia (feminine) and xenios (masculine) - (Greek: ξενία, xenía) The simple definition of xenía would be hospitality. Xenía is often defined as the guest/host relationship whereby hospitality is a religious duty protected by Zeus Xeneos. It is the reciprocal relationship between two xenoi. Xenoi is plural, xenos is singular. A xenos is defined in five ways: guest, host, stranger, foreigner, and friend. If you try to imagine a traveler approaching a home in a society that did not have hotels, you can see where these definitions come from. In its most basic meaning, xenía requires that we do not turn our back on one who is in need and in a vulnerable position. The individual who receives hospitality is then bound to the host in a reciprocal relationship. "For Homer says that all the Gods, and especially the God of strangers (ed. Zeus), are companions of the meek and just, and visit the good and evil among men." (Plato's Sophist, 216, translated by Benjamin Jowett, 1892; found in the 1937 Random House edition of The Dialogues of Plato, Volume II, p.221) Xóanon - (Gr. Ξόανον, ΞΟΑΝΟΝ. Plural: Ξόανα) The Xóanon is an archaic wooden statue of a God, rarely showing precise features, often believed to have fallen from heaven. There is the famous story of the Xóanon of Ártæmis Orthía, (Gr. Ἄρτεμις Ὀρθία), stolen from Taurikí (Taurica or Tauride; Gr. Ταυρικὴ) by Orǽstis (Orestes; Gr. Ὀρέστης) and his sister Iphiyǽneia (Iphigenia; Gr. Ἰφιγένεια). There are no original Xóana that have survived from antiquity. We know what they looked like because copies were created in stone which are extant. These copies were made when a new colony was created; they were sent off with the new colonists. Land Claims and the Power of Peace & War Land claims are a legal declaration of desired control over areas of property including bodies of water. The phrase is usually used only with respect to disputed or unresolved land claims. Some types of land claims include aboriginal (the term “aboriginal” is a creation of Canadian constitutional imposition, and oftentimes falsely applied, therefore most if not all claims from this office is relabeled and repackaged Canadian land claim) land claims, Antarctic land claims, and post-colonial land claims. This of course is a colonial concept of ownership propagated by the papal bulls of the Vatican, allowing Christians to stake claims on foreign lands and rid the lands of conflicting authorities. Romanus Pontifex, January 8, 1455 — …We bestow suitable favors and special graces on those Catholic kings and princes, …athletes and intrepid champions of the Christian faith… to invade, search out, capture, vanquish, and subdue all Saracens and pagans whatsoever, and other enemies of Christ wheresoever placed, and… to reduce their persons to perpetual slavery, and to apply and appropriate… possessions, and goods, and to convert them to… their use and profit The papal bull was a foreign claim on lands not within their realm, so this is the root of land claims of today and unlawful occupation on north America. Another Papal bull Unam Sanctum 18 November 1302 states that: The Roman pope wanted to be the father of all creatures (sons) of the earth. However when these ever-reaching offers made it to the Onkwehonwe they said in counter No, we shall not be like father and creature/son, but we shall be like brothers, this is recorded and confirmed in the Two Row Wampum. In pre-colonial history the six nations as it where, knew and had a shared concept of territory and war, to encroach onto a territory meant to risk certain death, however through Confederation of the six nations and territories, the end of land claims brought about the end of war. Sken:non kowa ken? (mohawk language) this means “is there still the great peace”, an unaffirmable question and greeting meant as a reminder to the peace between the people of the league of nations. To make a land claim is the act of waging war against the people that live on the land and all those who are outside of the claim. When six nations (53 nations/tributaries) buried the hatchet between the league of nations and uniting the territories they had ended war against one another. More info about the commonwealth can be gained by studying the Dish with Spoon wampum. Integrity & Hospitality Agreements The Two Row Wampum Treaty Belt: A fundamental belief of Onkwehonwe is coexistence. This is demonstrated in wampum belts, treaties or agreements drawn between two or more parties. The Two Row Wampum belt gives an accurate portrayal of what it means to coexist with nature. It comes from the Haudenosaunee peoples and is considered the Grandfather of all belts because there is no end to it. The agreement arose out of concern for the sustainability of Mother Earth. The two purple lines represent the separate and distinct paths of North America’s First Peoples and the settler society, each with their own culture to maintain while traveling along the same river of life. The three white lines represent the River of Life or the shared territory. Each nation is to keep their separate and distinct cultures, while working together to maintain the lands they share and the earth that sustains all. Biodiversity is a modern term for the same principles of coexistence found in the Two Row Wampum belt agreement. The Two Row applies to all relationships. Male and female. Humans and Natural World. Spirit and body. Families to other families. Peoples to other Peoples. In the case of Europeans, the Two Row symbolized mutual aid and defense based on Friendship. International Treaties: Quite often we hear about the Guswentah or Tekeni Teiohate more commonly referred to as The Two Row Wampum Treaty. This treaty made between the Nation of Holland and the Five Nations may have been the first treaty between Onkwehonwe and a European nation, but it was certainly not the first treaty that the Onkwehonwe had ever entered into, in fact Onkwehonwe nations have been engaged in the treaty making process between them for centuries. The formation of the Kaianerekowa (Great Law of Peace) is a fine example of possibly one of the most highly advanced Peace treaties ever negotiated between sovereign nations. The agreements reached between each of the Five Nations in order to put an end to the conflict that they had been embroiled in, as well as enable each nation to retain their sovereignty and jurisdiction is beyond anything seen in the history of mankind. The first 12 Wampum’s of the Kaianerekowa lays out the procedures and protocol for each nation to follow in order to resolve any issues that may threaten one or all of the Five Nations. These 12 articles also guarantees the jurisdiction and sovereignty of each individual nation so that each nation shall have a forum to voice their position. Visit to review the 117 Wampums of the Great Law of Peace and commentary. Doctrine of Discovery: Offer & Counter offer King John Concession of 1213 of England and Ireland to the Pope. Unam Sanctam Bull of 1302, Boniface VIII proclaimed that it “is absolutely necessary for salvation that every human creature be subject (son) to the Roman pontiff (father)”. Dum Diversas Bull of 1452, Pope Nicholas V, It authorized Afonso V of Portugal to conquer Saracens and pagans and consign them to “perpetual slavery.”Pope Calixtus III reiterated the bull in 1456 with Etsi cuncti, renewed by Pope Sixtus IV in 1481 and Pope Leo X in 1514 with Precelse denotionis. Romanus Pontifex Bull of 1455 has served as the basis of legal arguments for taking Native American lands by “discovery”. The logic of the rights of conquest and discovery were followed in all western nations including those that never recognised papal authority. These offers being carried to the new world (North America c. 1492 (A’nowara:kowa (Great Turtle Island))) where the Onkwehonwe responded to these offers by conditionally accepting the Bulls by issuance of the Kuswhenta (Two Row Wampum ( = ), a perpetual reservation of sovereignty, interest and law on A’nowara:kowa), a counter offer to not be like father and son but we shall be like brothers: “You say that you are our Father and I am your Son We will not be like Father and Son, but like Brothers. This wampum belt confirms our words. Neither of us will make compulsory laws or interfere in the internal affairs of the other. Neither of us will try to steer the other’s vessel/vassal. Never to outpace the other, for as long as the sun shines and the grass grows” Wampum 57: Legislative Designation Its important to understand what is being said here. If we do not understand what makes a strong Nation of people, we cannot begin to understand this part of what is being said. The people as a whole need to recognize their true place within the Kaianerekowa. This is where there is strength and Peace, righteousness and power. Divisions in every community prevent the full realization of the power of the people. If this is so then there is no representation of the people. Any future that must take place for the people must include all people within the understanding of their place within the Kaianerekowa. Understanding ones place in Creation, and with those who provide life, has fallen to the wayside and replaced with individualism, conditioning to that individualism and peoples need to place themselves above another. As one arrow by itself can be snapped as a twig, a bunch tied together cannot be easily broken. Individualism has existed because the arrow has fallen from its place and has now broken in a dozen or more pieces living all of us without strength and power that we should have within our place that has been established for us in the Kaianerekowa. Individualism is detrimental to the great peace as long as it is implemented in a selfish manner that does not recognize the true value of individualism is in the context of people being strong in themselves to be of excellent loyalty, strength and service to the whole. Five weak and poorly made arrows are just as weak as the one. In encompasses our notion of creation, which in itself lacks individualism. When we use our tobacco to talk with Sonkwaiatison, we are just one person, we are not whole and so, because we are not whole, we ask all onkwesonha to join with us, we ask the water, we ask the roots and green plant life etc, we then become one and whole. Within this way we have planted Tsioneratasekowa, as being the only way to truly be one, all the people are also of one mind and therefore contributing to the Original agreement. Individualism, as in me myself and I, is European in nature, as they have forgotten their original teachings. This is what I was referring to, and what has left those arrows loose and in pieces. There is not one of us that has lack of knowledge of something, we do not share this knowledge, for the most part because we do not deem the other worthy enough to know it. When we can walk outside and stare up at the stars and know something, and know that life is never dependent upon a handful of people but to those who had created it. Wampum History In Brief Atl Law is an ancient oral equality system of law and language emerging from the Mesolithic Period (25,000 to 9,500 BCE) around the regions of Mexico, Central Americas, and the northern half of South America. Origins of Atl Law Atl law is named after the Atl indigenous of the Andes (Antis) Mountains and northern half of South America, otherwise known at the Atlanteans who believed their laws were passed down directly from flesh and blood higher order beings. Atl Law evolved into the foundation of the laws of MesoAmerican Civilizations (Olmec, Zapotex, Aztec and Maya), Andean Civilizations (Inca, Moche, Chibcha and Canaris) and the Great Plains Civilizations of North America such as Wampum Law. Atl Law and Roman Law As the Roman Cult is an imposter system founded by fraud in the 11th Century with finance from Venice and never was the founders of the Catholic Church nor Christian Faith, all law based on the Roman Cult including Feudal Law, Common Law, Civil Law and International Law is null and void from the beginning for all the lands and seas of North America, Central America and South America. As Atl law was never legitimately replaced, nor the people of North America, Central America or South America lawfully conquered within the physical realm, the law of the land has remained unbroken the Atl Law of the indigenous nations. As Wampum Law descends from Atl Law and incorporates the laws and knowledge of its common ancestry with the peoples of Central America and South America, Wampum Law remains the unbroken legitimate system of law of the land of North America. Haudenosaunee: Civil History in Brief Established in either 1142 or 1451, the Five Nations Iroquois confederacy consisted of the Mohawks, the Oneidas, the Onondagas, the Cayugas, and the Senecas. When the Tuscaroras joined in 1712 the union adapted the name Haudenosaunee, which translates to mean “People building a longhouse”. In treaties and other colonial documents they were known as the “Six Nations.” While each tribe controlled its own domestic affairs, the council at Onondaga controlled matters that referred to the nation as a whole. Similarly, despite the fact that all spoke the same language, each tribe had a distinct dialect of its own. Thus not only did the Iroquois provide a strong government and military base to protect their farmland, they also formed one of the nation’s earliest and strongest diplomacies. In terms of spirituality the Iroquois practiced a religion of love. They believed that the Great Spirit Tarachiawagon, which literally means “Holder of the Heavens”, cared for his people and asked that they care for one another. Furthermore, Tarachiawagon had appointed to each of the Six Nations its own dwelling place, taught them how to use the corn and fruits of the earth, and could be approached by way of the woods. Their religion also contributed to their deep sense of brotherhood. Social grades did not exist because the tribe shared everything. Leaders were respected, but considered equals with their lowest members. Words for “your highness”, “your majesty” and “your excellency”, were nonexistent; the English governor was called “Brother” and Shikellamy, the “great pro-council at Shamokin”, died in rags. This sense of brotherhood examplefies further that in their minds the true strength of the Iroquois was not exhibited through military victories, but rather through the large number of allies they had. The Origin of Man peace[1]In the distant past, all the earth was covered by deep water, and the only living things there were water animals. There was no sun, moon, or stars, and the watery earth was in darkness. People lived above the great sky dome. A tree of life grew there in the cloud world. where it shaded the councils of the supernaturals. One day the Great Chief became ill, and he dreamed that if the tree were uprooted he would be cured. He further commanded that his pregnant daughter, Sky Woman, look down at the watery darkness. He told her to follow the roots of the tree, and to bring light and land to the world below. The animals of the sea saw Sky Woman as she fell from the sky world. Waterfowl rose to cushion Sky Woman’s descent with their wings. Beaver dove to find earth to make dry land for Sky Woman. But Beaver drowned and floated lifelessly to the surface. Loon, Duck, and others all tried and failed as well. Finally. Muskrat tried, and came back with a paw-full of dirt that would spread and grow. He placed the dirt on Turtle’s back where Sky Woman landed. The dirt on Turtle’s back grew and became the earth. Time passed and Sky Woman gave birth to a daughter. The daughter grew rapidly, and when she reached maturity she was visited by a man. He placed two arrows within her, one tipped with chert and the other not. The daughter in turn bore twins. The left-handed twin was “Sawiskera” (Mischevious One) and the right handed one was known as “Teharonhiawako” (Holder of the Heavens). The left handed twin forced himself out through his mother’s armpit, killing her in the process. Corn, beans, squash, and tobacco grew from her body and she became one with the earth. Teharonhiakwako created animals, medicine and flowers while Sawiskera created the thorns on the rose bush and the mountain lion to kill the deer his brother created. Teharonhiawako was the more righteous of the two. Sawiskera had a great capacity for evil and was deceitful enough to convince his grandmother that he was really the righteous one. When their grandmother died, the twins could not agree on what to do with her body. Sawiskera just wanted to discard it, but Teharonhiawako had other plans; he honoured his grandmother by placing her up in the night sky; this is how the Moon came to be. After much fighting the brothers decided to divide the world in half and the nighttime would belong to Sawiskera and Teharionhiawako would get the daytime. The Onkwehonwe (Original People) were created by Teharionhiawako out of red earth and were to watch over his creations on Earth. Black soil, tree bark, and salt water were used to create other beings. Teharionhiawako told the beings that he was to be called “Sonkwaiatison” (The Creator) and to be respectful of one another and all living creatures. He instructed the people to appreciate each other’s differences and to share the world. When Teharonhiawako created all the waters, plants, trees and animals of the world, he decided that he should create a being in his likeness from the natural world. He wanted this being to have a superior mind so it would have the responsibility of looking after his creations. Then he decided it would be better if he created more than one being and give to each similar instructions and see if over a period of time, they would carry them through. The first being Teharonhiawako made was from the bark of a tree; the second from the foam of the great salt water; the third from the black soil, and the fourth from the red earth. All this he did in one day. He started in the early morning as the sun greeted the new day by picking certain types of bark from the tree of life and created a human form, reflected against the sky the form gave a yellowish appearance. Teharonhiawako decided that this would be one type of human that would exist on this world. After Teharonhiawako finished his first human, he then went to the great salt waters and took from the sea some white foam, together with other elements of the natural world he created another being. This being appeared pale in contrast to the natural surroundings, but he was satisfied that he has [sic] created another special kind of human being. Next Teharonhiawako traveled to the thickest part of a large forest and brought out some black soil, again with other elements of the natural world he created another human being. This being was very dark in color and he was pleased that he had created still another type of being for the world. Now Teharonhiawako thought to himself, it is getting towards the end of the day and I have created three beings, since everything on this world exists in cycles of four, I will create one more being. Thus he again looked for something different within the natural world and this time he found some reddish-brown earth. With this he again combined other elements from the land and created a human form. When he finished he observed that this from blending very well with the natural surroundings, especially against the setting sun, which gave the form a reddish color. Teharonhiawako now gathered the four human forms into one area and said to himself, “I have been very careful in providing certain characteristics into each form that will reflect their own unique and strong qualities. I will now give life to each form and see if they benefit from their gifts.” As the beings came to life he observed just how evident their uniqueness became. The white being was the first one to move about, he was also the most curious, observing closely all his surroundings. Next, the black and yellow slowly started to move about. When the black being picked a brightly colored object that he was attracted to, the white being pounced on him and pushed him to the ground, taking over the object. At the same instant, the yellow being stood up for the black and soon, a fight broke out between the three. Teharonhiawako noticed that the fourth being was still sitting on the ground, camouflaged by his surroundings. Now it became clear to Teharonhiawako that there was no way these four could exist in the same environment and survive. Teharonhiawako stopped their quarreling and brought them back to one place and told them “There is a reason why you were not created in the same manner, just as there are birds and animals who look alike, they are different in their ways, so are you. They have their own language, their own songs but have learned to share their world,. It is for this reason that I have created you, that in time you will all learn to respect and appreciate your differences. It is very evident that I can not put you together to watch over my creations, for you would probably destroy it as well as yourselves. You need to learn how to get along with each other, as well as with other living things. I will help you do this, but first I will have to keep you apart. You will come back together after a time when I have sent a messenger to visit each of you and give you a way to be thankful for the good things, as well as respect for other living creatures.” Teharonhiawako then took the white, black and yellow beings across the salt waters and placed them far from each other. The red being he kept at his place of origin. Teharonhiawako told him, “You will be called Onkwehonwe (original being). You will call me Sonkwaiatison (The Creator). I have given you the gift of life. You were created from the earth of this Island. I now realize that you would not survive very long among the others, for you are too much a part of nature, which is good, but you will need time before you come in contact with the other beings. You will also be given a sacred way by a messenger who will visit you and your descendants.” Now Teharonhiawako thought to himself, “They will all have a chance to learn of the reason for their existence and of a good way to live.”
79d7ab850b58e442	Bose–Einstein condensate (Redirected from Bose–Einstein condensation) Schematic Bose–Einstein condensation versus temperature and the energy diagram A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero (-273.15 °C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth the density of normal air, to ultra-low temperatures. This state was first predicted, generally, in 1924–1925 by Satyendra Nath Bose and Albert Einstein. Velocity-distribution data (3 views) for a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate. Satyendra Nath Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924.[1] (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.[2]). Einstein then extended Bose's ideas to matter in two other papers.[3][4] The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now called bosons. Bosons, which include the photon as well as atoms such as helium-4 (4He), are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. In 1938 Fritz London proposed BEC as a mechanism for superfluidity in 4He and superconductivity.[5][6] On June 5, 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NISTJILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins (nK).[7] Shortly thereafter, Wolfgang Ketterle at MIT demonstrated important BEC properties. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics.[8] Many isotopes were soon condensed, then molecules, quasi-particles, and photons in 2010.[9] Critical temperatureEdit This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: is the critical temperature, is the particle density, is the mass per boson, is the reduced Planck constant, is the Boltzmann constant, and is the Riemann zeta function; [10] Interactions shift the value and the corrections can be calculated by mean-field theory. This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. Bose Einstein's non-interacting gasEdit Consider a collection of N non-interacting particles, which can each be in one of two quantum states, and . If the two states are equal in energy, each different configuration is equally likely. If we can tell which particle is which, there are different configurations, since each particle can be in or independently. In almost all of the configurations, about half the particles are in and the other half in . The balance is a statistical effect: the number of configurations is largest when the particles are divided equally. If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state , there are N − K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system. Suppose now that the energy of state is slightly greater than the energy of state by an amount E. At temperature T, a particle will have a lesser probability to be in state by . In the distinguishable case, the particle distribution will be biased slightly towards state . But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most-likely outcome is that most of the particles will collapse into state . In the distinguishable case, for large N, the fraction in state can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails. In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential: For large N, the normalization constant C is (1 − p). The expected total number of particles not in the lowest energy state, in the limit that , is equal to . It does not grow when N is large; it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. Consider now a gas of particles, which can be in different momentum states labeled . If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit, the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, p/(1 − p): When the integral is evaluated with factors of kB and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of negligible chemical potential μ. In Bose–Einstein statistics distribution, μ is actually still nonzero for BECs; however, μ is less than the ground state energy. Except when specifically talking about the ground state, μ can be approximated for most energy or momentum states as μ ≈ 0. Bogoliubov theory for weakly interacting gasEdit Nikolay Bogoliubov considered perturbations on the limit of dilute gas,[11] finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure (T = 0): . The original interacting system can be converted to a system of non-interacting particles with a dispersion law. Gross–Pitaevskii equationEdit In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate . For a system of this nature, is interpreted as the particle density, so the total number of atoms is Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean field theory, the energy (E) associated with the state is: Minimizing this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear Schrödinger equation): is the mass of the bosons, is the external potential, is representative of the inter-particle interactions. In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for ): The Gross-Pitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for . It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature. Numerical SolutionEdit The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and different numerical methods, such as split-step Crank-Nicolson [12] and Fourier spectral [13] methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction [14][15] and long-range dipolar interaction [16] which can be freely used. Weaknesses of Gross–Pitaevskii modelEdit The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BECs. By construction, the GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to self-energy.[17] These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction acquires the terms containing higher-order powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially non-polynomial. The examples where this could happen are the Bose–Fermi composite condensates,[18][19][20][21] effectively lower-dimensional condensates,[22] and dense condensates and superfluid clusters and droplets.[23] However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case. Superfluidity of BEC and Landau criterionEdit The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium-4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model. Experimental observationEdit Superfluid He-4Edit In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly believed that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many properties of superfluid helium also appear in gaseous condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium-4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium-4. Note that helium-3, a fermion, also enters a superfluid phase (at a much lower temperature) which can be explained by the formation of bosonic Cooper pairs of two atoms (see also fermionic condensate). The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium-87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium-23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements.[24] A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work.[25] Lithium has attractive interactions, causing the condensate to be unstable and collapse for all but a few atoms. Hulet's team subsequently showed the condensate could be stabilized by confinement quantum pressure for up to about 1000 atoms. Various isotopes have since been condensed. Velocity-distribution data graphEdit In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: spatially confined atoms have a minimum width velocity distribution. This width is given by the curvature of the magnetic potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein.[26] Bose–Einstein condensation also applies to quasiparticles in solids. Magnons, Excitons, and Polaritons have integer spin which means they are bosons that can form condensates. Magnons, electron spin waves, can be controlled by a magnetic field. Densities from the limit of a dilute gas to a strongly interacting Bose liquid are possible. Magnetic ordering is the analog of superfluidity. In 1999 condensation was demonstrated in antiferromagnetic Tl Cu Cl3,[27] at temperatures as large as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons small mass (near an electron) and greater achievable density. In 2006, condensation in a ferromagnetic yttrium-iron-garnet thin film was seen even at room temperature,[28][29] with optical pumping. Excitons, electron-hole pairs, were predicted to condense at low temperature and high density by Boer et al. in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in sub-kelvin Cu2O in 2005 on. Polariton condensation was firstly detected for exciton-polaritons in a quantum well microcavity kept at 5 K.[30] Peculiar propertiesEdit As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. These phenomena are allowed for by the non-linear term in the GPE.[disputed ] As the vortices must have quantized angular momentum the wavefunction may have the form where and are as in the cylindrical coordinate system, and is the angular number. This is particularly likely for an axially symmetric (for instance, harmonic) confining potential, which is commonly used. The notion is easily generalized. To determine , the energy of must be minimized, according to the constraint . This is usually done computationally, however in a uniform medium the analytic form: , where: is density far from the vortex, is healing length of the condensate. demonstrates the correct behavior, and is a good approximation. A singly charged vortex ( ) is in the ground state, with its energy given by where is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary .) For multiply charged vortices ( ) the energy is approximated by which is greater than that of singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes. Closely related to the creation of vortices in BECs is the generation of so-called dark solitons in one-dimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relatively long-lived dark solitons have been produced and studied extensively.[31] Attractive interactionsEdit Experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist up to a critical atom number. Quench cooling the gas, they observed the condensate to grow, then subsequently collapse as the attraction overwhelmed the zero-point energy of the confining potential, in a burst reminiscent of a supernova, with an explosion preceded by an implosion. Further work on attractive condensates was performed in 2000 by the JILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally attracting atoms of rubidium-85 (having negative atom–atom scattering length). Through Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among wave-like condensate atoms. When the JILA team raised the magnetic field strength further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, then exploded, expelling about two-thirds of its 10,000 atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not seen in the cold remnant or expanding gas cloud.[24] Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean field theories have been proposed to explain it. Most likely they formed molecules of two rubidium atoms;[32] energy gained by this bond imparts velocity sufficient to leave the trap without being detected. The process of creation of molecular Bose condensate during the sweep of the magnetic field throughout the Feshbach resonance, as well as the reverse process, are described by the exactly solvable model that can explain many experimental observations.[33] Current researchEdit Unsolved problem in physics: How do we rigorously prove the existence of Bose–Einstein condensates for general interacting systems? (more unsolved problems in physics) Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.[34] The slightest interaction with the external environment can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas.[citation needed] Nevertheless, they have proven useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an increase in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,[35] the study of superfluidity and quantized vortices, the creation of bright matter wave solitons from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency.[36] Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the laboratory. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator,[37] and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow one-dimensional optical lattice originally observed by Haller\|display-authors=et al[38] has been explored via a tweaking of the primary optical lattice by a secondary weaker one.[39] Thus for a resulting weak bichromatic optical lattice, it has been found that the pinning transition is robust against the introduction of the weaker secondary optical lattice. Studies of vortices in nonuniform Bose–Einstein condensates [40] as well as excitatons of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken.[41][42] Within this context, the conditions for order and chaos in the dynamics of a trapped Bose–Einstein condensate have been explored by the application of moving blue and red-detuned laser beams via the time-dependent Gross-Pitaevskii equation.[43] Bose–Einstein condensates composed of a wide range of isotopes have been produced.[44] Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. molecules or Cooper pairs). The first molecular condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate composed of Cooper pairs.[45] In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second,[clarification needed] using a superfluid.[46] Hau and her associates have since made a group of condensate atoms recoil from a light pulse such that they recorded the light's phase and amplitude, recovered by a second nearby condensate, in what they term "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates: details are discussed in Nature.[47] Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision atom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at a drop tower in Bremen, Germany by a consortium of researchers led by Ernst M. Rasel from Leibniz University of Hanover.[48] The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space[49] and it is also the subject of two upcoming experiments on the International Space Station.[50][51] Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.[52] In 1970, BECs were proposed by Emmanuel David Tannenbaum for anti-stealth technology.[53] Dark matterEdit P. Sikivie and Q. Yang showed that cold dark matter axions form a Bose–Einstein condensate by thermalisation because of gravitational self-interactions.[54] Axions have not yet been confirmed to exist. However the important search for them has been greatly enhanced with the completion of upgrades to the Axion Dark Matter Experiment(ADMX) at the University of Washington in early 2018. The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultra-low temperatures of 10−7 K or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkali metal, alkaline earth metal, and lanthanide atoms (7Li, 23Na, 39K, 41K, 85Rb, 87Rb, 133Cs, 52Cr, 40Ca, 84Sr, 86Sr, 88Sr, 174Yb, 164Dy, and 168Er). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'.[55] In contrast, the superfluid state of 4He below 2.17 K is not a good example, because the interaction between the atoms is too strong. Only 8% of atoms are in the ground state near absolute zero, rather than the 100% of a true condensate.[56] The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and half-integer total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values. See alsoEdit 1. ^ S. N. Bose (1924). "Plancks Gesetz und Lichtquantenhypothese". Zeitschrift für Physik. 26: 178–181. Bibcode:1924ZPhy...26..178B. doi:10.1007/BF01327326. 2. ^ "Leiden University Einstein archive". 27 October 1920. Retrieved 23 March 2011. 3. ^ A. Einstein (1925). "Quantentheorie des einatomigen idealen Gases". Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1: 3. 4. ^ Clark, Ronald W. (1971). Einstein: The Life and Times. Avon Books. pp. 408–409. ISBN 0-380-01159-X. 5. ^ F. London (1938). "The λ-Phenomenon of liquid Helium and the Bose–Einstein degeneracy". Nature. 141 (3571): 643–644. Bibcode:1938Natur.141..643L. doi:10.1038/141643a0. 6. ^ London, F. Superfluids Vol.I and II, (reprinted New York: Dover 1964) 7. ^ Bose-Einstein Condensate: A New Form of Matter, NIST, 9 October 2001 8. ^ Levi, Barbara Goss (2001). "Cornell, Ketterle, and Wieman Share Nobel Prize for Bose–Einstein Condensates". Search & Discovery. Physics Today online. Archived from the original on 24 October 2007. Retrieved 26 January 2008. 9. ^ J. Klaers; J. Schmitt; F. Vewinger & M. Weitz (2010). "Bose–Einstein condensation of photons in an optical microcavity". Nature. 468 (7323): 545–548. arXiv:1007.4088. Bibcode:2010Natur.468..545K. doi:10.1038/nature09567. 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20e7a32340e4b8a3	Skip to content • Research • Open Access Spatial non-adiabatic passage using geometric phases EPJ Quantum Technology20174:3 • Received: 8 November 2016 • Accepted: 15 March 2017 • Published: Quantum technologies based on adiabatic techniques can be highly effective, but often at the cost of being very slow. Here we introduce a set of experimentally realistic, non-adiabatic protocols for spatial state preparation, which yield the same fidelity as their adiabatic counterparts, but on fast timescales. In particular, we consider a charged particle in a system of three tunnel-coupled quantum wells, where the presence of a magnetic field can induce a geometric phase during the tunnelling processes. We show that this leads to the appearance of complex tunnelling amplitudes and allows for the implementation of spatial non-adiabatic passage. We demonstrate the ability of such a system to transport a particle between two different wells and to generate a delocalised superposition between the three traps with high fidelity in short times. • shortcuts to adiabaticity • geometric phases • complex tunnelling 1 Introduction Adiabatic techniques are widely used for the manipulation of quantum states. They typically yield high fidelities and possess a high degree of robustness. One paradigmatic example is stimulated Raman adiabatic passage (STIRAP) in three-level atomic systems [13]. STIRAP-like techniques have been successfully applied to a wide range of problems, and in particular, to the control of the centre-of-mass states of atoms in microtraps. This spatial analogue of STIRAP is called spatial adiabatic passage (SAP) and it relies on coupling different spatial eigenstates via a controllable tunnelling interaction [4]. It has been examined for cold atoms in optical traps [512] and for electrons trapped in quantum dots [13, 14]. The ability to control the spatial degrees of freedom of trapped particles is an important goal for using these systems in future quantum technologies such as atomtronics [9, 15, 16] and quantum information processing [17]. SAP has also been suggested for a variety of tasks such as interferometry [11], creating angular momentum [12], and velocity filtering [18]. It is also applicable to the classical optics of coupled waveguides [19, 20]. However, the high fidelity and robustness of adiabatic techniques comes at the expense of requiring long operation times. This is problematic as the system will therefore also have a long time to interact with an environment leading to losses or decoherence. To avoid this problem, we will show how one can speed-up processes that control the centre-of-mass state of quantum particles and introduce a new class of techniques which we refer to as spatial non-adiabatic passage. The underlying foundation for these are shortcuts to adiabaticity (STA) techniques, which have been developed to achieve high fidelities in much shorter total times, for a review see [21, 22]. Moreover, shortcuts are known to provide the freedom to optimise against undesirable effects such as noise, systematic errors or transitions to unwanted levels [2231]. Implementing the STA techniques for spatial control requires complex tunnelling amplitudes. However, tunnelling frequencies are typically real. To solve this, we show that the application of a magnetic field to a triple well system containing a single charged particle (which could correspond to a quantum dot system [3237]) can achieve complex tunnelling frequencies through the addition of a geometric phase. This then allows one to implement a counter-diabatic driving term [21, 22, 3840] or, more generally, to design dynamics using Lewis-Riesenfeld invariants [41]. The paper is structured as follows. In the next section, we present the model we examine, namely a charged particle in a triple well ring system with a magnetic field in the centre. In Section 3, we introduce the spatial adiabatic passage technique in a three-level system and show that making one of the couplings imaginary allows the implementation of transitionless quantum driving. We then show, in Section 3.3, how to create inverse-engineering protocols in this system using Lewis-Riesenfeld invariants. Results for two such protocols, namely transport and generation of a three-trap superposition, are given in Section 4. Section 5 presents a more realistic one-dimensional continuum model for the system, where the same schemes are implemented. Finally, in Section 6, we review and summarise the results. 2 System model We consider a charged particle trapped in a system of three localised potentials, between which the tunnel coupling can be changed in a time-dependent manner. In order to have coupling between all traps, they are assumed to be arranged along a ring and a magnetic field exists perpendicular to the plane containing the traps, see Figure 1. The particle will initially be located in one of the traps and we will show how to design spatial non-adiabatic passage protocols where a specific final state can be reached within a finite time and with high fidelity. Such a model could, for example, correspond to an electron trapped in an arrangement of quantum dots, where gate electrodes can be used to change the tunnelling between different traps [42]. Another option would be to use ion trapping systems [43], where ring configurations have been recently demonstrated [4446]. In these systems, tunnelling of an ion has already been observed (and controlled by manipulating the radial confinement), as well as the Aharonov-Bohm phase [47] acquired due to the presence of an external magnetic field [44]. Figure 1 Figure 1 Diagram of the system consisting of three coupled quantum wells and a localised magnetic field in the centre. The basis states and the couplings strengths used in the three-level approximation are indicated. The coordinate system for the continuous model in Section 5 is also shown. The distance between two traps along the ring is defined as l, so that the total circumference of the ring is 3l. Let us start by considering the single-particle Schrödinger equation $$ i\hbar\frac{\partial\psi}{\partial t} = \frac{1}{2m} (- i \hbar \nabla- q \vec{A} )^{2} \psi+ V \psi, $$ where m and q are the mass and charge of the particle, respectively, and V corresponds to the potential describing the trapping geometry. We assume that the vector potential is originating from an idealised point-like and infinitely long solenoid at the origin (creating a magnetic flux $\Phi_{B}$) and it is therefore given by $\vec {A} = \frac{\Phi_{B}}{2 \pi r} \hat{e}_{\varphi}$ (for $\vec{r} \neq0$). Here r, φ, z are cylindrical coordinates and $\hat{e}_{\varphi}$ is a unit vector in the φ direction. At low energies such a system can be approximated by a three-level (3L) model, where each basis state, $\|j\rangle$, corresponds to the localised ground state in one of the trapping potentials (see Figure 1). These states are isolated when a high barrier between them exists, but when the barrier is lowered the tunnelling amplitude $\Omega_{jk}$ between states $\|j\rangle$ and $\|k\rangle$ becomes significant. The presence of the magnetic field leads to the particle acquiring an Aharonov-Bohm phase [47] whenever it moves (tunnels) between two different positions (traps). This phase is given by $\phi _{j,k} = \frac{q}{\hbar}\int_{\vec{r}_{j}}^{\vec{r}_{k}}\vec{A}(\vec {r})\cdot d\vec{r}$, where $\vec{r}_{j}$ is the position of the jth trap, and for consistency, we always chose the direction of the path of the integration to be anti-clockwise around the pole of the vector potential (at $\vec{r} = 0$). The effects of this phase on the tunnelling amplitudes is given through the Peierls phase factors [4850], $\exp (i \phi_{j,k} )$, and the Hamiltonian for the 3L system can be written as $$ H = -\frac{\hbar}{2} \begin{pmatrix} 0 & \Omega_{12}e^{i\phi_{1,2}} & \Omega_{31}e^{-i\phi_{3,1}} \\ \Omega_{12}e^{-i\phi_{1,2}} & 0 & \Omega_{23}e^{i\phi_{2,3}} \\ \Omega_{31}e^{i\phi_{3,1}} & \Omega_{23}e^{-i\phi_{2,3}} & 0 \end{pmatrix} . $$ Here the $\Omega_{jk}$ are the coupling coefficients in the absence of any vector potential. The total phase around a closed path containing the three traps is then given by $$ \Phi\equiv\phi_{1,2}+\phi_{2,3}+\phi_{3,1} = \frac{q}{\hbar} \oint\vec {A}(\vec{r})\cdot d\vec{l} = \frac{q}{\hbar} \Phi_{B}, $$ and is non-zero due to the pole of the vector potential A⃗ at the origin. To simplify the Hamiltonian (2) one can use the following unitary transformation, which only employs local phases, $$ U= \begin{pmatrix} 1 & 0 & 0\\ 0 & e^{-i\phi_{1,2}} & 0\\ 0 & 0 & e^{-i (\phi_{1,2}+\phi_{2,3} )} \end{pmatrix} , $$ and transforms the Hamiltonian as $$ H \rightarrow U^{\dagger} H U= - \frac{\hbar}{2} \begin{pmatrix} 0 & \Omega_{12} & \Omega_{31}e^{- i \Phi}\\ \Omega_{12} & 0 & \Omega_{23}\\ \Omega_{31}e^{i \Phi} & \Omega_{23} & 0 \end{pmatrix} , $$ so that two of the tunnelling amplitudes become real-valued. A case of particular interest is when $\Phi=\pi/2$, i.e., when the magnetic flux is $\Phi_{B} = \pi\hbar/ 2 q$. In this case the Hamiltonian becomes $$ H = -\frac{\hbar}{2} (\Omega_{12} K_{1}+ \Omega_{23} K_{2}+\Omega_{31} K_{3} ) , $$ where each $K_{j}$ is a spin 1 angular momentum operator defined as $$ K_{1}= \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} ,\qquad K_{2}= \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix} ,\qquad K_{3}= \begin{pmatrix} 0 & 0 & -i\\ 0 & 0 & 0\\ i & 0 & 0 \end{pmatrix} , $$ satisfying $[K_{j}, K_{k}] = i \epsilon_{jkl} K_{l}$ and $\epsilon_{jkl}$ is the Levi-Civita symbol [51]. This means that the tunnel coupling between $\|3\rangle$ and $\|1\rangle$ becomes purely imaginary. We will show in the next section that this allows for the implementation of spatial non-adiabatic passage processes by either applying a transitionless quantum driving protocol or by using Lewis-Riesenfeld invariants. 3 Processes in the three-level approximation 3.1 Adiabatic methods A series of spatial adiabatic passage (SAP) techniques have been developed in recent years, which allows one to manipulate and control the external degrees of freedom of quantum particles in localised potentials with high fidelity [4]. The standard SAP protocol for the transport of a single particle in a triple well system [5, 13] is the spatial analogue of the quantum-optical STIRAP technique [13]. It involves three linearly arranged, degenerate trapping states, $\|j\rangle$ with $j = 1, 2\mbox{ and }3$, that can be coupled through tunnelling by either changing the distance between the traps or lowering the potential barrier between them. The system in the 3L approximation is described by the Hamiltonian $$ H_{0} = -\frac{\hbar}{2} (\Omega_{12} K_{1}+\Omega_{23} K_{2} ), $$ which has a zero-energy eigenstate of the form $$ \|\lambda_{0}\rangle = \cos\theta\|1\rangle - \sin \theta\|3\rangle\quad \text{with } \tan\theta= \Omega_{12}/ \Omega_{23} . $$ This state is often called the dark state and SAP consists of adiabatically following $\|\lambda _{0}\rangle$ from $\|1\rangle$ (at $t=0$) to $-\|3\rangle$ (at a final time $t=T$), effectively transporting the particle between the outer traps one and three. This corresponds to changing θ from 0 ($\Omega_{23} \gg \Omega_{12}$) to $\pi/2$ ($\Omega_{23} \ll\Omega_{12}$). Hence in the case of ideal adiabatic following, trap two (located in the middle) is never populated. 3.2 Transitionless quantum driving The main drawback of SAP is that it requires the process to be carried out adiabatically and therefore slowly compared to the energy gap [4]. If this requirement is not met, unwanted excitations will lead to imperfect transport. One way to specifically cancel possible diabatic transitions in STIRAP was discussed in [52] and a general approach for recovering adiabatic dynamics in a non-adiabatic regime is to use shortcuts to adiabaticity, such as transitionless quantum driving [3840]. This technique consists of adding a counter-diabatic term to the original Hamiltonian, whose particular form is given as $$ H_{\mathrm{CD}} = i \hbar\sum_{n} \bigl(\| \partial_{t} \lambda _{n}\rangle \langle\lambda _{n}\| - \langle\lambda_{n}\|\partial_{t} \lambda_{n}\rangle \|\lambda _{n}\rangle \langle\lambda _{n}\| \bigr), $$ where the $\|\lambda_{n}\rangle$ are the eigenstates of $H_{0}$. For the reference Hamiltonian in Eq. (8) this gives [40] $$ H_{\mathrm{CD}} =- \frac{\hbar\Omega_{31}(t)}{2} K_{3}, \quad \text{with } \Omega_{31}(t) = 2 \dot{\theta}(t) =2 \biggl( \frac{\Omega_{23} \dot {\Omega}_{12} - \Omega_{12} \dot{\Omega}_{23}}{\Omega_{12}^{2} + \Omega _{23}^{2}} \biggr). $$ We will see in Section 4.1 how this exact same scheme can also be obtained using Lewis-Riesenfeld invariants. Shortcuts to adiabaticity have been studied in the context of STIRAP [40, 53], i.e., population transfer between internal levels. Its spatial analogue is more challenging as it requires that the additional tunnelling coupling between sites one and three is imaginary (see the definition of $K_{3}$ in Eq. (7)). However, the system we have presented here is ideal for this, as the system Hamiltonian Eq. (6) is already equal to the total Hamiltonian $H_{0} + H_{\mathrm{CD}}$. Other methods to implement the imaginary coupling could be, for example, the use of artificial magnetic fields [54] or angular momentum states [55]. A heuristic but not rigorous explanation of why the coupling needs to be imaginary can be obtained by examining the two ‘paths’ the particle can take to move from trap one to trap three. The first is via SAP and leads to $\|1\rangle \to- \|3\rangle$ whereas the second is via the direct coupling the shortcut introduces, which leads to $\|1\rangle \to i e^{i \Phi} \|3\rangle$. One can then immediately see that for constructive interference of these two terms the phase needs to have the value $\Phi= \pi/2$, which corresponds to the required imaginary coupling between states $\|1\rangle$ and $\|3\rangle$. It is also interesting to note that the coupling between traps one and three in the shortcut has the form of a π-pulse $$ \int_{0}^{T} \Omega_{31}(t) \, dt = 2 \int_{0}^{T} \dot{\theta}(t) \, dt = 2 \bigl[ \theta(T) - \theta(0) \bigr] = \pi. $$ 3.3 Invariant-based inverse engineering Another method of designing shortcuts to adiabaticity is by means of inverse-engineering using Lewis-Riesenfeld (LR) invariants [41, 56]. In this section we will briefly review these methods and then apply them to our particular system to both transport the particle and create a superposition between the three wells. A LR invariant for a Hamiltonian $H(t)$ is a Hermitian operator $I(t)$ satisfying [41] $$ \frac{\partial I}{\partial t}+\frac{i}{\hbar} [H,I ]=0. $$ Since $I(t)$ is a constant of motion it can be shown that it has time-independent eigenvalues. It can be further shown that a particular solution of the Schrödinger equation, $$ i\hbar\partial_{t} \bigl\vert \psi(t) \bigr\rangle = H(t) \bigl\vert \psi(t) \bigr\rangle , $$ can be written as $$ \bigl\vert \psi_{k}(t) \bigr\rangle =e^{i \alpha_{k}(t)} \bigl\vert \phi_{k}(t) \bigr\rangle , $$ where the $\|\phi_{k}(t)\rangle$ are the instantaneous eigenstates of $H(t)$ and $$ \alpha_{k}(t)=\frac{1}{\hbar} \int_{0}^{t} \bigl\langle \phi_{k}(s) \bigr\vert \bigl[i\hbar \partial_{s}-H(s) \bigr] \bigl\vert \phi_{k}(s) \bigr\rangle \, ds $$ are the LR phases. Hence a general solution to the Schrödinger equation can be written as $$ \bigl\vert \psi(t) \bigr\rangle =\sum_{k} c_{k} \bigl\vert \psi _{k}(t) \bigr\rangle , $$ where the $c_{k}$ are independent of time. The idea behind inverse engineering using LR invariants is not to follow an instantaneous eigenstate of the $H(t)$ as one would in the adiabatic case, but rather follow an eigenstate of $I(t)$ (up to the LR phase). To guarantee that the eigenstates coincide at the beginning and the end of the process, it is necessary that the invariant and the Hamiltonian commute at these times, i.e., $$ \bigl[I(0),H(0) \bigr]= \bigl[I(T),H(T) \bigr]=0. $$ One is then free to choose how the state evolves in the intermediate time and once this is fixed, Eq. (13) determines how the Hamiltonian should vary with time to achieve those dynamics. A LR invariant for a three-level system described by Eq. (6) can be written as $$ I = -\sin\beta\sin\alpha K_{1}-\sin\beta\cos\alpha K_{2}+ \cos\beta K_{3} , $$ where α and β are time dependent functions which must fulfil the following relations (imposed by Eq. (13)) $$\begin{aligned}& \dot{\alpha} = \frac{\Omega_{12} \sin\alpha+ \Omega_{23} \cos \alpha }{2 \tan\beta} + \frac{\Omega_{31}}{2}, \end{aligned}$$ $$\begin{aligned}& \dot{\beta} = \frac{1}{2} (\Omega_{23} \sin \alpha- \Omega_{12} \cos \alpha). \end{aligned}$$ The eigenstates of this invariant are $$\begin{aligned}& \bigl\vert \phi_{0}(t) \bigr\rangle = \left ( \begin{matrix} -\sin\beta\cos\alpha\\ -i\cos\beta\\ \sin\beta\sin\alpha \end{matrix} \right ), \end{aligned}$$ $$\begin{aligned}& \bigl\vert \phi_{\pm}(t) \bigr\rangle = \frac{1}{\sqrt{2}}\left ( \begin{matrix} \cos\beta\cos\alpha\pm i\sin\alpha\\ -i\sin\beta\\ -\cos\beta\sin\alpha\pm i\cos\alpha \end{matrix} \right ), \end{aligned}$$ with respective eigenvalues $\mu_{0}=0$ and $\mu_{\pm}=\pm1$. One solution of the time-dependent Schrödinger equation is then given by $\|\Psi(t)\rangle = \|\phi_{0} (t)\rangle$ as the corresponding LR phase is zero in this case. Note that this invariant is a generalisation of the invariant considered in [57] where a third coupling $\Omega _{31}$ was not taken into account. After fixing the boundary conditions using Eq. (18), one is free to choose the functions $\alpha(t)$ and $\beta(t)$. Moreover, in this case, one is also free to directly choose the function $\Omega_{31}$. By inverting Eqs. (20) and (21), the other coupling coefficients are then given by $$\begin{aligned}& \Omega_{12} = 2 \dot{\alpha}\sin\alpha\tan\beta- 2 \dot{\beta}\cos \alpha- \Omega_{31} \sin\alpha\tan\beta, \end{aligned}$$ $$\begin{aligned}& \Omega_{23} = 2 \dot{\alpha}\cos\alpha\tan\beta+ 2 \dot{\beta}\sin \alpha- \Omega_{31} \cos\alpha\tan\beta. \end{aligned}$$ 4 Examples of spatial non-adiabatic passage schemes In the following we will discuss two examples of spatial non-adiabatic passage derived from LR invariant based inverse engineering in the 3L approximation. The first one is the transport between two different traps, which is shown to be equivalent to the transitionless quantum driving method from Section 3 in some cases. The second scheme will create an equal superposition of the particle in all three traps. 4.1 Transport The first example of control we examine is the population transfer determined by $$ \bigl\vert \Psi(0) \bigr\rangle =\vert 1 \rangle \rightarrow \vert \Psi_{\mathrm{target}} \rangle = \bigl\vert \Psi(T) \bigr\rangle =- \vert 3 \rangle, $$ which was considered in the optical regime in [40]. This can be achieved by choosing auxiliary functions that fulfil the boundary conditions $$ \beta(0)= \beta(T)= - \frac{\pi}{2},\qquad \alpha(0)=0, \quad \text{and} \quad \alpha(T)=\frac{\pi}{2}. $$ The experimentally required tunnelling frequencies are then explicitly given by Eqs. (24) and (25). For the special choice of $\beta(t) = -\pi/2$, one can show that $\langle2\|\Psi(t)\rangle = 0$ for all times, i.e. trap two is never occupied during the process. This choice then results in $$ \tan\alpha= \frac{\Omega_{12}}{\Omega_{23}} \quad \text{and}\quad \Omega_{31} = 2\dot{\alpha}. $$ By identifying α with θ (see Eq. (9)) one can immediately see that this is the same pulse as in the STA scheme derived in Section 3.2. The transport scheme can be implemented by the choosing the counterintuitive SAP pulses $\Omega_{12}$ and $\Omega_{23}$ to have a Gaussian profile [4] $$\begin{aligned}& \Omega_{12}(t) = \Omega_{0} \exp \bigl[-100 (t/T - 1/2 )^{2} \bigr], \end{aligned}$$ $$\begin{aligned}& \Omega_{23}(t) = \Omega_{0} \exp \bigl[-100 (t/T - 1/3 )^{2} \bigr], \end{aligned}$$ and then calculating $\Omega_{31}$ from Eq. (28). The resulting pulses and associated dynamical populations are shown in Figure 2. As expected the system follows exactly the dark state, transferring the population between states $\|1\rangle$ and $\|3\rangle$ without populating state $\|2\rangle$. Figure 2 Figure 2 Spatial non-adiabatic passage transport in the 3L approximation. $T/\tau =100$ for $\Omega _{0} \tau = 0.25$. (a) Modulus of the tunnelling amplitudes. (b) Evolution of the populations $P_{i}=\|\langle i\|\Psi (t)\rangle\|^{2}$. The time unit τ is defined as $\tau = m l^{2}/\hbar $. The fidelity of the transport process as a function of the total time and the phase Φ generated by the magnetic field is shown in Figure 3(a). Transport can be seen to occur with perfect fidelity for any value of the total time if the phase takes the appropriate value $\Phi= \pi/2$. It can also be seen that the shortcut is successful for any value of the phase in the limit of very short or very long times. The latter one is not surprising, as $\Omega_{31}$ can be neglected in the adiabatic limit, and hence its phase becomes irrelevant. A similar effect occurs for short total times, where the roles are reversed. In this limit $\Omega_{31}$ is the largest of all three couplings, and hence the phase relation between it and the other couplings becomes inconsequential. As $\Omega_{31}$ is a π pulse, perfect population transfer in this regime can be achieved regardless of the phase. Figure 3 Figure 3 Transport process $\pmb{\|1\rangle \to -\|3\rangle}$ in the 3L approximation. (a) Fidelity as a function of the total time and the total magnetic phase traversing the system. The green contour line is defined by $P_{3}=99\%$. (b) Probabilities of population in each of the traps for $T/\tau =48$ (indicated by a dashed white line in (a)) as a function of the total magnetic phase traversing the system. The dashed black line indicates the optimal value of the phase $\Phi =\pi /2$. However, in order to maintain this pulse area, a strong coupling is required for very short processes, as the strength of $\Omega_{31}$ is inversely proportional to T. This sets a bound on how fast this scheme can be implemented, as any physical implementation will have a maximum tunnelling amplitude. Setting the maximum value of $\Omega_{31}$ to $0.25/\tau$, the minimum process times T to achieve fidelities above 99% are approximately 880τ for SAP and 100τ for the shortcut scheme. These times are similar to the ones achievable in a spin-dependent transport scheme recently presented by Masuda et al. [58], however the setup in their work requires four traps and a constant and an AC magnetic field. It is worth noting that this system also allows for the possibility of measuring the magnetic flux $\Phi_{B}$, as the amount of transferred population oscillates as a function of the total phase Φ, which is directly related to the magnetic flux as $\Phi=\frac{q}{\hbar}\Phi_{B}$. As an example we show the occupation probabilities for $T/\tau= 48$ in each trap at the end of the process as a function of the phase in Figure 3(b). One can see that the populations strongly depend on the phase and over a large range of values one can therefore determine the magnetic flux. The exact relationship between the probabilities and the magnetic flux differs for different total times T. 4.2 Creation of a three-trap superposition The second scheme we discuss highlights the generality of the LR invariant based method. In this scheme we create an equal superposition state between the particles being in all three traps, which means that the initial and target states are $$ \bigl\vert \Psi(0) \bigr\rangle = \vert 1 \rangle \rightarrow \vert \Psi_{\mathrm{target}} \rangle = \bigl\vert \Psi (T) \bigr\rangle = \frac{1}{\sqrt{3}} \bigl( \vert 1 \rangle- i \vert 2 \rangle - \vert 3 \rangle\bigr). $$ This can be realised by imposing the boundary conditions $$\begin{aligned}& \beta(0)=-\frac{\pi}{2},\qquad \beta(T)=-\arctan\sqrt{2}, \end{aligned}$$ $$\begin{aligned}& \alpha(0)=0,\qquad \alpha(T)=\frac{\pi}{4}, \end{aligned}$$ on the auxiliary functions. A simple ansatz which fulfils these boundary conditions is a fourth order polynomial for $\beta(t)$ and third order polynomials for $\alpha (t)$ and $\Omega_{31}(t)$. The pulses are then obtained from Eqs. (24) and (25) and their form is shown in Figure 4(a). From Figure 4(b) it can be seen that this choice creates the target state at the final time with perfect fidelity. Figure 4 Figure 4 Spatial non-adiabatic superposition scheme $\pmb{\vert 1 \rangle \to\frac{1}{\sqrt{3}} ( \vert 1 \rangle - i \vert 2 \rangle - \vert 3 \rangle)}$ in the 3L approximation. $T/\tau =400$. Sub-figures are the same as in Figure 2 and the fidelity shown in (b) is defined as $F=\vert \langle \Psi _{\mathrm{target}}\|\Psi (t)\rangle \vert ^{2}$. 5 Spatial non-adiabatic passage in the continuum model While the 3L approximation discussed above gives a clear picture of the physics of the system, it does not include effects such as excitations to higher energy states that can occur during the process. We will therefore in the following test the approximation by numerically integrating the full Schrödinger equation in real space. For this, we will consider traps that are narrow enough to limit the system dynamics to an effectively one-dimensional setting along the azimuthal coordinate, $x = \varphi R$, i.e., around a circle of radius R, see Figure 1. Moreover, we will assume that the magnetic field is characterised by a vector potential in the azimuthal direction, $\vec{A} = A \hat {e}_{\varphi}$. We are therefore dealing with a one-dimensional system of length $2 \pi R$ with periodic boundary conditions, whose dynamics are described by the following Schrödinger equation $$ i \hbar\frac{\partial\psi}{\partial t} = \frac{1}{2m} \biggl(- i \hbar \frac{\partial}{\partial x} - q A \biggr)^{2} \psi+ V(x) \psi. $$ We assume a constant vector potential throughout the dynamical part of the protocols, as any time-varying vector potential would produce an unwanted force due to the electric field $\vec{E} = - \partial_{t} \vec{A}$. In order to be able to apply a well-defined phase we model the trapping sites as highly localised point-like potentials of depth $\epsilon_{j}$ at the positions $x_{j} = j l - l/2$ (see Figure 5). They are separated by square barriers of heights $V_{jk}(t)$ (and length l), giving a total potential $$ V(x,t) = - \sum_{j=1}^{3} \epsilon_{j}(t) \delta(x - x_{j})+ \textstyle\begin{cases} V_{31}(t) & \text{if } 0 < x < x_{1}, \\ V_{12}(t) & \text{if } x_{1} < x < x_{2}, \\ V_{23}(t) & \text{if } x_{2} < x < x_{3}, \\ V_{31}(t) & \text{if } x_{3} < x < 3l . \end{cases} $$ Since point-like potentials are difficult to implement numerically, in the simulations below they are implemented as narrow Gaussians. It is important to note that this model is not designed to give realistic estimates for the fidelities or exactly reproduce the dynamics of the 3L approximation. It is a toy model to validate the basic underlying processes and show that our schemes also make sense in the continuum. Figure 5 Figure 5 Schematic of the potential used in the numerical simulations (black line) with the localised states in each trap (coloured areas). The Gaussian shape of the traps is exaggerated here for clarity. As mentioned above, the tunnelling amplitudes $\Omega_{jk}(t)$ in the 3L approximation are related to the barrier heights $V_{jk}(t)$ of the continuum model, see the Appendix. However, changing the barrier heights in order to achieve tunnelling will also affect the energies of the localised states in the neighbouring traps. Therefore, in order to reproduce the resonance of the 3L approximation (where the diagonal elements of the Hamiltonian are always zero) in the continuum model, the depths of the delta potentials $\epsilon_{j}$ have to be adjusted as the barriers heights change, see Figure 5. Finally, to map the barrier heights $V_{jk}$ and trap depths $\epsilon _{j}$ parameters of the continuum model to the tunnelling amplitudes $\Omega_{jk}$ of the 3L approximation, we numerically calculate the overlaps of neighbouring delta-trap eigenstates. Results for transport of a particle using the shortcut scheme described in Section 4.1 are shown in Figure 6 and the barrier heights and trap depths used to match the pulses given in Figure 2 are shown in Figure 6(a), (b). The probability density during the process can be seen in Figure 6(c) and the populations in each trap are given in Figure 6(d). While the process is not perfect, one can see that the particle is transported to the final trap with a fidelity of 87%. The effect of the magnetic field can be seen in Figure 6(e), (f), where we show results for the same process but with an inverted magnetic field (using a total phase of $\Phi= -\pi/2$). In this case the interference between the adiabatic and shortcut paths is destructive, and almost no population ends up in the final trap. Figure 6 Figure 6 Spatial non-adiabatic transport process in the continuum model. $T/\tau =100$. (a), (b) Barrier heights and trap depths obtained by mapping the couplings in Figure 2(a). (c) Evolution of the particle density $\|\psi (x,t)\|^{2}$. (d) Corresponding populations $P_{i}=\vert \langle i\|\Psi (t)\rangle \vert ^{2}$ in each trap and of the target state. (e)(f) are the same as (c), (d) but with the magnetic flux flowing in the opposite direction. The width of the Gaussian traps is 10−4 l. The results for the creation of the superposition state discussed in Section 4.2 are shown in Figure 7. The observed dynamics are very similar to the one in the 3L approximation and the process reaches a final fidelity of the target state of 91%. Figure 7 Figure 7 Same as Figure 6 (a)-(d) but for the spatial non-adiabatic superposition scheme given in Eq. ( 31 ) in the continuum model. $T/\tau =400$. $F=\vert \langle \psi _{\mathrm{target}}\|\psi (t)\rangle \vert ^{2}$ is the fidelity of the process. Since the continuum model has many more degrees of freedom than the 3L model, it is not surprising that the fidelities obtained are lower. Nevertheless, the basic functioning of our spatial non-adiabatic techniques is clearly established from the calculations shown above. Optimising the fidelity in the continuum is an interesting task which, however, goes beyond the scope of the current work. 6 Conclusions and outlook We have shown how complex tunnel frequencies in single-particle systems allow one to develop spatial non-adiabatic passage techniques that can lead to fast and robust processes for quantum technologies. In particular, we have discussed the case of a single, charged particle in a microtrap environment. The complex tunnelling couplings are obtained from the addition of a constant magnetic field, and have allowed us to generalise adiabatic state preparation protocols beyond the usual spatial adiabatic passage techniques [4]. This demonstrates that non-adiabatic techniques can be as efficient as their adiabatic counterparts, without requiring the long operation times. In particular, we have discussed the implementation of the counter-diabatic term for spatial adiabatic passage transport via a direct coupling of all the traps. This was, in a second step, generalised to a flexible and robust method for preparing any state of the single-particle system by using Lewis-Riesenfeld invariants. As an example, we have shown that an equal spatial superposition state between the three wells can be created on a short time scale. Finally, we have presented numerical evidence that spatial non-adiabatic processes work also in a one-dimensional toy model by introducing a mapping between the discrete three-level approximation and a continuum model. While in this work we have focused on a three-trap system, an interesting extension would be to investigate similar schemes in larger systems, or in different physical settings (for example, superconducting qubits [59]). Often, if the transitionless quantum driving technique is directly applied to complex quantum systems, the additional counter-adiabatic terms become very complicated, hard to implement or even unphysical. Nevertheless, the steps outlined in our work (using a few-level approximation, applying the shortcut technique, and then mapping everything back to a continuous model) can in principle be applied to any trap configuration. These steps might lead to schemes which are much easier to implement experimentally than the direct application of the transitionless quantum driving. However, each of these generalised configuration would need to be studied on an individual basis. It would also be very interesting to see the effect of interactions in this system. For very strong interactions such that double occupancy of a site is suppressed and a single empty site is present, one might expect to observe similar dynamics but for the empty site [9]. In this case, spatial non-adiabatic ideas can be straightforwardly transferred. For intermediate interaction strengths (but stronger than the tunnelling couplings), repulsively-bound pair processes have been shown to dominate the dynamics and single-particle-like dynamics can be recovered for the pair [10, 60, 61]. In this case the presented techniques might be extended for a particle pair. 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2ed64bb0109dfb60	The non-endpoint Strichartz estimates for the (linear) Schrödinger equation: $$ \\|e^{i t \Delta/2} u_0 \\|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \\|u_0\\|_{L^2_x(\mathbb{R}^d)} $$ $$ 2 \leq q,r \leq \infty,\;\frac{2}{q}+\frac{d}{r} = \frac{d}{2},\; (q,r,d) \neq (2,\infty,2),\; q\neq 2 $$ are easily obtained using (mainly) the Hardy-Littlewood-Sobolev inequality, the endpoint case $q = 2$ is however much harder (see Keel-Tao for example.) Playing around with the Fourier transform one sees that estimates for the restriction operator sometimes give estimates similar to Strichartz's. For example, the Tomas-Stein restriction theorem for the paraboloid gives: $$ \\|e^{i t \Delta/2} u_0\\|_{L^{2(d+2)/d}_t L^{2(d+2)/d}_x} \lesssim \\|u_0\\|_{L^2_x}, $$ which, interpolating with the easy bound $$ \\|e^{i t \Delta/2} u_0\\|_{L^{\infty}_t L^{2}_x} \lesssim \\|u_0\\|_{L^2_x}, $$ gives precisely Strichartz's inequality but restricted to the range $$ 2 \leq r \leq 2\frac{d+2}{d} \leq q \leq \infty. $$ As far as I know, the Tomas-Stein theorem (for the whole paraboloid) gives the restriction estimate $R_S^*(q'\to p')$ for $q' = \bigl(\frac{dp'}{d+2}\bigr)'$ (this $q$ is different from the one above), so I'm guessing that this cannot be strengthened (?). So my question is: what's the intuition of what goes wrong when trying to prove Strichartz's estimates all the way down to the endpoints using only Fourier restriction theory? From my less-than-expert (where's Terry when you need him?) point of view, a possible reason seems to be the following (I wouldn't call it something going wrong or even a difficulty): The statement of restriction estimates only give you estimates where the left hand side is an isotropic Lebesgue space, in the sense that you get an estimate $L^q_tL^r_x$ with $q = r$. This naturally excludes the end-point, which requires $r > q$. Why is this? The reason is that the restriction theorems only care about the local geometry of the hypersurface, and not its global geometry. (For example, the versions given in Stein's Harmonic Analysis requires either the hypersurface to have non-vanishing Gaussian curvature for a weaker version, or that the hypersurface to be finite type for a slightly stronger version. Both of these conditions are assumptions on the geometry of the hypersurface locally as a graph over a tangent plane.) Now, on each local piece, you do have something more similar to the classical dispersive estimates with $r > q$, which is derived using the method of oscillatory integrals (see, for example, Chapter IX of Stein's book; the dispersive estimate (15) [which has, morally speaking $q = r = \infty$ but with a weight "in $t$", so actually implies something with $q < \infty$] is used to prove Theorem 1, which is then used to derive the restriction theorem). But once you try to piece together the various "local" estimates to get an estimate on the whole function, you have no guarantee of what the "normal direction" is over the entire surface. (The normal direction, in the case of the application to PDEs, is the direction of the Fourier conjugate of the "time" variable.) So in the context of the restriction theorem, it is most natural to write the theorem using the $q = r$ version, since in the more general context of restriction theorems, there is no guarantee that you would have a globally preferred direction $t$. (Note that Keel-Tao's contribution is not in picking out that time direction: that Strichartz estimates can be obtained from interpolation of a dispersive inequality and energy conservation is well known, and quite a bit of the non-endpoint cases are already available as intermediate consequences of the proof of restriction theorems. The main contribution is a refined interpolation method to pick out the end-point exponents.) Your Answer
9d6dd66096dca49c	web analytics Posts made in October, 2012 Stop Selling Sprouts – And Certainly Stop Eating Them Kroger, America's largest supermarket chain, announced it will stop selling sprouts because of their "potential food safety risk". It joins retail behemoth Walmart, which stopped selling them way back in 2010."After a thorough, science-based review, we have decided to voluntarily discontinue selling fresh sprouts," Payton Pruett, Kroger's vice president of food safety, said in a statement that... read more Organic Food Self-Deceit Is organic food grown without pesticides? Of course not, that would be silly, the yields would be 10%.Then why do so many organic food buyers think they have no pesticides? Mostly because a $29 billion industry relies on that sort of casual deception. And Dr. Oz helps. read more Physicists Take The Schrödinger Equation To The Streets The Schrödinger equation, devised in 1926 following a huge international effort by many scientists, describes the 'beautiful and surprising' ways that light and matter behave when they interact at the smallest scale. Ithas led to much of the technological development of the modern world, for example fiber optics that create the Internet's backbone, solar panels, GPS and electron... read more 5 Tips To Survive The Upcoming Ice Age read more The Paradox Of Millennials They are not buying into global warming except they care about the environment more than anyone ever did before. They will eat healthier than previous generations, provided the products are in pouches and not cans and can be purchased in vending machines and be...microwaveable. Except it needs to be slow food and locally grown.What's up with Millennials? More importantly, what is up with... read more
cc2edae2aa3e593e	Today’s post is about a personal revelation I recently had. You see, I spend a lot of time researching for this blog, making sure I understand what I’m talking about, and doing my best to explain it all clearly and concisely. And all this work, in theory, is supposed to benefit my science fiction writing. But I don’t want to write hard Sci-Fi. I used to think science fiction existed on a spectrum from hard science fiction, where everything is super scientifically accurate (and here’s a full chapter explaining the math to prove it), to soft science fiction, where everything’s basically space wizards and technobabble magic (lol, who cares if unobtainium crystals make sense?). I’ve since discovered another way to think about science fiction, and I find that to be more useful. But sometimes I’m still left wondering why am I doing all this extra work? What’s it all for if I’m not trying to write hard Sci-Fi? Recently, I was talking with a new friend, and somehow the conversation turned to quantum physics. I swear I wasn’t the one who brought it up! My friend had seen a video on YouTube, and I felt the need to disillusion him of the weird quantum mysticism he’d apparently been exposed to. I was doing my best to explain what the Heisenberg uncertainty principle actually means, and I ended up digging into what I remembered about the math. Mathematically speaking, the momentum of a quantum particle is represented by the variable p, its position by the variable q, and the relationship between p and q is often expressed as: pq ≠ qp I don’t have the math skills to explain how this non-equivalency equation works. I think it has something to do with matrices. My high school math teacher skipped that chapter. To this day, I still haven’t got a clue how a matrix works. I just know it’s an important concept in quantum theory. But by this point, my friend was staring at me with a sort of dumbstruck awe, and he said: “Wow, you really do understand this stuff!” That brought me up short. “No, not really,” I said, feeling slightly embarrassed. I couldn’t help but recollect the famous line attributed to Richard Feynman: If you think you understand quantum theory, you don’t understand quantum theory. So I told my friend about this blog and about my writing, and how I use the research I do for my blog to flesh out the story worlds in my science fiction. And then I said something that I don’t remember ever thinking before or being consciously aware of, but as soon as the words were out of my mouth I knew they were true: “I just want to make sure I know enough so that I don’t make a total fool of myself in my stories.” And that’s it. That’s the answer I needed. I’m okay with stretching the truth if it suits my story. I’m okay with leaving some scientific inaccuracies in there. I just don’t want to make a mistake so glaringly obvious to my readers (some of whom know way more about science than I do) that it ruins the believability of my story world. And now if you’ll excuse me, I have to get back to writing. The fiction kind of writing, I mean. And on Wednesday, we’ll have story time here on the blog. 15 responses » 1. Steve Morris says: Story time, yay! (BTW, physicists say that p and q do not commute, which means that if you measure p and q you get a different answer depending on which you measure first, and that this is a fundamental fact, not a limitation of any measuring equipment. It follows from the Schrödinger equation. You probably knew that.) Liked by 2 people • J.S. Pailly says: That all does sound familiar to me. I have to admit, my memory about this subject is a bit fuzzy. It’s been a while since I really read up on quantum theory. It’s probably time I did a refresher course on it. 2. I think there’s also something to be said for a sci-fi writer having a love of science, which I have to admit powers my own research more than story prep. On making mistakes, I’m resigned to the fact that those are going to happen. As I’ve learned more science, I’ve increasingly caught published sci-fi writers making those mistakes. But since their sales don’t appear to have taken any hits, it appears that as long as the mistakes aren’t basic ones, it’s okay. Although I’m sure those authors hear about them anyway. Actually, I suspect when sci-fi fans catch an author in an obscure error, it makes them feel smart, which is why they probably continue reading that author. Liked by 1 person • J.S. Pailly says: Yeah, it’s those basic mistakes that I really want to avoid. I also don’t want to reinforce popular misconceptions about science, so I want to make sure I don’t fall for those misconceptions myself. But I tend to fall into the trap of trying to be a perfectionist, and I need to stop doing that. Liked by 1 person 3. I found myself coming here to find a post that sparked an idea a few weeks back, just to make sure I got the science right. What ever your reasons, I’m glad you write this blog! Liked by 1 person 4. I can relate to this . This is the kind of thing I was talking about when I said I was feeling inspired by Ray Bradbury. It’s all about story and imagination. Who is anyone to say any kind of speculative futuristic technology is impossible to achieve? There’s a reason there’s almost always sound in space movies. It adds flourish as the reality is a little boring, even though most people know there’s no sound in space. Self-aware artistic license when it comes to the “sci” part of sci-fi, works. You just have to SOUND convincing to readers. If they buy it as plausible, I don’t believe it really matters that it bugs someone who knows better, if it served the story well. But again that’s why I’m really liking “speculative fiction” over “sci-fi” as an umbrella term for anything with futuristic/spacy elements. It robs nitpickers of their favorite gripe to toss at every single sci-fi property ever: “Well that’s not scientifically accurate. How can they call this ‘science fiction’?” I mean, look at Star Trek. There’s virtually no “real” science in it. There’s little tidbits here and there, nods to things like Dyson spheres, but it’s largely speculative fantasy. Fans ask scientists “Is the warp speed possible? Will a transporter work?” Etc. And the scientists go “Weeeeeeeell, I mean, theoretically, it is POSSIBLE, but—“ and then the fans go “Yes! Star Trek confirmed as scientifically accurate!” And they lord it over fans of “space fantasy” Star Wars. It’s all “space fantasy,” though, and it all has merit. Both sci-fantasy and hard sci-fi can be done badly. I could read Arthur C Clarke as a kid, and I didn’t have to be a scientist to do so. I understood the concepts he was putting out there. That’s because he told good stories and the scientific aspects of it served the story. They weren’t the focus, as far as I’m concerned. He knew his stuff, and that’s awesome, but it doesn’t mean that it’s more valid than say, Dune. I mean, what’s the science in Dune? It’s like He-Man for adults, lol. But it’s great, and rightfully considered a classic. I’m rambling now. My point is that the story is king and everything else exists to serve it, and if reality has to be twisted into a certain shape to fit a story, and it sounds good, I say go for it. Let your imagination fly free.🚀 Liked by 1 person 5. kutukamus says: Really, pq ≠ qp sure sounds/looks like the [ever-changing] relationship between [the same] two people. Then again, even the mere name quantum scares me. 🙂 Liked by 1 person That hits the nail on the head, all right. Sometimes it’s a balance between serious science and good storytelling (usually using the good storytelling to offset the lacks in science), and the successful stories strike that balance well–the reason why over-the-top adventure dramas like the Star Wars and superhero movies can get away with so much. A writer can be good with science, or not so good; but their primary job is to be a storyteller, and they have to make sure their knowledge of science (or any subject, for that matter) doesn’t throw the reader out of their story. Liked by 1 person • J.S. Pailly says: I’ve been told many times over that most readers won’t know the difference and don’t really care about scientific accuracy in science fiction. I guess that might be true for a general audience, but I’m pretty sure avid science fiction readers are a little more scientifically literate than most people. So I think it’s a matter of knowing your audience. You don’t want your readers to lose their suspension of disbelief. With Sci-Fi readers, that means getting the science right, or at least not making a total mess of it. Liked by 1 person Leave a Reply to J.S. Pailly Cancel reply You are commenting using your account. Log Out / Change ) Google photo Twitter picture Facebook photo Connecting to %s
061b8ecb3036043d	Science and Transcendence: An Excerpt from Creative Tension Science and Transcendence: An Excerpt from Creative Tension Print Friendly, PDF & Email Limits of Language and Common Sense We all are realists. If we were not, the surrounding world would soon destroy us. We must take seriously information given us by our senses. If, when crossing the street, we looked for extrasensory inspiration instead of watching the traffic lights, we would have been very quickly eliminated from this game. Poets and philosophers seem odd and impractical to others because abstract worlds of ideas divert their sight from earthly things. From our everyday contacts with the surrounding world (but also from many slips and bruises) our common sense is born—that is, the set of practical rules that tell us how to behave in order to minimize the damage the world could inflict upon us. We like to quote science to justify our common sense. The scientific method is but a sharpening of our common sense. Experience constitutes the base of every science, and measuring instruments we use in our laboratories are “prolongations” of our senses. The world of technology, from the computer on my desk to artificial satellites, testifies to the ability of our common sense, which has so efficiently conquered the world of matter. Such views, although flattering to our ears, are totally false. Widely spread imaginings concerning science do not match what science really is. Contemporary physics, this most advanced of all sciences, provides us an example fatally destroying these imaginings. What could be more in agreement with our common sense than the fact that we cannot go back to our childhood? Time is irreversible. It flows irrevocably from the past to the future. However, this is not that obvious in physics. We know that to every elementary particle there corresponds an antiparticle. Such an antiparticle has the same mass as the corresponding particle but the opposite electric charge. When a particle collides with its antiparticle, they both change into energy. These are the experimental facts, but the first information about the existence of antiparticles came from theory. Since 1926 it has been known that the motion of an electron is described by the Schrödinger equation. The discovery of this equation by Schrödinger was a major breakthrough. Together with the works of Heisenberg, it has created the foundations of modern quantum mechanics. However, the Schrödinger equation had a serious drawback: it did not take into account the laws of special relativity discovered by Einstein two decades earlier. Einstein’s theory is a physical theory of space and time. Although we can ignore it when dealing with the first approximation to the real world, if we want to be more precise in our investigation of the world we cannot avoid using a relativistic approach. The relativistic counterpart of Schrödinger’s equation was discovered by Dirac in 1928. It turned out that Dirac’s equation admitted two types of solutions. One of these types described well the elementary particles known at the time. The remaining solutions referred to similar particles but going back in time. How should this be understood? Dirac was audacious enough to claim that such particles really existed and coined the name “antiparticles.” This step was not an easy one. Our common sense had to be put upside down. To make this step easier, Dirac helped his imagination with the picture of the void with holes in it, and he interpreted these holes as antiparticles. It does not matter whether we would prefer holes in the void or time flowing backward; our common sense is jeopardized. Let us consider another example. An atom emits two photons (quanta of light). They travel in two different directions, and, after a certain lapse of time, they are far away from each other (it does not matter how far; they can even be at two opposite edges of the Galaxy). Photons have the property called spin by physicists. It can be measured, and quantum mechanics teaches us that the results of the measurements can assume only two values. Let us denote them symbolically by +1 and –1. However, the situation is much more delicate than our inert language allows us to express. Strictly speaking, we cannot claim that an electron possesses the spin in such a manner as we say that Mr. Smith is tall or has twenty dollars in his pocket. When we are measuring the photon’s spin, it behaves as if it were always there. In fact, before the act of measurement, the photon had no spin. Before the act of measurement, a probability existed that the act of measurement would yield, if performed, a given result with a given probability. Let us assume that we have performed the measurement obtaining the result +1. In such a case, on the strength of the laws of quantum mechanics, another photon acquires spin –1, even if it is at the other edge of the Galaxy. How does this photon instantaneously know about our measurement on the first photon and the result it yields? This experiment was invented as a purely Gedanken experiment by Einstein, Podolsky, and Rosen in 1935 in order to show that the laws of quantum mechanics lead to nonsensical conclusions. However, the physicists—against the opinion of Einstein and his two collaborators—were not much surprised when Allain Aspect, together with his team, performed Einstein’s Gedanken experiment in reality, and it has turned out that quantum mechanics was right. Aspect was able to perform this experiment owing to enormous progress in experimental methods, but also owing to a theoretical idea of John Bell that enabled him to express Einstein’s intuitions in the form of precise formulae (the so-called Bell inequalities), which could be compared with the results of measurements. What happens to photons in Aspect’s experiments? When our intuition fails, we must look for help from the mathematical structure of the theory. In quantum mechanics, two photons that once interacted with each other are described by the same vector of state. Strictly speaking, positions of elementary particles behave like spin; an elementary particle is nowhere in space until its position is measured. The state vector of a given quantum object contains information only about probabilities of outcomes of various measurements. We are met here not only with particles that live “backward in time” but also with particles for which space distances are no obstacles. It looks as if elementary particles did not exist in space and time—as if space and time were only our macroscopic concepts, the usual meaning of which breaks down as soon as we try to apply them to the quantum world. Moreover, can one speak about the individuality of a particle (before its properties are measured) that exists neither in space nor in time? If we agree to consider as a single object something that is described by a single vector of state, could we treat two photons (which previously interacted with each other) situated at two different edges of the Galaxy as the single object? Contemporary physics has questioned the very applicability to the quantum world of such fundamental concepts as space, time, and individuality. Is not our common sense put upside down? Some philosophers claim that what cannot be said clearly is meaningless. The intention of this claim is praiseworthy; its aim is to eliminate verbosity, which does not contain any substance. However, modern physics has taught us that the possibilities of our language are limited. There are domains of reality— such as the quantum world—at the borders of which our language breaks down. This does not mean that within such domains anything goes—far from it. It turns out that mathematics constitutes a much more powerful language than our everyday means of communication. Moreover, mathematics is not only a language that describes what is seen by our senses. Mathematics is also a tool that discloses those regions of reality that without its help would forever remain inaccessible for us. All interpretational problems of modern physics can be reduced to the following question: How can all these things that are disclosed by the mathematical method be translated into our ordinary language? I think that the greatest discovery of modern physics is that our common sense is limited to the narrow domain of our everyday experience. Beyond this domain a region extends to which our senses have no access. Schrödinger’s Question The world of classical mechanics seemed simple and obvious, but in fact it never was simple or obvious. The method discovered by Galileo and Newton did not consist in performing many experiments with pendulums and freely falling bodies, the result of which would later be described with the help of mathematical formulae. Newton, led by his genius, posed a few hazardous hypotheses that suggested to him the mathematical shape of the laws of motion and those of universal gravity. His formulae did not describe the results of experiments. Nobody ever saw a particle uniformly moving to infinity because it was not acted upon by any forces. Moreover, there is no such particle in the entire Universe. And it is exactly this statement that is at the very foundations of modern mechanics. The world of classical mechanics is doubtlessly richer than the world we penetrate with our senses. The most fundamental principle of physics was discovered within the domain of classical mechanics—a principle that could be reached only by mathematical analysis. It is called the principle of the least action, and its claim is indeed extraordinary. It asserts that every physical theory— from classical mechanics to the most modern quantum field theory—can be constructed in the same way. First, one must correctly guess a function called Lagrangian (which is different for different theories). Then, one computes an integral of this function, called action. And finally, one obtains the laws of this theory by postulating that the action assumes the extreme value (usually the least one, but sometimes the greatest one). Physicists often speak about a superunification of physics, that is, about such a theory that would contain everything in itself. We do not yet have such a theory, but its chances that we will are becoming greater and greater. In fact, we already have, in a sense, the unification of the method; all major physical theories are obtainable from the principle of the least action. With our senses we cannot grasp the fact that all bodies around us move in such a way that a certain simple mathematical expression (the action) assumes the minimal value. But the bodies move in this way. We live surrounded by things that cannot be seen, or heard, or touched. It was Schrödinger who once asked himself: Which achievements of science have best helped the religious outlook of the world? In his answer to this question he pointed to the results of Boltzmann and Einstein concerning the nature of time. Time, which can change its direction depending on the fluctuations of entropy, which can flow differently in different systems of reference, is no longer a tyrant Chronos, whose absolute regime destroys all our hopes for nontemporal existence, but a physical quantity with a limited region of applicability. If Schrödinger lived today, he could add many new items to his list of achievements that teach us the sense of Mystery. Personally, I think, however, that particular scientific achievements do not do this work best, but rather the scientific method itself. Spectacular results of the most recent physical theories are but examples of what was present in the method of physics for a long time, although it was understood only by a very few. Two Experiences of Humankind If we pause for a moment, in our competition for new achievements, to look backward on the progress of science during the last two centuries, we can see an interesting regularity. In the nineteenth century humankind went through the great experience of the efficiency of the scientific method. It was a deep experience. Today, we speak of the century of “vapor and electricity” with a touch of irony in our voice. We must know, however, that the road from a candle to the electric bulb, and from a horse-drawn carriage to the railroad train, was longer and more laborious than that from the propeller plane to the intercontinental jet. In the twentieth century technology made a great jump, but in the nineteenth century it had started almost from nothing. But even then it was obvious that it would change the shape of the civilized world. In the nineteenth century technology was treated, like never before or after, as a synonym of progress and of the approaching new era of overwhelming happiness. Positivistic philosophy, regarding science as the only valuable source of knowledge, and scientism, wanting to replace philosophy and religion with science, could be considered as a philosophical articulation of this great experience— the experience of the efficiency of the scientific method. In the nineteenth century, any suggestion that there could exist any limits beyond that the scientific method does not work, would have been regarded as a senseless heresy. Nobody would have taken it seriously. The nineteenth century came together with its wars and revolutions. In my opinion, the revolution that took place in the foundations of physics, in the first decades of the twentieth century (and which, I think, is still taking place), had more permanent results for our culture than the political turmoil that shaped the profile of our times. First of all, it turned out that classical mechanics— once believed to be the theory of everything—in fact has but a limited field of applicability. It is limited on two sides: from below—in the domain of atoms and elementary particles the Newtonian laws must be replaced by the laws of quantum mechanics; and from above—for objects moving with a speed comparable to that of light, classical physics breaks down and should be replaced by Einstein’s theory of relativity. Moreover, the new theories are also, in a sense, limited: the finite value of Planck’s constant essentially limits the questions that can be asked in quantum physics, and the finite velocity of light determines horizons of the information transfer in the theory of relativity and cosmology. The method physics used from the times of Galileo and Newton (and possibly even from the time of Archimedes) consists in applying mathematics to the investigation of the world. The certainty of mathematical deductions is transferred to physics, and it is one of the two sources of the efficiency of the physical method (the other one being controlled experiment). It came as a shock when, in the third decade of the twentieth century, Kurt Gödel proved his famous theorems which assert that limitations are inherent in mathematics itself: no system of axioms could be formulated from which entire mathematics could be deduced (or even a part of mathematics that is at least as rich as arithmetic). Such a system would be either incomplete or self-contradictory. Today, there is no doubt that the twentieth century has confronted us with the new great experience—the experience of limitations inherent in the scientific method. Philosophers have understood this relatively late. In the first half of the twentieth century, positivism, in its radical form of the logical empiricism, dominated the scene. Only in the 1960s did it become evident that one cannot philosophically support an outdated vision of science. I do not here have in mind those anti-scientific and anti-intellectual currents that nowadays so often fanatically fight science in the name of supposed interests of humanity. I have in mind a philosophy of science that recognizes the epistemological beauty of science and its rational applications in the service of man, but does this based on the correct evaluation of both scientific method and the limitations inherent in it. Science and Transcendence Science could be compared to a great circle. The points in its interior denote all scientific achievements. What is outside the circle represents not-yet discovered regions. Consequently, the circumference of the circle should be interpreted as a place in which what we know today meets with what is still unknown, that is, as a set of scientific questions and unsolved problems. As science progresses, the set of achievements increases and the circle expands; but, together with the area inside the circle, the number of unanswered questions and unsolved problems becomes bigger and bigger. It is historical truth that each resolved problem poses new questions calling for new solutions. If we agree to understand the term transcendence—as suggested by its etymology— as “something that goes beyond,” then what is outside the circle of scientific achievements is transcendent with respect to what is inside it. We can see that transcendence admits a graduation: something may go beyond the limits of this particular theory, or beyond the limits of all scientific theories known till now, or beyond the limits of the scientific method as such. Do such ultimate limits exist? Usually three domains are quoted as forever inaccessible to all attempts of the mathematico-empirical method: the domain of existence, the domain of ultimate rationality, and the domain of meaning and value. How does one justify the existence of the world? Why does something exist rather than nothing? Some more optimistic physicists believe that in the foreseeable future one will be able to create the Unique Theory of Everything. Such a theory would not only explain everything, but it would also be the only possible theory of that type. In this way, the entire Universe would be understood; there would be no further questions. Let us suppose that we have such a theory—the set of equations fully describing (modeling) the Universe. One problem would remain: How can one change from the abstract equations to the real world? What is the origin of those existents that are described by the equations? Who or what ignited the mathematical formulae with existence? Science investigates the world in a rational way. Knowledge is rational if it is rationally justified. Here new questions arise: Why should we rationally justify our convictions? Why is the strategy of rational justifications so efficient in investigating the world? One cannot give a rationally justified answer to the first of these questions. Let us try doing this; that is, let us try to rationally justify the statement that everything should be rationally justified. However, our justification (our proof ) cannot presuppose what it is supposed to justify (to prove). Therefore, we cannot assume that our convictions should be rationally justified. Consequently, when constructing our proof we cannot use rational means of proving (because they presuppose that we are to prove something); that is, the proof cannot be carried out. There is no other way out of this dilemma but to assume that the postulate to rationally justify our convictions is but our choice. We have two options, and we must choose one of them: either, when doing science, we do it in a rational way or we admit an irrational way of doing science. Rationality is a value. This can be easily seen if rationality is confronted with irrationality. We evaluate rationality as something good and irrationality as something bad. When choosing rationality we choose something good. It is, therefore, a moral choice. The conclusion cannot be avoided; at the very basis of science there is a moral option. This option was made by humankind when it first formulated questions addressed to the world and started to look for rationally justified answers to them. The entire subsequent history of science could be regarded as a confirmation of this option. Now follows the second question: Why is the strategy of rational justifications so efficient in studying the world? One could risk the following answer: The fact that our rational methods of studying the world lead to such wonderful results suggests that our choice of rationality is somehow consonant with the structure of the world. The world is not a chaos but an ordered rationality. Or: the rational method of science turns out to be so efficient because the world is permeated with meaning. We should not understand this in an anthropomorphic manner. Meaning, in this context, is not something connected with the human consciousness; it is this property of the world because of which the world discloses its ordered structure, provided it is investigated with the help of rational methods. Schrödinger’s Question Once More After all these considerations, it would be worthwhile to go back to Schrödinger’s question: Which achievements of science have best helped the religious outlook of the world? I think that contemporary science teaches us, as never before, the sense of mystery. In science, we are confronted with mystery on every step. Only outsiders and mediocre scientists believe that in science everything is clear and obvious. Every good scientist knows that he is dancing on the edge of a precipice between what is known and what is only feebly felt in just-formulated questions. He also knows that the newly born questions open vistas that go beyond the possibilities of our present imagination—imagination that has learned its art in contact with these pieces that we had so painfully extracted from the mysteries of the world. Let us imagine a very good scientist of the nineteenth century, for instance, Maxwell or Boltzmann, who is informed by his younger colleague coming to him from our twenty-first century about recent developments of general relativity or quantum mechanics. Maxwell or Boltzmann would never believe in such “nonsense.” Now consider this question: How would we behave if a physicist from the twenty-second century told us about his textbook physics? Only a very shortsighted scientist can be unaware of the fact that he is surrounded by mysteries. Of course, I have in mind relative mysteries, that is, such mysteries as now go beyond the limits of our knowledge but perhaps tomorrow will become well-digested truths. Do not such mysteries point toward the Mystery (with the capital M)? Does not what today transcends the limits of science suggest something that transcends the limits of all scientific methods? I have expressed these ideas in the form of questions on purpose. Plain assertions are too rigid; they assert something that is expressed by its words and syntactic connection between them, but remain silent about what is outside the linguistic stuff. Therefore, let us stick to questions that open our intuition for regions not constrained by grammatical rules. Are these unimaginable achievements of science, which revolutionize our vision of the world (time flowing backward, cured space-time, particles losing their individuality but communicating with each other with no interaction of space and time), not clear suggestions that the reality is not exhausted in what can be seen, heard, touched, measured, and weighed? • Does not the fact that there exists something rather than nothing excite our metaphysical anxiety? • Does the fact that the world is not only an abstract structure—never a written formula, an equation solved by nobody, yet something that can be seen, heard, touched, measured, and weighed—direct our thought to the Ultimate Source of Existence? • Does not the fact that the world can, after all, be put into abstract formulae and equations suggest to us that the abstract thought is more significant than concrete matter? • Does the rationality that is presupposed but never explained by every scientific investigation not express a reflection of the rational plan hidden in every scientific question addressed to the Universe? • Does not the moral choice of the rationality that underlies all science offer a sign of the Good that is in the background of every correct decision? These questions are not situated far away, “beyond the limits.” The concreteness of existence, the rationality of the laws of nature, the meaning touched by us when we make our decisions are present in every atom, in every quantum of energy, in every living cell, in every fiber of our brain. It is true that the Mystery is not in the theorems of science but in its horizon. Yet this horizon permeates everything. Buy Read the rest of Michael Heller’s Creative Tension by purchasing the book at
8062bb0aa6fccf2e	A photon is an excitation or a particle created in the electromagnetic field whereas an electron is an excitation or a particle created in the "electron" field, according to second-quantization. However, it is often said in the literature that the wave function of a photon doesn't exist whereas it exists for an electron. Why is it so? • $\begingroup$ There are some not-so-innocent ways in which you can defend writing down a wave function for a photon as well but the general underlying reason for the issues is that photon is no ordinary particle. It quite literally travels at the speed of light. $\endgroup$ – Dvij D.C. Feb 24 '19 at 1:01 • $\begingroup$ @DanYand have a look at the link in my answer $\endgroup$ – anna v Feb 27 '19 at 4:51 Saying that a photon doesn't have a wavefunction can be misleading. A more accurate way to say it is that a photon doesn't have a strict position observable. A photon can't be strictly localized in any finite region of space. It can be approximately localized, so that it might as well be restricted to a finite region for all practical purposes; but the language "doesn't have a wavefunction" is referring to the non-existence of a strict position observable. An electron doesn't have a strict position observable, either, except in strictly non-relativistic models. In relativistic quantum field theory, nothing has a strict position observable. This is a consequence of a general theorem called the Reeh-Schlieder theorem. The proof of this theorem is not trivial, but it's nicely explained in [1]. Relativistic quantum field theory doesn't have strict single-particle position observables, but it does have other kinds of strictly localized observables, such as observables corresponding to the magnitude and direction of the electric and magnetic fields inside an arbitrarily small region of space. However, those observables don't preserve the number of particles. Strictly localized observables necessarily turn single-particle states into states with an indefinite number of particles. (Actually, even ignoring the question of localization, "particle" is not easy to define in relativistic quantum field theory, but I won't go into that here.) For example, relativistic quantum electrodynamics (QED) has observables corresponding to the amplitudes of the electric and magnetic fields. These field operators can be localized. The particle creation/annihilation operators can be expressed in terms of the field operators, and vice versa, but the relationship is non-local. Technically, the Reeh-Schlieder theorem says that a relativistic quantum field theory can't have any strictly-localized operator that annihilates the vacuum state. Therefore, it can't have any strictly-localized operator that counts the number of particles. (The vacuum state has zero particles, so a strictly-localized particle-counting operator would annihilate the vacuum state, which is impossible according to the Reeh-Schlieder theorem.) Strictly non-relativistic models are exempt from this theorem. To explain what "strictly non-relativistic" means, consider the relativistic relationship between energy $E$ and momentum $p$, namely $E=\sqrt{(mc^2)^2+(pc)^2}$, where $m$ is the single-particle mass. If $p\ll mc$, then we can use the approximation $E\approx mc^2+p^2/2m$. A non-relativistic model is one that treats this approximate relationship as though it were exact. The most familiar single-particle Schrödinger equation is a model of this type. Such a model does have a strict position operator, and individual particles can be strictly localized in a finite region of space in such a model. Since photons are massles ($m=0$), we can't use a non-relativistic model for photons. We can use a hybrid model, such as non-relativistic QED (called NRQED), which includes photons but treats electrons non-relativistically. But even in that hybrid model, photons still can't be strictly localized in any finite region of space. Loosely speaking, the photons are still relativistic even though the electrons aren't. So in NRQED, we can (and do) have a single-electron position observable, but we still don't have a single-photon position observable. "Wavefunction" is a more general concept that still applies even when strict position observables don't exist. The kind of "wavefunction" used in relativistic quantum field theory is very different than the single-particle wavefunction $\psi(x,y,z)$ familiar from strictly non-relativistic quantum mechanics. In the relativistic case, the wavefunction isn't a function of $x,y,z$. Instead, it's a function of more abstract variables, and lots of them (nominally infinitely many), and it describes the state of the whole system, which generally doesn't even have a well-defined number of particles at all. People don't use this kind of wavefunction very often, because it's very difficult, but once in a while it's used. For example, Feynman used this kind of "wavefunction" in [2] to study a relativistic quantum field theory called Yang-Mills theory, which is a simplified version of quantum chromodynamics that has gluons but not quarks. In this generalized sense, a single photon can have a wavefunction. In the non-relativistic case, the $x,y,z$ in $\psi(x,y,z)$ correspond to the components of the particle's position observables. When physicists say that a photon doesn't have a wavefunction, they mean that it doesn't have a wavefunction that is a function of the eigenvalues of position observables, and that's because it doesn't have any strict position observables. Also see these very similar questions: Can we define a wave function of photon like a wave function of an electron? Wave function of a photon? EM wave function & photon wavefunction [1] Witten, "Notes on Some Entanglement Properties of Quantum Field Theory", http://arxiv.org/abs/1803.04993 [2] Feynman (1981), "The qualitative behavior of Yang-Mills theory in 2 + 1 dimensions", Nuclear Physics B 188: 479-512, https://www.sciencedirect.com/science/article/pii/0550321381900055 Here is the wavefunction of the photon, which is a solution of a quantized maxwell's equation: In quantum field theory it is necessary to have a plane wave function solution for the fields on which creation and annihilation operators operate. This blog post describes how the classical fields emerge from the quantum ones of QFT. • $\begingroup$ This is oversimplified and basically misleading as written. QFT does not allow us to have wavefunctions like this for a relativistic particle in a state of definite particle number. Although you can cook up something like this and call it the wavefunction of a photon, it doesn't have the standard interpretation in terms of the Born rule. A long review article on this is Iwo Bialynicki-Birula, "Photon wave function," 2005, arxiv.org/abs/quant-ph/0508202 $\endgroup$ – user4552 Feb 28 '19 at 22:52 • $\begingroup$ @BenCrowell from the abstract of your link "can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics". Please note that QFT by construction is relativistic and needs wavefunctions to define the fields.,and that the question is about QFT. $\endgroup$ – anna v Mar 1 '19 at 4:14 Your Answer
c0d9a3f18b268fa5	Wednesday, December 30, 2020 Well, Actually. 10 Physics Answers. [This is a transcript of the video embedded below.] Today I will tell you how to be just as annoying as a real physicist. And the easiest way to do that is to insist correcting people when it really doesn’t matter. 1. “The Earth Orbits Around the Sun.” Well, actually the Earth and the Sun orbit around a common center of mass. It’s just that the location of the center of mass is very close by the center of the sun because the sun is so much heavier than earth. To be precise, that’s not quite correct either because Earth isn’t the only planet in the solar system, so, well, it’s complicated. 2. “The Speed of Light is constant.” Well, actually it’s only the speed of light in vacuum that’s constant. The speed of light is lower when the light goes through a medium, and just what the speed is depends on the type of medium. The speed of light in a medium is also no longer observer-independent – as the speed of light in vacuum is – but instead it depends on the relative velocity between the observer and the medium. The speed of light in a medium can also depend on the polarization or color of the light, the former is called birefringence and the latter dispersion. 3. “Gravity Waves are Wiggles in Space-time” Well, actually gravity waves are periodic oscillations in gases and fluids for which gravity is a restoring force. Ocean waves and certain clouds are examples of gravity waves. The wiggles in space-time are called gravitational waves, not gravity waves. 4. “The Earth is round.” Well, actually the earth isn’t round, it’s an oblate spheroid, which means it’s somewhat thicker at the equator than from pole to pole. That’s because it rotates and the centrifugal force is stronger for the parts that are farther away from the axis of rotation. In the course of time, this has made the equator bulge outwards. It is however a really small bulge, and to very good precision the earth is indeed round. 5. “Quantum Mechanics is a theory for Small Things” Well, actually, quantum mechanics applies to everything regardless of size. It’s just that for large things the effects are usually so tiny you can’t see them. 6. “I’ve lost weight!” Well, actually weight is a force that depends on the gravitational pull of the planet you are on, and it’s also a vector, meaning it has a direction. You probably meant you lost mass. 7. “Light is both a particle and a wave.” Well, actually, it’s neither. Light, as everything else, is described by a wave-function in quantum mechanics. A wave-function is a mathematical object, that can both be sharply focused and look pretty much like a particle. Or it can be very smeared out, in which case it looks more like a wave. But really it’s just a quantum-thing from which you calculate probabilities of measurement outcomes. And that’s, to our best current knowledge, what light “is”. 8. “The Sun is eight light minutes away from Earth.” Well, actually, this is only correct in a particular coordinate system, for example that in which Planet Earth is in rest. If you move really fast relative to Earth, and use a coordinate system in rest with that fast motion, then the distance from sun to earth will undergo Lorentz-contraction, and it will take light less time to cross the distance. 9. “Water is blue because it mirrors the sky.” Well, actually, water is just blue. No, really. If you look at the frequencies of electromagnetic radiation that water absorbs, you find that in the visual part of the spectrum the absorption has a dip around blue. This means water swallows less blue light than light of other frequencies that we can see, so more blue light reaches your eye, and water looks blue. However, as you have certainly noticed, water is mostly transparent. It generally swallows very little visible light and so, that slight taint of blue is a really tiny effect. Also, what I just told you is for chemically pure water, H two O, and that’s not the water you find in oceans, which contain various minerals and salt, not to mention dirt. So the major reason the oceans look blue, if they do look blue, is indeed that they mirror the sky. 10. “Black Holes have a strong gravitational pull.” Well, actually the gravitational pull of a black hole with mass M is exactly as large as the gravitational pull of a star with mass M. It’s just that – if you remember newton’s one over r square law – the gravitational pull depends on the distance to the object. The difference between a black hole and a star is that if you fall onto a star, you’re burned to ashes when you get too close. For a black hole you keep falling towards the center, cross the horizon, and the gravitational pull continues to increase. Theoretically, it eventually becomes infinitely large. How many did you know? Let me know in the comments. You can join the chat on this video tomorrow, Thursday Dec 31, at 6pm CET or noon Eastern Time here. Saturday, December 26, 2020 What is radiation? How harmful is it? [This is a transcript of the video embedded below.] So, I hope you learned something new today! Wednesday, December 23, 2020 How to speak English like Einstein [This is a transcript of the video embedded above. Parts of the text won’t make sense without the accompanying audio.] Hi everybody, I’ve been thinking really hard about why you are here. Of course theoretical physics is awesome, but in my experience, that opinion is, sadly enough, not widely shared among the general population. So while I am thrilled to see you’re all super excited about the square well potential in Schrödinger’s equation, I am secretly convinced you’re just here to hear me try to pronounce difficult English words with a German accent. So, today, we’ll have a special feature about How To Speak English Like Einstein. Albert Einstein: “The scientific method itself would not have led anywhere, it would not even have been formed, without a passionate striving for a clear understanding. Perfection of means and confusion of goals seem, in my opinion, to characterize our age.” Don’t worry if you don’t speak German, you don’t need to know a single word of German to understand this video. But before we get started, let’s have a look at Professor Einstein’s name, Albert Einstein. How is that pronounced correctly? Most importantly, the German ST is not pronounced “st” as you would in English, for example in “first” or “start”. The “ST” in Einstein is pronounced “scht”. Einstein. The German “sch” is similar, but not exactly identical, to the English “sh”. If you are familiar with phonetic spelling, that’s this thing that looks like an integral. You find it in words like “push” or “machine”. It’s a good first approximation to the German “sch”, but if you want to get the German sound right, you have to pull the tongue back in your mouth. Listen, the English one is “push”. Now pull back your tongue, and you get pusch. Pusch. That’s the sound that goes into Einschtein. The rest is details. All the German vowels are shifted relative to the English ones, long story, big headache, but just try not to rely on the spelling, just listen. It’s Albert Einstein. Don’t worry about the “r” in Albert, just make that an “a”. Everyone does it and it’ll sound just fine. Albeat. Albeat Einstein. Ok, so now about that German accent. To speak with a German accent, you have to remember which English sounds do not exist in German. And that’s most importantly, the English “th”, the vanishing “w”, and the “r”. If you replace those with the next closest German sounds, you’ll immediately sound very German. Let’s use this sentence as example “I remember in February we were still thinking that this would be over relatively soon.” I hope I pronounced this correctly. Here’s the first step to a German accent. Replace all the “th”s, the “th” with a “z”. Why a “z”? Because that’s what comes out if you put your tongue in the wrong place. That’s what you mean, zat’s what it sounds like. So, you replace “this” with “zis”. And “either” with “eizer”. “Therefore” with “zerefore”, and so on. The example sentence then becomes: “I remember in February we were still zinking zat zis would be over relatively soon.” Mayday, mayday. Hello, can you hear us? Can you hear us? Over. We are sinking. We are sinking. Hello. Ziz is ze German coast guard. We’re sinking. We’re sinking. What are you zinking about? German humor. Second step. The vanishing English “w”. As in “what” or “wonderful”. That sound doesn’t exist in German either, so you make it a “v”. What becomes vat. Wonderful becomes vonderful. Would become vould, and so on. With that our example sentence now sounds like this “I remember in February ve vere still zinking zat zis vould be over relatively soon.” The third step is the probably most difficult one if you’re an English native speaker. It’s to replace the English “r” with a German r. The German “r” is a short rolling r. Think of a happy cat, it’s purring, it goes “rrrrrr” “rrrr”. Comes from the back of your throat. Like if you’re snoring. Rrrr. Try that. I’ll wait. Excellent. Now you launch from that into a word. Let’s take the word “right”. “rrrrrrrrrrrrrright” right. Right. There you have it. It sounds very German doesn’t it? We don’t, in German, actually do a lot of rolling with the r, so don’t make that too long. Right. Also, don’t trill the r at the tip of your tongue, like in trust me. No, don’t do that. It should be tRust me. Some more examples. Friend becomes “fRiend”. Direction becomes diRection. It’s actually a terrible sound. The example sentence is now: “I Remember in FebRuaRy ve vere still zinking zat zis vould be over Relatively soon.” Repeat after me, I’ll pause. Great. You are awesome. Have fun with your Einstein English, don’t forget to subscribe and check my Patreon page for more content. Zanks for vatching. Saturday, December 19, 2020 All you need to know about 5G The new 5G network technology is currently being rolled out in the United States, Germany, the United Kingdom, and many other countries all over the world. What’s new about it? Does it really use microwaves? Like in microwave ovens? Is that something you should worry about? I began looking into this fully convinced I’d tell you that nah, this is the usual nonsense about cellphones causing cancer. But having looked at it in some more detail, now I’m not so sure. First of all, what is 5 G? 5 G is the fifth generation of wireless networks. The installation of antennas is not yet completed, and it will probably take at least several more years to complete, but in some places 5G is already operating, and you can now buy cellphones that use it. What’s it good for? 5G promises to deliver more data, faster, by up to a factor one hundred, optimistically. It could catapult us into an era where driverless cars and the internet of things have become reality. How is that supposed to work? 5 G uses a variety of improvements on the data routing that makes it more efficient, but the biggest change that has attracted the most attention is that 5G uses a frequency range that the previous generations of wireless networks did not use. These are the millimeter waves. And, yes, these are the same waves that are being used in the scanners at airport security, the difference is that in the scanners you’re exposed for a second every couple of months or so, while with 5G you’d be sitting in it at low power but possibly for hours a day, depending on how close you live and work to one of the new antennas. As the name says, millimeter waves have wavelengths in the millimeter range, and the ones used for 5G correspond to frequencies of twenty-four to forty-eight Giga-Hertz. If that number doesn’t tell you anything, don’t worry, I will give you more context in a moment. For now, let me just say that the new frequencies are about a factor ten higher than the highest frequencies that were previously used for wireless networks. Another thing that’s new about 5G are directional phased-array antennas. Complicated word that basically means the antennas don’t just radiate the signal off into all directions, but they can target a particular direction. And that’s an important difference, if you want to know how the signal strength drops with distance to the antenna. Roughly speaking, it becomes more difficult to know what’s going on. Because of these new features, conspiracy theories have flourished around 5G and there have been about a hundred incidents, mostly in the Netherlands, Belgium, Ireland, and the UK, where people have burned down or otherwise damaged 5G telephone towers. Dozens of cities, counties, and nations have stopped the installing. There have been protests against the rollout of the 5G technology all over the world. And groups of concerned scientists have written open letters twice, once in 2017 and once in 2019. Each letter attracted about a few hundred signatures from scientists. Not a terrible lot, but not nothing either. Before we can move on, I need to give you some minimal background on the physics, so bear with me for a moment. Wireless technology uses electromagnetic radiation to encode and send information. Electromagnetic radiation is electric and magnetic fields oscillating around each other creating a freely propagating wave that can travel from one place to another. Electromagnetic radiation is everywhere. Light is electromagnetic radiation. Radio stations air music with electromagnetic radiation. If you open an oven and feel the heat, that’s also electromagnetic radiation. These seem to be different phenomena, but physically, they’re all the same thing. The only difference is the wavelength of the oscillation. Commonly, we use different names for electromagnetic radiation depending on that wavelength. If we can see it, we call it light. Visible light with long wavelengths is red, and at even longer wave-lengths when we can no longer see it, we call it infrared. We can’t see infrared light, but we often still feel that it’s warm. At even longer wavelengths we call the radiation microwaves, and if the wavelengths are even longer, they are called radio waves. On the other side of visible light, at wavelengths shorter than violet, we have the ultraviolet, and then the X-rays, and gamma-rays. The new millimeter waves are in the high frequency part of microwaves. Now, we may call electromagnetic radiation a “wave” but those waves are actually quantized, which means they are made of small packs of energy. These small packs of energy are the particles of light, which are called “photons”. You may think it’s an unnecessary complication, to talk about quantization here, but knowing that electromagnetic radiation is made of these particles, the photons, is extremely helpful to understand what the radiation can do. That’s because the energy of the photons is proportional to the frequency of the radiation, or equivalently, the energy is inversely proportional to the wavelength. So, a high frequency means a short wavelength, and a large energy per photon. A small frequency means a long wavelength, which means small energy. Again that’s energy per photon. That the frequency of electromagnetic radiation tells you the energy of the particles in the radiation is so useful because if you want to damage a molecule, you need a certain minimum amount of energy. You need this energy to break the bonds between the atoms that make up the molecule. And so, the most essential thing you need to know to gauge how harmful electromagnetic radiation is, is whether the energy per photon in the radiation is large enough to break molecular bonds, like the bonds that hold together the DNA. Breaking molecular bonds is not the only way electromagnetic radiation can be harmful, and I will get to the other ways in few minutes, but it is the most direct and important harm electromagnetic radiation can do. So how much energy do you need to damage a molecule? Damage begins happening just above the high-energy-end of visible light, with the ultraviolet radiation. That’s the light that gives you a sunburn and that you’ve been told to avoid. It has wavelengths that are just a little bit shorter than visible light, or frequencies and energies that are just a little bit higher. In terms of energy, ultraviolet radiation has about three to thirty electron volts per photon. An electron Volt is just a unit of energy. If that’s unfamiliar to you, doesn’t matter, you merely need to know that the binding energy of most molecules also lies in the range of a few electron volts. If you want to break a molecule, you need energies above that binding energy, so you need frequencies at or above the ultraviolet. That’s because the energy for the damage has to come with the individual photons in the radiation. If the individual photons do not have enough energy to actually damage the molecule, they either just go through or, sometimes, if they hit a resonance frequency, they’ll wiggle the molecule. If you wiggle molecules that means you warm them up. So, what matters for the question whether you can damage a molecule is the energy per photon in the radiation, which means the frequency of the radiation, not the total energy of all the particles in the radiation, of which there could be many. If you take more particles, but each of them has an energy below what’s necessary for damaging a molecule, you’ll just get more wiggling. All the radiation used for wireless networks, including 5G, uses frequencies way below those necessary to break molecular bonds. It is below even the infrared. So in this regard, there is clearly nothing to worry about. But. As I mentioned, breaking molecular bonds is not the only way that electromagnetic radiation can harm living tissue. Because tissue is complicated. It’s not just physics. You can also harm tissue just by warming it. And how much warming you can get from electromagnetic radiation is not determined by the energy per photon, it is determined by the total energy per time that is transferred by all the photons and on the fraction that is absorbed by the tissue. That total energy transfer per time is called the “power” and it’s commonly measured in Watts. So: The frequency tells you the energy per photon. The power tells you the total energy in photons per time. For example, if you look at your microwave oven, that probably operates at about 2 GigaHertz, which is a really small energy per photon, about a million times below the energy required to break molecular bonds. But a microwave oven operates at maybe four hundred or up to a thousand Watts. And that’s high in terms of power. So, a lot of photons per time. On the other hand, if you have a wireless router at home, it quite possibly operates at a similar frequency as your microwave oven. But a wireless router typically uses something like one hundred milli Watts, that’s ten thousand times less than the microwave oven, and the router radiates into space, not into a closed cavity. That’s a relevant difference for a simple geometric reason. If the photons in the electromagnetic radiation distribute in all of the directions, as they do for antennas like your wireless router, then the density of particles will thin out, meaning the power will drop very quickly with distance to the sender. This is why, in wireless communication, the highest power you’ll be exposed to is if you are close to the sender and that is usually your cell phone, not an antenna, because the antennas tend to be on a roof or a mast or in any case, not on your ear. Ok, to summarize: The frequency tells you the energy per particle and determines the what type of damage is possible. The power tells you the number of particles and it drops very quickly with distance to the source. The power alone does not tell you how much is absorbed by the human body. Back to 5G. What the 5G controversy is about is whether the electromagnetic radiation from the new antennas poses a health risk. 5G actually uses electromagnetic radiation in three different parts of the spectrum, called the low band, the mid band, and the high band. The frequency of the radiation in all these bands is below that which is required to damage molecules. The frequency of the mid band is indeed comparable to the one your microwave oven is using, but actually, there’s nothing new about this, microwaves have been used by wireless networks for more than two decades. The radiation in the high band are the new millimeter waves. This band has so far been largely unused for telecommunication purposes simply because it’s not very good for long-range transmission. The electromagnetic waves in this range do not travel very far and can get blocked by walls, trees, and even humans. Therefore, the idea behind 5G is to use a short-range network, made of the so-called “small cells” for the millimeter waves. These small cells have to be distributed at distances of about one hundred meters or so. The small cells communicate with macro cells that use the mid and low bands with antennas that operate at higher power and that do the long range transmission. So, a fully functional 5G network is likely to increase the exposure to millimeter waves, which have not before been used for cell phones. This means the people who are citing the lack of correlation between cell phone use and cancer incidence in the past 20 years missed the point. These studies don’t tell you anything about the 5G high band because that wasn’t previously in use. Now the thing is if you look what is known about the health risks from long-term exposure to the new millimeter waves band, there are basically no studies. We know that millimeter waves cannot penetrate deeply into the human body, but we know that at high power, they warm the skin and irritate eyes. Exactly what power is too much in the long run no one knows because there just hasn’t been enough research. Here is for example a Meta-review published about a year ago, which came to the conclusion: “The available studies do not provide adequate and sufficient information for a meaningful safety assessment.” And here we have Rob Waterhouse, vice president of a telecommunication company in the United States: Waterhouse admits that although millimeter waves have been used for many different applications—including astronomy and military applications—the effect of their use in telecommunications is not well understood… “The majority of the scientific community does not think there’s an issue. However, it would be unscientific to flat out say there are no reasons to worry.” That’s not very reassuring. And the World Health Organization writes: “no adverse health effect has been causally linked with exposure to wireless technologies… but, so far, only a few studies have been carried out at the frequencies to be used by 5G.” So the protests that you see against 5G, I am afraid to say, are not entirely unjustified. Don’t get me wrong, damaging other people’s property is certainly not a legitimate response. But I can understand the concern. We have no reason to think 5G is a health risk. Indeed, it is reasonable to think it is *not a health risk, given that this radiation is of low energy and scatters in the upper layers of the skin, but there is very little data on what the effects of long-term exposure may be. How should one proceed in such a situation? Depends on how willing you are to tolerate risk. And that’s not a question for science, that’s a question for politics. What do you think? Let me know in the comments. You can join the chat on this week's topic on Saturday, Dec 19, at noon Eastern Time/6pm CET here. Saturday, December 12, 2020 Are Singularities Real? There is one exception to this, and that’s black holes. Saturday, December 05, 2020 Is Infinity Real? [This is a transcript of the video embedded below] Is infinity real? Or is it just mathematical nonsense that you get when you divide by zero? If infinity is not real, does this mean zero also is not real? And what does it mean that infinity appears in physics? That’s what we will talk about today. Most of us encounter infinity the first time when we learn to count, and realize that you can go on counting forever. I know it’s not a terribly original observation, but this lack of an end to counting because you can always add one and get an even larger numbers is the key property of infinity. Infinity is the unbounded. It’s larger than any number you can think of. You could say it’s unthinkably large. Okay, it isn’t quite as simple because, odd as this may sound, there are different types of infinity. The amount of natural numbers, 1,2,3 and so on is just the simplest type of infinity, called “countable infinity”. And the natural numbers are in a very specific way equally infinite as other sets of numbers, because you can count these other sets using the natural numbers. Formally this means a set of numbers is similarly infinite as the natural numbers, if you have a one-to-one map from the natural numbers to that other set. If there is such a map, then the two sets are of the same type of infinity. For example, if you add the number zero to the natural numbers – so you get the set zero, one, two, three, and so on – then you can map the natural numbers to this by just subtracting one from each natural number. So the set of natural numbers and the set of the natural numbers plus the number zero are of the same type of infinity. It’s the same for the set of all integers Z, which is zero, plus minus one, plus minus two, and so on. You can uniquely assign a natural number to each integer, so the integers are also countably infinite. The rational numbers, that is the set of all fractions of integers, is also countably infinite. The real numbers, that contain all numbers with infinitely many digits after the point, is however not countably infinite. You could say it’s even more infinite than the natural numbers. There are actually infinitely many types of infinity, but these two, those which correspond to the natural and real numbers, are the two most commonly used ones. Now, that there are many different types of infinity is interesting, but more relevant for using infinity in practice is that most infinities are actually the same. As a consequence of this, if you add one to infinity, the result is still the same infinity. And if you multiply infinity with two, you just get the same infinity again. If you divide one by infinity, you get a number with an absolute value smaller than anything, so that’s zero. But you get the same thing if you divide two or fifteen or square root of eight by infinity. The result is always zero. I hope there are no mathematicians watching this, because technically one should not write down these relations as equations. Really they are statements about the type of infinity. The first, for example, just means if you add one to infinity, then the result is the same type of infinity. The problem with writing these relations as equations is that it can easily go wrong. See, you could for example try to subtract infinity on both sides of this equation, giving you nonsense like one equals zero. And why is that? It’s because you forgot that the infinity here really only tells you the type of infinity. It’s not a number. And if the only thing you know about two infinities is that they are of the same type, then the difference between them can be anything. It’s even worse if you do things like dividing infinity by infinity or multiplying infinity with zero. In this case, not only can the result be any number, it could also be any kind of infinity. This whole infinity business certainly looks like a mess, but mathematicians actually know very well how to deal with infinity. You just have to be careful to keep track of where your infinity comes from. For example, suppose you have a function like x square that goes to infinity when x goes to infinity. You divide it by an exponential function, that also goes to infinity with x. So you are dividing infinity by infinity. This sounds bad. But in this case you know how you get to infinity and therefore you can unambiguously calculate the result. In this case, the result is zero. The easiest way to see this is to plot this fraction as a function of x, as I have done here. If you know where your infinities come from, you can also subtract one from anther. Indeed, physicists do this all the time in quantum field theory. You may for example have terms like 1/epsilon, 1/epsilon square and the logarithm of epsilon. Each of these terms will give you infinity for epsilon to zero. But if you know that two terms are of the same infinity, so they are the same function of epsilon, then you can add or subtract them like numbers. In physics, usually the goal of doing this is to show that at the end of a calculation they all cancel each other and everything makes sense. So, mathematically, infinity it interesting, but not problematic. For what the math is concerned, we know how to deal with infinity just fine. But is infinity real? Does it exist? Well, it arguably exists in the mathematical sense, in the sense that you can analyze its properties and talk about it as we just did. But in the scientific sense, infinity does not exist. That’s because, as we discussed previously, scientifically we can only say that an element of a theory of nature “exists” if it is necessary to describe observations. And since we cannot measure infinity, we do not actually need it to describe what we observe. In science, we can always replace infinity with a very large but finite number. We don’t do this. But we could. Here is an example that demonstrates how mathematical infinities are not measurable in reality. Suppose you have a laser pointer and you swing it from left to right, and that makes a red dot move on a wall in a far distance. What’s the speed by which the dot moves on the wall? That depends on how fast you move the laser pointer and how far away the wall is. The farther away the wall, the faster the dot moves with the swing. Indeed, it will eventually move faster than light. This may sound perplexing, but note that the dot is not actually a thing that moves. It’s just an image which creates the illusion of a moving object. What is actually moving is the light from the pointer to the wall and that moves just with the speed of light. Nevertheless, you can certainly observe the motion of the dot. So, we can ask then, can the dot move infinitely fast, and can we therefore observe something infinite? It seems that for the dot to move infinitely fast you’d have to place the wall infinitely far away, which you cannot do. But wait. You could instead tilt the wall at an angle to you. The more you tilt it, the faster the dot moves across the surface of the wall as you swing the laser pointer. Indeed, if the wall is parallel to the direction of the laser beam, it seems the dot would be moving infinitely fast across the wall. Mathematically this happens because the value of the tangent function at pi over two is infinity. But does this happen in reality? In reality, the wall will never be perfectly flat, so there are always some points that will stick out and that will smear out the dot. Also, you could not actually measure that the dot is at exactly the same time on both ends of the wall because you cannot measure times arbitrarily precisely. In practice, the best you can do is to show that the dot moved faster than some finite value. This conclusion is not specific to the example with the laser pointer, this is generally the case. Whenever you try to measure something infinite, the best you can do in practice is to say it’s larger than something finite that you have measured. But to show that it was really infinite you would have to show the result was larger than anything you could possibly have measured. And there’s no experiment that can show that. So, infinity is not real in the scientific sense. Nevertheless, physicists use infinity all the time. Take for example the size of the universe. In most contemporary models, the universe is infinitely large. But this is a statement about a mathematical property of these models. The part of the universe that we can actually observe only has a finite size. And the issue that infinity is not measurable is closely related to the problem with zero. Take for example the mathematical abstraction of a point. Physicists use this all the time when they deal with point particles. A point has zero size. But you would have to measure infinitely precisely to show that you really have something of zero size. So you can only ever show it’s smaller than whatever your measurement precision allows. You can join the chat on this video on Saturday 12PM EST / 6PM CET Sunday 2PM EST / 8PM CET Saturday, November 28, 2020 Magnetic Resonance Imaging [This is a transcript of the video embedded below. Some of the text may not make sense without the animations in the video.] Magnetic Resonance Imaging is one of the most widely used imaging methods in medicine. A lot of you have probably had one taken. I have had one too. But how does it work? This is what we will talk about today. Magnetic Resonance Imaging, or MRI for short, used to be called Nuclear Magnetic Resonance, but it was renamed out of fear that people would think the word “nuclear” has something to do with nuclear decay or radioactivity. But the reason it was called “nuclear magnetic resonance” has nothing to do with radioactivity, it is just that the thing which resonates is the atomic nucleus, or more precisely, the spin of the atomic nucleus. Nuclear magnetic resonance was discovered already in the nineteen-forties by Felix Bloch and Edward Purcell. They received a Nobel Prize for their discovery in nineteen-fifty-two. The first human body scan using this technology was done in New York in nineteen-seventy-seven. Before I tell you how the physics of Magnetic Resonance Imaging works in detail, I first want to give you a simplified summary. If you put an atomic nucleus into a time-independent magnetic field, it can spin. And if does spin, it spins with a very specific frequency, called the Larmor frequency, named after Joseph Larmor. This frequency depends on the type of nucleus. Usually the nucleus does not spin, it just sits there. But if you, in addition to the time-independent magnetic field, let an electromagnetic wave pass by the nucleus at just exactly the right resonance frequency, then the nucleus will extract energy from the electromagnetic wave and start spinning. After the electromagnetic wave has travelled through, the nucleus will slowly stop spinning and release the energy it extracted from the wave, which you can measure. How much energy you measure depends on how many nuclei resonated with the electromagnetic wave. So, you can use the strength of the signal to tell how many nuclei of a particular type were in your sample. For magnetic resonance imaging in the human body one typically targets hydrogen nuclei, of which there are a lot in water and fat. How bright the image is then tells you basically the amount of fat and water. Though one can also target other nuclei and measure other quantities, so some magnetic resonsnce images work differently. Magnetic Resonance Imaging is particularly good for examining soft tissue, whereas for a broken bone you’d normally use an X-ray. In more detail, the physics works as follows. Atomic nuclei are made of neutrons and protons, and the neutrons and protons are each made of three quarks. Quarks have spin one half each and their spins combine to give the neutrons and protons also spin one half. The neutrons and protons then combine their spins to give a total spin to atomic nuclei, which may or may not be zero, depending on the number of neutrons and protons in the nucleus. If the spin is nonzero, then the atomic nucleus has a magnetic moment, which means it will spin in a magnetic field at a frequency that depends on the composition of the nucleus and the strength of the magnetic field. This is the Larmor frequency that nuclear spin resonance works with. If you have atomic nuclei with spin in a strong magnetic field, then their spins will align with the magnetic field. Suppose we have a constant and homogeneous magnetic field pointing into direction z, then the nuclear spins will preferably also point in direction z. They will not all do that, because there is always some thermal motion. So, some of them will align in the opposite direction, though this is not energetically the most favorable state. Just how many point in each direction depends on the temperature. The net magnetic moment of all the nuclei is then called the magnetization, and it will point in direction z. In an MRI machine, the z-direction points into the direction of the tube, so usually that’s from head to toe. Now, if the magnetization does for whatever reason not point into direction z, then it will circle around the z direction, or precess, as the physicists say, in the transverse directions, which I have called x and y. And it will do that with a very specific frequency, which is the previously mentioned Larmor frequency. The Larmor frequency depends on a constant which itself depends on the type of nucleus, and is proportional to the strength of the magnetic field. Keep this in mind because it will become important later. The key feature of magnetic resonance imaging is now that if you have a magnetization that points in direction z because of the homogenous magnetic field, and you apply an additional, transverse magnetic field that oscillates at the resonance frequency, then the magnetization will turn away from the z axis. You can calculate this with the Bloch-equation, named after the same Bloch who discovered nuclear magnetic resonance in the first place. For the following I have just integrated this differential equation. For more about differential equations, please check my earlier video. What you see here is the magnetization that points in the z-direction, so that’s the direction of the time-independent magnetic field. And now a pulse of an electromagnetic wave come through. This pulse is not at the resonance frequency. As you can see, it doesn’t do much. And here is a pulse that is at the resonance frequency. As you see, the magnetization spirals down. How far it spirals down depends on how long you apply the transverse magnetic field. Now watch what happens after this. The magnetization slowly returns to its original direction. Why does this happen? There are two things going on. One is that the nuclear spins interact with their environment, this is called spin-lattice relaxation and brings the z-direction of the magnetization back up. The other thing that happens is that the spins interact with each other, which is called spin-spin relaxation and it brings the transverse magnetization, the one in x and y direction, back to zero. Each of these processes has a characteristic decay time, usually called T_1 and T_2. For soft tissue, these decay times are typically in the range of ten milliseconds to one second. What you measure in an MRI scan is then roughly speaking the energy that is released in the return of the nuclear spins to the z-direction and the time that takes. Somewhat less roughly speaking, you measure what’s called the free induction decay. Another way to look at this process of resonance and decay is to look at the curve which the tip of the magnetization vector traces out in three dimensions. I have plotted this here for the resonant case. Again you see it spirals down during the pulse, and then relaxes back into the z-direction. So, to summarize, for magnetic resonance imaging you have a constant magnetic field in one direction, and then you have a transverse electromagnetic wave, which oscillates at the resonance frequency. For this transverse field, you only use a short pulse which makes the nuclear spins point in the transverse direction. Then they turn back to the z-direction, and you can measure this. I have left out one important thing, which is how do you manage to get a spatially resolved image and not just a count of all the nuclei. You do this by using a magnetic field with a strength that slightly changes from one place to another. Remember that I pointed out the resonance frequency is proportional to the magnetic field. Because of this, if you use a magnetic field that changes from one place to another, you can selectively target certain nuclei at a particular position. Usually one does that by using a gradient for the magnetic field, so then the images you get are slices through the body. The magnetic fields used in MRI scanners for medical purposes are incredibly strong, typically a few Tesla. For comparison, that’s about a hundred thousand times stronger than the magnetic field of planet earth, and only a factor two or three below the strength of the magnets used at the Large Hadron Collider. These strong magnetic fields do not harm the body, you just have to make sure to not take magnetic materials with you in the scanner. The resonance frequencies that fit to these strong magnetic fields are in the range of fifty to three-hundred Megahertz. These energies are far too small to break chemical bonds, which is why the electromagnetic waves used in Magnetic Resonance Imaging do not damage cells. There is however a small amount of energy deposited into the tissue by thermal motion, which can warm the tissue, especially at the higher frequency end. So one has to take care to not do these scans for a too long time. So if you have an MRI taken, remember that it literally makes your atomic nuclei spin. Saturday, November 21, 2020 Warp Drive News. Seriously! [This is a transcript of the video embedded below.] As many others, I became interested in physics by reading too much science fiction. Teleportation, levitation, wormholes, time-travel, warp drives, and all that, I thought was super-fascinating. But of course the depressing part of science fiction is that you know it’s not real. So, to some extent, I became a physicist to find out which science fiction technologies have a chance to one day become real technologies. Today I want to talk about warp drives because I think on the spectrum from fiction to science, warp drives are on the more scientific end. And just a few weeks ago, a new paper appeared about warp drives that puts the idea on a much more solid basis. But first of all, what is a warp drive? In the science fiction literature, a warp drive is a technology that allows you to travel faster than the speed of light or “superluminally” by “warping” or deforming space-time. The idea is that by warping space-time, you can beat the speed of light barrier. This is not entirely crazy, for the following reason. Einstein’s theory of general relativity says you cannot accelerate objects from below to above the speed of light because that would take an infinite amount of energy. However, this restriction applies to objects in space-time, not to space-time itself. Space-time can bend, expand, or warp at any speed. Indeed, physicists think that the universe expanded faster than the speed of light in its very early phase. General Relativity does not forbid this. There are two points I want to highlight here: First, it is a really common misunderstanding, but Einstein’s theories of special and general relativity do NOT forbid faster-than-light motion. You can very well have objects in these theories that move faster than the speed of light. Neither does this faster-than light travel necessarily lead to causality paradoxes. I explained this in an earlier video. Instead, the problem is that, according to Einstein, you cannot accelerate from below to above the speed of light. So the problem is really crossing the speed of light barrier, not being above it. The second point I want to emphasize is that the term “warp drive” refers to a propulsion system that relies on the warping of space-time, but just because you are using a warp drive does not mean you have to go faster than light. You can also have slower-than-light warp drives. I know that sounds somewhat disappointing, but I think it would be pretty cool to move around by warping spacetime at any speed. Warp drives were a fairly vague idea until in 1994, Miguel Alcubierre found a way to make them work in General Relativity. His idea is now called the Alcubierre Drive. The explanation that you usually get for how the Alcubierre Drive works, is that you contract space-time in front of you and expand it behind you, which moves you forward. That didn’t make sense to you? Just among us, it never made sense to me either. Because why would this allow you to break the speed of light barrier? Indeed, if you look at Alcubierre’s mathematics, it does not explain how this is supposed to work. Instead, his equations say that this warp drive requires large amounts of negative energy. This is bad. It’s bad because, well, there isn’t any such thing as negative energy. And even if you had this negative energy that would not explain how you break the speed of light barrier. So how does it work? A few weeks ago, someone sent me a paper that beautifully sorts out the confusion surrounding warp drives. To understand my problem with the Alcubierre Drive, I have to tell you briefly how General Relativity works. General Relativity works by solving Einstein’s field equations. Here they are. I know this looks somewhat intimidating, but the overall structure is fairly easy to understand. It helps if you try to ignore all these small Greek indices, because they really just say that there is an equation for each combination of directions in space-time. More important is that on the left side you have these R’s. The R’s quantify the curvature of space-time. And on the right side you have T. T is called the stress-energy tensor and it collects all kinds of energy densities and mass densities. That includes pressure and momentum flux and so on. Einstein’s equations then tell you that the distribution of different types of energy determines the curvature, and the curvature in return determines the how the distribution of the stress-energy changes. The way you normally solve these equations is to use a distribution of energies and masses at some initial time. Then you can calculate what the curvature is at that initial time, and you can calculate how the energies and masses will move around and how the curvature changes with that. So this is what physicists usually mean by a solution of General Relativity. It is a solution for a distribution of mass and energy. But. You can instead just take any space-time, put it into the left side of Einstein’s equations, and then the equations will tell you what the distribution of mass and energy would have to be to create this space-time. On a purely technical level, these space-times will then indeed be “solutions” to the equations for whatever is the stress energy tensor you get. The problem is that in this case, the energy distribution which is required to get a particular space-time is in general entirely unphysical. And that’s the problem with the Alcubierre Drive. It is a solution to a General Relativity, but in and by itself, this is a completely meaningless statement. Any space-time will solve the equations of General Relativity, provided you assume that you have a suitable distribution of masses and energies to create it. The real question is therefore not whether a space-time solves Einstein’s equations, but whether the distribution of mass and energy required to make it a solution to the equations is physically reasonable. And for the Alcubierre drive the answer is multiple no’s. First, as I already said, it requires negative energy. Second, it requires a huge amount of that. Third, the energy is not conserved. Instead, what you actually do when you write down the Alcubierre space-time, is that you just assume you have something that accelerates it beyond the speed of light barrier. That it’s beyond the barrier is why you need negative energies. And that it accelerates is why you need to feed energy into the system. Please check the info below the video for a technical comment about just what I mean by “energy conservation” here. Let me then get to the new paper. The new paper is titled “Introducing Physical Warp Drives” and was written by Alexey Bobrick and Gianni Martire. I have to warn you that this paper has not yet been peer reviewed. But I have read it and I am pretty confident it will make it through peer review. In this paper, Bobrick and Martire describe the geometry of a general warp-drive space time. The warp-drive geometry is basically a bubble. It has an inside region, which they call the “passenger area”. In the passenger area, space-time is flat, so there are no gravitational forces. Then the warp drive has a wall of some sort of material that surrounds the passenger area. And then it has an outside region. This outside region has the gravitational field of the warp-drive itself, but the gravitational field falls off and in the far distance one has normal, flat space-time. This is important so you can embed this solution into our actual universe. What makes this fairly general construction a warp drive is that the passage of time inside of the passenger area can be different from that outside of it. That’s what you need if you have normal objects, like your warp drive passengers, and want to move them faster than the speed of light. You cannot break the speed of light barrier for the passengers themselves relative to space-time. So instead, you keep them moving normally in the bubble, but then you move the bubble itself superluminally. As I explained earlier, the relevant question is then, what does the wall of the passenger area have to be made of? Is this a physically possible distribution of mass and energy? Bobrick and Martire explain that if you want superluminal motion, you need negative energy densities. If you want acceleration, you need to feed energy and momentum into the system. And the only reason the Alcubierre Drive moves faster than the speed of light is that one simply assumed it does. Suddenly it all makes sense! I really like this new paper because to me it has really demystified warp drives. Now, you may find this somewhat of a downer because really it says that we still do not know how to accelerate to superluminal speeds. But I think this is a big step forward because now we have a much better mathematical basis to study warp drives. For example, once you know how the warped space-time looks like, the question comes down to how much energy do you need to achieve a certain acceleration. Bobrick and Martire show that for the Alcubiere drive you can decrease the amount of energy by seating passengers next to each other instead of behind each other, because the amount of energy required depends on the shape of the bubble. The flatter it is in the direction of travel, the less energy you need. For other warp-drives, other geometries may work better. This is the kind of question you can really only address if you have the mathematics in place. Another reason I find this exciting is that, while it may look now like you can’t do superluminal warp drives, this is only correct if General Relativity is correct. And maybe it is not. Astrophysicists have introduced dark matter and dark energy to explain what they observe, but it is also possible that General Relativity is ultimately not the correct theory for space-time. What does this mean for warp drives? We don’t know. But now we know we have the mathematics to study this question. So, I think this is a really neat paper, but it also shows that research is a double-edged sword. Sometimes, if you look closer at a really exciting idea, it turns out to be not so exciting. And maybe you’d rather not have known. But I think the only way to make progress is to not be afraid of learning more. Note: This paper has not appeared yet. I will post a link here once I have a reference. You can join the chat on this video on Saturday 11/21 at 12PM EST / 6PM CET or on Sunday 11/22 at 2PM EST / 8PM CET. We will also have a chat on Black Hole Information loss on Tuesday 11/24 at 8PM EST / 2AM CET and on Wednesday 11/25 at 2PM EST / 8PM CET. Wednesday, November 18, 2020 The Black Hole information loss problem is unsolved. Because it’s unsolvable. Hi everybody, welcome and welcome back to science without the gobbledygook. I put in a Wednesday video because last week I came across a particularly bombastically nonsensical claim that I want to debunk for you. The claim is that the black hole information loss problem is “nearing its end”. So today I am here to explain why the black hole information loss problem is not only unsolved but will remain unsolved because it’s for all practical purposes unsolvable. First of all, what is the black hole information loss problem, or paradox, as it’s sometimes called. It’s an inconsistency in physicists’ currently most fundamental laws of nature, that’s quantum theory and general relativity. Stephen Hawking showed in the early nineteen-seventies that if you combine these two theories, you find that black holes emit radiation. This radiation is thermal, which means besides the temperature, that determines the average energy of the particles, the radiation is entirely random. This black hole radiation is now called Hawking Radiation and it carries away mass from the black hole. But the radius of the black hole is proportional to its mass, so if the black hole radiates, it shrinks. And the temperature is inversely proportional to the black hole mass. So, as the black hole shrinks, it gets hotter, and it shrinks even faster. Eventually, it’s completely gone. Physicists refer to this as “black hole evaporation.” When the black hole has entirely evaporated, all that’s left is this thermal radiation, which only depends on the initial mass, angular momentum, and electric charge of the black hole. This means that besides these three quantities, it does not matter what you formed the black hole from, or what fell in later, the result is the same thermal radiation. Black hole evaporation, therefore, is irreversible. You cannot tell from the final state – that’s the outcome of the evaporation – what the initial state was that formed the black holes. There are many different initial states that will give the same final state. The problem is now that this cannot happen in quantum theory. Processes in quantum theory are always time-reversible. There are certainly processes that are in practice irreversible. For example, if you mix dough. You are not going to unmix it, ever. But. According to quantum mechanics, this process is reversible, in principle. In principle, one initial state of your dough leads to exactly one final state, and using the laws of quantum mechanics you could reverse it, if only you tried hard enough, for ten to the five-hundred billion years or so. It’s the same if you burn paper, or if you die. All these processes are for all practical purposes irreversible. But according to quantum theory, they are not fundamentally irreversible, which means a particular initial state will give you one, and only one, final state. The final state, therefore, tells you what the initial state was, if you have the correct differential equation. For more about differential equations, please check my earlier video. So you set out to combine quantum theory with gravity, but you get some something that contradicts what you started with. That’s inconsistent. Something is wrong about this. But what? That’s the black hole information loss problem. Now, four points I want to emphasize here. First, the black hole information loss problem has actually nothing to do with information. John, are you listening? Really the issue is not loss of information, which is an extremely vague phrase, the issue is time irreversibility. General Relativity forces a process on you which cannot be reversed in time, and that is inconsistent with quantum theory. So it would better be called the black hole time irreversibility problem, but you know how it goes with nomenclature, it doesn’t always make sense. Peanuts aren’t nuts, vacuum cleaners don’t clean the vacuum. Dark energy is neither dark nor energy. And black hole information loss is not about information. Second, black hole evaporation is not an effect of quantum gravity. You do not need to quantize gravity to do Hawking’s calculation. It merely uses quantum mechanics in the curved background of non-quantized general relativity. Yes, it’s something with quantum and something with gravity. No, it’s not quantum gravity. The third point is that the measurement process in quantum mechanics does not resolve the black hole information loss problem. Yes, according to the Copenhagen interpretation a quantum measurement is irreversible. But the inconsistency in black hole evaporation occurs before you make a measurement. And related to this is the fourth point, it does not matter whether you believe time-irreversibility is wrong even leaving aside the measurement. It’s a mathematical inconsistency. Saying that you do not believe one or the other property of the existing theories does not explain how to get rid of the problem. So, how do you get rid of the black hole information loss problem. Well, the problem comes from combining a certain set of assumptions, doing a calculation, and arriving at a contradiction. This means any solution of the problem will come down to removing or replacing at least one of the assumptions. Mathematically there are many ways to do that. Even if you do not know anything about black holes or quantum mechanics, that much should be obvious. If you have a set of inconsistent axioms, there are many ways to fix that. It will therefore not come as a surprise to you that physicists have spent the past forty years coming up with always new “solutions” to the black hole information loss problem, yet they can’t agree which one is right. I have already made a video about possible solutions to the black hole information loss problem, so let me just summarize this really quickly. For details, please check the earlier video. The simplest solution to the black hole information loss problem is that the disagreement is resolved when the effects of quantum gravity become large, which happens when the black hole has shrunk to a very small size. This simple solution is incredibly unpopular among physicists. Why is that? It’s because we do not have a theory of quantum gravity, so one cannot write papers about it. Another option is that the black holes do not entirely evaporate and the information is kept in what’s left, usually called a black hole remnant. Yet another way to solve the problem is to simply accept that information is lost and then modify quantum mechanics accordingly. You can also put information on the singularity, because then the evaporation becomes time-reversible. Or you can modify the topology of space-time. Or you can claim that information is only lost in our universe but it’s preserved somewhere in the multiverse. Or you can claim that black holes are actually fuzzballs made of strings and information creeps out slowly. Or, you can do ‘t Hooft’s antipodal identification and claim what goes in one side comes out the other side, fourier transformed. Or you can invent non-local effects, or superluminal information exchange, or baby universes, and that’s not an exhaustive list. These solutions are all mathematically consistent. We just don’t know which one of them is correct. And why is that? It’s because we cannot observe black hole evaporation. For the black holes that we know exist the temperature is way, way too small to be observable. It’s below even the temperature of the cosmic microwave background. And even if it wasn’t, we wouldn’t be able to catch all that comes out of a black hole, so we couldn’t conclude anything from it. And without data, the question is not which solution to the problem is correct, but which one you like best. Of course everybody likes their own solution best, so physicists will not agree on a solution, not now, and not in 100 years. This is why the headline that the black hole information loss problem is “coming to an end” is ridiculous. Though, let me mention that I know the author of the piece, George Musser, and he’s a decent guy and, the way this often goes, he didn’t choose the title. What’s the essay actually about? Well, it’s about yet another proposed solution to the black hole information problem. This one is claiming that if you do Hawking’s calculation thoroughly enough then the evaporation is actually reversible. Is this right? Well, depends on whether you believe the assumptions that they made for this calculation. Similar claims have been made several times before and of course they did not solve the problem. The real problem here is that too many theoretical physicists don’t understand or do not want to understand that physics is not mathematics. Physics is science. A theory of nature needs to be consistent, yes, but consistency alone is not sufficient. You still need to go and test your theory against observations. The black hole information loss problem is not a math problem. It’s not like trying to prove the Riemann hypothesis. You cannot solve the black hole information loss problem with math alone. You need data, there is no data, and there won’t be any data. Which is why the black hole information loss problem is for all practical purposes unsolvable. The next time you read about a supposed solution to the black hole information loss problem, do not ask whether the math is right. Because it probably is, but that isn’t the point. Ask what reason do we have to think that this particular piece of math correctly describes nature. In my opinion, the black hole information loss problem is the most overhyped problem in all of science, and I say that as someone who has published several papers about it. On Saturday we’ll be talking about warp drives, so don’t forget to subscribe. Saturday, November 14, 2020 Understanding Quantum Mechanics #8: The Tunnel Effect [This is a transcript of the video embedded below. Parts of the text will not make sense without the graphics in the video.] Have you heard that quantum mechanics is impossible to understand? You know what, that’s what I was told, too, when I was a student. But twenty years later, I think the reason so many people believe one cannot understand quantum mechanics is because they are constantly being told they can’t understand it. But if you spend some time with quantum mechanics, it’s not remotely as strange and weird as they say. The strangeness only comes in when you try to interpret what it all means. And there’s no better way to illustrate this than the tunnel effect, which is what we will talk about today. Before we can talk about tunneling, I want to quickly remind you of some general properties of wave-functions, because otherwise nothing I say will make sense. The key feature of quantum mechanics is that we cannot predict the outcome of a measurement. We can only predict the probability of getting a particular outcome. For this, we describe the system we are observing – for example a particle – by a wave-function, usually denoted by the Greek letter Psi. The wave-function takes on complex values, and probabilities can be calculated from it by taking the absolute square. But how to calculate probabilities is only part of what it takes to do quantum mechanics. We also need to know how the wave-function changes in time. And we calculate this with the Schrödinger equation. To use the Schrödinger equation, you need to know what kind of particle you want to describe, and what the particle interacts with. This information goes into this thing labeled H here, which physicists call the “Hamiltonian”. To give you an idea for how this works, let us look at the simplest possible case, that’s a massive particle, without spin, that moves in one dimension, without any interaction. In this case, the Hamiltonian merely has a kinetic part which is just the second derivative in the direction the particle travels, divided by twice the mass of the particle. I have called the direction x and the mass m. If you had a particle without quantum behavior – a “classical” particle, as physicists say – that didn’t interact with anything, it would simply move at constant velocity. What happens for a quantum particle? Suppose that initially you know the position of the particle fairly well, so the probability distribution is peaked. I have plotted here an example. Now if you solve the Schrödinger equation for this initial distribution, what happens is the following. The peak of the probability distribution is moving at constant velocity, that’s the same as for the classical particle. But the width of the distribution is increasing. It’s smearing out. Why is that? That’s the uncertainty principle. You initially knew the position of the particle quite well. But because of the uncertainty principle, this means you did not know its momentum very well. So there are parts of this wave-function that have a somewhat larger momentum than the average, and therefore a larger velocity, and they run ahead. And then there are some which have a somewhat lower momentum, and a smaller velocity, and they lag behind. So the distribution runs apart. This behavior is called “dispersion”. Now, the tunnel effect describes what happens if a quantum particle hits an obstacle. Again, let us first look at what happens with a non-quantum particle. Suppose you shoot a ball in the direction of a wall, at a fixed angle. If the kinetic energy, or the initial velocity, is large enough, it will make it to the other side. But if the kinetic energy is too small, the ball will bounce off and come back. And there is a threshold energy that separates the two possibilities. What happens if you do the same with a quantum particle? This problem is commonly described by using a “potential wall.” I have to warn you that a potential wall is in general not actually a wall, in the sense that it is not made of bricks or something. It is instead just generally a barrier for which a classical particle would have to have an energy above a certain threshold. So it’s kind of like in the example I just showed with the classical particle crossing over an actual wall, but that’s really just an analogy that I have used for the purpose of visualization. Mathematically, a potential wall is just a step function that’s zero everywhere except in a finite interval. You then add this potential wall as a function to the Hamiltonian of the Schrödinger equation. Now that we have the equation in place, let us look at what the quantum particle does when it hits the wall. For this, I have numerically integrated the Schrödinger equation I just showed you. The following animations are slow-motion compared to the earlier one, which is why you cannot see that the wave-function smears out. It still does, it’s just so little that you have to look very closely to see it. It did this because it makes it easier to see what else is happening. Again, what I have plotted here is the probability distribution for the position of the particle. We will first look at the case when the energy of the quantum particle is much higher than the potential wall. As you can see, not much happens. The quantum particle goes through the barrier. It just gets a few ripples. Next we look at the case where the energy barrier of the potential wall is much, much higher than the energy of the particle. As you can see, it bounces off and comes back. This is very similar to the classical case. The most interesting case is when the energy of the particle is smaller than the potential wall but the potential wall is not extremely much higher. In this case, a classical particle would just bounce back. In the quantum case, what happens is this. As you can see, part of the wave-function makes it through to the other side, even though it’s energetically forbidden. And there is a remaining part that bounces back. Let me show you this again. Now remember that the wave-function tells you what the probability is for something to happen. So what this means is that if you shoot a particle at a wall, then quantum effects allow the particle to sometimes make it to the other side, when this should actually be impossible. The particle “tunnels” through the wall. That’s the tunnel effect. I hope that these little animations have convinced you that if you actually do the calculation, then tunneling is half as weird as they say it is. It just means that a quantum particle can do some things that a classical particle can’t do. But, wait, I forgot to tell you something... Here you see the solutions to the Schrödinger equation with and without the potential wall, but for otherwise identical particles with identical energy and momentum. Let us stop this here. If you compare the position of the two peaks, the one that tunneled and the one that never saw a wall, then the peak of the tunneled part of the wave-function has traveled a larger distance in the same time. If the particle was travelling at or very close by the speed of light, then the peak of the tunneled part of the wave-function seems to have moved faster than the speed of light. Oops. What is happening? Well, this is where the probabilistic interpretation of quantum mechanics comes to haunt you. If you look at where the faster-than light particles came from in the initial wave-function, then you find that they were the ones which had a head-start at the beginning. Because, remember, the particles did not all start from exactly the same place. They had an uncertainty in the distribution. Then again, if the wave-function really describes single particles, as most physicists today believe it does, then this explanation makes no sense. Because then only looking at parts of the wave-function is just not an allowed way to define the particle’s time of travel. So then, how do you define the time it takes a particle to travel through a wall? And can the particle really travel faster than the speed of light? That’s a question which physicists still argue about today. This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I hope this video has given you an idea how quantum mechanics works. But if you really want to understand the tunnel effect, then you have to actively engage with the subject. Brilliant is a great starting point to do exactly this. To get more background on this video’s content, I recommend you look at their courses on quantum objects, differential equations, and probabilities. To support this channel and learn more about Brilliant, go to and sign up for free. The first 200 subscribers using this link will get 20 percent off their annual premium subscription. You can join the chat on this week’s video here: • Saturday at 12PM EST / 6PM CET (link) • Sunday at 2PM EST / 8PM CET (link) Saturday, November 07, 2020 Understanding Quantum Mechanics #7: Energy Levels Today I want to tell you what these plots show. Has anybody seen them before? Yes? Atomic energy levels, right! It’s one of the most important applications of quantum mechanics. And I mean important both historically and scientifically. Today’s topic also a good opportunity to answer a question one of you asked on a previous video “Why do some equations even actually need calculating, as the answer will always be the same?” That’s a really good question. I just love it, because it would never have occurred to me. Okay, so we want to calculate what electrons do in an atom. Why is this interesting? Because what the electrons do determines the chemical properties of the elements. Basically, the behavior of the electrons explains the whole periodic table: Why do atoms come in particular groups, why do some make good magnets, why are some of them good conductors? The electrons tell you. How do you find out what the electrons do? You use quantum mechanics. Quantum mechanics, as we discussed previously, works with wave-functions, usually denoted Psi. Here is Psi. And you calculate what the wave-function does with the Schrödinger equation. Here is the Schrödinger equation. Now, the way I have written this equation here, it’s completely useless. We know what Psi is, that’s the thing we want to calculate, and we know how to take a time-derivative, but what is H? H is called the “Hamiltonian” and it contains the details about the system you want to describe. The Hamiltonian consists of two parts. The one part tells you what the particles do when you leave them alone and they don’t know anything of each other. So that would be in empty space, with no force acting on them, with no interaction. This is usually called the “kinetic” part of the Hamiltonian, or sometimes the “free” part. Then you have a second part that tells you how the particle, or particles if there are several, interact. In the simplest case, this interaction term can be written as a potential, usually denoted V. And for an electron near an atomic nucleus, the potential is just the Coulomb potential. So that’s proportional to the charge of the nucleus, and falls with one over r, where r is the distance to the center of the nucleus. There is a constant in front of this term that I have called alpha, but just what it quantifies doesn’t matter for us today. And the kinetic term, for a slow-moving particle is just the square of the spatial derivatives, up to constants. So, now we have a linear, partial differential equation that we need to solve. I don’t want to go through this calculation, because it’s not so relevant here just how to solve it, let me just say there is no magic involved. It’s pretty straight forward. But there some interesting things to learn from it. The first interesting thing you find when you solve the Schrödinger equation for electrons in a Coulomb potential is that the solutions fall apart in two different classes. The one type of solution is a wave that can propagate through all of space. We call these the “unbound states”. And the other type of solution is a localized wave, stuck in the potential of the nucleus. It just sits there while oscillating. We call these the “bound states”. The bound states have a negative energy. That’s because you need to put energy in to rip these electrons off the atom. The next interesting thing you find is that the bound states can be numbered, so you can count them. To count these states, one commonly uses, not one, but three numbers. These numbers are all integers and are usually called n, l, and m. “n” starts at 1 and then increases, and is commonly called the “principal” quantum number. “l” labels the angular momentum. It starts at zero, but it has to be smaller than n. So for n equal to one, you have only l equal to zero. For n equal to 2, l can be 0 or 1. For n equal to three, l can be zero, one or two, and so on. The third number “m” tells you what the electron does in a magnetic field, which is why it’s called the magnetic quantum number. It takes on values from minus l to l. And these three numbers, n l m, together uniquely identify the state of the electron. Let me then show you how the solutions to the Schrödinger equation look like in this case, because there are more interesting things to learn from it. The wave-functions give you a complex value for each location, and the absolute value tells you the probability of finding the electron. While the wave-function oscillates in time, the probability does not depend on time. I have here plotted the probability as a function of the radius, so I have integrated over all angular directions. This is for different principal quantum numbers n, but with l and m equal to zero. You can see that the wave-function has various maxima and minima, but with increasing n, the biggest maximum, so that’s the place you are most likely to find the electron, moves away from the center of the atom. That’s where the idea of electron “shells” comes from. It’s not wrong, but also somewhat misleading. As you can see here, the actual distribution is more complicated. A super interesting property of these probability distributions is that they are perfectly well-behaved at r equals zero. That’s interesting because, if you remember, we used a Coulomb potential that goes as 1 over r. This potential actually diverges at r equal zero. Nevertheless, the wave-functions avoids this divergence. Some people have argued that actually something similar can avoid that a singularity forms in black holes. Please check the information below the video for a reference. But these curves show only the radial direction, what about the angular direction? To show you how this looks like, I will plot the probability of finding the electron with a color code for slices through the sphere. And I will start with showing you the slices for the cases of which you just saw the curves in the radial direction, that is, different n, but with the other numbers at zero. The more red-white the color, the more likely you are to find the electron. I have kept the radius fix, so this is why the orbitals with small n only make a small blip when we scan through the middle. Here you see it again. Note how the location of the highest probability moves to a larger radius with increasing n. Then let us look at a case where l is nonzero. This is for example for n=3, l=1 and m equals plus minus 1. As you can see, the distribution splits up in several areas of high probability and now has an orientation. Here is the same for n=4, l=2, m equals plus minus 2. It may appear as if this is no longer spherically symmetric. But actually if you combine all the quantum numbers, you get back spherical symmetry, as it has to be. Another way to look at the electron probability distributions is to plot them in three dimensions. Personally I prefer the two-dimensional cuts because the color shading contains more information about the probability distribution. But since some people prefer the 3-dimensional plots, let me show you some examples. The surface you see here is the surface inside of which you will find the electron with a probability of 90%. Again you see that thinking of the electrons as sitting on “shells” doesn’t capture very well what is going on. Now that you have an idea how we calculate atomic energy levels and what they look like, let me then get to the question: Why do we calculate the same things over and over again? So, this particular calculation of the atomic energy levels was frontier research a century ago. Today students do it as an exercise. The calculations physicists now do in research in atomic physics are considerably more advanced than this example, because we have made a lot of simplifications here. First, we have neglected that the electron has a spin, though this is fairly easy to integrate. More seriously, we have assumed that the nucleus is a point. It is not. The nucleus has a finite size and it is neither perfectly spherically symmetric, nor does it have a homogeneous charge distribution, which makes the potential much more complicated. Worse, nuclei themselves have energy levels and can wobble. Then the electrons on the outer levels actually interact with the electrons in the inner levels, which we have ignored. There are further corrections from quantum field theory, which we have also ignored. Yet another thing we have ignored is that electrons in the outer shells of large atoms get corrections from special relativity. Indeed, fun fact: without special relativity, gold would not look gold. And then, for most applications it’s not energy levels of atoms that we want to know, but energy levels of molecules. This is a huge complication. The complication is not that we don’t know the equation. It’s still the Schrödinger equation. It’s also not that we don’t know how to solve it. The problem is, with the methods we currently use, doing these calculations for even moderately sized molecules, takes too long, even on supercomputers. And that’s an important problem. Because the energy levels of molecules tell you whether a substance is solid or brittle, what its color is, how good it conducts electricity, how it reacts with other molecules, and so on. This is all information you want to have. Indeed, there’s a whole research area devoted to this question, which is called “quantum chemistry”. It also one of the calculations physicists hope to speed up with quantum computers. So, why do we continue solving the same equation? Because we are improving how good the calculation is, we are developing new methods to solve it more accurately and faster, and we are applying it to new problems. Calculating the energy levels of electrons is not yesterday’s physics, it’s still cutting edge physics today. If you really want to understand how quantum mechanics works, I recommend you check out Brilliant, who have been sponsoring this video. Brilliant is a website that offers a large variety of interactive courses in mathematics and science, including quantum mechanics, and it’s a great starting point to dig deeper into the topic. For more background on what I just talked about, have a look for example at their courses on quantum objects, differential equations, and linear algebra. To support this channel and learn more about Brilliant go to and sign up for free. The first 200 subscribers using this link will get twenty percent off the annual premium subscription. Thanks for watching, see you next week. You can join the chat about this video today (Saturday, Nov 7) at 6pm CET or tomorrow at the same time.
b3f8c662c100f998	You are hereKM Course: Quantum Mechanics Department/Abbreviation: SLO/KM Year: 2020 Guarantee: 'prof. RNDr. Jan Peřina, Ph.D.' Annotation: Basic principles of quantum mechanics. Course review: 1. Photoeffect, Compton scattering, de Broglie hypothesis, plane waves 2. Schrödinger equation a. Postulates in v x-representation b. Free particle, group velocity, wave packet. 3. Time dependent and independent Schr. equation, separation of variables, stationary states a. Normalizacion, probabilistic interpretation, probability density conservation, continuity equation. b. Principle superposition 4. Simple systems and their solutions a. Infinite 1D well, generalization to 3D b. Finite square well, reflection, transmission, resonance energy. c. Finite barrier, tunneling d. Delta function potential, bound state 5. Postulates of quantum mechanics in bracket formalism, state, operators of observables, quantization. a. Observables, operators, Poisson brackets, commutation relations, superposition. b. Eigenstate and eigenvalues of position and momentum operators. X and P representations and their correspondence. c. Formal construction of QM, brackets, states, Hilbert space d. Postulates in brackets. e. Operators expectation values, matrix elements. f. Eigenstates of Hamiltonian g. Discrete and continuous spectra, normalization of momentum eigenstates to a delta function. h. Collapse of the wave function, measurement, probability amplitude. i. Ehrenfest theorems, virial theorem. 6. The uncertainty principle a. expectation values and quadratic fluctuations in statistics b. the unc. principle for non-commuting observables c. applications on a wave packed 7. Schroedinger equation revisited in brackets a. representations b. general state as a superposition c. system and observables in t > 0 8. Harmonic oscillator a. algebraic method, ladder operators, their commutators b. analytical method of Frobenius c. coherent states 9. WKM method, classical limit, alpha decay 10. Angular momentum and its addition a. commutation relations, ladder operators, the algebraic method b. spherical harmonics, parity 11. Central potential a. Correspondence of the angular momentum operator and Laplace operator b. Radial and angular equations, energy quantization c. Spherically symmetric infinite well, Bessel function, 3D harmonic oscillator. d. Hydrogen atom 12. Particle in a homogeneous electric field a. harmonic oscillator in a homog. el. field b. hydrogen atom in homog. field, Stark effect 13. Spin, particle in an electromagnetic field, Pauli equation a. Pauli matrices, spinors, spin operators to a given direction b. Larmore precession 14. Perturbation methods a. time-independent perturbation theory b. Stark effect, Zeeman effect c. correction to energy and wave function
fd97afe8323daed0	Terrifying 20m-tall rogue waves are actually real The #1 UFO Resource 1 UFOS - News - Books - Videos - Feeds 1 UFOS Search Engine is Powered by the 1 Search Project However, we now know that they are no maritime myths. A wave is a disturbance that moves energy between two points. The most familiar waves occur in water, but there are plenty of other kinds, such as radio waves that travel invisibly through the air. Although a wave rolling across the Atlantic is not the same as a radio wave, they both work according to the same principles, and the same equations can be used to describe them. This led scientists to altogether more difficult questions. Given that they exist, what causes rogue waves? More importantly for people who work at sea, can they be predicted? Until the 1990s, scientists’ ideas about how waves form at sea were heavily influenced by the work of British mathematician and oceanographer Michael Selwyn Longuet-Higgins. In work published from the 1950s onwards, he stated that, when two or more waves collide, they can combine to create a larger wave through a process called “constructive interference”. According to the principle of “linear superposition”, the height of the new wave should simply be the total of the heights of the original waves. A rogue wave can only form if enough waves come together at the same point according to this view. However, during the 1960s evidence emerged that things might not be so simple. The key player was mathematician and physicist Thomas Brooke Benjamin, who studied the dynamics of waves in a long tank of shallow water at the University of Cambridge. With his student Jim Feir, Benjamin noticed that while waves might start out with constant frequencies and wavelengths, they would change unexpectedly shortly after being generated. Those with longer wavelengths were catching those with shorter ones. This meant that a lot of the energy ended up being concentrated in large, short-lived waves. At first Benjamin and Feir assumed there was a problem with their equipment. However, the same thing happened when they repeated the experiments in a larger tank at the UK National Physical Laboratory near London. What’s more, other scientists got the same results. For many years, most scientists believed that this “Benjamin-Feir instability” only occurred in laboratory-generated waves travelling in the same direction: a rather artificial situation. However, this assumption became increasingly untenable in the face of real-life evidence. There are many more rogue waves in the oceans than linear theory predicts “Satellite measurements have shown there are many more rogue waves in the oceans than linear theory predicts,” says Amin Chabchoub of Aalto University in Finland. “There must be another mechanism involved.” In the last 20 years or so, researchers like Chabchoub have sought to explain why rogue waves are so much more common than they ought to be. Instead of being linear, as Longuet-Higgins had argued, they propose that rogue waves are an example of a non-linear system. A non-linear equation is one in which a change in output is not proportional to the change in input. If waves interact in a non-linear way, it might not be possible to calculate the height of a new wave by adding the originals together. Instead, one wave in a group might grow rapidly at the expense of others. Not everyone is convinced that Chabchoub has found the explanation When physicists want to study how microscopic systems like atoms behave over time, they often use a mathematical tool called the Schrödinger equation. It turns out that certain non-linear version of the Schrödinger equation can be used to help explain rogue wave formation. The basic idea is that, when waves become unstable, they can grow quickly by “stealing” energy from each other. Researchers have shown that the non-linear Schrödinger equation can explain how statistical models of ocean waves can suddenly grow to extreme heights, through this focusing of energy. In a 2016 study, Chabchoub applied the same models to more realistic, irregular sea-state data, and found rogue waves could still develop. “We are now able to generate realistic rogue waves in the laboratory environment, in conditions which are similar to those in the oceans,” says Chabchoub. “Having the design criteria of offshore platforms and ships being based on linear theory is no good if a non-linear system can generate rogue waves they can’t cope with.” “Chabchoub was examining isolated waves, without allowing for interference with other waves,” says optical physicist Günter Steinmeyer of the Max Born Institute in Berlin. “It’s hard to see how such interference can be avoided in real-world oceans.” In principle, it is possible to predict an ocean rogue wave Instead, Steinmeyer and his colleague Simon Birkholz looked at real-world data from different types of rogue waves. They looked at wave heights just before the 1995 rogue at the Draupner oil platform, as well as unusually bright flashes in laser beams shot into fibre optic cables, and laser beams that suddenly intensified as they exited a container of gas. Their aim was to find out whether these rogue waves were at all predictable. The pair divided their data into short segments of time, and looked for correlations between nearby segments. In other words, they tried to predict what might happen in one period of time by looking at what happened in the periods immediately before. They then compared the strengths of these correlations with those they obtained when they randomly shuffled the segments. The results, which they published in 2015, came as a surprise to Steinmeyer and Birkholz. It turned out, contrary to their expectations, that the three systems were not equally predictable. They found oceanic rogue waves were predictable to some degree: the correlations were stronger in the real-life time sequence than in the shuffled ones. There was also predictability in the anomalies observed in the laser beams in gas, but at a different level, and none in the fibre optic cables. “In principle, it is possible to predict an ocean rogue wave, but our estimate of the reliable forecast time needed is some tens of seconds, perhaps a minute at most,” says Steinmeyer. “Given that two waves in a severe North Sea storm could be separated by 10 seconds, to those who say they can build a useful device collecting data from just one point on a ship or oil platform, I’d say it’s already been invented. It’s called a window.” Most other attempts to predict rogue waves have attempted to model all the waves in a body of water and how they interact. This is an extremely complex and slow process, requiring immense computational power. Instead, Sapsis and Cousins found they could accurately predict the focusing of energy that can cause rogues, using only the measurements of the distance from the first to last waves in a group, and the height of the tallest wave in the pack. “Instead of looking at individual waves and trying to solve their dynamics, we can use groups of waves and work out which ones will undergo instabilities,” says Sapsis. He thinks his approach could allow for much better predictions. If the algorithm was combined with data from LIDAR scanning technology, Sapsis says, it could give ships and oil platforms 2-3 minutes of warning before a rogue wave formed. Others believe the emphasis on waves’ ability to catch other waves and steal their energy – which is technically called “modulation instability” – has been a red herring. “These modulation instability mechanisms have only been tested in laboratory wave tanks in which you focus the energy in one direction,” says Francesco Fedele of Georgia Tech in Atlanta. “There is no such thing as a uni-directional stormy sea. In real-life, oceans’ energy can spread laterally in a broad range of directions.” In a 2016 study, Fedele and his colleagues argued that more straightforward linear explanations can account for rogue waves after all. They used historic weather forecast data to simulate the spread of energy and ocean surface heights in the run up to the Draupner, Andrea and Killard rogue waves, which struck respectively in 1995, 2007 and 2014. Their models matched the measurements, but only when they factored in the irregular shapes of ocean waves. Because of the pull of gravity, real waves have rounded troughs and sharp peaks – unlike the perfectly smooth wave shapes used in many models. Once this was factored in, interfering waves could gain an extra 15-20% in height, Fedele found. What’s more, previous estimates of the chances of simple linear interference generating rogue waves only looked at single points in time and space, when in fact ships and oil rigs occupy large areas and are in the water for long periods. This point was highlighted in a 2016 report from the US National Transportation Safety Board, written by a group overseen by Fedele, into the sinking of an American cargo ship, the SS El Faro, on 1 October 2015, in which 33 people died. “If you account for the space-time effect properly, then the probability of encountering a rogue wave is larger,” Fedele says. Also in 2016, Steinmeyer proposed that linear interference can explain how often rogue waves are likely to form. As an alternative approach to the problem, he developed a way to calculate the complexity of ocean surface dynamics at a given location, which he calls the “effective” number of waves. “Predicting an individual rogue wave event might be hopeless or non-practical, because it requires too much data and computing power. But what if we could do a forecast in the meteorological sense?” says Steinmeyer. “Perhaps there are particular weather conditions that we can foresee that are more prone to rogue wave emergence.” Steinmeyer’s group found that rogue waves are more likely when low pressure leads to converging winds; when waves heading in different directions cross each other; when the wind changes direction over a wide range; and when certain coastal shapes and subsea topographies push waves together. They concluded that rogue waves could only occur when these and other factors combined to produce an effective number of waves of 10 or more. Steinmeyer also downplays the idea that anything other than simple interference is required for rogue wave formation, and agrees that wave shape plays a role. However, he disagrees with Fedele’s view that sharp peaks can have a significant impact on wave height. “Non-linearities have a role, but it’s a minor one,” he says. “Their main role is that ocean waves are not perfect sine waves, but have more spikey crests and depressed troughs. However, what we calculated for the Draupner wave is that the effect of non-linearities on wave height was in the order of a few tens of centimetres.” In fact, the argument over exactly why rogue waves form seems set to rumble on for some time. Part of the issue is that several kinds of scientists are studying them – experimentalists and theoreticians, specialists in optical waves and fluid dynamics – and they have not as yet done a good job of integrating their different approaches. There is no sign that a consensus is developing. 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55fce70f9ee20fde	Open main menu Faster-than-light (also superluminal or FTL) communication and travel are the conjectural propagation of information or matter faster than the speed of light. The special theory of relativity implies that only particles with zero rest mass may travel at the speed of light. Tachyons, particles whose speed exceeds that of light, have been hypothesized, but their existence would violate causality, and the consensus of physicists is that they cannot exist. On the other hand, what some physicists refer to as "apparent" or "effective" FTL[1][2][3][4] depends on the hypothesis that unusually distorted regions of spacetime might permit matter to reach distant locations in less time than light could in normal or undistorted spacetime. According to the current scientific theories, matter is required to travel at slower-than-light (also subluminal or STL) speed with respect to the locally distorted spacetime region. Apparent FTL is not excluded by general relativity; however, any apparent FTL physical plausibility is speculative. Examples of apparent FTL proposals are the Alcubierre drive and the traversable wormhole. Superluminal travel of non-informationEdit In the context of this article, FTL is the transmission of information or matter faster than c, a constant equal to the speed of light in a vacuum, which is 299,792,458 m/s (by definition of the meter) or about 186,282.397 miles per second. This is not quite the same as traveling faster than light, since: • Some processes propagate faster than c, but cannot carry information (see examples in the sections immediately following). • In some materials where light travels at speed c/n (where n is the refractive index) other particles can travel faster than c/n (but still slower than c), leading to Cherenkov radiation (see phase velocity below). Daily sky motionEdit For an earth-bound observer, objects in the sky complete one revolution around the Earth in one day. Proxima Centauri, the nearest star outside the Solar System, is about four light-years away.[5] In this frame of reference, in which Proxima Centauri is perceived to be moving in a circular trajectory with a radius of four light years, it could be described as having a speed many times greater than c as the rim speed of an object moving in a circle is a product of the radius and angular speed.[5] It is also possible on a geostatic view, for objects such as comets to vary their speed from subluminal to superluminal and vice versa simply because the distance from the Earth varies. Comets may have orbits which take them out to more than 1000 AU.[6] The circumference of a circle with a radius of 1000 AU is greater than one light day. In other words, a comet at such a distance is superluminal in a geostatic, and therefore non-inertial, frame. Light spots and shadowsEdit If a laser beam is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c.[7] Similarly, a shadow projected onto a distant object can be made to move across the object faster than c.[7] In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light.[7][8][9] An analogy can be made to pointing a water hose in one direction and then quickly moving the hose to point the stream of water in another direction. At no point does the water leaving the hose ever increase in velocity, but the endpoint of the stream can be moved faster than the water in the stream itself. Apparent FTL propagation of static field effectsEdit Closing speedsEdit Special relativity does not prohibit this. It tells us that it is wrong to use Galilean relativity to compute the velocity of one of the particles, as would be measured by an observer traveling alongside the other particle. That is, special relativity gives the correct velocity-addition formula for computing such relative velocity. It is instructive to compute the relative velocity of particles moving at v and −v in accelerator frame, which corresponds to the closing speed of 2v > c. Expressing the speeds in units of c, β = v/c: Proper speedsEdit Possible distance away from EarthEdit Since one might not travel faster than light, one might conclude that a human can never travel further from the Earth than 40 light-years if the traveler is active between the age of 20 and 60. A traveler would then never be able to reach more than the very few star systems which exist within the limit of 20–40 light-years from the Earth. This is a mistaken conclusion: because of time dilation, the traveler can travel thousands of light-years during their 40 active years. If the spaceship accelerates at a constant 1 g (in its own changing frame of reference), it will, after 354 days, reach speeds a little under the speed of light (for an observer on Earth), and time dilation will increase their lifespan to thousands of Earth years, seen from the reference system of the Solar System, but the traveler's subjective lifespan will not thereby change. If the traveler returns to the Earth, they will land thousands of years into the Earth's future. Their speed will not be seen as higher than the speed of light by observers on Earth, and the traveler will not measure their speed as being higher than the speed of light, but will see a length contraction of the universe in their direction of travel. And as the traveler turns around to return, the Earth will seem to experience much more time than the traveler does. So, while their (ordinary) coordinate speed cannot exceed c, their proper speed (distance as seen by Earth divided by their proper time) can be much greater than c. This is seen in statistical studies of muons traveling much further than c times their half-life (at rest), if traveling close to c.[10] Phase velocities above cEdit The phase velocity of an electromagnetic wave, when traveling through a medium, can routinely exceed c, the vacuum velocity of light. For example, this occurs in most glasses at X-ray frequencies.[11] However, the phase velocity of a wave corresponds to the propagation speed of a theoretical single-frequency (purely monochromatic) component of the wave at that frequency. Such a wave component must be infinite in extent and of constant amplitude (otherwise it is not truly monochromatic), and so cannot convey any information.[12] Thus a phase velocity above c does not imply the propagation of signals with a velocity above c.[13] Group velocities above cEdit The group velocity of a wave may also exceed c in some circumstances.[14][15] In such cases, which typically at the same time involve rapid attenuation of the intensity, the maximum of the envelope of a pulse may travel with a velocity above c. However, even this situation does not imply the propagation of signals with a velocity above c,[16] even though one may be tempted to associate pulse maxima with signals. The latter association has been shown to be misleading, because the information on the arrival of a pulse can be obtained before the pulse maximum arrives. For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind (distortion), the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than c without this effect.[17] However, group velocity can exceed c in some parts of a Gaussian beam in a vacuum (without attenuation). The diffraction causes the peak of the pulse to propagate faster, while overall power does not.[18] Universal expansionEdit History of the Universe - gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang.[19][20][21] Astronomical observationsEdit Apparent superluminal motion is observed in many radio galaxies, blazars, quasars, and recently also in microquasars. The effect was predicted before it was observed by Martin Rees[clarification needed] and can be explained as an optical illusion caused by the object partly moving in the direction of the observer,[28] when the speed calculations assume it does not. The phenomenon does not contradict the theory of special relativity. Corrected calculations show these objects have velocities close to the speed of light (relative to our reference frame). They are the first examples of large amounts of mass moving at close to the speed of light.[29] Earth-bound laboratories have only been able to accelerate small numbers of elementary particles to such speeds. Quantum mechanicsEdit There have been various reports in the popular press of experiments on faster-than-light transmission in optics — most often in the context of a kind of quantum tunnelling phenomenon. Usually, such reports deal with a phase velocity or group velocity faster than the vacuum velocity of light.[32][33] However, as stated above, a superluminal phase velocity cannot be used for faster-than-light transmission of information.[34][35] Hartman effectEdit The Hartman effect is the tunneling effect through a barrier where the tunneling time tends to a constant for large barriers.[36][37] This could, for instance, be the gap between two prisms. When the prisms are in contact, the light passes straight through, but when there is a gap, the light is refracted. There is a non-zero probability that the photon will tunnel across the gap rather than follow the refracted path. For large gaps between the prisms the tunnelling time approaches a constant and thus the photons appear to have crossed with a superluminal speed.[38] However, the Hartman effect cannot actually be used to violate relativity by transmitting signals faster than c, because the tunnelling time "should not be linked to a velocity since evanescent waves do not propagate".[39] The evanescent waves in the Hartman effect are due to virtual particles and a non-propagating static field, as mentioned in the sections above for gravity and electromagnetism. Casimir effectEdit EPR paradoxEdit An experiment performed in 1997 by Nicolas Gisin has demonstrated non-local quantum correlations between particles separated by over 10 kilometers.[40] But as noted earlier, the non-local correlations seen in entanglement cannot actually be used to transmit classical information faster than light, so that relativistic causality is preserved. The situation is akin to sharing a synchronized coin flip, where the second person to flip their coin will always see the opposite of what the first person sees, but neither has any way of knowing whether they were the first or second flipper, without communicating classically. See No-communication theorem for further information. A 2008 quantum physics experiment also performed by Nicolas Gisin and his colleagues has determined that in any hypothetical non-local hidden-variable theory, the speed of the quantum non-local connection (what Einstein called "spooky action at a distance") is at least 10,000 times the speed of light.[41] Delayed choice quantum eraserEdit The delayed-choice quantum eraser is a version of the EPR paradox in which the observation (or not) of interference after the passage of a photon through a double slit experiment depends on the conditions of observation of a second photon entangled with the first. The characteristic of this experiment is that the observation of the second photon can take place at a later time than the observation of the first photon,[42] which may give the impression that the measurement of the later photons "retroactively" determines whether the earlier photons show interference or not, although the interference pattern can only be seen by correlating the measurements of both members of every pair and so it can't be observed until both photons have been measured, ensuring that an experimenter watching only the photons going through the slit does not obtain information about the other photons in an FTL or backwards-in-time manner.[43][44] Superluminal communicationEdit • Either way, such acceleration requires infinite energy. • Some observers with sub-light relative motion will disagree about which occurs first of any two events that are separated by a space-like interval.[45] In other words, any travel that is faster-than-light will be seen as traveling backwards in time in some other, equally valid, frames of reference,[46] or need to assume the speculative hypothesis of possible Lorentz violations at a presently unobserved scale (for instance the Planck scale).[citation needed] Therefore, any theory which permits "true" FTL also has to cope with time travel and all its associated paradoxes,[47] or else to assume the Lorentz invariance to be a symmetry of thermodynamical statistical nature (hence a symmetry broken at some presently unobserved scale). • In special relativity the coordinate speed of light is only guaranteed to be c in an inertial frame; in a non-inertial frame the coordinate speed may be different from c.[48] In general relativity no coordinate system on a large region of curved spacetime is "inertial", so it is permissible to use a global coordinate system where objects travel faster than c, but in the local neighborhood of any point in curved spacetime we can define a "local inertial frame" and the local speed of light will be c in this frame,[49] with massive objects moving through this local neighborhood always having a speed less than c in the local inertial frame. Relative permittivity or permeability less than 1Edit The speed of light is related to the vacuum permittivity ε0 and the vacuum permeability μ0. Therefore, not only the phase velocity, group velocity, and energy flow velocity of electromagnetic waves but also the velocity of a photon can be faster than c in a special material has the constant permittivity or permeability whose value is less than that in vacuum.[50] Casimir vacuum and quantum tunnellingEdit The experimental determination has been made in vacuum. However, the vacuum we know is not the only possible vacuum which can exist. The vacuum has energy associated with it, called simply the vacuum energy, which could perhaps be altered in certain cases.[51] When vacuum energy is lowered, light itself has been predicted to go faster than the standard value c. This is known as the Scharnhorst effect. Such a vacuum can be produced by bringing two perfectly smooth metal plates together at near atomic diameter spacing. It is called a Casimir vacuum. Calculations imply that light will go faster in such a vacuum by a minuscule amount: a photon traveling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 1036.[52] Accordingly, there has as yet been no experimental verification of the prediction. A recent analysis[53] argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signalling. However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze. Other authors argue that Scharnhorst's original analysis, which seemed to show the possibility of faster-than-c signals, involved approximations which may be incorrect, so that it is not clear whether this effect could actually increase signal speed at all.[54] The physicists Günter Nimtz and Alfons Stahlhofen, of the University of Cologne, claim to have violated relativity experimentally by transmitting photons faster than the speed of light.[38] They say they have conducted an experiment in which microwave photons — relatively low-energy packets of light — travelled "instantaneously" between a pair of prisms that had been moved up to 3 ft (1 m) apart. Their experiment involved an optical phenomenon known as "evanescent modes", and they claim that since evanescent modes have an imaginary wave number, they represent a "mathematical analogy" to quantum tunnelling.[38] Nimtz has also claimed that "evanescent modes are not fully describable by the Maxwell equations and quantum mechanics have to be taken into consideration."[55] Other scientists such as Herbert G. Winful and Robert Helling have argued that in fact there is nothing quantum-mechanical about Nimtz's experiments, and that the results can be fully predicted by the equations of classical electromagnetism (Maxwell's equations).[56][57] Nimtz told New Scientist magazine: "For the time being, this is the only violation of special relativity that I know of." However, other physicists say that this phenomenon does not allow information to be transmitted faster than light. Aephraim Steinberg, a quantum optics expert at the University of Toronto, Canada, uses the analogy of a train traveling from Chicago to New York, but dropping off train cars from the tail at each station along the way, so that the center of the ever-shrinking main train moves forward at each stop; in this way, the speed of the center of the train exceeds the speed of any of the individual cars.[58] It was later claimed by Eckle et al. that particle tunneling does indeed occur in zero real time.[62] Their tests involved tunneling electrons, where the group argued a relativistic prediction for tunneling time should be 500–600 attoseconds (an attosecond is one quintillionth (10−18) of a second). All that could be measured was 24 attoseconds, which is the limit of the test accuracy. Again, though, other physicists believe that tunneling experiments in which particles appear to spend anomalously short times inside the barrier are in fact fully compatible with relativity, although there is disagreement about whether the explanation involves reshaping of the wave packet or other effects.[59][60][63] Give up (absolute) relativityEdit Spacetime distortionEdit Gerald Cleaver and Richard Obousy, a professor and student of Baylor University, theorized that manipulating the extra spatial dimensions of string theory around a spaceship with an extremely large amount of energy would create a "bubble" that could cause the ship to travel faster than the speed of light. To create this bubble, the physicists believe manipulating the 10th spatial dimension would alter the dark energy in three large spatial dimensions: height, width and length. Cleaver said positive dark energy is currently responsible for speeding up the expansion rate of our universe as time moves on.[66] Heim theoryEdit Lorentz symmetry violationEdit The possibility that Lorentz symmetry may be violated has been seriously considered in the last two decades, particularly after the development of a realistic effective field theory that describes this possible violation, the so-called Standard-Model Extension.[68][69][70] This general framework has allowed experimental searches by ultra-high energy cosmic-ray experiments[71] and a wide variety of experiments in gravity, electrons, protons, neutrons, neutrinos, mesons, and photons.[72] The breaking of rotation and boost invariance causes direction dependence in the theory as well as unconventional energy dependence that introduces novel effects, including Lorentz-violating neutrino oscillations and modifications to the dispersion relations of different particle species, which naturally could make particles move faster than light. In some models of broken Lorentz symmetry, it is postulated that the symmetry is still built into the most fundamental laws of physics, but that spontaneous symmetry breaking of Lorentz invariance[73] shortly after the Big Bang could have left a "relic field" throughout the universe which causes particles to behave differently depending on their velocity relative to the field;[74] however, there are also some models where Lorentz symmetry is broken in a more fundamental way. If Lorentz symmetry can cease to be a fundamental symmetry at Planck scale or at some other fundamental scale, it is conceivable that particles with a critical speed different from the speed of light be the ultimate constituents of matter. In current models of Lorentz symmetry violation, the phenomenological parameters are expected to be energy-dependent. Therefore, as widely recognized,[75][76] existing low-energy bounds cannot be applied to high-energy phenomena; however, many searches for Lorentz violation at high energies have been carried out using the Standard-Model Extension.[72] Lorentz symmetry violation is expected to become stronger as one gets closer to the fundamental scale. Superfluid theories of physical vacuumEdit In this approach the physical vacuum is viewed as the quantum superfluid which is essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of nature but rather the approximate description valid only for the small fluctuations of the superfluid background.[77] Within the framework of the approach a theory was proposed in which the physical vacuum is conjectured to be the quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger equation. It was shown that the relativistic gravitational interaction arises as the small-amplitude collective excitation mode[78] whereas relativistic elementary particles can be described by the particle-like modes in the limit of low momenta.[79] The important fact is that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one - they can reach the speed of light limit at finite energy; also, faster-than-light propagation is possible without requiring moving objects to have imaginary mass.[80][81] Time of flight of neutrinosEdit MINOS experimentEdit In 2007 the MINOS collaboration reported results measuring the flight-time of 3 GeV neutrinos yielding a speed exceeding that of light by 1.8-sigma significance.[82] However, those measurements were considered to be statistically consistent with neutrinos traveling at the speed of light.[83] After the detectors for the project were upgraded in 2012, MINOS corrected their initial result and found agreement with the speed of light. Further measurements are going to be conducted.[84] OPERA neutrino anomalyEdit On September 22, 2011, a preprint[85] from the OPERA Collaboration indicated detection of 17 and 28 GeV muon neutrinos, sent 730 kilometers (454 miles) from CERN near Geneva, Switzerland to the Gran Sasso National Laboratory in Italy, traveling faster than light by a relative amount of 2.48×10−5 (approximately 1 in 40,000), a statistic with 6.0-sigma significance.[86] On 17 November 2011, a second follow-up experiment by OPERA scientists confirmed their initial results.[87][88] However, scientists were skeptical about the results of these experiments, the significance of which was disputed.[89] In March 2012, the ICARUS collaboration failed to reproduce the OPERA results with their equipment, detecting neutrino travel time from CERN to the Gran Sasso National Laboratory indistinguishable from the speed of light.[90] Later the OPERA team reported two flaws in their equipment set-up that had caused errors far outside their original confidence interval: a fiber optic cable attached improperly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast.[91] Various theorists have suggested that the neutrino might have a tachyonic nature,[94][95][96][97] while others have disputed the possibility.[98] Exotic matterEdit Mechanical equations to describe hypothetical exotic matter which possesses a negative mass, negative momentum, negative pressure, and negative kinetic energy are[99] Considering and , the energy-momentum relation of the particle is corresponding to the following dispersion relation of a wave that can propagate in the negative-index metamaterial. The pressure of radiation pressure in the metamaterial is negative[100] and negative refraction, inverse Doppler effect and reverse Cherenkov effect imply that the momentum is also negative. So the wave in a negative-index metamaterial can be applied to test the theory of exotic matter and negative mass. For example, the velocity equals That is to say, such a wave can break the light barrier under certain conditions. General relativityEdit Variable speed of lightEdit In physics, the speed of light in a vacuum is assumed to be a constant. However, hypotheses exist that the speed of light is variable. The speed of light is a dimensional quantity and so cannot be measured.[clarification needed][102] Measurable quantities in physics are, without exception, dimensionless, although they are often constructed as ratios of dimensional quantities. For example, when the height of a mountain is measured, what is really measured is the ratio of its height to the length of a meter stick. The conventional SI system of units is based on seven basic dimensional quantities, namely distance, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.[103] These units are defined to be independent and so cannot be described in terms of each other. As an alternative to using a particular system of units, one can reduce all measurements to dimensionless quantities expressed in terms of ratios between the quantities being measured and various fundamental constants such as Newton's constant, the speed of light and Planck's constant; physicists can define at least 26 dimensionless constants which can be expressed in terms of these sorts of ratios and which are currently thought to be independent of one another.[104] By manipulating the basic dimensional constants one can also construct the Planck time, Planck length, and Planck energy which make a good system of units for expressing dimensional measurements, known as Planck units. 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b514a46513f30d4b	Journal cover Journal topic Annales Geophysicae An interactive open-access journal of the European Geosciences Union Journal topic ANGEO \| Articles \| Volume 36, issue 4 Ann. Geophys., 36, 1015–1026, 2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. Ann. Geophys., 36, 1015–1026, 2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. Regular paper 26 Jul 2018 Regular paper \| 26 Jul 2018 The mirror mode: a “superconducting” space plasma analogue The mirror mode: a “superconducting” space plasma analogue Rudolf A. Treumann1 and Wolfgang Baumjohann2 Rudolf A. Treumann and Wolfgang Baumjohann • 1International Space Science Institute, Bern, Switzerland • 2Space Research Institute, Austrian Academy of Sciences, Graz, Austria Correspondence: Wolfgang Baumjohann ( Back to toptop We examine the physics of the magnetic mirror mode in its final state of saturation, the thermodynamic equilibrium, to demonstrate that the mirror mode is the analogue of a superconducting effect in a classical anisotropic-pressure space plasma. Two different spatial scales are identified which control the behaviour of its evolution. These are the ion inertial scale λim(τ) based on the excess density Nm(τ) generated in the mirror mode, and the Debye scale λD(τ). The Debye length plays the role of the correlation length in superconductivity. Their dependence on the temperature ratio τ=T/T<1 is given, with T the reference temperature at the critical magnetic field. The mirror-mode equilibrium structure under saturation is determined by the Landau–Ginzburg ratio κD=λim/λD, or κρ=λim/ρ, depending on whether the Debye length or the thermal-ion gyroradius ρ – or possibly also an undefined turbulent correlation length turb – serve as correlation lengths. Since in all space plasmas κD≫1, plasmas with λD as the relevant correlation length always behave like type II superconductors, naturally giving rise to chains of local depletions of the magnetic field of the kind observed in the mirror mode. In this way they would provide the plasma with a short-scale magnetic bubble texture. The problem becomes more subtle when ρ is taken as correlation length. In this case the evolution of mirror modes is more restricted. Their existence as chains or trains of larger-scale mirror bubbles implies that another threshold, VA>υth, is exceeded. Finally, in case the correlation length turb instead results from low-frequency magnetic/magnetohydrodynamic turbulence, the observation of mirror bubbles and the measurement of their spatial scales sets an upper limit on the turbulent correlation length. This might be important in the study of magnetic turbulence in plasmas. 1 Introduction Back to toptop Under special conditions high-temperature collisionless plasmas may develop properties which resemble those of superconductors. This is the case with the mirror mode when the anisotropic pressure gives rise to local depletions of the magnetic field similar to the Meissner effect in metals where it signals the onset of superconductivity (Fetter and Walecka1971; Huang1987; Kittel1963; Lifshitz and Pitaevskii1998), i.e. the suppression of friction between the current and the lattice. In collisionless plasmas there is no lattice, the plasma is frictionless, and thus it already is ideally conducting which, however, does not mean that it is superconducting. To be superconducting, additional properties are required. These, as we show below, are given in the saturation state of the mirror mode. The mirror mode is a non-oscillatory plasma instability (Chandrasekhar1961; Gary1993; Hasegawa1969; Kivelson and Southwood1996; Southwood and Kivelson1993) which evolves in anisotropic plasmas (Sulem2011). It has been argued that it should readily saturate by quasilinear depletion of the temperature anisotropy (Noreen et al.2017). Observations do not support this conclusion. In fact, we recently argued (Treumann and Baumjohann2018a) that the large amplitudes of mirror-mode oscillations observed in the Earth's magnetosheath, magnetotail, and elsewhere, like other planetary magnetosheaths, in the solar wind and generally in the heliosphere (Constantinescu et al.2003; Czaykowska et al.1998; Lucek et al.1999a, b; Tsurutani et al.1982, 2011; Volwerk et al.2008; Zhang et al.2008, 2009, 1998), are a sign of the impotence of quasilinear theory in limiting the growth of the mirror instability. Instead, mirror modes should be subject to weak kinetic turbulence theory (Davidson1972; Sagdeev and Galeev1969; Tsytovich1977; Yoon2007, 2018; Yoon and Fang2007), which allows them to evolve until they become comparable in amplitude to the ambient magnetic field long before any dissipation can set in. This is not unreasonable, because all those plasmas where the mirror instability evolves are ideal conductors on the scales of the plasma flow. On the other hand, no weak turbulence theory of the mirror mode is available yet as it is difficult to identify the various modes which interact to destroy quasilinear quenching. The frequent claim that whistlers (lion roars) excited in the trapped electron component would destroy the bulk (global) temperature anisotropy is erroneous, because whistlers (Baumjohann et al.1999; Maksimovic et al.2001; Thorne and Tsurutani1981; Zhang et al.1998) grow at the expense of a small component of anisotropic resonant particles only (Kennel and Petschek1966). Depletion of the resonant anisotropy by no means affects the bulk temperature anisotropy that is responsible for the evolution of the mirror instability. On the other hand, construction of a weak turbulence theory of the mirror mode poses serious problems. One therefore needs to refer to other means of description of its final saturation state. Since measurements suggest that the observed mirror modes are about stationary phenomena which are swept over the spacecraft at high flow speeds (called Taylor's hypothesis, though, in principle, it just refers to the Galilei transformation), it seems reasonable to tackle them within a thermodynamic approach, i.e. assuming that in the observed large-amplitude saturation state they can be described as the stationary state of interaction between the ideally conducting plasma and magnetic field. This can be most efficiently done when the free energy of the plasma is known, which, unfortunately, is not the case. Magnetohydrodynamics does not apply, and the formulation of a free energy in the kinetic state is not available. For this reason we refer to some phenomenological approach which is guided by the phenomenological theory of superconductivity. There we have the similar phenomenon that the magnetic field is expelled from the medium due to internal quantum interactions, known as the Meissner effect. This resembles the evolution of the mirror mode, though in our case the interactions are not in the quantum domain. This is easily understood when considering the thermal length λ=2π2/meT and comparing it to the shortest plasma scale, viz. the inter-particle distance dNN-13. The former is, for all plasma temperatures T, in the atomic range, while the latter in space plasmas for all densities N is at least several orders of magnitude larger. Plasmas are classical. In their equilibrium state classical thermodynamics applies to them. In the following we boldly ask for the thermodynamic equilibrium state of a mirror unstable plasma. For other non-thermodynamical attempts at modelling the equilibrium configuration of magnetic mirror modes and application to multi-spacecraft observations, the reader may consult Constantinescu (2002) and Constantinescu et al. (2003). Such an approach is rather alien to space physics. It follows the path prescribed in solid-state physics but restricts itself to the domain of classical thermodynamics only. 2 Properties of the mirror instability Back to toptop The mirror instability evolves whence the magnetic field B in a collisionless magnetized plasma with an internal pressure–temperature anisotropy T>T, where the subscripts refer to the directions perpendicular and parallel to the ambient magnetic field, drops below a critical value where Θj=(T/T-1)j>0 is the temperature anisotropy of species j=e,i (for ions and electrons) and θ is the angle of propagation of the wave with respect to the ambient magnetic field (Treumann and Baumjohann2018a). Here any possible temperature anisotropy in the electron population has been included, but will be dropped below as it seems (Yoon and López2017) that it does not provide any further insight into the physics of the final state of the mirror mode. The important observation is that the existence of the mirror mode depends on the temperature difference T-T and the critical magnetic field. Commonly only the temperature anisotropy is reclaimed as being responsible for the growth of the mirror mode. Though this is true, it also implies the above condition on the magnetic field. To some degree this resembles the behaviour of magnetic fields under superconducting conditions. To demonstrate this, we take T as the reference – or critical – temperature. The critical magnetic field becomes a function of the temperature ratio τ=T/T. Once τ<1 and B<Bcrit the magnetic field will be pushed out of the plasma to give space to an accumulated plasma density and also weak diamagnetic surface currents on the boundaries of the (partially) field-evacuated domain. The τ dependence of the critical magnetic field can be cast into the form which indeed resembles that in the phenomenological theory of superconductivity. Here and the critical threshold vanishes for τ=1 where the range of possible unstable magnetic field values shrinks to zero; the limits T=0 or T= make no physical sense. Though the effects are similar to superconductivity, the temperature dependence is different from that of the Meissner effect in metals in their isotropic low-temperature super-conducting phase. In contrast, in an anisotropic plasma the effect occurs in the high-temperature phase only while being absent at low temperatures. Nevertheless, the condition τ<1 indicates that the mirror mode, concerning the ratio of parallel to perpendicular temperatures, is a low-temperature effect in the high-temperature plasma phase. This may suggest that even in metals high-temperature superconductivity might be achieved more easily for anisotropic temperatures, a point we will follow elsewhere (Treumann and Baumjohann2018b). Since the plasma is ideally conducting, any quasi-stationary magnetic field is subject to the penetration depth, which is the inertial scale λim=c/ωim, with ωim2=e2Nm/ϵ0mi based on the density Nm of the plasma component involved in the mirror effect. The mirror instability is a slow purely growing instability with real frequency ω≈0. At these low frequencies the plasma is quasi-neutral. In metallic superconductivity this length is the London penetration depth which refers to electrons as the ions are fixed to the lattice. Here, in the space plasma, it is rather the ion scale because the dominant mirror effect is caused by mobile ions in the absence of any crystal lattice. Such a “magnetic lattice” structure is ultimately provided under conditions investigated below by the saturated state of the mirror mode, where it collectively affects the trapped ion component on scales of an internal correlation length. 3 Free energy Back to toptop In the thermodynamic equilibrium state the quantity which describes the matter in the presence of a magnetic field B is the Landau–Gibbs free energy density where FL is the Landau free energy density (Kittel and Kroemer1980) which, unfortunately, is not known. In magnetohydrodynamics it can be formulated but becomes a messy expression which contains all stationary, i.e. time-averaged, nonlinear contributions of low-frequency electromagnetic plasma waves and thermal fluctuations. The total Landau–Gibbs free energy is the volume integral of this quantity over all space. In thermodynamic equilibrium this is stationary, and one has In order to restrict to our case we assume that FL in the above expression, which contains the full dynamics of the plasma matter, can be expanded with respect to the normalized density Nm<1 of the plasma component which participates in the mirror instability: with F0 the Helmholtz free energy density, which is independent of Nm corresponding to the normal (or mirror stable) state. Normalization is to the ambient density N0, thus attributing the dimension of energy density to the expansion coefficients a and b. An expansion like this one is always possible in the spirit of a perturbation approach as long as the total density N/N0=1+Nm with \|Nm\|<1. It is thus clear that Nm is not the total ambient plasma density N0, which is itself in pressure equilibrium with the ambient field B0 under static conditions expressed by N0T=B02/2μ0 under the assumption that no static current J0 flows in the medium. Otherwise its Lorentz force J0×B0=-TN0 is compensated for by the pressure gradient force already in the absence of the mirror mode and includes the magnetic stresses generated by the current. This case includes a stationary contribution of the free energy F0 around which the mirror state has evolved. Regarding the presence of the mirror mode, we know that it must also be in balance between the local plasma gradient Nm of the fluctuating pressure and the induced magnetic pressure (δB)2∕2μ0. Note that all quantities are stationary; the prefix δ refers to deviations from “normal” thermodynamic equilibrium, not to variations. Moreover, we have Maxwell's equations which in the stationary state reduce to accounting for the vanishing divergence by introducing the fluctuating vector potential A (where we drop the δ prefix on the vector potential). This enables us to write the kinetic part of the free energy of the particles involved in the canonical operator form referring to ions of positive charge q>0, and the constant α naturally has the dimension of a classical action. (There is a little problem as to what is meant by the mass m in this expression, to which we will briefly return below.) In this form the momentum acts on a complex dimensionless “wave function” ψ(x) whose square we below identify with the above-used normalized excess in plasma density known to be present locally in any of the mirror-mode bubbles. Unlike quantum theory, ψ(x) is not a single-particle wave function: it rather applies to a larger compound of trapped particles (ions) in the mirror modes which behave similarly and are bound together by some correlation length (a very important parameter, which is to be discussed later). It enters the expression for the free energy density, thus providing the units of energy density to the expansion coefficients a,b. In the quantum case (as for instance in the theory of superconductivity), we would have α=ℏ; in the classical case considered here, α remains undetermined until a connection to the mirror mode is obtained. Clearly, α≫ℏ cannot be very small because the gradient and the corresponding wave vector k involved in the operation are of the scale of the inverse ion gyroradius in the mirror mode. Hence, we suspect that αT/ωp, where T is a typical plasma temperature (in energy units), and ωp is a typical frequency of collective ion oscillations in the plasma. Any such oscillations naturally imply the existence of correlations which bind the particles (ions) to exert a collective motion and which give rise to the field A and density fluctuations δN. Such frequencies can be either plasma ωp=ωi=eN/ϵ0m, cyclotron ωc=eB/m frequencies, or some unknown average turbulent frequency ωturb on turbulence scales shorter than the typical average mirror-mode scale. For the ion mirror mode the choice is that q+e, and mmi. Inspecting Eq. (8) we will run into difficulties when assuming q=e and m=mi because with a large number of particles collectively participating, each contributing a charge e and mass m, the ratio q2m will be proportional to the number of particles. In superconductivity this provides no problem because pairing of electrons tells that mass and charge just double, which is compensated for in Eq. (8) by m→2m. Similarly, in the case of the mirror mode we have for the normalized density excess Nm=δN/Nζ<1, where 𝒩 is the total particle number, and δ𝒩 its excess. We thus identify an effective mass mΔmi, where Δ=1+ζ. Because of the restriction on ζ<1 this yields for the effective mass in mirror modes the preliminary range which is similar to the mass in metallic superconductivity. However, each mirror bubble contains a different number δ𝒩 of trapped particles. Hence ζ(x) becomes a function of space x which varies along the mirror chain, and Δ(x) then becomes a function of space. The restriction on ζ<1 makes this variation weak. For an observed chain of mirror modes one defines some mean effective mass meff by Averaging reduces Δ, making the effective mass closer to the lower bound mi, which is to be used below for mmeff wherever the mass appears. Retaining the quantum action and dividing by the charge q, the factor of the Nabla operator becomes /q=Φ0e/2πq. Hence, α is proportional to the number ν=Φ/Φ0 of elementary flux elements in the ion-gyro cross section, which in a plasma is a large number due to the high temperature T. This makes α≫ℏ. With these assumptions in mind we can write for the free energy density up to second order in Nm: Inserted into the Gibbs free energy density, the last term is absorbed by the Gibbs potential. Applying the Hamiltonian prescription and varying the Gibbs free energy with respect to A and ψ,ψ yields (for arbitrary variations) an equation for the “wave function” ψ(x): which is recognized as a nonlinear complex Schrödinger equation. Such equations appear in plasma physics whence waves undergo modulation instability and evolve towards the general family of solitary structures. It is known that the nonlinear Schrödinger equation can be solved by inverse scattering methods and, under certain conditions, yields either single solitons or trains of solitary solutions. To our knowledge, the nonlinear Schrödinger equation has not yet been derived for the mirror instability because no slow wave is known which would modulate its amplitude. Whether this is possible is an open question which we will not follow up here. Hence the quantity α remains undetermined for the mirror mode. Instead, we chose a phenomenological approach which is suggested by the similarity of both the mirror-mode effect in ideally conducting plasma and the above-obtained nonlinear Schrödinger equation to the phenomenological Landau–Ginzburg theory of metallic superconductivity. In the thermodynamic equilibrium state the above equation does not describe the mirror-mode amplitude itself. Rather it describes the evolution of the “wave function” of the compound of particles trapped in the mirror-mode magnetic potential A which it modulates. This differs from superconductivity where we have pairing of particles and escape from collisions with the lattice and superfluidity of the paired particle population at low temperatures. In the ideally conducting plasma we have no collisions, but, under normal conditions, also no pairing and no superconductivity, though in the presence of some particular plasma waves, attractive forces between neighbouring electrons can sometimes evolve (Treumann and Baumjohann2014). In superconductivity the pairing implies that the particles become correlated, an effect which in plasma must also happen whence the superconducting mirror-mode Meissner effect occurs, but it happens in a completely different way by correlating large numbers of particles, as we will exemplify further below. The wave function ψ(x) describes only the trapped particle component which is responsible for the maintenance of the pressure equilibrium between the magnetic field and plasma. In a bounded region one must add boundary conditions to the above equation which imply that the tangential component of the magnetic field is continuous at the boundary and the normal components of the electric currents vanish at the boundary because the current has no divergence. The current, normalized to N0, is then given by which shows that the main modulated contribution to the current is provided by the last term, the product of the mirror particle density \|ψ\|2=Nm times the vector potential fluctuation A, which is the mutual interaction term between the density and magnetic fields. One may note that the vector potential from Maxwell's equations is directly related to the magnetic flux Φ in the wave flux tube of radius R through its circumference A=Φ/2πR. One also observes that under certain conditions in the last expression for the current density the two gradient terms of the ψ function partially cancel. Assuming ψ=\|ψ(x)\|e-ikx, the current term becomes The first term is small in the long-wavelength domain kα≪1. Assuming that this is the case for the mirror mode, which implies that the first term is important only at the boundaries of the mirror bubbles where it comes up for the diamagnetic effect of the surface currents, the current is determined mainly by the last term, which can be written as This is to be compared to μ0δJ=-2A, thus yielding the penetration depth which is the ion inertial length based on the relevant temperature dependence of the particle density Nm(τ) for the mirror mode, where we should keep in mind that the latter is normalized to N0. Thus, identifying the reference temperature as T, we recover the connection between the mirror-mode penetration depth and its dependence on temperature ratio τ from thermodynamic equilibrium theory in the long wavelength limit with main density N0 constant on scales larger than the mirror-mode wavelength. 4 Magnetic penetration scale Back to toptop So far we considered only the current. Now we have to relate the above penetration depth to the plasma, the mirror mode. What we need is the connection of the mirror mode to the nonlinear Schrödinger equation. Because treating the nonlinear Schrödinger equation is very difficult even in two dimensions, this is done in one dimension, assuming for instance that the cross section of the mirror structures is circular with the relevant dimension the radius. In the presence of a magnetic wave field A≠0, Eq. (13) under homogeneous or nearly homogeneous conditions, with the canonical gradient term neglected, has the thermodynamic equilibrium solution which implies that either a or b is negative. In addition there is the trivial solution ψ=0 which describes the initial stable state when no instability evolves. The Helmholtz free energy density in this state is F=F0. Equation (12) shows that the thermodynamic equilibrium has free energy density where the last term is provided by the critical magnetic field, which is the external magnetic field. Thus b>0 and a<0, and the dependence on temperature τ can be freely attributed to a. Comparison with Eq. (2) then yields At critical field one still has A=0. Hence the density at critical field is which shows that the distortion of the density vanishes for τ=1, as it should be. This expression can be used in the magnetic penetration depth to obtain its critical temperature dependence which suggests that the critical penetration depth vanishes for τ=0. However, τ=0 is excluded by the argumentation following Eqs. (2) and (21), because it would imply infinite trapped densities. In principle, ττmin cannot become smaller than a minimum value which must be determined by other methods referring to measurements of the maximum density in thermodynamic equilibrium. One should, however, keep in mind that Bcrit0(θ)\|sinθ\| still depends on the angle θ which enters the above expressions. The last two expressions still contain the undetermined coefficient b. This can be expressed through the minimum value of the anisotropy τmin at maximum critical density Nm1 as (Note that for Nm>1 the above expansion of the free energy F becomes invalid. It is not expected, however, that the mirror mode will allow the evolution of sharp density peaks which locally double the density.) With this expression the inertial length becomes When the mirror mode saturates away from the critical field, the magnetic fluctuation grows until it saturates as well, and one has A≠0. The increased fractional density Nm is in perpendicular pressure equilibrium with the magnetic field distortion δB through There is also a small local contribution from the magnetic stresses which results from the surface currents at the mirror boundaries in which only a minor part of the trapped particles is involved. This is indicated by the approximate sign. The last two lines yield for the macroscopic penetration depth the expression Eq. (22). We thus conclude that Eq. (22) is also valid at saturation with τ=τsat. Measuring the saturation wavelength λsat and saturation temperature anisotropy τsat determines the unknown constant b through Eq. (23) with τmin replaced with τsat. Clearly as the mirror mode might saturate at temperature anisotropies larger than the permitted lowest anisotropy. Moreover, measurement of τsat at saturation, the state in which the mirror mode is actually observed, immediately yields the normalized saturation density excess Nm(τsat) from Eq. (21) which then from pressure balance yields the magnetic decrease, i.e. the mirror amplitude. To some extent this completes the theory of the mirror mode in as far as it relates the density at saturation to the saturated normalized temperature anisotropy at given T and determines the scale λim and δB(τsat). 5 The equivalent action α Back to toptop Since observations always refer to the final thermodynamic state, when the mirror mode is saturated, the anisotropy at saturation can be measured, and the value of the unknown constant α in the Schrödinger equation can also be determined. Expressed through b and λim at τsat, it becomes What is interesting about this number is that it is much larger than the quantum of action but at the same time is sufficiently small, which in retrospect justifies the neglect of the gradient term in the former section. It represents the elementary action in a mirror unstable plasma, where the characteristic length is given by the inertial scale α/2m=λsat or the maximum of the normalized density Nm. One may note that α is not an elementary constant like . It depends on the critical reference temperature T, and it depends on τ. Its constancy is understood in a thermodynamic sense. Our argument applies when A≠0. In this case Eq. (13) reads as and \|ψ\|=Nmax(x) is given by the maximum density excess in the centre x of the magnetic field decrease. Clearly this equation defines a natural scale length which is given by which, identifying it with λsat, yields the above expression for α. For x large the equation for f can be solved asymptotically when df/dx=0 for f2=1 corresponding to a maximum in Nm. It is then easy to show by multiplication by df∕dx that which has the Landau–Ginzburg solution This implies that the excess density assumes the shape It approaches Nmax for xx. The above condition on the vanishing gradient of f at x warrants the flat shape of the excess density at maximum (x) and the equally flat shape of the magnetic field in its minimum. At x=0 the amplitude f(x) starts increasing with finite slope f(0)=2λα. On the other hand, the initial slope of Nm is Nm(0)=0. The normalized excess density has a turning point at xt≈0.48λα with value Nm(xt)≈0.11Nmax. This behaviour is schematically shown in Fig. 1. Of course, these considerations apply strictly only to the one-dimensional case. It is, however, not difficult to generalize them to the cylindrical problem with radius r in place of x. The main qualitative properties are thereby retained. In the next section we will turn to the question of generation of chains of mirror-mode bubbles, as this is the case which is usually observed in space plasmas. Figure 1Shape of excess density in dependence on x/2λα. The shape of the magnetic field depression can be obtained directly from pressure balance. It mirrors the excess density. Since the quantum of action enters the magnetic quantum flux element Φ0=2π/e, we may also conclude that in a mirror-unstable plasma the relevant magnetic flux element is given by Φm=α/q. Identification of α is an important step. With its knowledge in mind the nonlinear Schrödinger equation for the hypothetical saturation state of the mirror mode is (up to the coefficient b, which, however, is defined in Eq. (23) and can be obtained from measurement) completely determined and thus ready for application of the inverse scattering procedure which solves it under any given initial conditions for the mirror mode. It thus opens up the possibility of further investigating the final evolution of the mirror mode. Executing this programme should, under various conditions, provide the different forms of the mirror mode in its final thermodynamic equilibrium state. This is left as a formally sufficiently complicated exercise which will not be treated in the present communication. Instead, we ask for the conditions under which the mirror mode evolves into a chain of separated mirror bubbles, which requires the existence of a microscopic though classical correlation length. 6 The problem of the correlation length Back to toptop The present phenomenological theory of the final thermodynamic equilibrium state of the mirror mode is modelled after the phenomenological Landau–Ginzburg theory of superconductivity as presented in the cited textbook literature. From the existence of λim we would conclude that, under mirror instability, the magnetic field inside the plasma volume should decay to a minimum value determined by the achievable minimum τsat of the temperature ratio. This conclusion would, however, be premature and contradicts observation where chains or trains (Zhang et al.2009) of mirror-mode fluctuations are usually observed (Luehr and Kloecker1987; Treumann et al.1990), which presumably are in their saturated state having had sufficient time to evolve beyond quasilinear saturation times and reached saturation amplitudes much in excess of any predicted quasilinear level. In fact, observations of mirror modes in their growth phase have to our knowledge not yet been reported. On the other hand, in no case known to us has a global reduction of the gross magnetic field in an anisotropic plasma been identified yet. Figure 2The Landau–Ginzburg parameter κρ/κρ,sat as a function of the anisotropy ratio τ=T/T<1 for the particular choice τsat=14. The parameter κρ refers to the thermal gyroradius as the short-scale correlation length, as explained in the text. It maximizes at saturation anisotropy τ=τsat and vanishes for τ=1 when no instability sets in. For any given ratio τ the value of κρ lies on a curve like the one shown. There is a threshold for the mirror mode to evolve into bubbles which it must overcome. It is given by the ratio κρ,sat>1 of the critical Alfvén speed to perpendicular thermal velocity. It is clear that in any real collisionless high-temperature plasmas neither can Nm become infinite nor can τ drop to zero. Since it is not known how and in which way, i.e. by which exactly known process mirror modes saturate in their final thermodynamic equilibrium state, their growth must ultimately become stopped when the particle correlation length comes into play. The nature of such a correlation length is unknown, nor is it precisely defined. There are at least three types of candidates for an effective correlation length, the Debye scale λD, the ion gyroradius ρ, and some turbulent correlation length turb. In a plasma the shortest natural correlation length is the Debye length λD which under all conditions is much shorter than the above-estimated penetration length λim. Referring to the Debye length, the Landau–Ginzburg parameter, i.e. the ratio of penetration to correlation lengths, in a plasma as a function of τ becomes a quantity that is large. Writing for the Debye length the Landau–Ginzburg parameter can be expressed in terms of τ, exhibiting only a weak dependence on the temperature ratio τ<1: Thus, κD is practically constant and about independent on the temperature anisotropy. Its value κD0=λi0/λD at τ=1, T=T refers to the isotropic case when no mirror instability evolves. This is an important finding because it implies that in a plasma the case that the magnetic field would be completely expelled from the volume of the plasma cannot be realized. Different regions of extension substantially larger than λD are (electrostatically) uncorrelated. They therefore behave separately, lacking knowledge about their (electrostatically) uncorrelated neighbours separated from them at distances substantially exceeding λD. Each of them experiences the penetration scale and adjusts itself to it. This is in complete analogy to Landau–Ginzburg theory. Thus, once the main magnetic field in an anisotropic plasma drops below threshold, the plasma will necessarily evolve into a chain of nearly unrelated mirror bubbles which interact with each other because each occupies space. In superconductivity this corresponds to a type II superconductor. Mirror unstable plasmas in this sense behave like type II superconductors. They decay into regions of normal magnetic field strength and embedded domains of spatial-scale λm(τ) with a reduced magnetic field. These regions contain an excess plasma population which is in pressure and stress balance with the magnetic field. Its diamagnetism (perpendicular pressure) keeps the magnetic field partially out and causes weak diamagnetic currents to flow along the boundaries of each of the partially field-evacuated domains. This trapped plasma behaves analogously to the pair plasma in metallic superconductivity, this time however at the high plasma temperature being bound together not by pairing potentials, but – in the case of the Debye length playing the role of the correlation length – by the Debye potential over the Debye correlation length. However, the Debye length is a very short scale, in fact the shortest collective scale in the plasma, and though it must have an effect on the collective evolution of particles in plasmas, it should be doubted that, on the mirror-mode saturation scale, it would have a substantial or even decisive effect. Instead, there could also be larger scales on which the particles are correlated. Such a scale is, for instance, the thermal-ion gyroradius ρ(τ). For the low frequencies of the mirror mode, the magnetic moment μ(τ)=T/B(τ)=const of the particles is conserved in their dynamics, which implies that all particles with the same magnetic moment μ(τ) behave about collectively, at least in the sense of a gyro-kinetic theory. However, though μ(τ) is a constant of motion, it still is a function of the anisotropy through the dependence of the magnetic field on τ. Expressing the thermal gyroradius through the magnetic moment it can be taken as another kind of collective correlation scale as on scales larger than ρ it collectively binds particles of the same magnetic moment which, in particular, are magnetically trapped like those which are active in the mirror instability. Below the gyroradius charged particles are magnetically free. ρ is the scale where the particles magnetize, start feeling the magnetic field effect, and collectively enter another phase in their dynamics. This scale is much larger than the Debye length and may be more appropriate for describing the saturated behaviour of the mirror mode. Thus one may argue that, as long as the penetration depth (inertial sale) exceeds ρ, the thermal gyroradius is the relevant correlation length. Only when it drops below the gyroradius does the Debye length take over. The Landau–Ginzburg parameter then becomes This ratio depends on the temperature anisotropy τ=T/T, which is a measurable quantity and the important parameter, while it saturates at κρ,sat=λi0/ρ0, the ratio of inertial length to gyroradius at the critical field. This ratio is not necessarily large. It can be expressed by the ratio of Alfvén velocity VA to perpendicular ion-thermal velocity υ⟂th: Hence, when referring to the thermal-ion gyroradius as the correlation length, the mirror mode would evolve and saturate into a chain of mirror bubbles only, when the Alfvén speed VA>υth exceeds the perpendicular thermal velocity of the ions. (Since Bcrit0\|sinθ\|, highly oblique angles are favoured. The range of optimum angles has recently been estimated in Treumann and Baumjohann2018a.) This is to be multiplied by the τ dependence, of which Fig. 2 gives an example. The value of this function is always smaller than one. For a chain of mirror bubbles to evolve in a plasma, the requirement κρ>1 can then be written as which is always satisfied for τsat<1 and κρ,sat>1, i.e. the Alfvén speed exceeding the perpendicular thermal speed, which indeed is the crucial condition for mirror modes to evolve into chains and become observable, with the gyroradius playing the role of a correlation length. Mirror-mode chains in the present case are restricted to comparably cool anisotropic plasma conditions, a prediction which can be checked experimentally to decide whether or not the gyroradius serves as a correlation length. Otherwise, when the above condition is not satisfied and τ<1 is below threshold, a very small and thus probably not susceptible reduction in the overall magnetic field is produced in the anisotropic pressure region over distances Lρ, much larger than the ion gyroradius. Observation of such domains of reduced magnetic field strengths under anisotropic pressure/temperature conditions would indicate the presence of a large-scale type I classical Meissner effect in the plasma. Such a reduction of the magnetic field would be difficult to explain otherwise and could only be understood as confinement of plasma by discontinuous boundaries of the kind of tangential discontinuities. The relative rarity of observations of mirror-mode chains or trains seems to support the case that the gyroradius, not the Debye length, plays the role of the correlation length in a magnetized plasma under conservation of the magnetic moments of the particles. From basic theory it cannot be decided which of the two correlation lengths, the Debye length λD or the ion gyroradius ρ, dominates the dynamics and saturation of the mirror mode. A decision can only be established by observations. However, the thermal-ion gyroradius, though the statistical average of the distribution of gyroscales, is itself just a plasma parameter which officially lacks the notion of a genuine correlation length. For this reason one would rather refer to the third possibility, a turbulent correlation length turb which evolves as the result of either high-frequency plasma or – in the case of mirror modes probably better suited – magnetic turbulence in the plasma. It is well known that, for instance, the solar wind or the magnetosheath carries a substantial level of turbulence which mixes plasmas of various properties and obeys a particular spectrum. In the solar wind such spectra have been shown to exhibit approximate Kolmogorov-type properties, at least in certain domains of frequencies or wave numbers, and similarly in the magnetosheath, where the conditions are more complicated because of the boundedness of the magnetosheath and the resulting spatial confinement of the plasma and its streaming. Such spectra imply that particles and waves are not independent but contain some information about their behaviour in different spatial and frequency domains; in other words, they are correlated. Unfortunately, the turbulent correlation length is imprecisely defined. No analytical expressions have been provided yet which would allow us to refer to it in the above determination of the Landau–Ginzburg parameter. This inhibits prediction of the range and parameter dependences of the turbulent Landau–Ginzburg ratio. Nonetheless, turbulent correlation scales might dominate the development of the mirror mode. The observation of a spectrum of mirror modes that is highly peaked around a certain wavelength not very much larger than the ion gyroradius may tell something about its nature. The above theory should open a way of relating a turbulent correlation length to the properties of a mirror unstable plasma. The condition is simply that the turbulent Landau–Ginzburg parameter is large, depending on the anisotropy parameter τ and the average transverse scales of the mirror bubbles. This expression yields an upper limit for the turbulent correlation length where λim(τ)〉 is known as a function of τ and the plasma parameters. Investigating this in further detail both observationally and theoretically should throw additional light on the nature of magnetic turbulence in high-temperature plasmas like those of the solar wind and magnetosheath. It would even contribute to a more profound understanding of magnetic turbulence in general as well as in view of its application to astrophysical problems. 7 Conclusions Back to toptop The mirror mode is a particular zero-frequency mesoscale plasma instability which provides some mesoscopic structure to an anisotropic plasma. It has been observed surprisingly frequently under various conditions in space, in the solar wind, cometary environments, near other planets and, in particular, behind the bow shock (Czaykowska et al.1998), such that one also believes that they occur in shocked plasmas if the shock causes a temperature anisotropy τ<1 (Balogh and Treumann2013). Since mirror modes are long scale, they provide the plasma with a very particular spatial texture. Mirror unstable plasmas are apparently built of a large number of magnetic bottles which contain a trapped particle population. This makes mirror modes most interesting even in magnetohydrodynamic terms as a kind of long-wavelength source of turbulence. In addition, their boundaries are surfaces which separate the bottles and thus have the character or tangential discontinuities or surfaces of diamagnetic currents which are produced by the internal interaction between the plasma and magnetic field. We have shown above that such an interaction resembles superconductivity, i.e. a classical Meissner effect. Mirror modes in the anisotropic collisionless space plasma apparently represent a classical thermodynamic analogue to a “superconducting” equilibrium state. One should, however, not exaggerate this analogy. This equilibrium state is no macroscopic quantum state. It is a classical effect. The analogy is just formal, even though it allows us to conclude about the final mirror equilibrium. Sometimes such an analogue helps in understanding the underlying physics1 like here, where it paves the way to a global understanding of the final saturation state of the mirror mode even though this does not release us from understanding in which way this final state is dynamically achieved. In contrast to metallic superconductivity which is described by the Landau–Ginzburg theory to which we refer here or, on the microscopic quantum level, by BCS-pairing theory, the problem of circumventing friction and resistance is of no interest in ideally conducting space plasmas which evolve towards mirror modes. High temperature plasmas are classical systems in which no pairing occurs and BCS theory is not applicable. Those plasmas are already ideally conducting. In contrast, there is a vital interest in the opposite problem, how a finite sufficiently large resistance can develop under conditions when collisions and friction among the particles are negligible. This is the problem of generating anomalous resistivity which may develop from high-frequency kinetic instabilities or turbulence and is believed to be urgently needed, for instance causing dissipation in reconnection. In the zero-frequency mirror mode it is of little importance even asymptotically, in the long-term thermodynamic limit, where such an anomalous resistance may contribute to decay of the mirror-surface currents which develop and flow along the boundaries of the mirror bubbles. The times when this happens are very long compared with the saturation time of the mirror instability and transition to the thermodynamic quasi-equilibrium which has been considered here. The more interesting finding concerns the explanation why at all, in an ideally conducting plasma, mirror bubbles can evolve. Fluid and simple kinetic theories demonstrate that mirror modes occur in the presence of temperature anisotropies, thereby identifying the linear growth rate of the instability. Trapping of large numbers of charged particles (ions, electrons) in accidentally forming magnetic bottles/traps causes the mirror instability to grow. The present theory contributes to clarification of this mechanism and its final thermodynamic equilibrium state as a nonlinear effect which is made possible by the available free energy which leads to a particular nonlinear Schrödinger equation. The perpendicular temperature in this theory plays the role of a critical temperature. When the parallel temperature drops below it, which means that 1>τ>τmin, mirror modes can evolve. Interestingly the anisotropy is restricted from below. The parallel temperature cannot drop below a minimum value. This value is open to determination by observations. The observation of chains of mirror bubbles, for instance in the magnetosheath, which provide the mirror-unstable plasma with a particular intriguing magnetic texture, suggests that the plasma, in addition to being mirror unstable, is subject to some correlation length which determines the spatial structure of the mirror texture in the saturated thermodynamic quasi-equilibrium state. This correlation length can be either taken as the Debye scale λD, which then naturally makes it plausible that many such mirror bubbles evolve, because in all magnetized plasmas the magnetic penetration depth by far exceeds the Debye length and makes the Landau–Ginzburg parameter based on the Debye length κD≫1. This, however, should lead to rather short-scale mirror bubbles. Otherwise, the role of a correlation length could also be played by the thermal-ion gyroradius ρ. In this case the conditions for the evolution of the mirror mode with the many observed bubbles become more subtle, because then κρ1 occurs under additional restrictions, implying that the Alfvén speed exceeds the perpendicular thermal speed. This prediction has to be checked and possibly verified experimentally. A particular case of the dependence of the gyroradius-based Landau–Ginzburg parameter κρ is shown graphically in Fig. 2. It may be noted that the Debye length and the ion gyroradius are fundamental plasma scales. Correlations can of course also be provided by other means, in particular by any form of turbulence. In that case a turbulent correlation length would play a similar role in the Landau–Ginzburg parameter, whether shorter or larger than the above-identified penetration scale. Regarding mirror modes in the magnetosheath to which we referred (Treumann and Baumjohann2018a), it is well known that the magnetosheath hosts a broad turbulence spectrum in the magnetic field as well as in the dynamics of the plasma (fluctuations in the velocity and density). Though this makes it highly probable that turbulence intervenes and affects the evolution of mirror modes, any “turbulent correlation length” is, unfortunately, rather imprecisely defined as some average quantity. To our knowledge, though referring to multi-spacecraft missions is not impossible, it has even not yet been precisely identified in any observations of turbulence in space plasmas. Even when identified, its functional dependence on temperature and density is required for application in our theory. If these functional dependencies are not available, it becomes difficult to include any turbulent correlation length. In addition, one expects that its turbulent nature would make the theory nonlocal. Attempts in that direction must, at this stage of the investigation, be relegated to future efforts. Finally, it should be noted that the magnetic penetration depth λm which lies at the centre of our investigation is rather different from the ordinary inertial length scale of the plasma. It is based on the excess density Nm<1 less than the bulk plasma density N0. It thus gives rise to an enhanced (excess) plasma frequency ωm=ωiNm+1=ωi1+ζ2ωi, which implies that L>c/ωi>λm is shorter than the typical scale of the volume L and (slightly) shorter than the bulk inertial length cωi. This becomes clear when recognizing that the mirror mode evolves inside the plasma from some thermal fluctuation (Yoon and López2017), which causes the magnetic field locally to drop below its critical value – Eq. (2). Then λm identifies the local perpendicular scale of a mirror bubble after it has saturated and is in thermodynamic equilibrium. One expects that the transverse diameter of a single mirror bubble in the ideal case would be roughly 2λm. However, since each bubble occupies real space, in a mirror-saturated plasma state the bubbles compete for space and distort each other (Treumann and Baumjohann1997), thereby providing the plasma with an irregular magnetic texture of some, probably narrow, spectrum of transverse scales which peaks around some typical transverse wavelength and resembles a strongly distorted crystal lattice that is elongated along the ambient magnetic field. It also relates the measurable saturated magnetic amplitudes of mirror modes to the saturated anisotropy τsat and the Landau–Ginzburg parameter κ, transforming both into experimentally accessible quantities. These should be of use in the development of a weak-kinetic turbulence theory of magnetic mirror modes as the result of which mirror modes can grow to the observed large amplitudes which are known to far exceed the simple quasilinear saturation limits. It also paves the way to the determination of a (possibly turbulent) correlation length in mirror unstable plasmas of which so far no measurements have been provided. To the space plasma physicist the present investigation may look a bit academic. However, it provides some physical understanding of how mirror modes do really saturate, why they assume such large amplitudes and evolve into chains of many bubbles or magnetic holes, and what the conditions are when this happens. Moreover, since the mirror mode in some sense resembles superconductivity, which also implies that some population of particles involved behaves like a superfluid, it would be of interest to infer whether such a population exhibits properties of a superfluid. One suggestion is that the untrapped ions and electrons which escape from the magnetic bottles along the magnetic field resemble such a superfluid population. This also suggests that other high-temperature plasma effects like the formation of purely electrostatic electron holes in beam–plasma interaction may exhibit superfluid properties. In conclusion, the unexpected working of the thermodynamic treatment in the special case of the magnetic mirror mode shows once more the enormous explanatory power of thermodynamics. Data availability Back to toptop Data availability. No data sets were used in this article. Competing interests Back to toptop Competing interests. The authors declare that they have no conflict of interest. Back to toptop This work was part of a Visiting Scientist Programme at the International Space Science Institute (ISSI) Bern. We acknowledge the interest of the ISSI directorate and the hospitality of the ISSI staff. We thank the referees Karl-Heinz Glassmeier (TU Braunschweig, DE), Ovidiu Dragos Constantinescu (U Bukarest, RO), and Hans-Reinhard Mueller (Dartmouth College, Hanover, NH, USA) for their critical remarks on the manuscript, in particular the non-trivial questions on the effective mass and the role of turbulence. The topical editor, Anna Milillo, thanks Karl-Heinz Glassmeier, Dragos Constantinescu, and Hans-Reinhard Müller for help in evaluating this paper. Back to toptop Balogh, A. and Treumann, R. A.: Physics of Collisionless Shocks: Space plasma shock waves, Springer-Verlag, New-York, chap. 4, 149–220,, 2013. a Baumjohann, W. and Treumann, R. A.: Basic Space Plasma Physics, London 1996, Revised and enlarged edition, Imperial College Press, London, chap. 11, 357–359,, 2012. Baumjohann, W., Treumann, R. A., Georgescu, E., Haerendel, G., Fornaçon, K. H., and Auster, U.: Waveform and packet structure of lion roars, Ann. 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A. and Baumjohann, W.: Electron mirror branch: Probable observational evidence from “historical” AMPTE-IRM and Equator-S measurements, available at: arXiv:1804.01131 [], 2018a. a, b, c, d Treumann, R. A. and Baumjohann, W.: Possible increased critical temperature Tc in the ideal anisotropic boson gas, available at: arXiv:1806.00626 [cond-mat.stat-mech], 2018b. a Treumann, R. A. and Baumjohann, W.: Classical Higgs mechanism in plasma, available at: arXiv:1804.003463 [physics.plasm-ph], 2018c. a Treumann, R. A., Brostrom, L., LaBelle, J., and Sckopke, N.: The plasma wave signature of a “magnetic hole” in the vicinity of the magnetopause, J. Geophys. Res., 95, 19099–19114,, 1990. a Treumann, R. A., Jaroschek, C. H., Constantinescu, O. D., Nakamura, R., Pokhotelov, O. A., and Georgescu, E.: The strange physics of low frequency mirror mode turbulence in the high temperature plasma of the magnetosheath, Nonlin. Proc. Geophys., 11, 647–657,, 2004. Tsurutani, B. T., Smith, E. J., Anderson, R. R., Ogilvie, K. W., Scudder, J. D., Baker, D. N., and Bame, S. J.: Lion roars and nonoscillatory drift mirror waves in the magnetosheath, J. Geophys. Res., 87, 6060–6072,, 1982. a Tsurutani, B. T., Lakhina, G. S., Verkhoglyadova, O. P., Echer, E., Guarnieri, F. L., Narita, Y., and Constantinescu, D. O.: Magnetosheath and heliosheath mirror mode structures, interplanetary magnetic decreases, and linear magnetic decreases: Differences and distinguishing features, J. Geophys. Res.-Space, 116, A02103,, 2011. a Tsytovich, V. N.: Theory of Plasma Turbulence, Consultants Bureau, New York, 1977. a Volwerk, M., Zhang, T. L., Delva, M., Vörös, Z., Baumjohann, W., and Glassmeier, K.-H.: Mirror-mode-like structures in Venus' induced magnetosphere, J. Geophys. Res., 113, E00B16,, 2008. a Yoon, P. H.: Kinetic theory of hydromagnetic turbulence, I. Formal results for parallel propagation, Phys. Plasmas, 14, 102302,, 2007. a Yoon, P. H.: Weak turbulence theory for beam-plasma interaction, Phys. Plasmas, 25, 011603-1,, 2018. a Yoon, P. H. and Fang, T. M.: Kinetic theory of hydromagnetic turbulence, II. Susceptibilities, Phys. Plasmas, 14, 102303,, 2007. a Yoon, P. H. and López, R. A.: Spontaneous emission of electromagnetic fluctuations in magnetized plasmas, Phys. Plasmas, 24, 022117,, 2017. a, b Zhang, T. L., Russell, C. T., Baumjohann, W., Jian, L. K., Balikhin, M. A., Cao, J. B., Wang, C., Blanco-Cano, X., Glassmeier, K.-H., Zambelli, W., Volwerk, M., Delva, M., and Vörös, Z.: Characteristic size and shape of the mirror mode structures in the solar wind at 0.72 AU, Geophys. Res. Lett., 35, L10106,, 2008. a Zhang, T. L., Baumjohann, W., Russell, C. T., Jian, L. K., Wang, C., Cao, J. B., Balikhin, M. A., Blanco-Cano, X., Delva, M., and Volwerk, M.: Mirror mode structures in the solar wind at 0.72 AU, J. Geophys. Res., 114, A10107,, 2009. a, b Zhang, Y., Matsumoto, H., and Kojima, H.: Lion roars in the magnetosheath: The Geotail observations, J. Geophys. Res., 103, 4615–4626,, 1998. a, b In a recent paper (Treumann and Baumjohann2018c), we have shown that a classical Higgs mechanism is responsible for bending the free space O-L and X-R electromagnetic modes in their long-wavelength range away from their straight vacuum shape when passing a plasma. The plasma in that case acts like a Higgs field and attributes a tiny mass to the photons, making them heavy. This is interesting because it shows that any bosons become heavy only in permanent interaction with a Higgs field and only in a certain energy–momentum–wavelength range. It also shows that earlier attempts at measuring a permanent photon mass by observing scintillations of radiation (and also by other means) have just measured this effect. Their interpretations as upper limits for a real permanent photon mass are incorrect because they missed the action of the plasma as a classical Higgs field. Publications Copernicus Short summary The physics of the magnetic mirror mode in its final state of saturation, the thermodynamic equilibrium, is re-examined to demonstrate that the mirror mode is the classical analogue of a superconducting effect in an anisotropic-pressure space plasma. Three different spatial correlation scales are identified which control the behaviour of its evolution into large-amplitude chains of mirror bubbles. The physics of the magnetic mirror mode in its final state of saturation, the thermodynamic...
22c479abc67754b0	How to make: a Möbius Surprise You will need • A3 paper • scissors • glue 1. Cut your A3 paper into 4 strips. 5. Be surprised. Prize crossnumber, Issue 08 Our original prize crossnumber is featured on pages 54 and 55 of Issue 08. • One randomly selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Word DVD, a dodecaplex puzzle and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide. Find out more at • To enter, submit the sum of the across clues via this form by 2 February 2019. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 16 February 2019. Continue reading On the cover: Hydrogen orbitals Quantum mechanics has a reputation. It’s notorious for being obtuse, difficult, confusing, and unintuitive. That reputation is… entirely deserved. I work on quantum systems full time for my job and I feel like I’ve barely scratched the surface of the mysteries it contains. But one other feature of quantum mechanics that’s often overlooked is how beautiful it can be. So, for the cover of this issue, I wanted to share one aspect of quantum mechanics that I think is stunning. It’s a certain set of solutions to a differential equation: the orbitals of an electron in a hydrogen atom. In school, you’re taught that electrons orbit the nucleus of an atom like a planet orbiting a star. This is mostly wrong. The main problem is that electrons, protons and neutrons aren’t little billiard balls, they exist as ‘clouds’ of probability. To understand what a hydrogen atom really looks like, imagine a cloud of something whizzing around a single proton. The proton’s positive charge attracts and traps the negatively-charged something in what we call the proton’s potential well. Imagine that cloud is denser in some places and sparser in others. That cloud of something can be just one electron whose position has been smeared out. The density of the cloud at a point represents the probability of finding the electron at that point in space. The electron’s position may be smeared out over all space, but it has different odds of being found at different points in space. In fact, it’s usually exponentially less likely to be found outside the small, confined volume of the potential well. The mathematical explanation for this is that our system is obeying the Schrödinger equation. For our case, it looks like this: \Big(\dfrac{-\hbar^{\hspace{0.3mm}2}}{2m}\nabla^2 + V(\mathbf{r})\Big)\psi(\mathbf{r}) = E\psi(\mathbf{r}). The Schrödinger equation is the foundational equation of quantum mechanics. It’s used to determine the wavefunction, $\psi(\mathbf{r})$, and the energy, $E$, of the components of the system. In this case the wavefunction represents the electron (with mass $m$) trapped in the electric potential well of the proton, which is represented by $V(\mathbf{r})$. The reduced Planck constant, $\hbar$, (often called “h-bar”) is a fundamental physical constant, and $\nabla^2$ is the Laplacian operator, which sums second derivatives over all the coordinates. The modulus squared of the wavefunction, $\vert\psi(\mathbf{r})\vert^2$, tells you what the density of that probability cloud is like: where are you more likely to find the electron? Most people who do quantum mechanics for a living spend their time solving this equation and its variants, myself included. The problem is that this is really, really hard. The Schrödinger equation for a hydrogen atom has analytic solutions you can write down, but with almost all other physical systems, you aren’t so lucky. Once you have more than one electron, the complexity skyrockets. Understanding the analytic solutions form an important part of a physics undergraduate’s introduction to quantum mechanics, especially in my field of research. I work on finding approximate solutions to the Schrödinger equation for more complex systems. To solve the Schrödinger equation, you can separate the wavefunction to get a radial part which is a function of the distance from the nucleus, $r$, and an angular part which is a function of the angles $(\theta, \phi)$. Both parts have multiple solutions, and it turns out that you need three labels to identify these solutions. We call these labels quantum numbers. Here, the three are called $n$, $l$, and $m$. Putting these two concepts together, we can say: E\psi_{nlm}(\mathbf{r}) = R_n(r)Y_{lm}(\theta, \phi). Plots of the solution with $n=0$, $l=5$ and $m=0$ to $m=5$ There’s lots of constraints on the allowed values of $n$, $l$, and $m$, but the most important one is that each number take whole number values only. This is where the ‘quantum’ in quantum mechanics comes from! The quantum numbers each have physical interpretations: they loosely correspond to the three spatial coordinates. Here, $n$ corresponds to energy. Higher values of $n$ mean the electron has a larger amount of energy, which, due to how electric fields work, also exactly corresponds to a larger distance from the nucleus. That means $n$ is associated with the radial coordinate: the higher $n$ is, the further from the nucleus the electron can be. Meanwhile, $l$ and $m$ correspond to angular momentum, and so they are associated with the angular coordinates. Roughly speaking, higher values of $l$ correspond to the electron ‘orbiting’ around the nucleus with greater energy (in a weird, quantum mechanical way that doesn’t really look like a planet orbiting a star). Changing $m$ means changing exactly how it orbits for a given value of $l$. What this all means in practice is that by varying the three quantum numbers you get a huge variety of electron distributions. For instance, $n=1$, $l=0$, $m=0$ means that the electron isn’t orbiting the nucleus at all, so it’s most likely to be found right on top of the nucleus – opposite charges attract! When $n$, $l$, and $m$ are all large you get things like concentric sets of lobes of varying shapes and sizes. Bringing it back to the cover, the pictures were all generated by making a 2D slice through the full 3D distribution at $y=0$. The brighter a given point is shaded, the higher the value of $\vert\psi(\mathbf{r})\vert^2$ is there—the higher the odds of finding the electron there are. The full 3D versions look like spheres, balloons, lobes, and other wild shapes. The 2D slices have a different sort of haunting beauty to them. The distributions can be concentric rings, orange slices, weird lobes, insect-like segments, and more. The front cover is the 2D slice of the solution for $n=9$, $l=4$, $m=1$. The back cover contains all the allowed solutions from $n=1$, $l=0$, $m=0$ up to $n=9$, $l=7$, $m=7$. These orbitals are beautiful by themselves as pieces of abstract maths, but they also provide profound insights into the strange quantum nature of our reality. They’re a testament to the amazing power physics and mathematics can have when they work together to help us understand our universe. Dear Dirichlet, Issue 08 Dear Dirichlet, The annual village fete is fast approaching, and every year I embarrass myself at `guess the number of sweets in the jar’. My exasperated wife ends up telling me to just say a number, and I always panic. Last year my guess was $\mathrm{i} – \text{π}$. Maybe I was just hungry. — Hungry hungry hippo, Gospel Oak Continue reading Top Ten: Units of measurement This issue, Top Ten features the top ten units of measurement! Then vote here on the top ten Chalkdust regulars for issue 09! At 10, it’s the Zappa. At 9, and selling one tenth of the number of copies that number 6 sold: it’s a millimetre. At 8, and not receiving much radio play due to being far longer than the rest of the top ten: it’s a furlong. Following warm reviews from critics, degrees Celsius enters the top ten at 7. The new single by no-one’s favourite rapper 50 Centimetre is at 6. At 5, it’s the Yardbirds tribute act whose members are all 8.5cm taller than the originals: the Metrebirds. Following the release of its 51st anniversary deluxe edition, The Velvet Underground and Picometre is at 4. At 3, and selling 273.15 more copies than this issue’s number 7: it’s Kelvin Harris. Image: Wikimedia commons user Martinvl, CC BY-SA 3.0. Forced up two places from last issue, it’s the Newton. Topping the pops this issue, it’s dimensionless constants. Top ten vote issue 08 What is the best Chalkdust regular? View Results Loading ... Loading ...
27de85a25637d23a	He showed that there is a limit to how accurately two quantities – for instance a particle’s speed and its position – can be measured simultaneously. When quantum physics is used to calculate other. Nov 16, 2011 · Where n= 1, 2, 3… Is called the Quantum number. As E depends on n, we shall denote the energy of particle ar E n. Thus. E n = n 2 π 2 Ћ 2 /2mL 2 (10) This is the eigen value or energy value of the particle in a box. Physics; Quantum Mechanics and Applications (Video) Modules / Lectures. Introduction and Basic Mathematical Preliminary. Basic Quantum Mechanics I: Wave Particle Duality; Basic Quantum Mechanics II: The Schrodinger Equation and The Dirac Delta Function. The 1-Dimensional Potential Wall & Particle in a Box ; Particle in a Box and Density of. User Guide Quantum Espresso User’s Guide for Quantum ESPRESSO (version 5.0.2) Contents 1 Introduction 1. notably the FAQs and the User Guide(s). The answer to most questions is already there. • Reply to both the mailing list and the author or the post, using “Reply to all” (not User’s Guide for Quantum ESPRESSO (version 5.1) Contents. attempt to search In classical physics, this might appear as a billiard ball struck by a cue, traveling in a line. But in the quantum. sealed in a box might remain both dead and alive until its status is monitored. Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary. The pioneering quantum physicist Niels Bohr was something of a quote machine, with numerous pithy comments about the philosophical foundations of quantum physics that are quoted. a scientist doing. Particle physics is the branch of physics that looks at the elementary constituents. In quantum theory if a particle is placed in the box, the particle can be in any. Uncertainty Principle Important steps on the way to understanding the uncertainty principle are wave-particle duality and the DeBroglie hypothesis.As you proceed downward in size to atomic dimensions, it is no longer valid to consider a particle like a hard sphere, because the smaller the dimension, the more wave-like it becomes. Aug 1, 2017. Quantum mechanics (QMs) is a foundational subject in many science and engineering fields. It is difficult to teach, however, as it requires a. The 1950s and 60s were a Golden Age of particle physics, as accelerators produced a plethora. Mann did later regret. Often, they are simply referred to as "quantum mechanics." Schrödinger. The first three allowed de Broglie wave modes for a particle in a box. Figure 17: The. 5) What is/are the most probable position/s of the particle in the box? We will find the answers to these questions as we go along. Quantum mechanics allows us. October 09. Modern Physics. Particle in a box. Consider a particle confined to a 3 dimensional infinitely deep potential well – a. “box”. Outside the wave function. Quantum Field Theory and the Standard Model of particle physics are theories that can provide answers to some of the most. DENTON, Texas , May 20, 2019 /CNW/ — Albert Einstein did not accept the impossibility for quantum mechanics to identify the position of a particle with classical precision. For that reason. It is relatively straightforward to solve for simple systems such as a single quantum particle in a box and predicts that these systems. computational complexity theory (and perhaps in all of. Particle in a Box: An Experiential Environment for. Learning Introductory Quantum Mechanics. Aditya Anupam, Ridhima Gupta, Azad Naeemi, Senior Member, C The intermediate state can be bypassed if particles within the molecule are transferred via quantum tunneling, where a. Does quantum physics melt your brain? First. The electron is not a point particle, but a smear of electron-ness spread out in space. Particle/wave duality Quantum objects (like photons and. QUANTUM MECHANICAL PARTICLE IN A BOX Summary so far: ∞ ∞ V (x < 0, x > a)= ∞ ψ (x < 0, x > a)= 0 V (0 ≤ x ≤ a)= 0 ψ (0 ≤ x ≤ a)= B sin ⎝⎜ ⎛ x nπ a ⎠⎟ ⎞ n V(x) 0 n 2 h nπ 2a a E n = 8ma2 k = a λ = n 0 n = 1,2,3, x What is the “wavefunction” ψ (x)? Max Born interpretation: ( ) 2 x x is a probability distribution or probability density for the particle A French and an American physicist are sharing the 2012 Nobel Prize in Physics for, in a sense, letting a cat out of the box. And not just any. and even controlling the weird, quantum properties of. Quantum mechanics is based on the concept of wave-particle duality, which for. well potential (PIB, particle in a box) of width a must form standing waves. How does quantum physics work, you may ask, what is it, and where does it come from? In this article we discuss a very brief and simplified history of Quantum Mechanics and will quote what the founding fathers of this branch of science had to say about Vedic influence on the development of their theories. its wave function collapses and the particle will appear in only one spot, falling back under the laws of conventional physics. This makes studying quantum particles extremely difficult, because the. One of the central concepts in quantum physics is "wave-particle duality." As briefly as possible, here’s what it means. Any particle, such as an electron or proton, has a wave associated with it. This wave will typically extend over only a small region, and the particle will be somewhere in that region. Exactly where in that region the particle is, is unknown. May 19, 2016. Quantum mechanics was developed in just two years, 1925 and. As an example, imagine a single particle moving around in a closed box. Theories of Quantum Mechanics(QM) have been central to the philosophical and technological advances in physics and related fields. Some of the most. May 20, 2019 · Introduction to Quantum Physics concepts with an activity demonstrating Heisenberg’s Uncertainty Principle, wave/particle duality, Planck’s Constant, de Broglie wavelength, and how Newton’s Laws go right out the window on a quantum level. Used By Geographers To Determine Absolute Location The invention would map the user’s location every time he took. This information could then be used to market to them products or services they don’t want or used against them to determine such. Charles E. Schmidt College of Science Course Descriptions Biological Sciences Chemistry and Biochemistry Complex Systems and Brain Sciences What Are Promotional Application of Quantum Mechanics: Translational Motion of a Particle in a Box. Particle trapped in. 1-D box. Boundary condition: Particle wave must fit into the. Quantum Physics for Babies (Baby University) [Chris Ferrie] on Amazon.com. FREE shipping on qualifying offers. Simple explanations of complex ideas for your future genius! Written by an expert So let’s get back to our particle in a box. Another rule of quantum physics is that the shorter the wavelength, the higher the particle’s energy. And so, since only certain wavelengths can occur for the electron in a box, that means that the energy of the electron can only take on certain values. Oct 18, 2010 · Renowned theoretical physicist Nima Arkani-Hamed delivered the first in his series of five Messenger lectures on ‘The Future of Fundamental Physics’ Oct. 4. Quantum mechanics, science dealing with the behaviour of matter and light on the atomic. The gradual recognition by scientists that radiation has particle-like properties. The square of the wave function, Ψ2, has a physical interpretation. Quantum Mechanics. To solve the Schrödinger equation for a one-dimensional “particle in a box”; To study the behavior of a quantum-mechanical particle in a. A breakthrough that got credit for "simplifying" quantum physics could wipe away some. They’re approximations of things like where a particle might be right at the moment and what it might be doing. In some ways, the particles of quantum theory are like little tiny points of matter, as the. by considering a very simple case, a particle/wave trapped in a box. Therefore, I’m restricted to a specific-sized ball – say a tennis ball – and the box can only be filled with this “energy quantum. physics will bring us a theory of light that can be understood as. Apr 25, 2019. If you were to perform this same experiment with a quantum particle, square well) in classical mechanics (A) and quantum mechanics (B-F). Once the observer takes his measurement, convention says that the cat will be discovered to be dead or alive. But Schrodinger reasoned that quantum physics describes an outcome in which the cat is both dead and alive. This is because the atom, in its wave function, was, at one time, in either box… Assume the potential U(x) in the time-independent Schrodinger equation to be zero inside a one-dimensional box of length L and infinite outside the box. For a particle inside the box a free particle wavefunction is appropriate, but since the probability of finding the particle outside the box is zero, the wavefunction must go to zero at the walls. This constrains the form of the solution to Dec 04, 2017 · A Particle in a Box- 1 Faheel Hashmi Particle in a box , Quantum mechanics , Uncategorized December 4, 2017 December 13, 2017 5 Minutes In the previous example we have seen that quantum mechanics allows us to calculate the experimentally verifiable probabilities. Unlike relativity theory, the birth of quantum theory was slow and required many hands.It emerged in the course of the first quarter of the twentieth century with contributions from. Otherwise you saw a realization of one of the classic formal problems in quantum mechanics: the particle in a box. Imagine that a very small, light particle like an. Ecological Factors In Criminology Crime is seldom considered as an outcome in public health research. Yet major theoretical and empirical developments in the field of criminology during the past 50 years suggest that the same social environmental factors which predict geographic variation in crime rates may also be relevant for explaining community variations in health and wellbeing. What environmental Quantum mechanics is also giving rise to the areas of quantum information, quantum 1. 2 Quantum Mechanics Made Simple communication, quantum cryptography, and quantum computing. It is seen that the richness of quantum physics will greatly a ect the future generation technologies in many aspects. In day to day life, we intuitively understand how the world works. Drop a glass and it will smash to the floor. Push a wagon and it will roll along. Walk to a wall and you can’t walk through it. Dark matter is theorized to be a particle. a set of laws of physics, but provides proverbial boundary conditions that describe the spectrum of particles that can exist. Because the Standard Model. A cat is trapped in a box with a vial of poison that is. it becomes very hard to get simple answers to questions from quantum physics, such as the mass of a neutrino, an electrically neutral. Social Scientists Generally Collect Data Earthwatch combines volunteer opportunities for individuals from all walks of life with scientific research expeditions to conserve wildlife and the environment. As part of the collaboration, the initiative will join a 305-acre science and technology campus in Lakewood. s infrastructure within Lakewood Ranch, and begin collecting data from residents. Phase. For example, students from the Dec 25, 2017. But quantum physics, which seeks to explain how life works at the subatomic. under quantum theory, objects can exist as both waves and particles, A cat in a closed box that also contains a vial of poison could be thought. Richard Dawkins Creationism Interview Take Richard Dawkins, the famous evolutionary biologist and atheist, for example. In a recent interview with Der Spiegel, Dawkins was asked if he ever experienced a religious.(Read Full Article). Latest breaking news, including politics, crime and celebrity. Find stories, updates and expert opinion. 22 December 2012. A post in 2006, now removed, criticised the organisation [How Quantum Entanglement Works (Infographic)] Here’s how it goes: Quantum physics dictates that, under particular conditions, a particle can have two contradictory. He imagined an opaque steel box. But in the case of quantum physics, the standard rules are. is a classic example of quantum weirdness. Inside the box, the cat will be either alive or dead, depending on whether a radioactive. In physics, a sufficiently strong barrier will prevent any incoming object from passing through it. But at the quantum level, this isn’t strictly true. If you replace a tennis ball with a quantum. In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers.The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a.
af82bfb1c6de2e1f	All-Order Methods for Relativistic Atomic Structure Calculations 1 Please download to get full document. View again of 21 Information Report Science & Technology Views: 0 \| Pages: 21 Extension: PDF \| Download: 0 Related documents All-Order Methods for Relativistic Atomic Structure Calculations 1 M. S. Safronova a,2 W. R. Johnson b,3 a Department of Physics and Astronomy, 223 Sharp Lab, University of Delaware, Newark, Delaware 19716 All-Order Methods for Relativistic Atomic Structure Calculations 1 M. S. Safronova a,2 W. R. Johnson b,3 a Department of Physics and Astronomy, 223 Sharp Lab, University of Delaware, Newark, Delaware b Department of Physics, University of Notre Dame, Notre Dame, IN Abstract All-order extensions of relativistic atomic many-body perturbation theory are described and applied to predict properties of heavy atoms. Limitations of relativistic many-body perturbation theory are first discussed and the need for all-order calculations is established. An account is then given of relativistic all-order calculations based on a linearized version of the coupled-cluster expansion. This account is followed by a review of applications to energies, transition matrix elements, and hyperfine constants. The need for extensions of the linearized coupled-cluster method is discussed in light of accuracy limits, the availability of new computational resources, and precise modern experiments. For monovalent atoms, calculations that include nonlinear terms and triple excitations in the coupled-cluster expansion are described. For divalent atoms, results from second- and third-order perturbation theory calculations are given, along with results from configuration-interaction calculations and mixed configuration interaction many-body perturbation theory calculations. Finally, applications of all-order methods to atomic parity nonconservation, polarizabilities, C 3 and C 6 coefficients, and isotope shifts are given. Key words: relativistic atomic structure, many-body perturbation theory, coupled-cluster theory, electron correlation calculations, isotope shift, polarizability, hyperfine structure, oscillator strengths, lifetimes, transition moments, weak-interaction effects in atoms 1 AMO review paper: version 7 12/02/ Preprint submitted to Elsevier Preprint 7 December 2006 1 Introduction and Overview The nonperturbative treatment of relativity in atomic many-body calculations can be traced back to the formulation of relativistic self-consistent field (SCF) equations with exchange by Swirles [1] in The SCF equations, also referred to as Dirac-Hartree-Fock (DHF) equations, are based on a manyelectron Hamiltonian in which the electron kinetic and rest energies are from the Dirac equation and the electron-electron interaction is approximated by the Coulomb potential. Numerical solutions of the DHF equations without exchange were obtained during the years by Williams [2], Mayers [3], and Cohen [4]. The formulation of relativistic SCF theory by Swirles was reexamined by Grant [5] in 1961 and the DHF equations were brought into a compact and easily used form. Numerical solutions to the DHF equations with exchange were published in 1963 by Coulthard [6], by Kim [7], and Smith and Johnson [8]. The Breit interaction was included in the later two calculations [7, 8]. In 1973, Desclaux [9] published complete DHF studies of atoms with Z = and Mann and Waber [10] published DHF studies of the lanthanides, including effects of the Breit interaction. The DHF equations remain as the starting point for relativistic many-body studies of atoms and versatile multiconfiguration DHF codes are publically available; notably the codes of Desclaux [11] and Grant et al. [12]. Extensions of the DHF approximation have been developed over the past three decades driven by advances in several areas of experimental atomic physics. Of particular importance in this regard are the precise measurements of energy levels and transition moments for highly-charged ions produced in beam-foil experiments, electron beam ion trap (EBIT) experiments, tokamak plasmas, and astrophysical plasmas [13]. These measurements have reached such a high level of precision that it has become possible to detect two-loop Lamb-shift corrections to levels in lithiumlike U [14], putting very tight constraints on the accuracy of the underlying atomic structure calculations. An equally important motivating factor in the development of extensions of the DHF approximation are measurements of parity nonconserving (PNC) amplitudes in heavy atoms, especially those designed to test the standard model of the electroweak interaction and to set limits on its possible extensions [15]. For the case of cesium, measurements of PNC amplitudes have reached an accuracy of 0.4% [16]. To make meaningful tests of the standard model, calculations of the amplitudes must be carried out for heavy neutral atoms to a similar level of accuracy. One systematic extension of the DHF approximation is relativistic many-body perturbation theory (MBPT). Relativistic MBPT studies of atomic structure start from a lowest-order approximation in which the electron-electron interaction is the frozen core DHF potential and include an order-by-order 2 perturbation expansion (in powers of the residual interaction) of energies and wave functions. Relativistic MBPT was used to predict properties of alkalimetal atoms from Li to Cs in Ref. [17], where energy levels for the ground state and the first few excited states were calculated to second order. In [17], electric-dipole matrix elements for the principal transitions and hyperfine constants were calculated through second order and included dominant third-order corrections. Although accurate values for energies, transition matrix elements, and hyperfine constants were obtained for Li, results for heavier alkali-metal atoms were significantly less accurate. The ground-state energy for Cs was accurate to 1.5%, while the Cs transition and hyperfine matrix elements were accurate to about 5% as determined by comparisons with precise experimental data. Later, complete third-order calculations of electric-dipole matrix elements, including all third-order terms were carried out in Ref. [18] for alkali-metal atoms and for Li-like and Na-like ions. The agreement with available experiments was very good for lighter atoms (within experimental precision for Li and Na), but decreased significantly for Cs and Fr. To achieve the accuracy required for tests of the standard model in heavy atoms, it is imperative to include contributions beyond third order in MBPT. Although extensions to fourth order represent one possibility, the resulting calculations are formidable; for each first-order matrix element there are four terms in second order, 60 terms in third order, and 3072 terms in fourth order [19]. Owing to this very rapid increase in computational effort with MBPT order, one seeks alternatives to MBPT beyond third order. One such alternative is the coupled-cluster singles-doubles (CCSD) method in which single and double excitations of the DHF ground state are included to all orders of perturbation theory. A nonrelativistic version of this method was used to calculate precise values of energies and hyperfine constants of 2s and 2p states of Li by Lindgren [20]. A linearized, but relativistic, version of the coupled-cluster method was later used to obtain energy levels, finestructure intervals, and dipole matrix elements in Li and Be + in Ref. [21]. These all-order calculations substantially improved the accuracy of energies and matrix elements compared to older MBPT results [17]. A nonrelativistic CCSD calculation for Na was reported in [22], where energies and hyperfine constants of 3s and 3p states and the 3s 3p electric-dipole matrix elements were calculated. Partial contributions to the 3s energy and hyperfine constant from triple excitations were also included in [22]; the resulting 3s energy was accurate to 0.01% and the 3s hyperfine constant to 0.2%. A relativistic version of the CCSD method was applied to calculate energy levels of alkali-metal atoms in [23] and excellent agreement with experiment was found. A linearized version of the coupled-cluster formalism, including single, double, and partial triple excitations (SDpT) was used to determine atomic properties of Cs in Ref. [24], where removal energies agreed with experiment to 0.5% and matrix elements agreed with measurements to better than 1%. Properties of Na-like 3 ions (Z = 11 16), such as energies, transition matrix elements, and hyperfine constants were studied using the linearized CCSD method in Ref. [25], and similar studies of alkali-metal atoms including polarizabilities were reported in [26]. Although we concentrate on relativistic all-order coupled-cluster methods in this review, it should be noted that perturbation theory in the screened Coulomb interaction (PTSI) developed by Dzuba et al. [27, 28], in which important classes of MBPT corrections are summed to all-orders, is an alternative method that has been successfully applied to atomic structure calculations for heavy neutral atoms. Moreover, for atoms with more than one valence electron, relativistic configuration-interaction (CI) calculations in an effective Hamiltonian extracted from the linearized SD theory, which has been developed and applied to small systems by Kozlov [29], is a promising alternative to CCSD methods for large systems. 2 Relativistic Many-Body Perturbation Theory In the simplest picture of a relativistic many-electron atom, each electron moves independently in a central potential U(r) produced by the remaining electrons. The one-electron orbitals φ a (r) describing the motion of an electron with quantum numbers a = (n a, κ a, m a ) satisfy the one-electron Dirac equation where h(r)φ a (r) = ɛ a φ a (r), (1) h(r) = c α p + βmc 2 Z r + U(r). (2) The quantities α and β in Eq. (2) are 4 4 Dirac matrices. The Dirac eigenvalues ɛ a range through values: ɛ a mc 2 for electron scattering states, mc 2 ɛ a 0 for electron bound states, and mc 2 ɛ a for positron states. The point of departure for our discussions of many-electron atoms is the no-pair Hamiltonian obtained from QED by Brown and Ravenhall [30] and illuminated in Refs. [31 34]. In this Hamiltonian, the electron kinetic and rest energies are from the Dirac equation and the potential energy is the sum of Coulomb and Breit interactions. Contributions from negative-energy (positron) states are projected out of this Hamiltonian. The no-pair Hamilto- 4 nian can be written in second-quantized form as H = H 0 + V, where H 0 = i ɛ i [a ia i ], (3) V = 1 (g ijkl + b ijkl ) [a 2 ia ja l a k ] ijkl + (V HF + B HF U) ij [a ia j ] + 1 (V HF + B HF 2 U) aa. (4) ij 2 a In Eqs. (3-4), a i and a i are creation and annihilation operators for an electron state i, and the summation indices range over electron bound and scattering states only, since, as mentioned above, contributions from negative energy states are absent in the no-pair Hamiltonian. Products of operators enclosed in brackets, such as [a ia ja l a k ], designate normal products with respect to a closed core. The summation index a in the last term in (4) ranges over states in the closed core. The quantity ɛ i in Eq. (3) is the eigenvalue of the Dirac equation (1). The quantities g ijkl and b ijkl in Eq. (4) are two-electron Coulomb and Breit matrix elements, respectively 1 g ijkl = ij b ijkl = ij r kl 12, (5) α 1 α 2 + (α 1 ˆr 12 )(α 2 ˆr 12 ) 2r 12 kl. (6) In Eq. (4), the core DHF potential is designated by V HF and its Breit counterpart is designated by B HF ; thus, (V HF ) ij = b (B HF ) ij = b [g ibjb g ibbj ], (7) [b ibjb b ibbj ], (8) where b ranges over core states. For neutral atoms, the Breit interaction is often a small perturbation that can be ignored compared to the Coulomb interaction. In such cases, it is particularly convenient to choose the starting potential U(r) to be the core DHF potential U = V HF, since with this choice, the second term in Eq. (4) vanishes. The third term in (4) is, of course, a c-number and provides an additive constant to the energy of the atom. It should be noted that, although the no-pair Hamiltonian is a useful starting point for relativistic many-body calculations, certain small contributions to wave functions and energies, including frequency-dependent corrections to the Breit interaction, self-energy and vacuum-polarization corrections, and corrections from crossed-ladder diagrams, are omitted in this approach. Perturbation theory based directly on the Furry representation of QED includes all such omitted effects [35]. In calculations based on the no-pair Hamiltonian, 5 contributions from these omitted terms are usually estimated and added as an afterthought. Recently, however, an energy-dependent formulation of MBPT that includes QED corrections completely has been developed by Lindgren et al. [36] and applied to heliumlike ions. Let us return to MBPT and concentrate on the simplest atoms, those with a single valence electron. For monovalent atoms, we write the lowest-order state vector as Ψ (0) v = a v 0 c, (9) where 0 c = a aa b a n 0 is the state vector for the closed core, 0 being the vacuum state vector and a v being a valence-state creation operator. If we ignore the Breit interaction and start our calculation using DHF wave functions for one-electron states (U = V HF ), then the lowest-order energy of the atom, obtained from H 0 Ψ (0) v = E (0) Ψ (0) v, is E (0) = ɛ v + a ɛ a, (10) and the first-order energy is E (1) = Ψ (0) v V Ψ (0) v = 1 2 (V HF ) aa. (11) a We see that through first order, the energy separates into a core contribution and a valence contribution, with E (0+1) core = a ɛ a 1 (V HF ) aa = (h 0 ) aa + 1 (g abab g abba ), (12) 2 a a 2 E (0+1) v = ɛ v. (13) The summation indices a and b in Eqs. (11) and (12) range over core states. The quantity (h 0 ) aa is the matrix element in state a of the sum of the kinetic energy and nuclear potential terms in the Dirac Hamiltonian (2). The sum of zeroth- plus first-order energies in (12) is precisely the DHF energy of the core. The energy of a one-electron atom splits order-by-order into core and valence contributions E (k) = E core (k) + E v (k). Since the core contribution is the same for each valence states, it is sufficient to consider valence contributions when studying excitation or ionization energies of one-electron atoms using MBPT. The second-order contribution to the valence energy is found to be [37] E (2) v = g abvn g vnab nab ɛ v + ɛ n ɛ a ɛ b mnb ab g vbmn g mnvb ɛ m + ɛ n ɛ v ɛ b. (14) Here and in the following sections, we adopt the convention that letters near the start of the alphabet (a, b, c, ) designate core states, letters in the middle 6 of the alphabet (m, n, o, ) designate virtual states, and letters near the end of the alphabet (v, w, x, ) designate valence states. We let the letters (i, j, k, ) designate either core or virtual (general) states. In Eq. (14), we have also used the notation g ijkl = g ijkl g ijlk to designate anti-symmetrized two-particle matrix elements. The much longer expression for the third-order contribution to the valence energy for a monovalent atom E v (3) is given in Ref. [37] and will not be repeated here. To evaluate the expressions for second- and third-order energies, we first sum over magnetic quantum numbers analytically to obtain expressions involving radial Dirac wave functions and angular momentum coupling coefficients, then we sum over the remaining principal and angular quantum numbers numerically. To aid in the numerical work, we replace the spectrum of the radial Dirac equation, which consists of bound states, a positive-energy continuum of scattering states, and a negative-energy continuum of positron states, by a finite pseudospectrum. For the calculations discussed in this review, the pseudospectrum was constructed from B-splines confined to a large but finite cavity, as described in Ref. [38]. In Table 1, we give a breakdown of the zeroth-order, second-order, and thirdorder MBPT contributions to ionization energies of alkali-metal atoms and compare the sum with various all-order calculations and with experiment. Differences between third-order MBPT calculations and experiment range from fractions of 1% for Li and Na to about 3% for Cs. Moreover, for Cs, including third-order corrections actually worsens the agreement with measured energies found in second order, emphasizing the need for all-order methods. 3 Relativistic SD All-Order Method As an introduction to relativistic all-order calculations, we briefly describe the relativistic singles-doubles (SD) method, a linearized version of coupledcluster theory; a more detailed description can be found in [21, 25]. In the coupled-cluster theory, the exact many-body wave function is represented in the form [39] Ψ = exp(s) Ψ (0), (15) where Ψ (0) is the lowest-order atomic state vector. The operator S for an N-electron atom consists of cluster contributions from one-electron, twoelectron,, N-electron excitations of the lowest-order state vector Ψ (0) : S = S 1 + S S N. (16) 7 The exponential in Eq. (15), when expanded in terms of the n-body excitations S n, becomes Ψ = { 1 + S 1 + S 2 + S S2 1 + S 1 S } 2 S2 2 + Ψ (0). (17) In the linearized coupled-cluster method, all non-linear terms are omitted and the wave function takes the form Ψ = {1 + S 1 + S 2 + S S N } Ψ (0). (18) The SD method is the linearized coupled-cluster method restricted to single and double excitations only. The all-order singles-doubles-partial triples (SDpT) method is an extension of the SD method in which the dominant part of S 3 is treated perturbatively. A detailed description of the SDpT method is given in Refs. [24, 26]. Inclusion of the non-linear terms in the relativistic SD formalism and a more complete treatment of the triple excitations is given in [40, 41] and will be considered later. Restricting the sum in Eq. (18) to single and double excitations yields the following expansion for the SD state vector of a monovalent atom in state v: Ψ v = [ 1 + ma ρ ma a ma a mnab ρ mnab a ma na b a a + + ρ mv a ma v + ρ mnva a ma na a a v Ψ (0) v, (19) m v mna where Ψ (0) v is the lowest-order atomic state vector given in Eq. (9). In Eq. (19), the indices m and n range over all possible virtual states while indices a and b range over all occupied core states. The quantities ρ ma, ρ mv are singleexcitation coefficients for core and valence electrons and ρ mnab and ρ mnva are double-excitation coefficients for core and valence electrons, respectively. It should be noted that the operator products in Eq. (19) are normally ordered as they stand. To derive equations for the excitation coefficients, the state vector Ψ v is substituted into the many-body Schrödinger equation H Ψ v = E Ψ v, and terms on the left- and right-hand sides are matched, based on the number and type of operators they contain, leading to the following equations for the single and double valence excitation coefficients: 8 (ɛ v ɛ m + δe v )ρ mv = g mbvn ρ nb + g mbnr ρ nrvb g bcvn ρ mnbc, (20) bn bnr bcn (ɛ vb ɛ mn + δe v )ρ mnvb = g mnvb + g cdvb ρ mncd + g mnrs ρ rsvb cd rs [ + g mnrb ρ rv g cnvb ρ mc + ] g cnrb ρ mrvc + v b, (21) r c rc m n where δe v = E v ɛ v, the correlation correction to the energy of the state v, is given in terms of the excitation coefficients by δe v = ma g vavm ρ ma + g abvm ρ mvab + g vbmn ρ mnvb. (22) mab mna In Eq. (21), we use the abbreviation ɛ ij = ɛ i + ɛ j, and in Eq. (22), we use the notation ρ mnvb = ρ mnvb ρ nmvb. Equations for core excitation coefficients ρ ma and ρ mnab are obtained from the above equations by removing δe v from the left-hand side of the equations and replacing the valence index v by a core index a. The core correlation energy is given by δe c = 1 2 mnab g abmn ρ mnab. (23) After removing the dependence on magnetic quantum numbers, Eqs. (20) and (21) are solved iteratively. To this end, states a, b, m, and n are represented in a finite B-spline basis, identical to that used in the MBPT calculations discussed in Sec. 2. As a first step, equations for the core single- and doubleexcitation coefficients ρ ma and ρ mnab are solved iteratively; the core excitation coefficients are stored after the core correlation energy has converged to a specified accuracy. View more... We Need Your Support Thanks to everyone for your continued support. No, Thanks More details... Sign Now! We are very appreciated for your Prompt Action!
242e3c640c87f4c6	Open main menu Field electron emission (also known as field emission (FE) and electron field emission) is emission of electrons induced by an electrostatic field. The most common context is field emission from a solid surface into vacuum. However, field emission can take place from solid or liquid surfaces, into vacuum, air, a fluid, or any non-conducting or weakly conducting dielectric. The field-induced promotion of electrons from the valence to conduction band of semiconductors (the Zener effect) can also be regarded as a form of field emission. The terminology is historical because related phenomena of surface photoeffect, thermionic emission (or Richardson–Dushman effect) and "cold electronic emission", i.e. the emission of electrons in strong static (or quasi-static) electric fields, were discovered and studied independently from the 1880s to 1930s. When field emission is used without qualifiers it typically means "cold emission". Field emission in pure metals occurs in high electric fields: the gradients are typically higher than 1 gigavolt per metre and strongly dependent upon the work function. While electron sources based on field emission have a number of applications, field emission is most commonly an undesirable primary source of vacuum breakdown and electrical discharge phenomena, which engineers work to prevent. Examples of applications for surface field emission include construction of bright electron sources for high-resolution electron microscopes or the discharge of induced charges from spacecraft. Devices which eliminate induced charges are termed charge-neutralizers. Field emission was explained by quantum tunneling of electrons in the late 1920s. This was one of the triumphs of the nascent quantum mechanics. The theory of field emission from bulk metals was proposed by Ralph H. Fowler and Lothar Wolfgang Nordheim.[1] A family of approximate equations, "Fowler–Nordheim equations", is named after them. Strictly, Fowler–Nordheim equations apply only to field emission from bulk metals and (with suitable modification) to other bulk crystalline solids, but they are often used – as a rough approximation – to describe field emission from other materials. Terminology and conventionsEdit Field electron emission, field-induced electron emission, field emission and electron field emission are general names for this experimental phenomenon and its theory. The first name is used here. Fowler–Nordheim tunneling is the wave-mechanical tunneling of electrons through a rounded triangular barrier created at the surface of an electron conductor by applying a very high electric field. Individual electrons can escape by Fowler-Nordheim tunneling from many materials in various different circumstances. Cold field electron emission (CFE) is the name given to a particular statistical emission regime, in which the electrons in the emitter are initially in internal thermodynamic equilibrium, and in which most emitted electrons escape by Fowler-Nordheim tunneling from electron states close to the emitter Fermi level. (By contrast, in the Schottky emission regime, most electrons escape over the top of a field-reduced barrier, from states well above the Fermi level.) Many solid and liquid materials can emit electrons in a CFE regime if an electric field of an appropriate size is applied. Fowler–Nordheim-type equations are a family of approximate equations derived to describe CFE from the internal electron states in bulk metals. The different members of the family represent different degrees of approximation to reality. Approximate equations are necessary because, for physically realistic models of the tunneling barrier, it is mathematically impossible in principle to solve the Schrödinger equation exactly in any simple way. There is no theoretical reason to believe that Fowler-Nordheim-type equations validly describe field emission from materials other than bulk crystalline solids. For metals, the CFE regime extends to well above room temperature. There are other electron emission regimes (such as "thermal electron emission" and "Schottky emission") that require significant external heating of the emitter. There are also emission regimes where the internal electrons are not in thermodynamic equilibrium and the emission current is, partly or completely, determined by the supply of electrons to the emitting region. A non-equilibrium emission process of this kind may be called field (electron) emission if most of the electrons escape by tunneling, but strictly it is not CFE, and is not accurately described by a Fowler-Nordheim-type equation. Care is necessary because in some contexts (e.g. spacecraft engineering), the name "field emission" is applied to the field-induced emission of ions (field ion emission), rather than electrons, and because in some theoretical contexts "field emission" is used as a general name covering both field electron emission and field ion emission. Historically, the phenomenon of field electron emission has been known by a variety of names, including "the aeona effect", "autoelectronic emission", "cold emission", "cold cathode emission", "field emission", "field electron emission" and "electron field emission". Equations in this article are written using the International System of Quantities (ISQ). This is the modern (post-1970s) international system, based around the rationalized-meter-kilogram-second (rmks) system of equations, which is used to define SI units. Older field emission literature (and papers that directly copy equations from old literature) often write some equations using an older equation system that does not use the quantity ε0. In this article, all such equations have been converted to modern international form. For clarity, this should always be done. Since work function is normally given in electronvolts (eV), and it is often convenient to measure fields in volts per nanometer (V/nm), values of most universal constants are given here in units involving the eV, V and nm. Increasingly, this is normal practice in field emission research. However, all equations here are ISQ-compatible equations and remain dimensionally consistent, as is required by the modern international system. To indicate their status, numerical values of universal constants are given to seven significant figures. Values are derived using the 2006 values of the fundamental constants. Early history of field electron emissionEdit Field electron emission has a long, complicated and messy history. This section covers the early history, up to the derivation of the original Fowler–Nordheim-type equation in 1928. In retrospect, it seems likely that the electrical discharges reported by Winkler[2] in 1744 were started by CFE from his wire electrode. However, meaningful investigations had to wait until after J.J. Thomson's[3] identification of the electron in 1897, and until after it was understood – from thermal emission[4] and photo-emission[5] work – that electrons could be emitted from inside metals (rather than from surface-adsorbed gas molecules), and that – in the absence of applied fields – electrons escaping from metals had to overcome a work function barrier. It was suspected at least as early as 1913 that field-induced emission was a separate physical effect.[6] However, only after vacuum and specimen cleaning techniques had significantly improved, did this become well established. Lilienfeld (who was primarily interested in electron sources for medical X-ray applications) published in 1922[7] the first clear account in English of the experimental phenomenology of the effect he had called "autoelectronic emission". He had worked on this topic, in Leipzig, since about 1910. Kleint describes this and other early work.[8][9] After 1922, experimental interest increased, particularly in the groups led by Millikan at the California Institute of Technology (Caltech) in Pasadena, California,[10] and by Gossling at the General Electric Company in London.[11] Attempts to understand autoelectronic emission included plotting experimental current–voltage (i–V) data in different ways, to look for a straight-line relationship. Current increased with voltage more rapidly than linearly, but plots of type log(i) vs. V were not straight.[10] Schottky[12] suggested in 1923 that the effect might be due to thermally induced emission over a field-reduced barrier. If so, then plots of log(i) vs. V should be straight, but they were not.[10] Nor is Schottky's explanation compatible with the experimental observation of only very weak temperature dependence in CFE[7] – a point initially overlooked.[6] A breakthrough came when Lauritsen[13] (and Oppenheimer independently[14]) found that plots of log(i) vs. 1/V yielded good straight lines. This result, published by Millikan and Lauritsen[13] in early 1928, was known to Fowler and Nordheim. Oppenheimer had predicted[14] that the field-induced tunneling of electrons from atoms (the effect now called field ionization) would have this i(V) dependence, had found this dependence in the published experimental field emission results of Millikan and Eyring,[10] and proposed that CFE was due to field-induced tunneling of electrons from atomic-like orbitals in surface metal atoms. An alternative Fowler–Nordheim theory[1] explained both the Millikan-Lauritsen finding and the very weak dependence of current on temperature. Fowler–Nordheim theory predicted both to be consequences if CFE were due to field-induced tunneling from free-electron-type states in what we would now call a metal conduction band, with the electron states occupied in accordance with Fermi–Dirac statistics. In fact, Oppenheimer (although right in principle about the theory of field ionization) had mathematical details of his theory seriously incorrect.[15] There was also a small numerical error in the final equation given by Fowler–Nordheim theory for CFE current density: this was corrected in the 1929 paper of (Stern, Gossling & Fowler 1929).[16] Strictly, if the barrier field in Fowler-Nordheim 1928 theory is exactly proportional to the applied voltage, and if the emission area is independent of voltage, then the Fowler-Nordheim 1928 theory predicts that plots of the form (log(i/V2) vs. 1/V) should be exact straight lines. However, contemporary experimental techniques were not good enough to distinguish between the Fowler-Nordheim theoretical result and the Millikan-Lauritsen experimental result. Thus, by 1928 basic physical understanding of the origin of CFE from bulk metals had been achieved, and the original Fowler-Nordheim-type equation had been derived. The literature often presents Fowler-Nordheim work as a proof of the existence of electron tunneling, as predicted by wave-mechanics. Whilst this is correct, the validity of wave-mechanics was largely accepted by 1928. The more important role of the Fowler-Nordheim paper was that it was a convincing argument from experiment that Fermi–Dirac statistics applied to the behavior of electrons in metals, as suggested by Sommerfeld[17] in 1927. The success of Fowler–Nordheim theory did much to support the correctness of Sommerfeld's ideas, and greatly helped to establish modern electron band theory.[18] In particular, the original Fowler-Nordheim-type equation was one of the first to incorporate the statistical-mechanical consequences of the existence of electron spin into the theory of an experimental condensed-matter effect. The Fowler-Nordheim paper also established the physical basis for a unified treatment of field-induced and thermally induced electron emission.[18] Prior to 1928 it had been hypothesized that two types of electrons, "thermions" and "conduction electrons", existed in metals, and that thermally emitted electron currents were due to the emission of thermions, but that field-emitted currents were due to the emission of conduction electrons. The Fowler-Nordheim 1928 work suggested that thermions did not need to exist as a separate class of internal electrons: electrons could come from a single band occupied in accordance with Fermi–Dirac statistics, but would be emitted in statistically different ways under different conditions of temperature and applied field. The ideas of Oppenheimer, Fowler and Nordheim were also an important stimulus to the development, by Gamow,[19] and Gurney and Condon,[20][21] later in 1928, of the theory of the radioactive decay of nuclei (by alpha particle tunneling).[22] Practical applications: past and presentEdit Field electron microscopy and related basicsEdit As already indicated, the early experimental work on field electron emission (1910–1920) [7] was driven by Lilienfeld's desire to develop miniaturized X-ray tubes for medical applications. However, it was too early for this technology to succeed. After Fowler-Nordheim theoretical work in 1928, a major advance came with the development in 1937 by Erwin W. Mueller of the spherical-geometry field electron microscope (FEM) [23] (also called the "field emission microscope"). In this instrument, the electron emitter is a sharply pointed wire, of apex radius r. This is placed, in a vacuum enclosure, opposite an image detector (originally a phosphor screen), at a distance R from it. The microscope screen shows a projection image of the distribution of current-density J across the emitter apex, with magnification approximately (R/r), typically 105 to 106. In FEM studies the apex radius is typically 100 nm to 1 μm. The tip of the pointed wire, when referred to as a physical object, has been called a "field emitter", a "tip", or (recently) a "Mueller emitter". When the emitter surface is clean, this FEM image is characteristic of: (a) the material from which the emitter is made: (b) the orientation of the material relative to the needle/wire axis; and (c) to some extent, the shape of the emitter endform. In the FEM image, dark areas correspond to regions where the local work function φ is relatively high and/or the local barrier field F is relatively low, so J is relatively low; the light areas correspond to regions where φ is relatively low and/or F is relatively high, so J is relatively high. This is as predicted by the exponent of Fowler-Nordheim-type equations [see eq. (30) below]. The adsorption of layers of gas atoms (such as oxygen) onto the emitter surface, or part of it, can create surface electric dipoles that change the local work function of this part of the surface. This affects the FEM image; also, the change of work-function can be measured using a Fowler-Nordheim plot (see below). Thus, the FEM became an early observational tool of surface science.[24][25] For example, in the 1960s, FEM results contributed significantly to discussions on heterogeneous catalysis.[26] FEM has also been used for studies of surface-atom diffusion. However, FEM has now been almost completely superseded by newer surface-science techniques. A consequence of FEM development, and subsequent experimentation, was that it became possible to identify (from FEM image inspection) when an emitter was "clean", and hence exhibiting its clean-surface work-function as established by other techniques. This was important in experiments designed to test the validity of the standard Fowler-Nordheim-type equation.[27][28] These experiments deduced a value of voltage-to-barrier-field conversion factor β from a Fowler-Nordheim plot (see below), assuming the clean-surface φ–value for tungsten, and compared this with values derived from electron-microscope observations of emitter shape and electrostatic modeling. Agreement to within about 10% was achieved. Only very recently[29] has it been possible to do the comparison the other way round, by bringing a well-prepared probe so close to a well-prepared surface that approximate parallel-plate geometry can be assumed and the conversion factor can be taken as 1/W, where W is the measured probe-to emitter separation. Analysis of the resulting Fowler-Nordheim plot yields a work-function value close to the independently known work-function of the emitter. Field electron spectroscopy (electron energy analysis)Edit Energy distribution measurements of field-emitted electrons were first reported in 1939.[30] In 1959 it was realized theoretically by Young,[31] and confirmed experimentally by Young and Mueller[32] that the quantity measured in spherical geometry was the distribution of the total energy of the emitted electron (its "total energy distribution"). This is because, in spherical geometry, the electrons move in such a fashion that angular momentum about a point in the emitter is very nearly conserved. Hence any kinetic energy that, at emission, is in a direction parallel to the emitter surface gets converted into energy associated with the radial direction of motion. So what gets measured in an energy analyzer is the total energy at emission. With the development of sensitive electron energy analyzers in the 1960s, it became possible to measure fine details of the total energy distribution. These reflect fine details of the surface physics, and the technique of Field Electron Spectroscopy flourished for a while, before being superseded by newer surface-science techniques.[33][34] Field electron emitters as electron-gun sourcesEdit To achieve high-resolution in electron microscopes and other electron beam instruments (such as those used for electron beam lithography), it is helpful to start with an electron source that is small, optically bright and stable. Sources based on the geometry of a Mueller emitter qualify well on the first two criteria. The first electron microscope (EM) observation of an individual atom was made by Crewe, Wall and Langmore in 1970,[35] using a scanning electron microscope equipped with an early field emission gun. From the 1950s onwards, extensive effort has been devoted to the development of field emission sources for use in electron guns.[36][37][38] [e.g., DD53] Methods have been developed for generating on-axis beams, either by field-induced emitter build-up, or by selective deposition of a low-work-function adsorbate (usually Zirconium oxide - ZrO) into the flat apex of a (100) oriented Tungsten emitter.[39] Sources that operate at room temperature have the disadvantage that they rapidly become covered with adsorbate molecules that arrive from the vacuum system walls, and the emitter has to be cleaned from time to time by "flashing" to high temperature. Nowadays, it is more common to use Mueller-emitter-based sources that are operated at elevated temperatures, either in the Schottky emission regime or in the so-called temperature-field intermediate regime. Many modern high-resolution electron microscopes and electron beam instruments use some form of Mueller-emitter-based electron source. Currently, attempts are being made to develop carbon nanotubes (CNTs) as electron-gun field emission sources.[40][41] The use of field emission sources in electron optical instruments has involved the development of appropriate theories of charged particle optics,[37][42] and the development of related modeling. Various shape models have been tried for Mueller emitters; the best seems to be the "Sphere on Orthogonal Cone" (SOC) model introduced by Dyke, Trolan. Dolan and Barnes in 1953.[43] Important simulations, involving trajectory tracing using the SOC emitter model, were made by Wiesener and Everhart.[44][45][46] Nowadays, the facility to simulate field emission from Mueller emitters is often incorporated into the commercial electron-optics programmes used to design electron beam instruments. The design of efficient modern field-emission electron guns requires highly specialized expertise. Atomically sharp emittersEdit Nowadays it is possible to prepare very sharp emitters, including emitters that end in a single atom. In this case, electron emission comes from an area about twice the crystallographic size of a single atom. This was demonstrated by comparing FEM and field ion microscope (FIM) images of the emitter.[47] Single-atom-apex Mueller emitters also have relevance to the scanning probe microscopy and helium scanning ion microscopy (He SIM).[48] Techniques for preparing them have been under investigation for many years.[47][49] A related important recent advance has been the development (for use in the He SIM) of an automated technique for restoring a three-atom ("trimer") apex to its original state, if the trimer breaks up.[48] Large-area field emission sources: vacuum nanoelectronicsEdit Materials aspectsEdit Large-area field emission sources have been of interest since the 1970s. In these devices, a high density of individual field emission sites is created on a substrate (originally silicon). This research area became known, first as "vacuum microelectronics", now as "vacuum nanoelectronics". One of the original two device types, the "Spindt array",[50] used silicon-integrated-circuit (IC) fabrication techniques to make regular arrays in which molybdenum cones were deposited in small cylindrical voids in an oxide film, with the void covered by a counterelectrode with a central circular aperture. This overall geometry has also been used with carbon nanotubes grown in the void. The other original device type was the "Latham emitter".[51][52] These were MIMIV (metal-insulator-metal-insulator-vacuum) – or, more generally, CDCDV (conductor-dielectric-conductor-dielectric-vacuum) – devices that contained conducting particulates in a dielectric film. The device field-emits because its microstructure/nanostructure has field-enhancing properties. This material had a potential production advantage, in that it could be deposited as an "ink", so IC fabrication techniques were not needed. However, in practice, uniformly reliable devices proved difficult to fabricate. Research advanced to look for other materials that could be deposited/grown as thin films with suitable field-enhancing properties. In a parallel-plate arrangement, the "macroscopic" field FM between the plates is given by FM = V/W, where W is the plate separation and V is the applied voltage. If a sharp object is created on one plate, then the local field F at its apex is greater than FM and can be related to FM by The parameter γ is called the "field enhancement factor" and is basically determined by the object's shape. Since field emission characteristics are determined by the local field F, then the higher the γ-value of the object, then the lower the value of FM at which significant emission occurs. Hence, for a given value of W, the lower the applied voltage V at which significant emission occurs. For a roughly ten year-period from the mid-1990s, there was great interest in field emission from plasma-deposited films of amorphous and "diamond-like" carbon.[53][54] However, interest subsequently lessened, partly due to the arrival of CNT emitters, and partly because evidence emerged that the emission sites might be associated with particulate carbon objects created in an unknown way during the deposition process: this suggested that quality control of an industrial-scale production process might be problematic. The introduction of CNT field emitters,[41] both in "mat" form and in "grown array" forms, was a significant step forward. Extensive research has been undertaken into both their physical characteristics and possible technological applications.[40] For field emission, an advantage of CNTs is that, due to their shape, with its high aspect ratio, they are "natural field-enhancing objects". In recent years there has also been massive growth in interest in the development of other forms of thin-film emitter, both those based on other carbon forms (such as "carbon nanowalls[55] ") and on various forms of wide-band-gap semiconductor.[56] A particular aim is to develop "high-γ" nanostructures with a sufficiently high density of individual emission sites. Thin films of nanotubes in form of nanotube webs are also used for development of field emission electrodes,.[57][58][59] It is shown that by fine-tuning the fabrication parameters, these webs can achieve an optimum density of individual emission sites[57] Double-layered electrodes made by deposition of two layers of these webs with perpendicular alignment towards each other are shown to be able to lower the turn-on electric field (electric field required for achieving an emission current of 10 μA/cm2) down to 0.3 V/μm and provide a stable field emission performance.[58] Common problems with all field emission devices, particularly those that operate in "industrial vacuum conditions" is that the emission performance can be degraded by the adsorption of gas atoms arriving from elsewhere in the system, and the emitter shape can be in principle be modified deleteriously by a variety of unwanted subsidiary processes, such as bombardment by ions created by the impact of emitted electrons onto gas-phase atoms and/or onto the surface of counter-electrodes. Thus, an important industrial requirement is "robustness in poor vacuum conditions"; this needs to be taken into account in research on new emitter materials. At the time of writing, the most promising forms of large-area field emission source (certainly in terms of achieved average emission current density) seem to be Spindt arrays and the various forms of source based on CNTs. The development of large-area field emission sources was originally driven by the wish to create new, more efficient, forms of electronic information display. These are known as "field emission displays" or "nano-emissive displays". Although several prototypes have been demonstrated,[40] the development of such displays into reliable commercial products has been hindered by a variety of industrial production problems not directly related to the source characteristics [En08]. Other proposed applications of large-area field emission sources[40] include microwave generation, space-vehicle neutralization, X-ray generation, and (for array sources) multiple e-beam lithography. There are also recent attempts to develop large-area emitters on flexible substrates, in line with wider trends towards "plastic electronics". The development of such applications is the mission of vacuum nanoelectronics. However, field emitters work best in conditions of good ultrahigh vacuum. Their most successful applications to date (FEM, FES and EM guns) have occurred in these conditions. The sad fact remains that field emitters and industrial vacuum conditions do not go well together, and the related problems of reliably ensuring good "vacuum robustness" of field emission sources used in such conditions still await better solutions (probably cleverer materials solutions) than we currently have. Vacuum breakdown and electrical discharge phenomenaEdit As already indicated, it is now thought that the earliest manifestations of field electron emission were the electrical discharges it caused. After Fowler-Nordheim work, it was understood that CFE was one of the possible primary underlying causes of vacuum breakdown and electrical discharge phenomena. (The detailed mechanisms and pathways involved can be very complicated, and there is no single universal cause)[60] Where vacuum breakdown is known to be caused by electron emission from a cathode, then the original thinking was that the mechanism was CFE from small conducting needle-like surface protrusions. Procedures were (and are) used to round and smooth the surfaces of electrodes that might generate unwanted field electron emission currents. However the work of Latham and others[51] showed that emission could also be associated with the presence of semiconducting inclusions in smooth surfaces. The physics of how the emission is generated is still not fully understood, but suspicion exists that so-called "triple-junction effects" may be involved. Further information may be found in Latham's book[51] and in the on-line bibliography.[60] Internal electron transfer in electronic devicesEdit In some electronic devices, electron transfer from one material to another, or (in the case of sloping bands) from one band to another ("Zener tunneling"), takes place by a field-induced tunneling process that can be regarded as a form of Fowler-Nordheim tunneling. For example, Rhoderick's book discusses the theory relevant to metal-semiconductor contacts.[61] Fowler–Nordheim tunnelingEdit The next part of this article deals with the basic theory of cold field electron emission from bulk metals. This is best treated in four main stages, involving theory associated with: (1) derivation of a formula for "escape probability", by considering electron tunneling through a rounded triangular barrier; (2) an integration over internal electron states to obtain the "total energy distribution"; (3) a second integration, to obtain the emission current density as a function of local barrier field and local work function; (4) conversion of this to a formula for current as a function of applied voltage. The modified equations needed for large-area emitters, and issues of experimental data analysis, are dealt with separately. Fowler–Nordheim tunneling is the wave-mechanical tunneling of an electron through an exact or rounded triangular barrier. Two basic situations are recognized: (1) when the electron is initially in a localized state; (2) when the electron is initially not strongly localized, and is best represented by a travelling wave. Emission from a bulk metal conduction band is a situation of the second type, and discussion here relates to this case. It is also assumed that the barrier is one-dimensional (i.e., has no lateral structure), and has no fine-scale structure that causes "scattering" or "resonance" effects. To keep this explanation of Fowler-Nordheim tunneling relatively simple, these assumptions are needed; but the atomic structure of matter is in effect being disregarded. Motive energyEdit For an electron, the one-dimensional Schrödinger equation can be written in the form where Ψ(x) is the electron wave-function, expressed as a function of distance x measured from the emitter's electrical surface,[62] ħ is the reduced Planck constant, m is the electron mass, U(x) is the electron potential energy, En is the total electron energy associated with motion in the x-direction, and M(x) = [U(x) − En] is called the electron motive energy.[63] M(x) can be interpreted as the negative of the electron kinetic energy associated with the motion of a hypothetical classical point electron in the x-direction, and is positive in the barrier. The shape of a tunneling barrier is determined by how M(x) varies with position in the region where M(x) > 0. Two models have special status in field emission theory: the exact triangular (ET) barrier and the Schottky–Nordheim (SN) barrier.[64][65] These are given by equations (2) and (3), respectively: Here h is the zero-field height (or unreduced height) of the barrier, e is the elementary positive charge, F is the barrier field, and ε0 is the electric constant. By convention, F is taken as positive, even though the classical electrostatic field would be negative. The SN equation uses the classical image potential energy to represent the physical effect "correlation and exchange". Escape probabilityEdit For an electron approaching a given barrier from the inside, the probability of escape (or "transmission coefficient" or "penetration coefficient") is a function of h and F, and is denoted by D(h,F). The primary aim of tunneling theory is to calculate D(h,F). For physically realistic barrier models, such as the Schottky-Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way. The following so-called "semi-classical" approach can be used. A parameter G(h,F) can be defined by the JWKB (Jeffreys-Wentzel-Kramers-Brillouin) integral:[66] where the integral is taken across the barrier (i.e., across the region where M > 0), and the parameter g is a universal constant given by Forbes has re-arranged a result proved by Fröman and Fröman, to show that, formally – in a one-dimensional treatment – the exact solution for D can be written[67] where the tunneling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in complex space.[67][68] In the CFE regime we have (by definition) G ≫ 1. Also, for simple models P ≈ 1. So eq. (6) reduces to the so-called simple JWKB formula: For the exact triangular barrier, putting eq. (2) into eq. (4) yields GET = bh3/2/F, where This parameter b is a universal constant sometimes called the second Fowler–Nordheim constant. For barriers of other shapes, we write where ν(h,F) is a correction factor that in general has to be determined by numerical integration, using eq. (4). Correction factor for the Schottky–Nordheim barrierEdit The Schottky-Nordheim barrier, which is the barrier model used in deriving the standard Fowler-Nordheim-type equation,[69] is a special case. In this case, it is known that the correction factor is a function of a single variable fh, defined by fh = F/Fh, where Fh is the field necessary to reduce the height of a Schottky–Nordheim barrier from h to 0. This field is given by The parameter fh runs from 0 to 1, and may be called the scaled barrier field, for a Schottky-Nordheim barrier of zero-field height h. For the Schottky–Nordheim barrier, ν(h,F) is given by the particular value ν(fh) of a function ν(ℓ′). The latter is a function of mathematical physics in its own right and has been called the principal Schottky–Nordheim barrier function. An explicit series expansion for ν(ℓ′) is derived in a 2008 paper by J. Deane.[70] The following good simple approximation for ν(fh) has been found:[69] Decay widthEdit The decay width (in energy), dh, measures how fast the escape probability D decreases as the barrier height h increases; dh is defined by: When h increases by dh then the escape probability D decreases by a factor close to e ( ≈ 2.718282). For an elementary model, based on the exact triangular barrier, where we put ν = 1 and P ≈ 1, we get The decay width dh derived from the more general expression (12) differs from this by a "decay-width correction factor" λd, so: Usually, the correction factor can be approximated as unity. The decay-width dF for a barrier with h equal to the local work-function φ is of special interest. Numerically this is given by: For metals, the value of dF is typically of order 0.2 eV, but varies with barrier-field F. A historical note is necessary. The idea that the Schottky-Nordheim barrier needed a correction factor, as in eq. (9), was introduced by Nordheim in 1928,[65] but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, Kroemer and Houston[71] in 1953, and its mathematics was developed further by Murphy and Good in 1956.[72] This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable y known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ℓ′ ( = y2). And only recently has it been possible to complete the definition of ν(ℓ′) by developing and proving the validity of an exact series expansion for this function (by starting from known special-case solutions of the Gauss hypergeometric differential equation). Also, approximation (11) has been found only recently. Approximation (11) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course. The following summary brings these results together. For tunneling well below the top of a well-behaved barrier of reasonable height, the escape probability D(h,F) is given formally by: where ν(h,F) is a correction factor that in general has to be found by numerical integration. For the special case of a Schottky-Nordheim barrier, an analytical result exists and ν(h,F) is given by ν(fh), as discussed above; approximation (11) for ν(fh) is more than sufficient for all technological purposes. The pre-factor P is also in principle a function of h and (maybe) F, but for the simple physical models discussed here it is usually satisfactory to make the approximation P = 1. The exact triangular barrier is a special case where the Schrödinger equation can be solved exactly, as was done by Fowler and Nordheim;[1] for this physically unrealistic case, ν(fh) = 1, and an analytical approximation for P exists. The approach described here was originally developed to describe Fowler–Nordheim tunneling from smooth, classically flat, planar emitting surfaces. It is adequate for smooth, classical curved surfaces of radii down to about 10 to 20 nm. It can be adapted to surfaces of sharper radius, but quantities such as ν and D then become significant functions of the parameter(s) used to describe the surface curvature. When the emitter is so sharp that atomic-level detail cannot be neglected, and/or the tunneling barrier is thicker than the emitter-apex dimensions, then a more sophisticated approach is desirable. As noted at the beginning, the effects of the atomic structure of materials are disregarded in the relatively simple treatments of field electron emission discussed here. Taking atomic structure properly into account is a very difficult problem, and only limited progress has been made.[33] However, it seems probable that the main influences on the theory of Fowler-Nordheim tunneling will (in effect) be to change the values of P and ν in eq. (15), by amounts that cannot easily be estimated at present. All these remarks apply in principle to Fowler Nordheim tunneling from any conductor where (before tunneling) the electrons may be treated as in travelling-wave states. The approach may be adapted to apply (approximately) to situations where the electrons are initially in localized states at or very close inside the emitting surface, but this is beyond the scope of this article. Total-energy distributionEdit The energy distribution of the emitted electrons is important both for scientific experiments that use the emitted electron energy distribution to probe aspects of the emitter surface physics[34] and for the field emission sources used in electron beam instruments such as electron microscopes.[42] In the latter case, the "width" (in energy) of the distribution influences how finely the beam can be focused. The theoretical explanation here follows the approach of Forbes.[73] If ε denotes the total electron energy relative to the emitter Fermi level, and Kp denotes the kinetic energy of the electron parallel to the emitter surface, then the electron's normal energy εn (sometimes called its "forwards energy") is defined by Two types of theoretical energy distribution are recognized: the normal-energy distribution (NED), which shows how the energy εn is distributed immediately after emission (i.e., immediately outside the tunneling barrier); and the total-energy distribution, which shows how the total energy ε is distributed. When the emitter Fermi level is used as the reference zero level, both ε and εn can be either positive or negative. Energy analysis experiments have been made on field emitters since the 1930s. However, only in the late 1950s was it realized (by Young and Mueller[31] [,YM58]) that these experiments always measured the total energy distribution, which is now usually denoted by j(ε). This is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface.[34] To see how the total energy distribution can be calculated within the framework of a Sommerfeld free-electron-type model, look at the P-T energy-space diagram (P-T="parallel-total"). This shows the "parallel kinetic energy" Kp on the horizontal axis and the total energy ε on the vertical axis. An electron inside the bulk metal usually has values of Kp and ε that lie within the lightly shaded area. It can be shown that each element dεdKp of this energy space makes a contribution to the electron current density incident on the inside of the emitter boundary.[73] Here, zS is the universal constant (called here the Sommerfeld supply density): and is the Fermi–Dirac distribution function: where T is thermodynamic temperature and kB is Boltzmann's constant. This element of incident current density sees a barrier of height h given by: The corresponding escape probability is D(h,F): this may be expanded (approximately) in the form[73] where DF is the escape probability for a barrier of unreduced height equal to the local work-function φ. Hence, the element dεdKp makes a contribution to the emission current density, and the total contribution made by incident electrons with energies in the elementary range dε is thus where the integral is in principle taken along the strip shown in the diagram, but can in practice be extended to ∞ when the decay-width dF is very much less than the Fermi energy KF (which is always the case for a metal). The outcome of the integration can be written: where and are values appropriate to a barrier of unreduced height h equal to the local work function φ, and is defined by this equation. For a given emitter, with a given field applied to it, is independent of F, so eq. (21) shows that the shape of the distribution (as ε increases from a negative value well below the Fermi level) is a rising exponential, multiplied by the FD distribution function. This generates the familiar distribution shape first predicted by Young.[31] At low temperatures, goes sharply from 1 to 0 in the vicinity of the Fermi level, and the FWHM of the distribution is given by: The fact that experimental CFE total energy distributions have this basic shape is a good experimental confirmation that electrons in metals obey Fermi–Dirac statistics. Cold field electron emissionEdit Fowler–Nordheim-type equationsEdit Fowler–Nordheim-type equations, in the J-F form, are (approximate) theoretical equations derived to describe the local current density J emitted from the internal electron states in the conduction band of a bulk metal. The emission current density (ECD) J for some small uniform region of an emitting surface is usually expressed as a function J(φ,F) of the local work-function φ and the local barrier field F that characterize the small region. For sharply curved surfaces, J may also depend on the parameter(s) used to describe the surface curvature. Owing to the physical assumptions made in the original derivation,[1] the term Fowler-Nordheim-type equation has long been used only for equations that describe the ECD at zero temperature. However, it is better to allow this name to include the slightly modified equations (discussed below) that are valid for finite temperatures within the CFE emission regime. Zero-temperature formEdit Current density is best measured in A/m2. The total current density emitted from a small uniform region can be obtained by integrating the total energy distribution j(ε) with respect to total electron energy ε. At zero temperature, the Fermi–Dirac distribution function fFD = 1 for ε<0, and fFD = 0 for ε>0. So the ECD at 0 K, J0, is given from eq. (18) by where is the effective supply for state F, and is defined by this equation. Strictly, the lower limit of the integral should be –KF, where KF is the Fermi energy; but if dF is very much less than KF (which is always the case for a metal) then no significant contribution to the integral comes from energies below KF, and it can formally be extended to –∞. Result (23) can be given a simple and useful physical interpretation by referring to Fig. 1. The electron state at point "F" on the diagram ("state F") is the "forwards moving state at the Fermi level" (i.e., it describes a Fermi-level electron moving normal to and towards the emitter surface). At 0 K, an electron in this state sees a barrier of unreduced height φ, and has an escape probability DF that is higher than that for any other occupied electron state. So it is convenient to write J0 as ZFDF, where the "effective supply" ZF is the current density that would have to be carried by state F inside the metal if all of the emission came out of state F. In practice, the current density mainly comes out of a group of states close in energy to state F, most of which lie within the heavily shaded area in the energy-space diagram. Since, for a free-electron model, the contribution to the current density is directly proportional to the area in energy space (with the Sommerfeld supply density zS as the constant of proportionality), it is useful to think of the ECD as drawn from electron states in an area of size dF2 (measured in eV2) in the energy-space diagram. That is, it is useful to think of the ECD as drawn from states in the heavily shaded area in Fig. 1. (This approximation gets slowly worse as temperature increases.) ZF can also be written in the form: where the universal constant a, sometimes called the First Fowler–Nordheim Constant, is given by This shows clearly that the pre-exponential factor a φ−1F2, that appears in Fowler-Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model. Non-zero temperaturesEdit To obtain a result valid for non-zero temperature, we note from eq. (23) that zSdFDF = J0/dF. So when eq. (21) is integrated at non-zero temperature, then – on making this substitution, and inserting the explicit form of the Fermi–Dirac distribution function – the ECD J can be written in the form: where λT is a temperature correction factor given by the integral. The integral can be transformed, by writing and , and then , into the standard result:[74] This is valid for w>1 (i.e., dF/kBT > 1). Hence – for temperatures such that kBT<dF: where the expansion is valid only if (πkBT /dF) << 1. An example value (for φ= 4.5 eV, F= 5 V/nm, T= 300 K) is λT= 1.024. Normal thinking has been that, in the CFE regime, λT is always small in comparison with other uncertainties, and that it is usually unnecessary to explicitly include it in formulae for the current density at room temperature. The emission regimes for metals are, in practice, defined, by the ranges of barrier field F and temperature T for which a given family of emission equations is mathematically adequate. When the barrier field F is high enough for the CFE regime to be operating for metal emission at 0 K, then the condition kBT<dF provides a formal upper bound (in temperature) to the CFE emission regime. However, it has been argued that (due to approximations made elsewhere in the derivation) the condition kBT<0.7dF is a better working limit: this corresponds to a λT-value of around 1.09, and (for the example case) an upper temperature limit on the CFE regime of around 1770 K. This limit is a function of barrier field.[33][72] Note that result (28) here applies for a barrier of any shape (though dF will be different for different barriers). Physically complete Fowler–Nordheim-type equationEdit Result (23) also leads to some understanding of what happens when atomic-level effects are taken into account, and the band-structure is no longer free-electron like. Due to the presence of the atomic ion-cores, the surface barrier, and also the electron wave-functions at the surface, will be different. This will affect the values of the correction factor , the prefactor P, and (to a limited extent) the correction factor λd. These changes will, in turn, affect the values of the parameter DF and (to a limited extent) the parameter dF. For a real metal, the supply density will vary with position in energy space, and the value at point "F" may be different from the Sommerfeld supply density. We can take account of this effect by introducing an electronic-band-structure correction factor λB into eq. (23). Modinos has discussed how this factor might be calculated: he estimates that it is most likely to be between 0.1 and 1; it might lie outside these limits but is most unlikely to lie outside the range 0.01<λB<10.[75] By defining an overall supply correction factor λZ equal to λT λB λd2, and combining equations above, we reach the so-called physically complete Fowler-Nordheim-type equation:[76] where [= (φ,F)] is the exponent correction factor for a barrier of unreduced height φ. This is the most general equation of the Fowler–Nordheim type. Other equations in the family are obtained by substituting specific expressions for the three correction factors , PF and λZ it contains. The so-called elementary Fowler-Nordheim-type equation, that appears in undergraduate textbook discussions of field emission, is obtained by putting λZ→1, PF→1, →1; this does not yield good quantitative predictions because it makes the barrier stronger than it is in physical reality. The so-called standard Fowler-Nordheim-type equation, originally developed by Murphy and Good,[72] and much used in past literature, is obtained by putting λZtF−2, PF→1, vF, where vF is v(f), where f is the value of fh obtained by putting h=φ, and tF is a related parameter (of value close to unity).[69] Within the more complete theory described here, the factor tF−2 is a component part of the correction factor λd2 [see,[67] and note that λd2 is denoted by λD there]. There is no significant value in continuing the separate identification of tF−2. Probably, in the present state of knowledge, the best approximation for simple Fowler-Nordheim-type equation based modeling of CFE from metals is obtained by putting λZ→1, PF → 1, v(f). This re-generates the Fowler-Nordheim-type equation used by Dyke and Dolan in 1956, and can be called the "simplified standard Fowler-Nordheim-type equation". Recommended form for simple Fowler–Nordheim-type calculationsEdit Explicitly, this recommended simplified standard Fowler-Nordheim-type equation, and associated formulae, are: where Fφ here is the field needed to reduce to zero a Schottky-Nordheim barrier of unreduced height equal to the local work-function φ, and f is the scaled barrier field for a Schottky-Nordheim barrier of unreduced height φ. [This quantity f could have been written more exactly as fφSN, but it makes this Fowler-Nordheim-type equation look less cluttered if the convention is adopted that simple f means the quantity denoted by fφSN in,[69] eq. (2.16).] For the example case (φ= 4.5 eV, F= 5 V/nm), f≈ 0.36 and v(f) ≈ 0.58; practical ranges for these parameters are discussed further in.[77] Note that the variable f (the scaled barrier field) is not the same as the variable y (the Nordheim parameter) extensively used in past field emission literature, and that " v(f) " does NOT have the same mathematical meaning and values as the quantity " v(y) " that appears in field emission literature. In the context of the revised theory described here, formulae for v(y), and tables of values for v(y) should be disregarded, or treated as values of v(f1/2). If more exact values for v(f) are required, then[69] provides formulae that give values for v(f) to an absolute mathematical accuracy of better than 8×10−10. However, approximation formula (30c) above, which yields values correct to within an absolute mathematical accuracy of better 0.0025, should gives values sufficiently accurate for all technological purposes.[69] A historical note on methods of deriving Fowler-Nordheim-type equations is necessary. There are several possible approaches to deriving these equations, using free-electron theory. The approach used here was introduced by Forbes in 2004 and may be described as "integrating via the total energy distribution, using the parallel kinetic energy Kp as the first variable of integration".[73] Basically, it is a free-electron equivalent of the Modinos procedure[33][75] (in a more advanced quantum-mechanical treatment) of "integrating over the surface Brillouin zone". By contrast, the free-electron treatments of CFE by Young in 1959,[31] Gadzuk and Plummer in 1973[34] and Modinos in 1984,[33] also integrate via the total energy distribution, but use the normal energy εn (or a related quantity) as the first variable of integration. There is also an older approach, based on a seminal paper by Nordheim in 1928,[78] that formulates the problem differently and then uses first Kp and then εn (or a related quantity) as the variables of integration: this is known as "integrating via the normal-energy distribution". This approach continues to be used by some authors. Although it has some advantages, particularly when discussing resonance phenomena, it requires integration of the Fermi–Dirac distribution function in the first stage of integration: for non-free-electron-like electronic band-structures this can lead to very complex and error-prone mathematics (as in the work of Stratton on semiconductors).[79] Further, integrating via the normal-energy distribution does not generate experimentally measured electron energy distributions. In general, the approach used here seems easier to understand, and leads to simpler mathematics. It is also closer in principle to the more sophisticated approaches used when dealing with real bulk crystalline solids, where the first step is either to integrate contributions to the ECD over constant energy surfaces in a wave-vector space ( k -space),[34] or to integrate contributions over the relevant surface Brillouin zone.[33] The Forbes approach is equivalent either to integrating over a spherical surface in k -space, using the variable Kp to define a ring-like integration element that has cylindrical symmetry about an axis in a direction normal to the emitting surface, or to integrating over an (extended) surface Brillouin zone using circular-ring elements. CFE theoretical equationsEdit The preceding section explains how to derive Fowler-Nordheim-type equations. Strictly, these equations apply only to CFE from bulk metals. The ideas in the following sections apply to CFE more generally, but eq. (30) will be used to illustrate them. For CFE, basic theoretical treatments provide a relationship between the local emission current density J and the local barrier field F, at a local position on the emitting surface. Experiments measure the emission current i from some defined part of the emission surface, as a function of the voltage V applied to some counter-electrode. To relate these variables to J and F, auxiliary equations are used. The voltage-to-barrier-field conversion factor β is defined by: The value of F varies from position to position on an emitter surface, and the value of β varies correspondingly. For a metal emitter, the β−value for a given position will be constant (independent of voltage) under the following conditions: (1) the apparatus is a "diode" arrangement, where the only electrodes present are the emitter and a set of "surroundings", all parts of which are at the same voltage; (2) no significant field-emitted vacuum space-charge (FEVSC) is present (this will be true except at very high emission current densities, around 109 A/m2 or higher [27][80]); (3) no significant "patch fields" exist,[63] as a result of non-uniformities in local work-function (this is normally assumed to be true, but may not be in some circumstances). For non-metals, the physical effects called "field penetration" and "band bending" [M084] can make β a function of applied voltage, although – surprisingly – there are few studies of this effect. The emission current density J varies from position to position across the emitter surface. The total emission current i from a defined part of the emitter is obtained by integrating J across this part. To obtain a simple equation for i(V), the following procedure is used. A reference point "r" is selected within this part of the emitter surface (often the point at which the current density is highest), and the current density at this reference point is denoted by Jr. A parameter Ar, called the notional emission area (with respect to point "r"), is then defined by: where the integral is taken across the part of the emitter of interest. This parameter Ar was introduced into CFE theory by Stern, Gossling and Fowler in 1929 (who called it a "weighted mean area").[81] For practical emitters, the emission current density used in Fowler-Nordheim-type equations is always the current density at some reference point (though this is usually not stated). Long-established convention denotes this reference current density by the simple symbol J, and the corresponding local field and conversion factor by the simple symbols F and β, without the subscript "r" used above; in what follows, this convention is used. The notional emission area Ar will often be a function of the reference local field (and hence voltage),[30] and in some circumstances might be a significant function of temperature. Because Ar has a mathematical definition, it does not necessarily correspond to the area from which emission is observed to occur from a single-point emitter in a field electron (emission) microscope. With a large-area emitter, which contains many individual emission sites, Ar will nearly always be very very[clarification needed] much less than the "macroscopic" geometrical area (AM) of the emitter as observed visually (see below). Incorporating these auxiliary equations into eq. (30a) yields This is the simplified standard Fowler-Nordheim-type equation, in i-V form. The corresponding "physically complete" equation is obtained by multiplying by λZPF. Modified equations for large-area emittersEdit The equations in the preceding section apply to all field emitters operating in the CFE regime. However, further developments are useful for large-area emitters that contain many individual emission sites. For such emitters, the notional emission area will nearly always be very very[clarification needed] much less than the apparent "macroscopic" geometrical area (AM) of the physical emitter as observed visually. A dimensionless parameter αr, the area efficiency of emission, can be defined by Also, a "macroscopic" (or "mean") emission current density JM (averaged over the geometrical area AM of the emitter) can be defined, and related to the reference current density Jr used above, by This leads to the following "large-area versions" of the simplified standard Fowler-Nordheim-type equation: Both these equations contain the area efficiency of emission αr. For any given emitter this parameter has a value that is usually not well known. In general, αr varies greatly as between different emitter materials, and as between different specimens of the same material prepared and processed in different ways. Values in the range 10−10 to 10−6 appear to be likely, and values outside this range may be possible. The presence of αr in eq. (36) accounts for the difference between the macroscopic current densities often cited in the literature (typically 10 A/m2 for many forms of large-area emitter other than Spindt arrays[50]) and the local current densities at the actual emission sites, which can vary widely but which are thought to be generally of the order of 109 A/m2, or possibly slightly less. A significant part of the technological literature on large-area emitters fails to make clear distinctions between local and macroscopic current densities, or between notional emission area Ar and macroscopic area AM, and/or omits the parameter αr from cited equations. Care is necessary in order to avoid errors of interpretation. It is also sometimes convenient to split the conversion factor βr into a "macroscopic part" that relates to the overall geometry of the emitter and its surroundings, and a "local part" that relates to the ability of the very-local structure of the emitter surface to enhance the electric field. This is usually done by defining a "macroscopic field" FM that is the field that would be present at the emitting site in the absence of the local structure that causes enhancement. This field FM is related to the applied voltage by a "voltage-to-macroscopic-field conversion factor" βM defined by: In the common case of a system comprising two parallel plates, separated by a distance W, with emitting nanostructures created on one of them, βM = 1/W. A "field enhancement factor" γ is then defined and related to the values of βr and βM by With eq. (31), this generates the following formulae: where, in accordance with the usual convention, the suffix "r" has now been dropped from parameters relating to the reference point. Formulae exist for the estimation of γ, using classical electrostatics, for a variety of emitter shapes, in particular the "hemisphere on a post".[82] Equation (40) implies that versions of Fowler-Nordheim-type equations can be written where either F or βV is everywhere replaced by . This is often done in technological applications where the primary interest is in the field enhancing properties of the local emitter nanostructure. However in some past work, failure to make a clear distinction between barrier field F and macroscopic field FM has caused confusion or error. More generally, the aims in technological development of large-area field emitters are to enhance the uniformity of emission by increasing the value of the area efficiency of emission αr, and to reduce the "onset" voltage at which significant emission occurs, by increasing the value of β. Eq. (41) shows that this can be done in two ways: either by trying to develop "high-γ" nanostructures, or by changing the overall geometry of the system so that βM is increased. Various trade-offs and constraints exist. In practice, although the definition of macroscopic field used above is the commonest one, other (differently defined) types of macroscopic field and field enhancement factor are used in the literature, particularly in connection with the use of probes to investigate the i-V characteristics of individual emitters.[83] In technological contexts field emission data are often plotted using (a particular definition of) FM or 1/FM as the x-coordinate. However, for scientific analysis it usually better not to pre-manipulate the experimental data, but to plot the raw measured i-V data directly. Values of technological parameters such as (the various forms of) γ can then be obtained from the fitted parameters of the i-V data plot (see below), using the relevant definitions. Modified equations for nanometrically sharp emittersEdit Most of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky-Nordheim form eq. (3). However, this barrier form is not valid for emitters with radii of curvature comparable to the length of the tunnelling barrier. The latter depends on the work function and the field, but in cases of practical interest, the SN barrier approximation can be considered valid for emitters with radii , as explained in the next paragraph. The main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form in the tunnelling region. The latter has been proved to hold only if [84]. Therefore, if the tunnelling region has a length , for all that determines the tunnelling process; thus if eq. (1) holds and the SN barrier approximation is valid. If the tunnelling probability is high enough to produce measurable field emission, L does not exceed 1-2nm. Hence, the SN barrier is valid for emitters with radii of the order of some tens of nm. However, modern emitters are much sharper than this, with radii that of the order of a few nm. Therefore, the standard FN equation, or any version of it that assumes the SN barrier, leads to significant errors for such sharp emitters. This has been both shown theoretically [85][86] and confirmed experimentally [87]. The above problem was tackled in ref. [84]. The SN barrier was generalized taking into account the curvature of the emitter. It can be proven that the electrostatic potential in the vicinity of any metal surface with radius of curvature can be asymptotically expanded as Furthermore, the image potential for a sharp emitter is better represented by the one corresponding to a spherical metal surface rather than a planar one. After neglecting all terms, the total potential barrier takes the form found by Kyritsakis and Xanthakis [84] If the JWKB approximation (4) is used for this barrier, the Gamow exponent takes a form that generalizes eq. (5) where is defined by (30d), is given by (30c) and is a new function that can be approximated in a similar manner as (30c): Given the expression for the Gamow exponent as a function of the field-free barrier height , the emitted current density for cold field emission can be obtained from eq. (23). It yields where the functions and are defined as In equation (46), for completeness purposes, is not approximated by unity as in (29) and (30a), although for most practical cases it is a very good approximation. Apart from this, equations (43), (44) and (46) coincide with the corresponding ones of the standard Fowler-Nordheim theory (3), (9), and (30a), in the limit ; this is expected since the former equations generalise the latter. Finally, note that the above analysis is asymptotic in the limit , similarly to the standard Fowler-Nordheim theory using the SN barrier. However, the addition of the quadratic terms renders it significantly more accurate for emitters with radii of curvature in the range ~5-20nm. For sharper emitters there is no general approximation for the current density. In order to obtain the current density, one has to calculate the electrostatic potential and evaluate the JWKB integral numerically. For this purpose, scientific computing software-libraries have been developed [88] . Empirical CFE iV equationEdit At the present stage of CFE theory development, it is important to make a distinction between theoretical CFE equations and an empirical CFE equation. The former are derived from condensed matter physics (albeit in contexts where their detailed development is difficult). An empirical CFE equation, on the other hand, simply attempts to represent the actual experimental form of the dependence of current i on voltage V. In the 1920s, empirical equations were used to find the power of V that appeared in the exponent of a semi-logarithmic equation assumed to describe experimental CFE results. In 1928, theory and experiment were brought together to show that (except, possibly, for very sharp emitters) this power is V−1. It has recently been suggested that CFE experiments should now be carried out to try to find the power (κ) of V in the pre-exponential of the following empirical CFE equation:[89] where B, C and κ are treated as constants. From eq. (42) it is readily shown that In the 1920s, experimental techniques could not distinguish between the results κ =0 (assumed by Millikan and Laurtisen)[13] and κ=2 (predicted by the original Fowler-Nordheim-type equation).[1] However, it should now be possible to make reasonably accurate measurements of dlni/d(1/V) (if necessary by using lock-in amplifier/phase-sensitive detection techniques and computer-controlled equipment), and to derive κ from the slope of an appropriate data plot.[50] Following the discovery of approximation (30b), it is now very clear that – even for CFE from bulk metals – the value κ=2 is not expected. This can be shown as follows. Using eq. (30c) above, a dimensionless parameter η may be defined by For φ = 4.50 eV, this parameter has the value η = 4.64. Since f = F/Fφ and v(f) is given by eq (30b), the exponent in the simplified standard Fowler-Nordheim-type equation (30) can be written in an alternative form and then expanded as follows:[69] Provided that the conversion factor β is independent of voltage, the parameter f has the alternative definition f = V/Vφ, where Vφ is the voltage needed, in a particular experimental system, to reduce the height of a Schottky-Nordheim barrier from φ to zero. Thus, it is clear that the factor v(f) in the exponent of the theoretical equation (30) gives rise to additional V-dependence in the pre-exponential of the empirical equation. Thus, (for effects due to the Schottky-Nordheim barrier, and for an emitter with φ=4.5 eV) we obtain the prediction: Since there may also be voltage dependence in other factors in a Fowler-Nordheim-type equation, in particular in the notional emission area[30] Ar and in the local work-function, it is not necessarily expected that κ for CFE from a metal of local work-function 4.5 eV should have the value κ = 1.23, but there is certainly no reason to expect that it will have the original Fowler-Nordheim value κ = 2.[90] A first experimental test of this proposal has been carried out by Kirk, who used a slightly more complex form of data analysis to find a value 1.36 for his parameter κ. His parameter κ is very similar to, but not quite the same as, the parameter κ used here, but nevertheless his results do appear to confirm the potential usefulness of this form of analysis.[91] Use of the empirical CFE equation (42), and the measurement of κ, may be of particular use for non-metals. Strictly, Fowler-Nordheim-type equations apply only to emission from the conduction band of bulk crystalline solids. However, empirical equations of form (42) should apply to all materials (though, conceivably, modification might be needed for very sharp emitters). It seems very likely that one way in which CFE equations for newer materials may differ from Fowler-Nordheim-type equations is that these CFE equations may have a different power of F (or V) in their pre-exponentials. Measurements of κ might provide some experimental indication of this. Fowler–Nordheim plots and Millikan–Lauritsen plotsEdit The original theoretical equation derived by Fowler and Nordheim[1] has, for the last 80 years, influenced the way that experimental CFE data has been plotted and analyzed. In the very widely used Fowler-Nordheim plot, as introduced by Stern et al. in 1929,[81] the quantity ln{i/V2} is plotted against 1/V. The original thinking was that (as predicted by the original or the elementary Fowler-Nordheim-type equation) this would generate an exact straight line of slope SFN. SFN would be related to the parameters that appear in the exponent of a Fowler-Nordheim-type equation of i-V form by: Hence, knowledge of φ would allow β to be determined, or vice versa. [In principle, in system geometries where there is local field-enhancing nanostructure present, and the macroscopic conversion factor βM can be determined, knowledge of β then allows the value of the emitter's effective field enhancement factor γ to be determined from the formula γ = β/βM. In the common case of a film emitter generated on one plate of a two-plate arrangement with plate-separation W (so βM = 1/W) then Nowadays, this is one of the most likely applications of Fowler-Nordheim plots.] It subsequently became clear that the original thinking above is strictly correct only for the physically unrealistic situation of a flat emitter and an exact triangular barrier. For real emitters and real barriers a "slope correction factor" σFN has to be introduced, yielding the revised formula The value of σFN will, in principle, be influenced by any parameter in the physically complete Fowler-Nordheim-type equation for i(V) that has a voltage dependence. At present, the only parameter that is considered important is the correction factor relating to the barrier shape, and the only barrier for which there is any well-established detailed theory is the Schottky-Nordheim barrier. In this case, σFN is given by a mathematical function called s. This function s was first tabulated correctly (as a function of the Nordheim parameter y) by Burgess, Kroemer and Houston in 1953;[71] and a modern treatment that gives s as function of the scaled barrier field f for a Schottky-Nordheim barrier is given in.[69] However, it has long been clear that, for practical emitter operation, the value of s lies in the range 0.9 to 1. In practice, due to the extra complexity involved in taking the slope correction factor into detailed account, many authors (in effect) put σFN = 1 in eq. (49), thereby generating a systematic error in their estimated values of β and/or γ, thought usually to be around 5%. However, empirical equation (42) – which in principle is more general than Fowler-Nordheim-type equations - brings with it possible new ways of analyzing field emission i-V data. In general, it may be assumed that the parameter B in the empirical equation is related to the unreduced height H of some characteristic barrier seen by tunneling electrons by (In most cases, but not necessarily all, H would be equal to the local work-function; certainly this is true for metals.) The issue is how to determine the value of B by experiment. There are two obvious ways. (1) Suppose that eq. (43) can be used to determine a reasonably accurate experimental value of κ, from the slope of a plot of form [–dln{i}/d(1/V) vs. V]. In this case, a second plot, of ln(i)/Vκ vs. 1/V, should be an exact straight line of slope –B. This approach should be the most accurate way of determining B. (2) Alternatively, if the value of κ is not exactly known, and cannot be accurately measured, but can be estimated or guessed, then a value for B can be derived from a plot of the form [ln{i} vs. 1/V]. This is the form of plot used by Millikan and Lauritsen in 1928. Re-arranging eq. (43) gives Thus, B can be determined, to a good degree of approximation, by determining the mean slope of a Millikan-Lauritsen plot over some range of values of 1/V, and by applying a correction, using the value of 1/V at the midpoint of the range and an assumed value of κ. The main advantages of using a Millikan-Lauritsen plot, and this form of correction procedure, rather than a Fowler-Nordheim plot and a slope correction factor, are seen to be the following. (1) The plotting procedure is marginally more straightforward. (2) The correction involves a physical parameter (V) that is a measured quantity, rather than a physical parameter (f) that has to be calculated [in order to then calculate a value of s(f) or, more generally σFN(f)]. (3) Both the parameter κ itself, and the correction procedure, are more transparent (and more readily understood) than the Fowler-Nordheim-plot equivalents. (4) This procedure takes into account all physical effects that influence the value of κ, whereas the Fowler-Nordheim-plot correction procedure (in the form in which it has been carried out for the last 50 years) takes into account only those effects associated with barrier shape – assuming, furthermore, that this shape is that of a Schottky-Nordheim barrier. (5) There is a cleaner separation of theoretical and technological concerns: theoreticians will be interested in establishing what information any measured values of κ provide about CFE theory; but experimentalists can simply use measured values of κ to make more accurate estimates (if needed) of field enhancement factors.[citation needed] This correction procedure for Millikan-Lauritsen plots will become easier to apply when a sufficient number of measurements of κ have been made, and a better idea is available of what typical values actually are. At present, it seems probable that for most materials κ will lie in the range -1<κ<3.[citation needed] Further theoretical informationEdit Developing the approximate theory of CFE from metals above is comparatively easy, for the following reasons. (1) Sommerfeld's free-electron theory, with its particular assumptions about the distribution of internal electron states in energy, applies adequately to many metals as a first approximation. (2) Most of the time, metals have no surface states and (in many cases) metal wave-functions have no significant "surface resonances". (3) Metals have a high density of states at the Fermi level, so the charge that generates/screens external electric fields lies mainly on the outside of the top atomic layer, and no meaningful "field penetration" occurs. (4) Metals have high electrical conductivity: no significant voltage drops occur inside metal emitters: this means that there are no factors obstructing the supply of electrons to the emitting surface, and that the electrons in this region can be both in effective local thermodynamic equilibrium and in effective thermodynamic equilibrium with the electrons in the metal support structure on which the emitter is mounted. (5) Atomic-level effects are disregarded.[citation needed] The development of "simple" theories of field electron emission, and in particular the development of Fowler-Nordheim-type equations, relies on all five of the above factors being true. For materials other than metals (and for atomically sharp metal emitters) one or more of the above factors will be untrue. For example, crystalline semiconductors do not have a free-electron-like band-structure, do have surface states, are subject to field penetration and band bending, and may exhibit both internal voltage drops and statistical decoupling of the surface-state electron distribution from the electron distribution in the surface region of the bulk band-structure (this decoupling is known as "the Modinos effect").[33][92] In practice, the theory of the actual Fowler-Nordheim tunneling process is much the same for all materials (though details of barrier shape may vary, and modified theory has to be developed for initial states that are localized rather than are travelling-wave-like). However, notwithstanding such differences, one expects (for thermodynamic equilibrium situations) that all CFE equations will have exponents that behave in a generally similar manner. This is why applying Fowler-Nordheim-type equations to materials outside the scope of the derivations given here often works. If interest is only in parameters (such as field enhancement factor) that relate to the slope of Fowler-Nordheim or Millikan-Lauritsen plots and to the exponent of the CFE equation, then Fowler-Nordheim-type theory will often give sensible estimates. However, attempts to derive meaningful current density values will usually or always fail. Note that a straight line in a Fowler-Nordheim or Millikan-Lauritsen plot does not indicate that emission from the corresponding material obeys a Fowler-Nordheim-type equation: it indicates only that the emission mechanism for individual electrons is probably Fowler-Nordheim tunneling.[citation needed] Different materials may have radically different distributions in energy of their internal electron states, so the process of integrating current-density contributions over the internal electron states may give rise to significantly different expressions for the current-density pre-exponentials, for different classes of material. 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Bibcode:1991ApPhL..59.1644H. doi:10.1063/1.106257. 86. ^ Fursey, G. N.; Glazanov, D. V. (1998). "Deviations from the Fowler–Nordheim theory and peculiarities of field electron emission from small-scale objects". Journal of Vacuum Science and Technology B. 16: 910. Bibcode:1998JVSTB..16..910F. doi:10.1116/1.589929. 87. ^ Cabrera, H.; et al. (2013). "Scale invariance of a diodelike tunnel junction". Physical Review B. 87: 115436. arXiv:1303.4985. Bibcode:2013PhRvB..87k5436C. doi:10.1103/PhysRevB.87.115436. 88. ^ Kyritsakis, A.; Djurabekova, F. (2017). "A general computational method for electron emission and thermal effects in field emitting nanotips". Computational Material Science. 128: 15. arXiv:1609.02364. doi:10.1016/j.commatsci.2016.11.010. 89. ^ Forbes, Richard G. (2008). "Call for experimental test of a revised mathematical form for empirical field emission current-voltage characteristics". Applied Physics Letters. 92 (19): 193105. Bibcode:2008ApPhL..92s3105F. doi:10.1063/1.2918446. 90. ^ Jensen, K. L. (1999). "Exchange-correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height". Journal of Applied Physics. 85 (5): 2667. Bibcode:1999JAP....85.2667J. doi:10.1063/1.369584. 91. ^ T. Kirk, 21st Intern. Vacuum Nanoelectronics Conf., Wrocław, July 2008. 92. ^ Modinos, A (1974). "Field emission from surface states in semiconductors". Surface Science. 42: 205–227. Bibcode:1974SurSc..42..205M. doi:10.1016/0039-6028(74)90013-2. Further readingEdit General information • W. Zhu, ed. (2001). Vacuum Microelectronics. Wiley, New York. • G.N. Fursey (2005). Field Emission in Vacuum Microelectronics. Kluwer Academic, New York. ISBN 0-306-47450-6. Field penetration and band bending (semiconductors) • Seiwatz, Ruth; Green, Mino (1958). "Space Charge Calculations for Semiconductors". Journal of Applied Physics. 29 (7): 1034. Bibcode:1958JAP....29.1034S. doi:10.1063/1.1723358. • A. Many, Y. Goldstein, and N.B. Grover, Semiconductor Surfaces (North Holland, Amsterdam, 1965). • W. Mönsch, Semiconductor Surfaces and Interfaces (Springer, Berlin, 1995). • Peng, Jie; Li, Zhibing; He, Chunshan; Chen, Guihua; Wang, Weiliang; Deng, Shaozhi; Xu, Ningsheng; Zheng, Xiao; Chen, GuanHua; Edgcombe, Chris J.; Forbes, Richard G. (2008). "The roles of apex dipoles and field penetration in the physics of charged, field emitting, single-walled carbon nanotubes". Journal of Applied Physics. AIP Publishing. 104 (1): 014310. arXiv:cond-mat/0612600. doi:10.1063/1.2946449. ISSN 0021-8979. Field emitted vacuum space-charge Field emission at high temperatures, and photo-field emission Field-induced explosive electron emission • G.A. Mesyats, Explosive Electron Emission (URO Press, Ekaterinburg, 1998),
803955717eb79bc2	Formation of matter-wave soliton trains by modulational instability See allHide authors and affiliations Science 28 Apr 2017: Vol. 356, Issue 6336, pp. 422-426 DOI: 10.1126/science.aal3220 Imaging an atomic soliton train Solitons—waveforms that keep their shape as they travel—can form in various environments where waves propagate, such as optical media. In a one-dimensional tube of bosonic atoms, solitons are formed when the interaction between the atoms is suddenly switched from repulsive to attractive. This causes the atoms to clump together into a “train” of solitons. Nguyen et al. used a nearly nondestructive imaging technique to follow the dynamics of this train. The solitons repulsed each other and underwent collective oscillations known as breathing modes. Science, this issue p. 422 Nonlinear systems can exhibit a rich set of dynamics that are inherently sensitive to their initial conditions. One such example is modulational instability, which is believed to be one of the most prevalent instabilities in nature. By exploiting a shallow zero-crossing of a Feshbach resonance, we characterize modulational instability and its role in the formation of matter-wave soliton trains from a Bose-Einstein condensate. We examine the universal scaling laws exhibited by the system and, through real-time imaging, address a long-standing question of whether the solitons in trains are created with effectively repulsive nearest-neighbor interactions or rather evolve into such a structure. Modulational instability (MI) is a process in which broadband perturbations spontaneously seed the nonlinear growth of a nearly monochromatic wave disturbance (1). Owing to its generality, MI plays a role in a variety of different physical systems such as water waves, where it is known as a Benjamin-Feir instability (2); plasma waves; nonlinear optics (35); and ultracold atomic gases (6). The nonlinear interaction resulting in MI also supports solitons, which are localized waves whose dispersion is exactly balanced by the nonlinearity (7, 8). Thus, the rapid growth of fluctuations from MI, which leads to the breakup of the wave, is seen as a natural precursor to the formation of soliton trains. In optical systems, this was first observed in the temporal domain (911) and, subsequently, in the spatial domain (12). Analogously, in an atomic Bose-Einstein condensate (BEC), MI drives the spontaneous formation of bright matter-wave solitons when the interaction between atoms is rapidly quenched from repulsive to attractive. These systems are well described in most respects by the Gross-Pitaevskii equation, which is equivalent to the nonlinear Schrödinger equation with the addition of a harmonic trapping potential. Here, the nonlinearity is determined by the s-wave scattering length, which is positive for a repulsive, defocusing nonlinearity and negative for an attractive, focusing one. We will see that dissipation plays an important role in the matter-wave system, as it does in optical media. Bright matter-wave solitons were first observed by applying an interaction ramp traversing a zero-crossing of the interaction parameter in a quasi–one-dimensional (quasi-1D) BEC (13, 14). Several experiments have produced trains of up to Embedded Image solitons (14, 15). Because these solitons are harmonically confined, they are not truly 1D and are susceptible to collapse resulting from the attractive nonlinearity. This has the effect of limiting the number of atoms a single soliton can stably support (1618). Additionally, solitons themselves may interact, exhibiting an effectively attractive or repulsive force that, according to mean-field theory, can be ascribed to a relative phase between solitons of Embedded Image or Embedded Image, respectively (19). These phase-dependent interactions were first observed in optical solitons (20, 21). In the case of matter-wave solitons, the peak density increases for in-phase collisions Embedded Image, which can produce annihilations and mergers, whereas out-of-phase collisions Embedded Image are expected to be more stable against collapse (2225). These effects have been observed experimentally (26). Solitons created in trains were found to be surprisingly stable, persisting for many cycles of oscillation in a harmonic trap despite being near the threshold for collapse (14, 15). From this observation, it was inferred that an alternating-phase (0-π-0) structure was present, protecting the structure against collapse (14, 15). Detailed theoretical investigations have studied the formation of matter-wave soliton trains and attempted to explain the origins of the observed repulsive interaction between neighboring solitons (25, 2733). We address these issues in the experiments described here. For MI, there is a positive-feedback–driven exponential growth that is largest for the wave number Embedded Image (29, 30). Here, Embedded Image is the characteristic confinement length in the radial direction, ħ is Planck’s constant divided by 2π, m is the atomic mass, ωr is the radial frequency of the harmonic trap, af is the (negative) scattering length after the quench, and n1D is the line density of the condensate before the quench. The healing length, Embedded Image, naturally lends itself as the characteristic length scale for MI in this system; correspondingly, the rate at which fluctuations grow sets a characteristic time scale given by γ–1, where Embedded Image. Once the scattering length is quenched from positive to negative, the effects of MI manifest as density modulations of the gas (Fig. 1A). The atoms first clump together into regions of increased density, owing to the nonlinear focusing of the attractive interaction. Regions of high density, separated by a spatial distance of 2πξ, appear on a time scale given by γ–1. These density clumps evolve into solitons whose dispersion is balanced by the nonlinear attraction between atoms. Fig. 1 Soliton-train formation from modulational instability (MI). (A) Schematic representation of the effects of a scattering length quench. At short times, the condensate has not responded to changes in scattering length. MI results in rapid growth of fluctuations at a length scale of 2πξ. Atoms flow toward regions of high density on a time scale of γ–1, owing to a nonlinear focusing from attractive interactions. Solitons are formed for t > γ–1. (B) Column density images for af = –0.18a0. Immediately after the quench, there is no discernible change in Na, nor is there any change in shape from that of the original condensate at ai = +3a0. Solitons form at later times and undergo breathing and dipole oscillations. (C) Similar to (B), except with af = –2.5a0. Modulations appear much earlier, as do gaps near the center where the density of the original condensate was high, which we attribute to primary collapses. A reduction in Ns is evident at longer th. Each image corresponds to a different experimental run, and hence, real-time dynamics cannot be directly inferred from these images. Here, z is the position along the axial coordinate. Although it is clear that MI is crucial to the formation of matter-wave soliton trains (2732), the identification of the mechanism responsible for their stability has remained elusive. Several theories have been proposed. In the simulations of (27), the authors determined the spectrum of the phase of the wave function produced by quantum fluctuations when the scattering length was suddenly quenched. They imprinted the condensate wave function with this phase and found the subsequent development of an alternating-phase structure and dynamics that match those of the experiment (14). In another study (28), similar dynamics were calculated with the use of an effective time-dependent 1D nonpolynomial Schrödinger equation, but an alternating-phase structure was simply imprinted onto the solitons. In a subsequent paper (29), imprinting the condensate with an ad hoc phase structure was shown to be unnecessary; a nearly alternating-phase structure emerged in numerical simulations by allowing the phase of the condensate to evolve self-consistently according to a Gross-Pitaevskii equation that included a dissipative three-body term. In (30), self-interference, rather than quantum fluctuations, served to seed MI. Exponential growth of these fringes first led to primary collapse in cases where the atom number of an individual soliton exceeded a critical value during the early part of MI. The resultant solitons in the train were found to have arbitrary phases. To acquire an alternating-phase structure, it was proposed that a stage of secondary collapses occurred, wherein binary collisions between solitons resulted in annihilations and mergers of near in-phase soliton pairs. These collisions would serve to distill the soliton train, resulting in the eventual formation of alternating phases but accompanied by the loss of a large number of atoms (30). In a subsequent comparison between MI seeded by noise and self-interference (31), it was determined that both should contribute to MI at comparable time scales. By varying the noise added into their simulations, the authors were able to identify regimes dominated by self-interference or noise. Notably, with MI seeded by self-interference, soliton formation occurred at the edges of the condensate first, because the fringes from self-interference first achieve a sufficiently long wavelength there (30, 31). In contrast, MI seeded by noise led to the development of solitons first in the center, where the density and the rate of MI was highest, and finally at the edges (31). In the mean-field approaches discussed thus far, the observed stability against secondary collapses is attributed to an alternating-phase structure, although whether this alternating-phase structure is initially present or evolves out of the mutual annihilation of attractively interacting solitons has been debated (2731). Approaches extending beyond mean-field theory, such as the truncated Wigner approximation (TWA) in one dimension (32) or the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method (3335), suggest that quantum effects may produce effectively repulsive interactions, independent of the relative phase. The extension of the TWA to three dimensions (32), however, resulted in a rapid loss of solitons that contradicts observations (14, 15). Furthermore, convergence with the MCTDHB method has been shown to be pathological for bosons with attractive interactions, which may have affected previous conclusions (36). We address these issues with a degenerate gas of 7Li atoms [our methods have been described elsewhere (26, 37)]. A BEC of atoms in the Embedded Image state (where F and mF are the quantum numbers of the total atomic angular momentum and its projection, respectively) is confined in a cylindrically symmetric harmonic trap with radial and axial oscillation frequencies of ωr/2π = 346 Hz and ωz/2π = 7.4 Hz, respectively. The interaction between atoms is magnetically controlled via a broadly tunable Feshbach resonance (38, 39) and is initially set to a scattering length of ai ≈ 3a0 (where a0 is the Bohr radius) (37). We quench the interaction to a final scattering length, af < 0, in a linear ramp time of tr = 1 ms. After waiting a variable hold time, th, we take an in situ polarization phase-contrast image (PPCI) (40). Our PPCI method can be minimally destructive, resulting in the loss of <2% of atoms per image, thus enabling a sequence of images of the same soliton train. The formation of a soliton train is shown in Fig. 1B for a scattering length of af = –0.18a0, with th from 0 to 20 ms (each of these images corresponds to a different experimental run). The images in Fig. 1C depict the formation for a scattering length of af = –2.5a0, highlighting some key differences between smaller and larger Embedded Image. For larger Embedded Image, we find that the formation occurs on a faster time scale, and we also see a reduction in the number of solitons remaining with increasing th. We characterize the effect of MI on the density profile of the BEC by defining a contrast parameter, η, which is a measure of the deviation in the density from a Thomas-Fermi profile (37). We observe rapid growth of η in the central region as compared with the sides of the condensate (fig. S2). According to (31), this implies that the seed for MI is dominated by noise, which may be technical, thermal, or quantum in origin, rather than self-interference. The loss of total atom number (Na) versus th is plotted in Fig. 2A. We observe an initial plateau where Na changes little, followed by a period of rapid atom loss. The plateau and subsequent atom loss are reminiscent of experiments exploring the collapse of an attractive condensate of 85Rb atoms (41, 42). MI provides a simple and intuitive explanation for this initial plateau. When tr is fast compared with γ–1, the dynamics are initially frozen out. This time scale is indicated by the arrows in Fig. 2A, calculated for several values of af. As Embedded Image is increased, γ–1 is predicted to become smaller, in agreement with the data. Solitons are formed for times longer than γ–1. Fig. 2 Postquench evolution of atom number. (A) Na versus th for various af. The arrows indicate the calculated γ–1 for each value of af, which is determined using the peak value of n1D. The black dashed line corresponds to half of a breathing period (tbr = 68 ms). We observe a plateau in Na for each af, followed by a rapid decrease in atoms starting shortly after th ≈ γ–1. We attribute the lack of a plateau for af = –2.5a0 to tr > γ–1. (B) Data replotted versus thγ. The data collapse onto a single curve, except for af = –2.5a0. The data are fit to a power law, Embedded Image, shown as a solid black line, where κ = –0.35(1) for both fits. For all af, points for th > tbr/2 have been omitted from the fit. Error bars are the SD of the mean of up to 30 shots. The universality of the MI time scale and of atom loss becomes evident when th is rescaled by γ–1 (Fig. 2B). We find that the data collapse onto a single curve, with the exception of af = –2.5a0. Because tr = 1 ms > γ–1 = 0.42 ms for this scattering length, the plateau is notably absent. For all other scattering lengths, the onset of atom loss begins shortly after thγ = 1. We fit the data (Fig. 2B) for th > γ–1 to a power law decay, where Embedded Image with κ = –0.35(1) (here, N0 is the total initial number of atoms and κ is the power law exponent). Scaling laws within the system also provide us with a simple yet surprisingly accurate estimate of the number of solitons, Ns, formed by MI. Assuming an initial condensate length of 2RTF, where RTF is the Thomas-Fermi radius, we estimate Embedded Image from simple length-scale arguments (27, 29). Because the dynamics of the system are frozen for fast tr (as compared with γ–1), the initial conditions are entirely determined by Embedded Image. MI produces a modulation of the density, with the density of defects set by Embedded Image. In our experiments, RTF is held constant, whereas ξ is controlled by changing af. In Fig. 3A we plot Ns versus af and find excellent agreement with this simple model for Embedded Image. For larger Embedded Image, Ns is limited by primary collapses that arise when the number of atoms for a single soliton exceeds the critical number for collapse, Embedded Image, where the factor of Embedded Image accounts for the aspect ratio of the trapping potential (1618). Furthermore, solitons are able to undergo primary collapse during the quench for tr > γ–1. Fig. 3 Postquench evolution of soliton number and strength of nonlinearity. (A) Ns versus af. The dashed line corresponds to a fit of the data to the model (see text), where an overall scaling of 1.04(2) is the only fit parameter. Data for Embedded Image are omitted from the fit. We attribute the suppression in Ns for Embedded Image to primary collapse, resulting in a reduction in the number of solitons formed. (B) Ns versus th. Ns does not change with th for the two smallest Embedded Image, whereas for larger Embedded Image, Ns decays with th. Dashed lines correspond to the initial number of solitons. (C) Δ versus th. The initial value of Δ = Na/(NsNc) increases as Embedded Image is increased and is consistent with an expected Embedded Image scaling. This trend continues up to Δ = 1, above which the solitons are unstable against primary collapse. Error bars are the SD of the mean of up to 30 shots. To examine whether primary collapses, or secondary collapses that arise from annihilations or mergers, contribute to the observed decrease in Na, we plot Ns versus th in Fig. 3B. We find that for the two smallest Embedded Image, af = –0.18a0 and –0.42a0, Ns remains constant with increasing th, indicating that neither primary nor secondary collapses have occurred. The fact that Ns remains constant indicates that the interactions between neighboring solitons are dominantly repulsive, thus suppressing secondary collapses. Ns decreases for larger values of Embedded Image, indicating the effect of collapse. Because the collisional time scale is expected to be on the order of the breathing-mode period, tbr = 68 ms, we attribute the initial rapid (th < 20 ms) soliton loss to primary collapses. Secondary collapses are likely to play a role at later times, particularly for the af = –2.5a0 data, for which soliton loss is observed until Ns ≈ 2. Additional insight into the appearance of collapse may be obtained by examining the strength of the nonlinearity, Δ, which we define as the number of atoms per soliton, normalized to the critical number, Δ = Na/(NsNc) (Fig. 3C). For both af = –0.18a0 and –0.42a0, the initial Δ < 0.6, and Δ decays only because of the loss of atoms from each independent soliton, not by losing solitons. On the other hand, the large initial value of Δ for larger Embedded Image explains the relative instability to collapse exhibited by these solitons. To gain further insight into the nature of atom loss, we fit the decay in Na to a function that assumes that atoms are lost from independent solitons by three-body recombination (fig. S3). This assumption yields a power law decay, as observed, but with κ = –0.5 or –0.25, depending on assumptions regarding the soliton length (37). These values bracket the measured exponent of –0.35. We extract a three-body loss coefficient, L3, from this analysis and find that it ranges between 10–26 and 10–25 cm6/s, depending on the initial assumptions (37). This is much greater than values previously measured for small positive scattering lengths of 10–28 cm6/s (39). Additionally, when the scattering length is ramped slowly (>250 ms) rather than suddenly quenched, the loss rate is below our ability to measure (L3 < 10–28 cm6/s) (fig. S3). We conclude that the much larger rate of loss arises from dynamical changes in the density induced by the sudden quench. One consequence is the excitation of a breathing mode that periodically modulates the density and, thus, the rate of three-body loss. The loss-rate plateau seen for th > tbr/2 in Fig. 2A is likely a manifestation of this effect. The quench may also induce partial collapses that originate in localized high-density regions of a soliton. The resulting atom loss can self-arrest the collapse, thus resulting in a series of intermittent, partial collapses (43, 44). Our minimally destructive imaging technique allows us to take multiple images of the same soliton train to directly observe the dynamics. These images for the small Embedded Image data confirm the expected repulsive soliton-soliton interactions. Two such examples are shown for af = –0.18a0 in Fig. 4, A and B. We find that the solitons remain well-separated from one another at all times, from which we infer dominantly repulsive interactions, even as the soliton train first emerges. Fig. 4 Soliton-train dynamics. (A) Multiple images of the same soliton train, for af = –0.18a0. Beginning at th = 10 ms, a new image was taken every 2 ms. We infer dominantly repulsive interactions, although occasional attractive collisions occur between neighbors. The reduction in the overall size of the train is caused by a breathing mode excited by the quench, and a dipole oscillation is also evident. (B) Similar to (A), starting with th = 40 ms. The effects of the breathing mode in its expansion phase are evident. We have examined MI in detail, elucidating its universal role in the spontaneous formation of matter-wave soliton trains. Our results indicate that MI in this context is driven by noise and that, for small Embedded Image, neighboring solitons already interact repulsively during the initial formation of the soliton train, independent of secondary collisions. This may also be the case for larger Embedded Image, but primary collapse dominates the dynamics in this case, and the soliton train quickly dissipates as a result. Similar phase and wavelength correlations have been observed in optical MI experiments (3). We have also demonstrated natural scaling laws for atom loss. The scaling behavior is similar to systems that are described by the Kibble-Zurek mechanism (4547), although a key difference in our system is the presence of dissipation and collapse, which is not part of the Kibble-Zurek scenario. The ability to finely control the interaction between atoms and the relatively slow time scale for dynamics point toward the study of rogue matter-waves (48, 49), analogous to the rogue waves observed in optical systems (50), as a natural extension of this work. Our methods are additionally amenable to studying the formation and propagation of higher-order solitons, such as breathers (51, 52). Note added in proof: A manuscript reporting modulational instability in 85Rb (53) was posted after the submission of this manuscript. Supplementary Materials Materials and Methods Supplementary Text Figs. S1 to S3 References and Notes 1. See the supplementary materials. Acknowledgments: We thank K. Hazzard, L. Carr, E. Mueller, and B. Malomed for helpful discussions. This work was supported by the NSF (grants PHY-1408309 and PHY-1607215), the Welch Foundation (grant C-1133), the Army Research Office Multidisciplinary University Research Initiative (grant W911NF-14-1-0003), and the Office of Naval Research. View Abstract Navigate This Article
c702e3d8aad618d3	Friday, October 29, 2010 Magnetic monopoles at sixties Old age is usually associated with wisdom and similar virtues. In my case this association unfortunately fails and therefore the first morning at sixties gives me authority to free associations about everything under the heaven, and magnetic monopoles are a good place to start from. The evidence for condensed matter monopoles is accumulating (see this and this) and the question is whether they really represent some new physics. Perhaps this is the case. Dirac monopoles are mathematically singular and cannot be therefore tolerated in elite circles of theoretical physics appreciating good manners coded by gauge invariance. Since I frantically want to belong to the elite, I am happy that TGD provides me with homological monopoles, which can exist gracefully because of the homological non-triviality of CP2. Homological non-triviality means that CP2 has non-contractible 2-surfaces such as spheres: this does not mean that it would have a hole as a lazy popularizer usually says. Rather, this kind of sphere is a 2-dimensional analog of a circle around torus not allowing contraction to a point without cutting. The imbedding of CP2 to some higher dimensional space would contain a hole in some sense. The weak form of electric-magnetic duality- a purely TGD based notion- implies that all elementary particles correspond to pairs of wormhole contacts. Each contact has two throats carrying magnetic monopole charge and each throat is connected to the corresponding throat of second contact. This makes altogether four wormhole throats so that graviton can be constructed in this manner. The length of the magnetic flux tube defining string like object corresponds to the weak length scale about 10-17 m. All particles would be this kind of string like objects, "weak" strings. Emergence gives excellent hopes about the realization of exact Yangian invariance and twistor Grassmannian program without infrared and UV divergences (see this). Emergence states that at the fundamental level there are only massless(!) wormhole throats carrying many-fermion states identifiable in terms of representations for the analog of space-time super-symmetry algebra with generators identified as fermionic oscillator operators. Masslessness applies also to the building blocks of virtual particles meaning a totally new interpretation of loop corrections and manifest UV finiteness. Also the vibrational degrees of freedom of partonic 2-surfaces are present as bosonic degrees of freedom and correspond to orbital degrees of freedom for the spinor fields of world of classical worlds (WCW) whereas fermionic degrees of freedom define WCW spin degrees of freedom. The dark variants of the elementary particles having large value of hbar have zoomed up size and in living matter these scaled up elementary particles would be the key players in the drama of life. Quite recently I realized that dark variants of elementary particles identified in this manner are more or less the same thing as the wormhole magnetic fields that I introduced for more than decade ago (see this) and suggested that their Bose-Einstein condensates and coherent states could be crucial for understanding living matter. At that time I did not of course realize the connection with ordinary elementary particle physics and proposed these exotics as new kind of particle like objects. These flux tubes have become the basic structures in TGD inspired quantum biology. For instance, the model for DNA as topological quantum computer assumes that the nucleotides of DNA and lipids of cell membrane are connected by this kind of flux tubes with quarks at their ends and the braiding of the flux tubes codes for topological quantum computations. If this picture is correct, quantum biology might be to high degree a collection of zoomed up variants of elementary particle physics at very high density! Also the super-partners and scaled up hadrons would be present. If this is true we would be able to study elementary particle interiors by zooming up them to the scales of living matter! There would be no need for the followers of LHC! Living matter could be an enormous particle physics laboratory providing physicists with incredibly refined research facilities;-). But are these facilities meant for us after we finally have realized that we ourselves are the most refined laboratory? Or are the physicists already there? If so, who these physicists from higher levels of self hierarchy might be;-)? By the way, this crazy speculation might have been inspired also by the earlier finding that the model of dark nucleons allows to map the spectrum of nucleon states to RNA, DNA, tRNA triplets and aminoacids and also reproduces vertebrate genetic code in a very natural manner (see this and this). Thursday, October 28, 2010 Tau-pions again but now in galactic center The standard view about dark matter is that it has only gravitatitonal interactions with ordinary matter so that high densities of dark matter are required to detect its signatures. On the average the density of dark matter is about 80 per cent of ordinary matter. Clearly, Milky Way's center is an excellent place for detecting the signatures of dark matter. The annihilation of pairs of dark matter particles to gamma rays is one possible signature and one could study the anomalous features of gamma ray spectrum from the galactic center (a region with radius about 100 light years). Europe's INTEGRAL satellite launched in 2002 indeed found bright gamma ray radiations coming from the center of galaxy with energy of .511 MeV, which is slightly above electron mass (see the references below). The official interpretation is that the gammas are produced in the annihilations of particles of positrons and electrons in turn created in dark matter annihilations. TGD suggests much simpler mechanism. Gamma rays would be produced in the decay of what I call electropions having mass which is slightly larger than m=2me. The news of the day was that the data from Fermi Gamma Ray telescope give analyzed by Dan Hooper and Lisa Goodenough gives evidence for a dark matter candidate with mass between 7.3-9.2 GeV decaying predominantly into a pair of τ leptons. The estimate for the mass region is roughly 4 times τ mass. What puts bells ringing that a mass of a charged lepton appears again! Explanation in TGD framework The new finding fits nicely to a bigger story based on TGD. 1. TGD predicts that both quarks and leptons should have colored excitations (see the chapter devoted to the leptohadron model). In the case of leptons lowest excitations are color octets. In the case of electro-pion this hypothesis finds support from the anomalous production of electron positron pairs in heavy ion collisions discovered already at seventies but forgotten for long ago since the existence of light particle at this mass scale simply was in total complete with standard model and what was known about the decay widths of intermediate gauge bosons. Also ortopositronium decay width anomaly -forgotten also-has explanation in terms of leptopion hypothesis (see the references below) 2. The colored leptons would be dark in TGD sense, which means that they live in dark sector of the "world of classical worlds" (WCW) meaning that they have no direct interactions (common vertices of Feynman diagrams) with ordinary matter. They simply live at different space-time sheets. A phase transition which is geometrically a leakage between dark sector and ordinary sector are possible and make possible interactions between ordinary and dark matter based on exchanged particles suffering this phase transition. Therefore the decay widths of intermediate gauge bosons do not kill the model. TGD based model of dark matter in terms of hierarchy of values of Planck constants coming as multiples of its smallest possible value (the simplest option) need not to be postulated separately and can be regarded as a prediction of quantum TGD reflecting directly the vacuum degenerarcy and extreme non-linearity of Kähler action (Maxwell action for induced CP2 Kähler form). 3. CDF anomaly which created a lot of discussion in blogs for two years ago can be understood in terms of taupion. Taupion and its p-adically scaled up versions with masses about 2kmτ, k=1,2,3 and mτ≈ 1.8 GeV explains the findings reported by CDF in TGD framework. The masses of taupions would be 3.6 GeV, 7.2 GeV, and 14.2 GeV in good approximation and come as octaves of the mass of tau-lepton pair. The mass estimate for the dark matter particle suggests by Fermi Gamma Ray telescope corresponds to k=2 octave for taupion and the predict mass is about 7.2 GeV which at the lower boundary of the range 7.3-9.2 GeV. Also dark matter particles decaying to tau pairs and having masses 3.6 GeV and 14.2 GeV should be found. Also muo-pion should exist there and should have mass slightly above 2mμ= 210.4 MeV so that a gamma rays peak slightly above the energy μ=105.2 MeV should be discovered. Also octaves of this mass are possible. There is also evidence also for the existence of muopion (around 2007, see the links below). LHC should provide excellent opportunities to test tau-pion and muo-pion hypothesis. Electro-pion was discovered in heavy ion collisions and also at LHC they study have heavy ion collisions but at much higher energies generating the required very strong non-orthogonal electric and magnetic fields for which the "instanton density" defined as the inner product of electric and magnetic fields is large and rapidly varying. I do not of course consider for a second the possibility that the mighty ones at LHC would take seriously what some ridiculed TGD guy without any academic affiliation suggests. As an optimist I hope that muo-pion and tau-pion could be discovered despite the fact that their decay signatures are very different from those for ordinary particles and despite that fact that at these energies one must know precisely what one is trying to find in order to disentangle it from the enormous background. Also DAMA, CoGeNT, and PAMELA give indications for tau-pion Note that also DAMA suggests the existence of dark matter particle in this mass range but it is not clear whether it can have anything to do with tau-pion state. One could of course imagine that dark tau-pions are created in the collisions of highly energetic cosmic rays with the nuclei of atmosphere. Also Coherent Germanium Neutrino Technology (CoGeNT) experiment has released data that are best explained in terms of a dark matter particle with mass in the range 7-11 GeV. The decay of tau-pions produce lepton pairs, mostly tau but also muons and electrons. The subsequent decays of tau-leptons to muons and electrons produce also electrons and positrons. This relates interestingly to the positron excess reported by PAMELA collaboration at the same time as CDF anomaly was reported (my second birth days gift;-). The anomaly started at positron energy about 3.6 GeV, which is one just one half of 7. 2 GeV for tau-pion mass! What was remarkable that no antiproton excess predicted by standard dark matter candidates was observed. Therefore the interpretation as decay products of tau-pions seems to make sense! A short comment about sociology of science By the way, CDF anomaly published two years ago meant quite an intensive drama in my life as a lonely dissident. The announcement of CDF about the anomaly happened to come just at the eve of my birth day and I took it as a birth day gift;-). Amusingly, also this news deserves to be called a birthday gift (New Scientists dates the article at October 28 and I will be 60 years old October 30. Note however that the eprint has been added to arXiv October 13). The explanation of the CDF anomaly was of course a great victory for TGD and meant a period of intense work lasting for several months. I had an excellent reason to participate blog discussions and this induced an extremely hostile attacks from the besserwissers of science in Resonaances. Probably also because the first evidence for electropions is from seventies and the neglect of all this data for a period of decades just because it does not conform with standard moded is a scandal. To put it mildly. Also the powerholders of Finnish theoretical physics decided to give their own birthday gift: I lost sthe right to use the memory of university computer for my homepage which had served as a symbolic support hoped to keep my silent! Small nuisance after all but a nuisance in any case since I had to be quick since the deadline was absolute. The situation today in Finnish theoretical physics has become rather surreal. I am mentioned in the list of fifty world-wide known finnish scientists in Wikipedia among two other living finnish physicists but absolutely no one in the academic environment dares to know about my existence publicly! An excellent opportunity for a gifted writer to create a brilliant satire about the madness of the academic world. For the details of leptohadron hypothesis see the chapter Recent Status of Leptohadron Hypothesis of "p-Adic length Scale Hypothesis and Dark Matter Hierarchy". I have listed below publications related to lepto-pion anomaly. 1. Electropion anomaly 1. W. Koenig et al(1987), Zeitschrift fur Physik A, 3288, 1297. 2. A.T. Goshaw et al(1979), Phys. Rev. Lett. 43, 1065. 3. P.V. Chliapnikov et al(1984), Phys. Lett. B 141, 276. 4. K. Dantzman et al (1989), Phys. Rev. Lett., 62, 2353. 5. C. I. Westbrook ,D. W Kidley, R. S. Gidley, R. S Conti and A. Rich (1987), Phys. Rev. Lett. 58 , 1328. 6. S. Barshay (1992) , Mod. Phys. Lett. A, Vol 7, No 20, p. 1843. 7. J.Schweppe et al.(1983), Phys. Rev. Lett. 51, 2261. 8. H.Tsertos et al. (1985) , Phys. Lett. 162B, 273, H.Tsertos et al.(1987) , Z. Phys. A 326, 235. 9. P. Salabura et al (1990), Phys. Lett. B 245, 2, 153. 10. A. Chodos (1987) , Comments Nucl. Part. Phys., Vol 17, No 4, pp. 211, 223. 11. L. Kraus and M. Zeller (1986), Phys. Rev. D 34, 3385. 12. M. Clemente et al. (1984), Phys. Rev. Lett. 137B, 41. 13. S. Judge et al (1990) , Phys.Rev. Lett., 65(8), 972. 14. T. Cowan et al.(1985), Phys. Rev. Lett. 54, 1761 and T. Cowan et al.(1986), Phys. Rev. Lett. 56, 444. 2. Electro-pions as a candidate for dark matter in galactic center 1. G. Weidenspointner et al (2006), The sky distribution of positronium annihilation continuum emission measured with SPI/INTEGRAL, Astron. Astrophys. 450, 1013, astro-ph/0601673. 2. E. Churazov, R. Sunyaev, S. Sazonov, M. Revnivtsev, and D. Varshalovich, Positron annihilation spectrum from the Galactic Center region observed by SPI/INTEGRAL, Mon. Not. Roy. 17. Astron. Soc. 357, 1377 (2005), astro-ph/0411351. 3. Ortopositronium anomaly R. Escribabno,E. Masso, R. Toldra (1995), Phys. Lett. B. 356, 313-318. 4. Muopion anomaly 1. X.-G. He, J. Tandean, G. Valencia (2007), Has HyperCP Observed a Light Higgs Boson?,Phys. Rev. D74. . 2. X.-G. He, J. Tandean, G. Valencia (2007), Light Higgs Production in Hyperon Decay, Phys. Rev. Lett. 98. 5. Taupion anomaly 1. CDF: T. Daniels et al (1994), Fermilab-Conf-94/136-E; Fermilab-Conf-94/212-E. 2. CDF Collaboration (2008), Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV. 3. T. Dorigo (2008), Some notes on the multi-muon analysis - part I. 6. Taupions as a candidate for dark matter in galactic center D. Hooper and L. Goodenough (2010), Dark Matter Annihilation in The Galactic Center As Seen by the Fermi Gamma Ray Space Telescope. DAMA collaboration (2010), Results from DAMA/LIBRA at Gran Sasso, Found. Phys. 40, p. 900. CoGENT collaboration (2010), Results from a Search for Light-Mass Dark Matter with a P-type Point Contact Germanium Detector. PAMELA Collaboration (2008), Observation of an anomalous positron abundance in the cosmic radiation. M. Boexio (2008), talk represented at IDM 2008, Stockholm, Sweden. Topological explanation of family replication phenomenon One of the basic ideas of TGD approach has been genus-generation correspondence: boundary components of the 3-surface should be carriers of elementary particle numbers and the observed particle families should correspond to various boundary topologies. Last summer meant quite a progress in the understanding of quantum TGD, which forced also the updating of the views about the topological explanation of family replication phenomenon. With the advent of zero energy ontology the original picture changed somewhat. It is the wormhole throats identified as light-like 3-surfaces at with the induced metric of the space-time surface changes its signature from Minkowskian to Euclidian, which correspond to the light-like orbits of partonic 2-surfaces. One cannot of course exclude the possibility that also boundary components could allow to satisfy boundary conditions without assuming vacuum extremal property of nearby space-time surface. The intersections of the wormhole throats with the light-like boundaries of causal diamonds (CDs) identified as intersections of future and past directed light cones (CD × CP2 is actually in question but I will speak about CDs) define special partonic 2-surfaces and it is the comformal moduli of these partonic 2-surfaces which appear in the elementary particle vacuum functionals naturally. The first modification of the original simple picture comes from the identification of physical particles as bound states of pairs of wormhole contacts and from the assumption that for generalized Feynman diagrams stringy trouser vertices are replaced with vertices at which the ends of light-like wormhole throats meet. In this picture the interpretation of the analog of trouser vertex is in terms of propagation of same particle along two different paths. This interpretation is mathematically natural since vertices correspond to 2-manifolds rather than singular 2-manifolds which are just splitting to two disjoint components. Second complication comes from the weak form of electric-magnetic duality forcing to identify physical particles as weak strings with magnetic monopoles at their ends and one should understand also the possible complications caused by this generalization. These modifications force to consider several options concerning the identification of light fermions and bosons and one can end up with a unique identification only by making some assumptions. Masslessness of all wormhole throats- also those appearing in internal lines- and dynamical SU(3) symmetry for particle generations are attractive general enough assumptions of this kind. This means that bosons and their super-partners correspond to wormhole contacts with fermion and antifermion at the throats of the contact. Free fermions and their superpartners could correspond to CP2 type vacuum extremals with single wormhole throat. It turns however that dynamical SU(3) symmetry forces to identify massive (and possibly topologically condensed) fermions as (g,g) type wormhole contacts. Do free fermions correspond to single wormhole throat or (g,g) wormhole? The original interpretation of genus-generation correspondence was that free fermions correspond to wormhole throats characterized by genus. The idea of SU(3) as a dynamical symmetry suggested that gauge bosons correspond to octet and singlet representations of SU(3). The further idea that all lines of generalized Feynman diagrams are massless poses a strong additional constraint and it is not clear whether this proposal as such survives. 1. Twistorial program assumes that fundamental objects are massless wormhole throats carrying collinearly moving many-fermion states and also bosonic excitations generated by super-symplectic algebra. In the following consideration only purely bosonic and single fermion throats are considered since they are the basic building blocks of physical particles. The reason is that propagators for high excitations behave like p-n, n the number of fermions associated with the wormhole throat. Therefore single throat allows only spins 0,1/2,1 as elementary particles in the usual sense of the word. 2. The identification of massive fermions (as opposed to free massless fermions) as wormhole contacts follows if one requires that fundamental building blocks are massless since at least two massless throats are required to have a massive state. Therefore the conformal excitations with CP2 mass scale should be assignable to wormhole contacts also in the case of fermions. As already noticed this is not the end of the story: weak strings are required by the weak form of electric-magnetic duality. 3. If free fermions corresponding to single wormhole throat, topological condensation is an essential element of the formation of stringy states. The topological condensation of fermions by topological sum (fermionic CP2 type vacuum extremal touches another space-time sheet) suggest (g,0) wormhole contact. Note however that the identification of wormhole throat is as 3-surface at which the signature of the induced metric changes so that this conclusion might be wrong. One can indeed consider also the possibility of (g,g) pairs as an outcome of topological conensation. This is suggested also by the idea that wormhole throats are analogous to string like objects and only this option turns out to be consistent with the BFF vertex based on the requirement of dynamical SU(3) symmetry to be discussed later. The structure of reaction vertices makes it possible to interpret (g,g) pairs as SU(3) triplet. If bosons are obtained as fusion of fermionic and antifermionic throats (touching of corresponding CP2 type vacuum extremals) they correspond naturally to (g1,g2) pairs. 4. p-Adic mass calculations distinguish between fermions and bosons and the identification of fermions and bosons should be consistent with this difference. The maximal p-adic temperature T=1 for fermions could relate to the weakness of the interaction of the fermionic wormhole throat with the wormhole throat resulting in topological condensation. This wormhole throat would however carry momentum and 3-momentum would in general be non-parallel to that of the fermion, most naturally in the opposite direction. p-Adic mass calculations suggest strongly that for bosons p-adic temperature T=1/n, n>1, so that thermodynamical contribution to the mass squared is negligible. The low p-adic temperature could be due to the strong interaction between fermionic and antifermionic wormhole throat leading to the "freezing" of the conformal degrees of freedom related to the relative motion of wormhole throats. 5. The weak form of electric-magnetic duality forces second wormhole throat with opposite magnetic charge and the light-like momenta could sum up to massive momentum. In this case string tension corresponds to electroweak length scale. Therefore p-adic thermodynamics must be assigned to wormhole contacts and these appear as basic units connected by Kähler magnetic flux tube pairs at the two space-time sheets involved. Weak stringy degrees of freedom are however expected to give additional contribution to the mass, perhaps by modifying the ground state conformal weight. A nice implication is that all elementary particles -not only gravitons- correspond to pairs of wormhole throats connected by magnetic flux tubes to form "weak strings". This has obvious implications at LHC. Dynamical SU(3) fixes the identification of fermions and bosons and fundamental interaction vertices For 3 light fermion families SU(3) suggests itself as a dynamical symmetry with fermions in fundamental N=3-dimensional representation and N× N=9 bosons in the adjoint representation and singlet representation. The known gauge bosons have same couplings to fermionic families so that they must correspond to the singlet representation. The first challenge is to understand whether it is possible to have dynamical SU(3) at the level of fundamental reaction vertices. This is a highly non-trivial constraint. For instance, the vertices in which n wormhole throats with same (g1,g2) glued along the ends of lines are not consistent with this symmetry. The splitting of the fermionic worm-hole contacts before the proper vertices for throats might however allow the realization of dynamical SU(3). The condition of SU(3) symmetry combined with the requirement that virtual lines resulting also in the splitting of wormhole contacts are always massless, leads to the conclusion that massive fermions correspond to (g,g) type wormhole contacts transforming naturally like SU(3) triplet. This picture conformsl with the identification of free fermions as throats but not with the naive expectation that their topological condensation gives rise to (g,0) wormhole contact. The argument leading to these conclusions runs as follows. 1. The question is what basic reaction vertices are allowed by dynamical SU(3) symmetry. FFB vertices are in principle all that is needed and they should obey the dynamical symmetry. The meeting of entire wormhole contacts along their ends is certainly not possible. The splitting of fermionic wormhole contacts before the vertices might be however consistent with SU(3) symmetry. This would give two a pair of 3-vertices at which three wormhole lines meet along partonic 2-surfaces (rather than along 3-D wormhole contacts). 2. Note first that crossing gives all possible reaction vertices of this kind from F(g1)Fbar(g2)→ B(g1,g2) annihilation vertex, which is relatively easy to visualize. In this reaction F(g1) and Fbar(g2) wormhole contacts split first. If one requires that all wormhole throats involved are massless, the two wormhole throats resulting in splitting and carrying no fermion number must carry light-like momentum so that they cannot just disappear. The ends of the wormhole throats of the boson must glued together with the end of the fermionic wormhole throat and its companion generated in the splitting of the wormhole. This means that fermionic wormhole first splits and the resulting throats meet at the partonic 2-surface. This requires that topologically condensed fermions correspond to (g,g) pairs rather than (g,0) pairs. The reaction mechanism allows the interpretation of (g,g) pairs as a triplet of dynamical SU(3). The fundamental vertices would be just the splitting of wormhole contact and 3-vertices for throats since SU(3) symmetry would exclude more complex reaction vertices such as n-boson vertices corresponding the gluing of n wormhole contact lines along their 3-dimensional ends. The couplings of singlet representation for bosons would have same coupling to all fermion families so that the basic experimental constraint would be satisfied. 3. Both fermions and bosons cannot correspond to octet and singlet of SU(3). In this case reaction vertices should correspond algebraically to the multiplication of matrix elements eij: eij ekl = δjk eil allowing for instance F(g1,g2) +Fbar(g2,g3)→ B(g1,g3) . Neither the fusion of entire wormhole contacts along their ends nor the splitting of wormhole throats before the fusion of partonic 2-surfaces allows this kind of vertices so that BFF vertex is the only possible one. Also the construction of QFT limit starting from bosonic emergence led to the formulation of perturbation theory in terms of Dirac action allowing only BFF vertex as fundamental vertex. 4. Weak electric-magnetic duality brings in an additional complication. SU(3) symmetry poses also now strong constraints and it would seem that the reactions must involve copies of basic BFF vertices for the pairs of ends of weak strings. The string ends with the same Kähler magnetic charge should meet at the vertex and give rise to BFF vertices. For instance, FFbarB annihilation vertex would in this manner give rise to the analog of stringy diagram in which strings join along ends since two string ends disappear in the process. If one accepts this picture the remaining question is why the number of genera is just three. Could this relate to the fact that g≤ 2 Riemann surfaces are always hyper-elliptic (have global Z2 conformal symmetry) unlike g>2 surfaces? Why the complete bosonic de-localization of the light families should be restricted inside the hyper-elliptic sector? Does the Z2 conformal symmetry make these states light and make possible delocalization and dynamical SU(3) symmetry? Could it be that for g>2 elementary particle vacuum functionals vanish for hyper-elliptic surfaces? If this the case and if the time evolution for partonic 2-surfaces changing g commutes with Z2 symmetry then the vacuum functionals localized to g≤ 2 surfaces do not disperse to g>2 sectors. These and many other questions are discussed in the chapters of p-Adic length scale hypothesis and dark matter hierarchy, in particular in the chapter Elementary Particle Vacuum Functionals. By the way, I have performed and updating of several books about TGD in order to achieve a more coherent representation. I have also added three new chapters to the book Topological Geometrodynamics: an Overview discussing TGD from particle physics perspective (see this, this, and this). Also the chapters of p-Adic length scale hypothesis and dark matter hierarchy are heavily updated. Monday, October 25, 2010 Quark compositeness nowhere near: what about weak strings? We are living exciting times. At least I have full reason to feel like this;-). LHC has already given evidence for deviations from QCD possibly due to the fact that QCD plasma resides at long entangled color magnetic flux tubes. Then came first rumors about indications for supersymmetric partners. As I saw Tommaso's posting about quark compositeness I was for a moment absolutely sure that quark compositeness in the sense of TGD has been discovered. Unfortunately my wishful thinking (or rather feeling!) was wrong. What has been found that there is no substructure at energy scales below 4 TeV. In any case it is worth of summarizing what compositeness would mean in TGD framework since the concept of substructure is a delicate notion. The weak form of electric-magnetic duality, last summer's big theoretical discovery in TGD, forces to conclude that elementary particles in TGD Universe correspond to "weak strings", which are essentially magnetic flux tubes carrying opposite magnetic charges at their ends. The fermion at the first end is accompanied by a neutrino antineutrino pair at second end. The neutrino pair neutralizes weak isospin and in this manner causes weak confinement and screening which closely relates to TGD counterpart for particle massivation. I have explained at my blog gauge boson massivation based on this picture: see this. One highly suggestive conclusion is that also photon gets massive by eating the remaining component of Higgs ( consisting of SU(2) triplet and singlet as gauge bosons rather than complex doublet) so that there would be no Higgs to be found at LHC. What should be found (among other things) would be compositeness of both quarks, leptons, and intermediate gauge bosons. All of them would be string like objects -magnetic flux tubes with wormhole contacts with two throats at their ends of length of order weak scale. The weak string tension is the crucial parameter which does not however make itself visible through the masses of elementary particles which correspond to the lowest states. The first guess is in terms of weak mass scale in which case new physics would be easy to observe and might have been already observed. The second natural guess is that Mersenne prime M_89 characterizing weak bosons determines the tension. If so the tension would be 29=512 times hadronic string tension and by p-adic length scale hypothesis would correspond to about 512 times 1 GeV = .5 TeV. I have also proposed that ordinary hadron physics characterized by Mersenne prime M107 has a scaled up variant of characterized by M89 with about 512 GeV string tension. The proposal is inspired by the observation that Mersenne primes seem to correspond to hadron like physics in TGD Universe: leptons e and tau correspond to Mersenne primes M127 and M107 and muon to Gaussian Mersenne with k=113 and there is evidence for leptopion like states formed by color octet excitations of these states for all three leptons. For electron evidence comes from seventies, for tau CDF anomaly provides the evidence, and there is also evidence in case of muon. It remains to be seen if both M89 hadronic physics and/or weak stringy physics is or neither of them are there. For details see this. What makes the situation exciting since I do not have enough understanding to conclude whether the results say anything about the notion of weak string. One can say that below the length scale one would see quarks and leptons as particles without the weak screening is this what we have seen already for a long time above weak energy scale. Only time will show. Thursday, October 21, 2010 What before Big Bang? Both Phil Gibbs and Lubos have commented a BBC documentary in which the familiar old names and also two younger not so namy cosmologists told about their answers to the question "What before Big Bang". I must admit that I enjoyed the aggressive rhetoric of Lubos's commentary although I do not share his ultra-conservative views and belief in inflation. Most of these approaches shared something with my own approach although all of them are conceptually primitive and involve a lot of hand waving. The reason is that these theoreticians remain in the framework of General Relativity where the new ideas do not have a natural place. 1. Probably Penrose was the only one who raised the question whether the question "What before Big Bang" makes sense at all. His earlier answer to the question had been negative in general relativity context but unfortunately he had changed his view. If one leaves GRT framework, the situation changes. For instance, if one decides to take TGD seriously and identifies space-times as 4-D surfaces of M4× CP2, it takes only five years to end up with the notion of world of classical worlds (WCW), and only 27 years with zero energy ontology (ZEO);-). In ZEO WCW decomposes to a union of sub-WCWs consisting of space-time surfaces located inside causal diamonds (CD, essentially the intersection of future and past directed light-cones) carrying zero energy states with positive and negative energy parts of the state at the light-like boundaries of the causal diamond. One can form unions of CDs and CDs can also intersect. In this framework one has a hierarchy of CDs beginning from elementary particle level and extending up to Russian doll hierarchy of cosmologies. I would have been happy if at least one of the visionaries had said something about the relationship between experienced time and geometric time of physicist. These times are not one and the same thing as even child realizes. Unfortunately the academic habit is to think that they are. I have become convinced that the proper understanding of this difference will mean enormous progress both in the quantum theory of consciousness and in quantum physics defined in standard manner (the extension of physics to a quantum theory of consciousness is natural in the wider framework). Unfortunately Penrose's arguments were so popular that I could not get any idea about the mathematics behind it. My approximation for what Penrose said is that when the density of matter gets sufficiently low the space-time somehow begins to look like a good candidate for the first moment of a new Big Bang. I failed to understand. Note however that in TGD framework the mass per comoving volume for critical and string dominated cosmologies goes to zero as linear function of the scaling factor of 3-metric and identified as the light-cone proper time in TGD framework. I have talked about a silent whisper amplified to big bang as a more approriate description of TGD inspired cosmology than Big Bang which is a mathematical singularity. Penrose's intuition can be actually justified in TGD context. The canonical imbedding of empty Minkowski space to M4× CP2 is maximally critical in the sense that Kähler action is fourth order in small deformations so that perturbative quantum field theory is impossible: this was the problem which lead to the notion of WCW and eventually to the notion of hierarchy of Planck constants. Criticality has also interpretation as criticality against deformations assignable to zero energy states representing sub-cosmologies in very long length scales. Note also that there is analogy with Higgs potential in the sense that the point at the origin of Mexican hat potential is replaced with the infinite-dimensional space of vacuum extremals. 2. As a full day zero energy ontologists I liked Michio Kaku's vision about the fusion of Buddhist's vision about complete emptiness as source of everything and of the Christian "Let there be light" idea. ZEO solves many deep philosophical problems. For instance, the classical question about what was the initial state and the quantal question about what where the values of the conserved net quantum numbers associated with the initial state becomes irrelevant. ZEO is also consistent with crossing symmetry of quantum field theories and leads to an elegant generalization of thermal quantum field theories. At practical level one ends up to an opening of the black box of virtual particle and a manifestly finite version of Feynman diagrammatics emerges with massless fermions serving as fundamental building bricks of all particles, including stringy objects. Twistor approach is absolutely essential element of this approach. As a representative of Christian culture I find it amusing that the basic objects would be light-like 3-surfaces so that the statement "Let there be light" receives an additional hidden meaning! Maybe Christian God is Great Humorist after all although Bible does not suggest this. Of course, this is not the only manner to say it. By general coordinate invariance one can equivalently speak about space-like 3-surfaces. This implies effective 2-dimensionality and strong form of holography: partonic 2-surfaces and the 4-D tangent space data of the space-time surfaces at them code for the quantum physics. 3. Linde is an inflationary theorist wanting to give up the notion of Big Bang altogether and replace it with eternal inflation."What before Big Bang?" transforms to "What before Inflation?" so that not much has been gained. The basic problem of inflationary scenarios is that it involves GUTs and thus arbitrary amounts of Higgs like stuff with a lot Higgs potentials with a lot of parameters so that everything can be fitted but nothing predicted. Some of us -even Lubos- regard this as a success. Linde tested the limits of plausibility by claiming that their calculations have led to some gigantic number involving many exponents equal to 10. The highest exponent in the impressive tower of exponents was - surprise surprise- number 7! Why just 7? Sensitive listener could perhaps argue that the number seven as the number of mystic world views must be coded to the basic laws of physics and this is how it achieved;-). This number was supposed to be number of possible universes if I got it correctly. What makes me astonished that theoretical cosmologists still fail to realize that the flatness of 3-space could be also seen as a correlate of quantum criticality. Quantum criticality means universality and one can forget all fiddling with Higgs potentials. Indeed, in TGD framework criticality plus imbeddability to M4× CP2 fixes the cosmology apart from the value of the parameter fixing its duration as I have repeatedly tried to tell. A model for critical periods involving only a single parameter would be easy to kill or shown to be the cosmological counterpart of Nordström metric. One prediction is a fractal hierarchy of long range correlations in cosmological scales reflecting the hierarchy of Planck constants having gigantic values in astrophysical systems and assignable to dark matter and to the counterpart of dark energy. What made me happy that one experimentalist involved is interested in testing of the presence of this kind of correlations! There is actually already indications for these correlations: for instance, copies of astrophysical object appearing at same line of sight. If they are actual this suggest lattice like structures if cosmological scales. They could be also artefacts resulting from a circulation of the light coming from the object around circular path several times before being detected. In any case, all hope is not lost since the experimentalists are still among us! 4. Neil Turok criticized inflation and proposed an M-theory inspired model of pre Big Bang era assuming the presence of two branes which then collided. These kind of models are of course non-predictive but if cosmologists get interested they can produce endless number of fits and conclude that on basis of the amount of literature written on the subject this is the only game in the town. What connects this with TGD is that if one necessarily wants so, one can call 3-surfaces and 4-surfaces branes also in TGD framework. I still do not know how much of inspiration for the second superstring revolution came from TGD and whether the hope was that M-theory would work and TGD as a predecessor of the idea could be safely buried in sands of time. This hope was not realized. TGD is making detailed predictions to LHC whereas M-theorists remain remarkably silent. 5. Param Singh was second non-namy cosmologist allowed to tell about his views. He proposed that instead of big bang there is a series of bounces: almost big crunch followed by almost big bang. Planck scale would be the scale where GRT based cosmology would fail and super-string models would somehow come in rescue. I am afraid that super-string models do not have time to help since they are fighting with their very severe personal problems. In TGD framework CD could be visualized as big bang followed by a big crush (or better to say, a silent whisper amplified to a lot of noise eventually calming down and ending with a silent last breath). In ZEO a more approriate manner to interpret the big crush would be as a big bang in reversed time direction. It is also quite possible that partonic 2-surfaces at boundaries of CDs can continue as light-like 3-surfaces in both directions and this is essential for generalized Feynman diagrammatics. Could this define something which could be regarded as the analog of the bounce? 6. Lee Smolin represented his idea of cosmological evolution and suggested that the collapse of star to black hole is somehow followed by a creation of new cosmology inside black hole. The idea about natural selection in cosmological scales is quite interesting and I ended up with it fifteen years ago through the p-adic calculations of elementary particle masses. The calculations made one key assumption or better to say observation: elementary particles correspond to p-adic primes which are near to powers of two and Mersenne primes and their Gaussian counterparts turned out to be especially important. Zero energy cosmology combined with number theoretical universality can give at least a partial justification for this hypothesis. The proper time distances between the tips of CDs would come as octaves of CP2 time and correspond to what I am used to call secondary p-adic length/time scales. For instance, in the case of electron one obtains .1 second which is fundamental biological length scale! The idea that there is natural selection also in elementary particle length scales selecting p-adic length scales characterized by favored p-adic primes as those for which particles are long lived looks very natural. Also TGD inspired quantum biology and theory of consciousness imply evolution in all length and time scales. Mersenne primes emerge also in quantum information theory as special ones. 7. Laura Mersini-Houghton talked about "waves" in cosmology. I was unable to understand a single word of it but looked at web and found that she is proposing that the notion of wave function could make sense in M-theory landscape. Probably she had realized that string landscape is not a very sexy word nowadays and decided to avoid its use. It seems M-theorists have finally begun to think of the possibility that one could speak about quantum states in landscape. Wheeler talked about wave functions in super space for aeons ago and I talked about wave functions in the space of 3-surfaces already in my thesis around 1982, and ended up to the notion of configuration space (WCW) geometry and the modes of classical configuration space spinor field as a general representation of the quantun states of Universe around 1985. Around 1990 I ended up with the realization that general coordinate invariance forces to identify Kähler function of configuration space as Kähler action for a preferred extremal defining the counterpart of Bohr orbit and realizing holography. This almost incredibe delay in the natural evolution of ideas is an excellent lesson about how dangerous it is to censor out a bottle neck ideas. Wednesday, October 13, 2010 First rumors about super partners in LHC Lubos reports the first rumors from LHC concerning super-partners. The estimates for the masses are 200 GeV for scalar super partner (higgsino) and 160 GeV for fermion superpartner (I guess selectron). Being an incurable optimist I suppose that the rumors from LHC are more trustworthy than the physics blog rumors usually. If so, can one understand these masses in TGD framework and what can one conclude about them? Also this posting has been replaced with a new one since I finally ended up with the understaning of how the TGD based variant of gauge boson massivation could explain how gauge bosons get their longitudinal components and how the ratio of W and Z masses could result in this framework in terms of weak string picture. Consider first the theoretical background in light p-adic mass calculations, the weak form of electric-magnetic duality, and TGD based view about supersymmetry. 1. The simplest possibility is that the p-adic length scale of the super-partner differs from that of partner but the p-adic thermodynamical contributions to the mass squared obey the same formula. 2. If the p-adic prime p≈ 2k of super-partner is smaller than M89=289-1, the weak length scale must be scaled down and M61=289-1 is the next Mersenne prime. Scaled up variant of QCD for M89 would naturally correspond to M61 weak physics and would have hadronic string tension about 218 GeV2 by scaling the ordinary hadronic string tension of about 1 GeV2. This scaled up variant of hadronic physics is an old prediction of TGD. As noticed, also weak string tension could have the same value. Quite generally, the pairs of weak and hadronic scales predicted to form a hierarchy could correspond to pairs of subsequent (possibly Gaussian) Mersenne primes. 3. What happens for k=89? Can the particle topologically condense at the same p-adic scale that characterizes its weak flux tube? Or should one assume that the p-adic prime corresponds to k< 89 assuming that the particle has standard weak interactions. If so then the superpartners of light fermions would have k< 89. This is a strong prediction if superpartners obey the same mass formula as particles. In the case of weak gluinos and also QCD gluinos the bound would be k≤ 89 and even stronger bound would be k=89 so that the masses of wino and zino would be same as W and Z. One must be however very cautious with this kind of arguments since one is dealing with quantum theory. For instance, quarks inside proton have masses in 10 MeV scale and their Compton lengths are much larger than the Compton size of proton and even atomic nucleus. The interpretation is that for the corresponding space-time sheets is in terms of the color magnetic body of quark. These large space-time sheets are essential in the model of the Lamb shift anomaly of muonic hydrogen. 4. In TGD framework Higgs and its pseudo-scalar companion define electroweak triplet and singlet and Higgs could be eaten completely by electro-weak gauge bosons if the TGD based mechanism of massivation is correct. The condition of exact Yangian symmetry demands the cancellation of IR divergences requiring a small mass for all gauge bosons and graviton. The twistorially natural assumption that gauge bosons are bound states of massless fermion and antifermion implies that the three-momenta of fermion and antifermion are in opposite directions so that all gauge bosons -even photon- and graviton would be massive. Super-symmetry strongly suggests that gauginos eat Higgsinos as they become massive so that only massive gauge bosons and gauginos and possible pseudoscalar Higgs and its superpartner would remain to be discovered at LHC. Similar mechanism can indeed work also in the case of gluons expected to have colored scalar counterparts. Gluon would be massless below the scale corresponding to QCD Λ and massive above this scale. What does this picture give when compared with the rumors about super-partners of fermion and scalar. If selectron corresponds to the not necessarily allowed M89=289-1, and obeys otherwise the same mass formula as electron, the mass should be 250 GeV, which is too large. For k=88 which is the smallest value allowed by the above argument, one would obtain 177 GeV not far from 160 GeV. Therefore the interpretation as selectron could make sense. In the case of super-partner of scalar one can consider several options. 1. The first observation is that 200 GeV mass does not satisfy the proposed upper bound k> 89 for higgsinos and gauginos suggested by the condition that the weak string cannot have p-adic length scale longer than the p-adic length scale at which the particle condensed topologically. Hence neither higgsino nor longitudinal polarization of gaugino can be in question. 2. If one gives up the upper bound mZ=91.2 GeV on mass but takes the twistorially motivated and mathematically beautiful horror scenario for LHC seriously, the 200 GeV particle can only correspond to a longitudinal polarization of Zino or photino. One can of course forget the upper bound on mass and give up the horror scenario for a moment and look what one obtains. 1. If photonic Higgs is not eaten by photon, one would obtain k(Higgs)= k(Higgsino)+n. n=1,2,3 would give Higgs mass equal to (141,100, 71) GeV for m(Higgsino)= 200 GeV. On basis of experimental data mildly suggesting that neutral Higgs appears in two mass scales I have considered the possibility that Higgs indeed appears at two p-adic length scales corresponding to about 130 GeV and 92 GeV related by square root of two factor. 130 GeV would give m(Higgsino)= 184 GeV: I dare guess that this is consistent with the estimate 200 GeV. 2. For W and Z0 Higgsinos the mass mass would be p-adically scaled up variant of W or Z0 mass and for Z0 mass about 91.2 GeV Z0 Higgsino mass would be 182.4 GeV for n=2. For W Higgsino the mass would be around 160.8 GeV. I have already earlier considered the predictions of p-adic length scale hypothesis for super partners on basis of single very strange scattering event (see the section "Experimental indication for space-time supersymmetry"). This kind of considerations must of course be taken as a mere blog entertainment. The hypothesis assuming that the mass formulas for particles and sparticles are same but p-adic length scale is possibly different, combined with kinematical constraints fixes the masses of TGD counterparts of selectron, higgsino, and Z0-gluino to be 131 GeV (just at the upper bound allowed kinematically), 45.6 GeV, and 91.2 GeV (Z0 mass) respectively. The masses are consistent with the bounds predicted by the MSSM inspired model. Selectron mass would be by a factor factor 2-1/2 smaller than 177 GeV and presumably consistent with the 160 GeV rumor. Higgsino mass would be one half of Z0 mass and would satisfy the proposed constraint k< 89. Z0 gluino mass would be equal to Z0 mass also in accordance with the proposed constraint. W gluino is predicted to have same mass as W. In the case of photino the upper bound to the mass would be given by weak boson mass scale. Could it be that the life would be so simple? Could these predictions make it easy to discover super partners at LHC? Well-informed reader might be able to answer these questions. For background see the new section of p-Adic Mass Calculations: New Physics. Tuesday, October 12, 2010 Higgs and massivation in TGD framework The view about about particle massivation in TGD Universe has evolved considerably during the last half year thanks to the discovery of the weak form of electric-magnetic duality and in the following I try to explain it. The piece of text is actually a reply to a question by Ulla in Kea's blog. As I started to write the response my thoughts about Higgs mechanism in TGD framework were considerably different and this has forced to replace the posting with a new one. The core message is that one can really do without Higgs bosons and that it is quite possible and perhaps even unavoidable that photon eats the neutral Higgs boson getting very small mass so that only pseudoscalar counterpart of Higgs and Higgsinos would remain in the spectrum. This would mean that the search for Higgs at LHC would fail. In TGD framework p-adic thermodynamics gives the dominating contribution to fermion masses, which is something completely new. In the case of gauge bosons thermodynamic contribution is small since the inverse integer valued p-adic temperature is T=1/2 for bosons or even lower: for fermions one has T=1. Whether Higgs can contribute to the masses is not completely clear. In TGD framework Mexican hat potential however looks like trick. One must however keep in mind that any other mechanism must explain the ratio of W and Z0 masses and how these bosons receive their longitudinal polarizations. One must also consider seriously the possibility that all components for the TGD counterpart of Higgs boson are transformed to the longitudinal polarizations of the gauge bosons. Twistorial approach to TGD indeed strongly suggests that also the gauge bosons regarded usually as massless have a small mass guaranteing cancellation of IR singularities. As I started write to write this piece of text I believed that photon does not eat Higgs but had to challenge my beliefs. Maybe there is no Higgs to be found at LHC! Only pseudo-scalar partner of Higgs would and super partners of Higgs and pseudoscalar Higgs would remain to be discovered. The weak form of electric magnetic duality implying that each wormhole throat carrying fermionic quantum numbers is accompanied by a second wormhole throat carrying opposite magnetic charge and neutrino pair screening weak isospin and making gauge bosons massive. Concerning the implications the following view looks the most plausible one at this moment. 1. Neutral Higgs-if not eaten by photon- could develop a coherent state meaning vacuum expectation value and this is naturally proportional to the inverse of the p-adic length scale as are boson masses. This contribution can be assigned to the magnetic flux tube mentioned above since it screens weak force - or equivalently - makes them massive. Higgs expectation would not cause boson massivation. Rather, massivation and Higgs vacuum expectation would be caused by the presence of the magnetic flux tubes. Standard model would suffer from a causal illusion. Even a worse illusion is possible if the photon eats the neutral Higgs. 2. The "stringy" magnetic flux tube connecting fermion wormhole throat and the wormhole throat containing neutrino pair would give to the vacuum conformal weight a small contribution and therefore to the mass squared of both fermions and gauge bosons (dominating one for the latter). This contribution would be small in the p-adic sense (proportional 1/p2 rather than 1/p). I cannot calculate this "stringy" contribution but stringy formula in weak scale is very suggestive. 3. In the case of light fermions and massless gauge bosons the stringy contribution must vanish and therefore must correspond to n=0 string excitation (string does not vibrate at all) : otherwise the mass of fermion would be of order weak boson mass. For weak bosons n=1 would look like a natural identification but also n=0 makes sense since h+/- 1 states corresponds opposite three-momenta for massless fermion and antifermion so that the state is massive. The mechanism bringing in the h=0 helicity of gauge boson would be the TGD counterpart for the transformation of Higgs component to a longitudinal polarization. n> 0 excited states of fermions and n> 1 excitations of bosons having masses above weak boson masses are predicted and would mean new physics becoming possibly visible at LHC. Consider now the identification of Higgs in TGD framework. 1. In TGD framework Higgs particles do not correspond to complex SU(2) doublets but to triplet and singlet having same quantum numbers as gauge bosons. Therefore the idea that photon eats neutral Higgs is suggestive. Also a pseudo-scalar variant of Higgs is predicted. Let us see how these states emerge from weak strings. 2. The two kinds of massive states corresponding to n=0 and n=1 give rise to massive spin 1 and spin 2 particles. First of all, the helicity doublet (1,-1) is necessarily massive since the 3-momenta for massless fermion and anti-fermion are opposite. For n=L=0 this gives two states but helicity zero component is lacking. For n=L=1 one has tensor product of doublet (1,-1) and angular momentum triplet formed by L=1 rotational state of the weak string. This gives 2× 3 states corresponding to J=0 and J=2 multiplets. Note however than in spin degrees of freedom the Higgs candidate is not a genuine elementary scalar particle. 3. Fermion and antifermion can have parallel three momenta summing up to a massless 4-momentum. Spin vanishes so that one has Higgs like particle also now. This particle is however pseudo-scalar being group theoretically analogous to meson formed as a pair of quark and antiquark. p-Adic thermodynamics gives a contribution to the mass squared. By taking a tensor product with rotational states of strings one would obtain Regge trajectory containing pseudoscalar Higgs as the lowest state. Consider now the problem how the gauge bosons can eat the Higgs boson to get their longitudinal component. 1. (J=0,n=1) Higgs state can be combined with n=0 h=+/- 1 doublet to give spin 1 massive triplet provided the masses of the two states are same. This will be discussed below. 2. Also gauge bosons usually regarded as massless can eat the scalar Higgs so that Higgs like particle could disappear completely. There would be no Higgs to be discovered at LHC! But is this a real prediction? Could it be that it is not possible to have exactly massless photons and gluons? The mixing of M4 chiralities for Chern-Simons Dirac equation implies that also collinear massless fermion and antifermion can have helicity +/- 1. The problem is that the mixing of the chiralities is a signature of massivation! Could it really be that even the gauge bosons regarded as massless have a small mass characterized by the length scale of the causal diamond defining the physical IR cutoff and that the remaining Higgs component would correspond to the longitudinal component of photon? This would mean the number of particles in the final states for a particle reaction with a fixed initial state is always bounded from above. This is important for the twistorial aesthetics of generalized Feynman diagrammatics implied by zero energy ontology. Also the vanishing of IR divergences is guaranteed by a small physical mass. Maybe internal consistency allows only pseudo-scalar Higgs. The weak form of electric-magnetic duality suggests strongly the existence of weak Regge trajectories. 1. The most general linear mass squared formula with spin-orbit interaction term M2L-SL• S reads as M2= nM12+ M02 +M2L-SL• S , n=0,2,4 or n=1,3,5,... . M12 corresponds to string tension and M02 corresponds to the thermodynamical mass squared and possible other contributions. For a given trajectory even (odd) values of n have same parity and can correspond to excitations of same ground state. From ancient books written about hadronic string model one vaguely recalls that one can have several trajectories (satellites) and if one has something called exchange degeneracy, the even and odd trajectories define single line in M2-J plane. As already noticed TGD variant of Higgs mechanism combines together n=0 states and n=1 states to form massive gauge bosons so that the trajectories are not independent. 2. For fermions, possible Higgs, and pseudo-scalar Higgs and their super partners also p-adic thermodynamical contributions are present. M02 must be non-vanishing also for gauge bosons and be equal to the mass squared for the n=L=1 spin singlet. By applying the formula to h=+/- 1 states one obtains M02= M2(boson) . The mass squared for transversal polarizations with (h,n,L)=(+/- 1,n=L=0,S=1) should be same as for the longitudinal polarization with (h=0, n=L=1, S=1, J=0) state. This gives M12+M02+ M2L-SL• S= M02 . From L• S= [ J(J+1)-L(L+1)-S(S+1)]/2= -2 for J=0, L=S=1 one has ML-S2= -M12/2 . Only the value of weak string tension M12 remains open. 3. If one applies this formula to arbitrary n=L one obtains total spins J= L+1 and L-1 from the tensor product. For J=L-1 one obtains M2= (2n+1)M12+ M02. For J=L+1 only M02 contribution remains so that one would have infinite degeneracy of the lightest states. Therefore stringy mass formula must contain a non-linear term making Regge trajectory curved. The simplest possible generalization which does not affect n=0 and n=1 states is of from M2= n(n-1)M22+ (n-L• S/2)M12+ M02. The challenge is to understand the ratio of W and Z0 masses, which is purely group theoretic and provides a strong support for the massivation by Higgs mechanism. 1. The challenge is to understand the ratio of W and Z0 masses, which is purely group theoretic and provides a strong support for the massivation by Higgs mechanism. The above formula and empirical facts require M02(W)/M02(Z)= cos2W) . Since this parameter measures the interaction energy of the fermion and antifermion decomposing the gauge boson depending on the net quantum numbers of the pair, it would look very natural that one would have M02(W)= gW2MSU(2)2 , M02(Z)= gZ2MSU(2)2 . Here MSU(2)2 would be the fundamental mass squared parameter for SU(2) gauge bosons. p-Adic thermodynamics of course gives additional contribution which is vanishing or very small for gauge bosons. 2. The required mass ratio would result in an excellent approximation if one assumes that the mass scales associated with SU(2) and U(1) factors suffer a mixing completely analogous to the mixing of U(1) gauge boson and neutral SU(2) gauge boson W3 leading to γ and Z0. Also Higgs, which consists of SU(2) triplet and singlet in TGD Universe, would very naturally suffer similar mixing. Hence M0(B) for gauge boson B would be analogous to the vacuum expectation of corresponding mixed Higgs component. More precisely, one would have M0(W)= MSU(2) , M0(Z)= cos(θW) MSU(2)+ sin(θW) MU(1) , M0(γ)= -sin(θW) MSU(2)+ cos(θW) MU(1) . The condition that photon mass is very small and corresponds to IR cutoff mass scale gives M0(γ)=ε cos(θW)MSU(2), where ε is very small number, and implies MU(1)/M(W)=tan(θW) +ε , M(γ)/M(W)= ε× cos(θW) , M(Z)/M(W)= [1+ε × sin(θW)cos(θW)]/cos(θW) . There is a small deviation from the prediction of the standard model for W/Z mass ratio but by the smallness of photon mass the deviation is so small that there is no hope of measuring it. One can of course keep mind open for ε=0. The formulas allow also an interpretation in terms of Higgs vacuum expectations as it must. The vacuum expectation would most naturally correspond to interaction energy between the massless fermion and antifermion with opposite 3-momenta at the throats of the wormhole contact and the challenge is to show that the proposed formulas characterize this interaction energy. Since CP_2 geometry codes for standard model symmetries and their breaking, it woul not be surprising if this would happen. One cannot exclude the possibility that p-adic thermodynamics contributes to M02(boson). For instance, ε might characterize the p-adic thermal mass of photon. If the mixing applies to the entire Regge trajectories, the above formulas would apply also to weak string tensions, and also photons would belong to Regge trajectories containing high spin excitations. 3. What one can one say about the value of the weak string tension M12? The naive order of magnitude estimate is M12≈ mW2≈ 104 GeV2 is by a factor 1/25 smaller than the direct scaling up of the hadronic string tension about 1 GeV2scaled up by a factor 218. The above argument however allows also the identification as the scaled up variant of hadronic string tension in which case the higher states at weak Regge trajectories would not be easy to discover since the mass scale defined by string tension would be 512 GeV to be compared with the recent beam energy 7 TeV. Weak string tension need of course not be equal to the scaled up hadronic string tension. Weak string tension - unlike its hadronic counterpart- could also depend on the electromagnetic charge and other characteristics of the particle. Wednesday, October 06, 2010 Some thoughts inspired by graphene In viXra log there has been some discussion inspired by Phil's posting about the Nobel prize of physics received by Andre Geim and Konstantin Novoselov for discovering graphene. The discussion had the effect that I clicked "graphene" in Wikipedia to refresh my mental images about graphene. By looking at Wikipedia article, one realizes that graphene is an extremely interesting from the perspective of theoretical physicist willing to challenge the reductionistic belief that everything above weak length scale is perfectly understood by recent day physics (for a really extreme position bringing in mind the days before quantum mechanics see the article of Sean Carroll and a reaction to it by Johannes Koelman). Addition: I have made some corrections to the text below afer listening the excellent lecture straightenint out some mis-understandings due to the rather informal style of the Wikipedia article. Quantum Hall effect and graphene From Wikipedia one learns that quantum Hall effect (QHE) in graphene corresponds to the multiples N= 4× (2r+1)/2 of minimal transversal conductivity σxy. This could be understood as integer quantum Hall effect (QHE) allowing only even integers. Why even integers? This one should understand. This is possible. I learned from a nice lecture about graphene by Eva Andrei here that the formula for N is well-understood. The overall factor g=4 corresponds to the degeneracy of edge states and 1/2 in half odd integer comes from the effective masslessness of electrons at the lowest Landau level meaning that only second chirality for a given momentum is possible. This is so called γ5 anomaly having analog in particle physics. From the lecture one learns that also FQHE has been observed by Eva Andrei and her group for n=1/3 and there are excellent reasons to expect that it will be found also for other values of n. Also the prospects for graphene super-conductivity are excellent. Therefore the following TGD based explanation of FQHE in terms of quantization of Planck constant is well motivated. I have considered several variants for the quantization of Planck constant in TGD framework. The first option postulates the quantization of Planck constant as a first principle and in this case the spectrum of Planck constants would be given by rational numbers: hbar= q× hbar0 in the most general case but their are arguments favoring rationals for which the quantum phases exp(iq2π) are algebraically simple, say those representable in terms of square root operation alone (rules and compass integers as denominators of q). For q= 1/2 so that Planck constant would be hbar0/2, one would obtain even integer QHE but this explanation is not needed by the above facts from Eva Andrei's lecture. There is a slight indication for fractional quantization of Planck constant from hydrino atom of Mills for which the energy levels of hydrogen are claimed to be scaled up by a square of integer. Since energies are proportional to 1/hbar2 this would follow from rational quantization of hbar. One can however explain the anomaly also by replacing the Laguerre equation for radial parts of the solutions of Schrödinger equation for hydrogen atom with is q-counterpart. Therefore there is no pressing need to assume fractional values of hbar. This makes me happy since I have a competing argument reducing the quantization of Planck constant to the basic TGD without introducing it as a separate postulate. This option is of course the one which is more attractive since minimalism is an excellent guideline for a theoretician. This option is highly attractive also from the point of view of biology since integer valuedness means that it is possible to understand evolution in terms of drifting in the space of Planck constants to ever larger Planck constants. This is like difficusion in half-space. For rational values one would have analogy with diffusion along real axis to the directions of both small and large Planck constants and no direction of evolution. For this option the hierarchy of Planck constants gives a straightforward explanation for FQHE since integer multiple hbar=n× hbar0 implies that the transversal conductivity σxy proportional to alpha proportional to 1/hbar is proportional to 1/n and thus fractionized as multiples of 1/n. The argument giving quantization of Planck constant as integer multiples of ordinary Planck constant goes as follows. 1. Kähler action is extremely nonlinear and possesses enormous vacuum degeneracy since any space-time surface with CP2 projection which is Lagrange sub-manifold (maximum dimension 2) is vacuum extremal (Kähler gauge potential is pure gauge). The U(1) gauge symmetry realized as symplectic transformations of CP2 is not gauge symmetry but spin glass degeneracy and not present for non-vacuum extremals. TGD Universe would be 4-D spin glass and thus possess extremely rich structure of ground states. The failure of classical non-determinism for vacuum solutions would make possible to generalized quantum classical correspondence so that one would have space-time correlates also for quantum jump sequences and thus symbolic representations at space-time level for contents of consciousness (quantum jumps as moment of consciousness). Preferred extremal property guarantees both holography and generalized Bohr orbit property for space-time surfaces. 2. As a consequence, the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to -many: 1-to-infinite for vacuum extremals. This spoils all hopes about canonical quantization and path integral approach and led within 6 years or so to the realization that quantum physics as geometry of the world of classical worlds vision generalizing Einstein's geometrization program is the only way out of the situation. Much later -during last summer- I realized that this 1-to-many correspondence could allow to understand the quantization of Planck constant as a consequence of quantum TGD rather than as independent postulate. 3. Different roots for the values of time derivatives in the extremely non-linear formulas for canonical momentum densities correspond to same values of canonical momentum densities and therefore also conserved currents and of Kähler action if the weak form of electric-magnetic duality is accepted reducing Kähler action to Chern-Simons term. It is convenient to introduce n-sheeted covering of imbedding space as a convenient tool to describe the situation. hbar= n× hbar0 is the effective value of Planck constant at the sheets of covering. 4. Fractionization means simply division of Kähler action and various conserved charges between the n sheets. In this manner the amount of charge at given sheet is reduced by a factor 1/n and perturbation theory applies. One could say that the space-time sheet is unstable against this kind of splitting and in zero energy ontology the space-time sheets split at the boundaries of the causal diamond (intersection of future and past directed light-cones) to n sheets of the covering. One particular consequence is fractional quantum Hall effect. A very pleasant news for theoretician is that Mother Nature loves her theoreticing children and takes care that perturbative approach works! Kähler Dirac equation and graphene: a useful mis-understanding As I looked at Wikipedia article, I found that Dirac equation is applied by treating electron as a massless particle and by replacing light velocity with Fermi velocity. I must say, that I find it very difficult to believe that this description could be deduced from first principles. This skeptic thought led to the realization that here might be the natural physical interpretation of formally massless Kähler Dirac equation in space-time interior. Addition: Here again Eva's lecture clarified a lot. The spinors in question are not genuine Dirac spinors. There are two sub-lattices in graphene such that the wave functions of electron are localized to either of them. This is conveniently described in terms of spinors: the value of spin corresponds to a localization to either sub-lattice. Condensed matter physics uses rather informal Wikipedia terminology! "Schrödinger spinor" mentioned in the lecture would help enormously the random Wikipedia visitor. To avoid possible confusions let us stress that the linear dispersion relation has absolutely nothing to do with the dispersion relation of real electron in relativistic theory and and reflects only the dependence of electrons non-relativistic energy on momentum. Also spin is only a formal concept in this context. This irritatingly informal use of the notion of spinor caused very useful mis-understanding since it forced to ask whether these strange spinors describing effectively massless electrons could have a first principle counterpart in TGD. They do not and there is not need for this that but one ends up with a proposal for the physical interpretation of the Kähler Dirac equation for the induced spinor fields in the interior of space-time surface. To begin with, Dirac equation appears in three forms in TGD. 1. The Dirac equation in world of classical worlds codes for the super Virasoro conditions for the super Kac-Moody and similar representations formed by the states of wormhole contacts forming the counterpart of string like objects (throats correspond to the ends of the string. This Dirac generalizes the Dirac of 8-D imbedding space by bringing in vibrational degrees of freedom. This Dirac equation should gives as its solutions zero energy states and corresponding M-matrices generalizing S-matrix and their collection defining the unitary U-matrix whose natural application appears in consciousness theory as a coder of what Penrose calls U-process. 2. There is generalized eigenvalue equation for Chern-Simons Dirac operator at light-like wormhole throats. The generalized eigenvalue is pslash. The interpretation of pseudo-momentum p has been a problem but twistor Grassmannian approach suggests strongly that it can be interpreted as the counterpart of equally mysterious region momentum appearing in momentum twistor Grassmannian approach to N=4 SYM. The pseudo-/region momentum p is quantized (this does not spoil the basics of Grasssmannian residues integral approach) and 1/pslahs defines propagator in lines of generalized Feynman diagrams. The Yangian symmetry discovered generalizes in a very straightforward manner and leads alsoto the realization that TGD could allow also a twistorial formulation in terms of product CP3 ×CP3 of two twistor spaces. General arguments lead to a proposal for explicit form for the solutions of field equation represented identified as holomorphic 6-surfaces in this space subject to additional partial different equations for homogenenous functions of projective twistor coordinates suggesting strongly the quantal interpretation as analogs of partial waves. Therefore quantum-classical correspondence would be realize in beatiful manner. 3. There is Kähler Dirac equation in the interior of space-time. In this equation the gamma matrices are replaced with modified gamma matrices defined by the contractions of canonical momentum currents T&alphak = ∂ L/∂α hk with imbedding space gamma matrices γk. This replacement is required by internal consistency and by super-conformal symmetries. Could Kähler Dirac equation provide a first principle justification for the light-hearted use of effective mass and the analog of Dirac equation in condensed manner physics? This would conform with the holographic philosophy. Partonic 2-surfaces with tangent space data and their light-like orbits would give hologram like representation of physics and the interior of space-time the 4-D representation of physics. Holography would have in the recent situation interpretation also as quantum classical correspondence between representations of physics in terms of quantized spinor fields at the light-like 3-surfaces on one hand and in terms of classical fields on the other hand. The resulting dispersion relation for the square of the Kähler-Dirac operator assuming that induced like metric, Kähler field, etc. are very slowly varying contains quadratic and linear terms in momentum components plus a term corresponding to magnetic moment coupling. In general massive dispersion relation is obtained as is also clear from the fact that Kähler Dirac gamma matrices are combinations of M4 and CP2 gammas so that modified Dirac mixes different M4 chiralities (basic signal for massivation). If one takes into account the dependence of the induced geometric quantities on space-time point dispersion relations become non-local. Let us however add again that this dispersion relation has nothing to do with the dispersion relation for Schrödinger spinors in graphene. Does energy metric provided the gravitational dual for condensed matter systems? The modified gamma matrices define an effective metric via their anticommutators which are quadratic in components of energy momentum tensor (canonical momentum densities). This effective metric vanishes for vacuum extremals. Note that the use of modified gamma matrices guarantees among other things internal consistency and super-conformal symmetries of the theory. The physical interpretation has remained obscure hitherto although corresponding effective metric for Chern-Simons Dirac action has now a clear physical interpretation. If the above argument is on the right track, this effective metric should have applications in condensed matter theory. In fact, energy metric has a natural interpretation in terms of effective light velocities which depend on direction of propagation. One can diagonalize the energy metric geαβ (contravariant form results from the anticommutators) and one can denote its eigenvalues by (v0,vi) in the case that the signature of the effective metric is (1,-1,-1,-1). The 3-vector vi/v0 has interpretation as components of effective light velocity in various directions as becomes clear by thinking the d'Alember equation for the energy metric. This velocity field could be interpreted as that of hydrodynamic flow. The study of the extremals of Kauml;hler action shows that if this flow is actually Beltrami flow so that the flow parameter associated with the flow lines extends to global coordinate, Kähler action reduces to a 3-D Chern-Simons action and one obtains effective topological QFT. The conserved fermion current has interpretation as incompressible hydrodynamical flow. This would give also a nice analogy with AdS/CFT correspondence allowing to describe various kinds of physical systems in terms of higher-dimensional gravitation and black holes are introduced quite routinely to describe condensed matter systems: probably also graphene has already fallen in some 10-D black hole or even many of them. In TGD framework one would have an analogous situation but with 10-D space-time replaced with the interior of 4-D space-time and the boundary of AdS representing Minkowski space with the light-like 3-surfaces carrying matter. The effective gravitation would correspond to the "energy metric". One can associate with it curvature tensor, Ricci tensor and Einstein tensor using standard formulas and identify effective energy momentum tensor associated as Einstein tensor with effective Newton's constant appearing as constant of proportionality. Note however that the besides ordinary metric and "energy" metric one would have also the induced classical gauge fields having purely geometric interpretation and action would be Kähler action. This 4-D holography would provide a precise, dramatically simpler, and also a very concrete dual description. This cannot be said about model of graphene based on the introduction of 10-dimensional black holes, branes, and strings chosen in more or less ad hoc manner. This raises questions. Does this give a general dual gravitational description of dissipative effects in terms of the "energy" metric and induced gauge fields? Does one obtain the counterparts of black holes? Do the general theorems of general relativity about the irreversible evolution leading to black holes generalize to describe analogous fate of condensed matter systems caused by dissipation? Can one describe non-equilibrium thermodynamics and self-organization in this manner? One might argue that the incompressible Beltrami flow defined by the dynamics of the preferred extremals is dissipationless and viscosity must therefore vanish locally. The failure of complete non-determinism of Kähler action however means generation of entropy since the knowledge about the state decreases gradually. This in turn should have a phenomenological local description in terms of viscosity which characterizes the transfer of energy to shorter scales and eventually to radiation. The deeper description should be non-local and basically topological and might lead to quantization rules. For instance, one can imagine the quantization of the ratio η/s of the viscosity to entropy density as multiples of a basic unit defined by its lower bound (note that this would be analogous to Quantum Hall effect). For the first M-theory inspired derivation of the lower bound of η/s see this. The lower bound for η/s is satisfied in good approximation by what should have been QCD plasma but found to be something different (RHIC and the first evidence for new physics from LHC: I have discussed TGD based understanding of these anomalies in previous posting). An encouraring sign comes from the observation that for so called massless extremals representing classically arbitrarily shaped pulses of radiation propagating without dissipation and dispersion along single direction the canonical momentum currents are light-like. The effective contravariant metric vanishes identically so that fermions cannot propate in the interior of massless extremals! This is of course the case also for vacuum extremals. Massless extremals are purely bosonic and represent bosonic radiation. Many-sheeted space-time decomposes into matter containing regions and radiation containing regions. Note that when wormhole contact (particle) is glued to a massless extremal, it is deformed so that CP2 projection becomes 4-D guaranteing that the weak form of electric magnetic duality can be satisfied. Therefore massless extremals can be seen as asymptotic regions. Perhaps one could say that dissipation corresponds to a decoherence process creating space-time sheets consisting of matter and radiation. Those containing matter might be even seen as analogs blackholes as far as energy metric is considered. Could warped imbeddings relate to graphene? An interesting question is whether the reduction of light-velocity to Fermi velocity could be interpreted as an actual reduction of light-velocity at space-time surface. I have discussed this possibility for some years in some blog posting and the argument is also buried in some chapter of some of the seven books about TGD. The proposed interpretation of energy metric in terms of hydrodynamic velocities does not allow this interpretation. Rather, the velocity in question should be assigned to the ordinary radiation. TGD allows infinite family of warped imbeddings of M4 to M4xCP2. They are analogous to different imbeddings of flat plane to 3-D space. In real world the warped imbeddings of 2-D flat space are obtained spontaneously when you have a thin plane of metal or just a sheet of paper: it gets spontaneously warped. The resulting induced geometry is flat as long as no stretching occurs. A very simple example of this kind of imbedding is obtained as graph of a map from M4 to the geodesic circle S1 of CP2 with angle coordinate Φ linear in M4 time coordinate t: Φ= &omega× t. What is interesting is that although their is no gravitation in the standard sense, the light velocity is in this simple situation reduced to v =(gtt)1/2c= (1-R2ω2)1/2c in the sense that it takes time T=L/v to move from point A to B along light-like geodesic of warped space-time surface whereas along non-warped space-time surface the time would be only T= L/c. The reason is of course that the imbedding space distance travelled is longer due to the warping. One particular effect is anomalous time dilation which could be much larger than the usual special relativistic and general relativistic time dilations. Suppose that Kähler Dirac equation and Kähler action itself can be used as a possible first principle counterpart for the phenomenological Dirac equation and Maxwell's equations in the modeling of condensed matter systems. This is kind of description might make sense for so called slow photons with very slow group velocity. These surfaces could provide a holographic description for the reduction of the light-velocity also in di-electrics caused by interactions between particles described in terms of light-like 3-surfaces. Strongly warped space-time surfaces obtained as deformations of warped imbeddings of flat Minkowski geometries (vacuum extremals) do not seem to provide a natural model for graphene. The basic objection is that electrons are in question and this light velocity is associated with genuinely massless particles. As already proposed, one could however assign effective light-velocity also to the "energy" metric. This velocity could be assigned to electrons in condensed matter. Monday, October 04, 2010 Is the new physics at LHC "approximately unavoidable"? Tommaso Dorigo has written a summary about a highly interesting conference talk by Guido Altarelli in 2010 LHC Days in Split (slides can be found here). The talk begins with the question "Is it possible that Higgs will not be found?". The general conclusion is that if Higgs is not found then some other new physics is "approximately unavoidable". One very general reason is that the unitarity of electroweak theory is otherwise spoiled. Altarelly saw also a reason for worry. The new physics should should emerge rather abruptly: the general view is that there is no evidence for it existence from the previous experimental work. How can it is possible that the new physics lurking just behind the corner manages to hide itself so completely? TGD predict Higgs and supersymmetry and also weak confinement This touched something inside me since the questions whether TGD predicts Higgs and standard space-time super-symmetry have shadowed my life for a long time. When the notions of bosonic emergence and understanding of super-conformal symmetry in terms of partons identified as wormhole throats emerged, it became clear that boson with quantum numbers of Higgs identified as wormhole contact with opposite throats carrying fermion and antifermion quantum numbers is bound to exist. Also an appropriate generalization of broken space-time supersymmetry exists and reduces to N=1 super-symmetry at low energy limit. The emergence of the weak form of electric-magnetic duality during this year led to the realization that the wormhole throats behave like magnetic monopoles since the CP2 projections of these 2-surfaces are homologically non-trivial. The only manner to avoid macroscopic magnetic monopole fields is magnetic confinement appearing as a side product of electro-weak symmetry breaking and possibly also of color confinement. In the case of electroweak symmetry breaking this would mean that a wormhole throat carrying lepton or quark quantum numbers is accompanied by second throat with opposite Kähler magnetic charge and carrying quantum numbers of neutrino and antineutrino neutralizing the weak charge of the elementary fermion and screening of weak force. One can speak of weak confinement. For quarks the neutralization of magnetic charge need not be complete and valence quarks could be Kähler magnetic monopoles giving rise to hadrons which have neither magnetic nor color charges. Physical elementary particles would be string like objects with length of order weak length scale. This would certainly represent new physics which could become visible at LHC. This piece of new physics (TGD predicts also many other pieces) would resemble the good old hadron physics for which the predecessor of the recent super string theory provided a satisfactory description. Regge trajectories would be one striking signature of this physics both at the level of states and scattering amplitudes. The string tension of these trajectories would be enormous: in the first estimate 2107-89=18 times higher than that for low energy hadrons. Mass scale would be about .512 TeV to be compared with the collision energy of 7 TeV of LHC. The proton of this physics would have mass of about .512 TeV (if one believes on naive p-adic scaling) and is expected to be unstable against decay to ordinary hadrons. Lifetime should be long since otherwise also the ordinary proton is expected to be unstable against decay to scaled down hadrons with say p-adic length scale of electron (, which corresponds to the largest Mersenne prime which does not define super-astrophysical p-adic length scale). p-Adic thermodynamics and the emergence of string like objects from massless partons While reading the summary about Guido's representation I realized that I have been talking for years about scaled up copy of hadron physics at electroweak length scale. What distinguishes the string like objects of this hadronic physics from those of electroweak physics? Or do they represent two different aspects of something more general? The obvious answer would be that color confinement is not involved with weak strings and that this is the basic distinction. This answer seems to be correct. 1. Dirac equation in M4×CP2 predicts that free fermions -also leptons- in general correspond to in non-trivial color partial waves of CP2 and that the correlation between color and electroweak quantum numbers is wrong although quarks correspond to triality t=1 and leptons to triality t=0. This was a strong objection against TGD until I realized that super-conformal invariance could resolve the problem. The lightest leptonic (quark) states are color singlets (triplets) and colored super-conformal generators can generated the anomalous color so that lightest leptons and quarks are colors singlets and triplets. p-Adic mass calculations are consistent with this picture. The contributions from enormous bare mass squared (conformal weight) whose values are dictated by the color partial waves of quarks and leptons are compensated by negative tachyonic mass squared (conformal weight) of the vacuum state. 2. p-Adic thermodynamics assumes that elementary particles correspond to representations of super-conformal algebra characterized by enormous string tension. Elementary particle mass scales emerge thermodynamically from a fundamental mass scale which corresponds to CP2 mass, which is roughly 10-4-10-3 times Planck mass. Massless states with vanishing conformal weight are thermally mixed with those with non-vanishing conformal weight and enormous value of mass squared given by string mass formula. 3. Weak form of electric-magnetic duality, the basic facts about modified Dirac equation, and also twistorialization of quantum TGD force to conclude that both strings and bosons and their super-counterparts emerge from massless fermions moving collinearly at partonic two-surfaces. Stringy mass spectrum is consistent with this only if p-adic thermodynamics describes wormhole contacts. For instance, the three-momenta of massless wormhole throats could be in opposite direction so that wormhole contact would become massive. String like objects would therefore correspond to the wormhole contacts with size scale of order CP2 length. Wormhole contacts would be the fundamental stringy objects and already these have the correct correlation between color and electroweak quantum numbers. 4. One can of course ask whether the anomalous color could be neutralized in the weak scale? This is not possible. p-Adic thermodynamics with string tension defined by electro-weak length scale would make completely unrealistic predictions. How the new physics around the corner manages to hide so well? The basic worry of Guido Altarelly is expressed by the question of the title and it seem that the new physics predicted TGD might provide a satisfactory answer to the question. 1. What seems to be a prediction is that the weak length scale serves as the confinement scale for the string like objects with second end containing neutrino pairs with electroweak isospin. Regge trajectories of weak bosons and Higgs is one consequence. The new physics would behind the corner would be made virtually invisible by weak confinement. The replacement of these neutrino pairs with more general states would give a lot of new physics. 2. Of course, Nature could choose to scale up the weak scale to say Mersenne prime M61 meaning weak bosons with mass scale 512 higher than weak scale. This would be more or less equivalent with the disappearance of weak interactions and the new weak physics would emergence in discontinuous manner via phase transition. That an entire weak physics would just disappear from existence without any warning sounds of course weird! In reality of course the phase transition would take place for a small portion of the stuff created in the collisions. The scaled up weak bosons would also decay in time scale which is by a factor 1/512 shorter than than the life time of weak bosons. The challenge is therefore to detect very small signals from background. 3. Whether a scaled up counterpart of hadron physics exists at weak scale remains an open question. There is evidence for scaled up variants of leptohadrons for which both ends would contain charged leptons in color partial waves. For these states at p-adic mass scales characterizing ordinary leptons there indeed exists experimental evidence.
2a5a0c45604620ea	Skip to main content Chemistry LibreTexts 3.1: The Schrödinger Equation • Page ID • Learning Objectives • To be introduced to the general properties of the Schrödinger equation and its solutions. De Broglie’s doctoral thesis, defended at the end of 1924, created a lot of excitement in European physics circles. Shortly after it was published in the fall of 1925 Pieter Debye, a theorist in Zurich, suggested to Erwin Schrödinger that he give a seminar on de Broglie’s work. Schrödinger gave a polished presentation, but at the end Debye remarked that he considered the whole theory rather childish: why should a wave confine itself to a circle in space? It wasn’t as if the circle was a waving circular string, real waves in space diffracted and diffused, in fact they obeyed three-dimensional wave equations, and that was what was needed. This was a direct challenge to Schrödinger, who spent some weeks in the Swiss mountains working on the problem and constructing his equation. There is no rigorous derivation of Schrödinger’s equation from previously established theory, but it can be made very plausible by thinking about the connection between light waves and photons, and construction an analogous structure for de Broglie’s waves and electrons (and, later, other particles). The Schrödinger Equation: A Better Quantum Approach While the Bohr model is able to predict the allowed energies of any single-electron atom or cation, it by no means, a general approach. Moreover, it relies heavily on classical ideas, clumsily grafting quantization onto an essentially classical picture, and therefore, provides no real insights into the true quantum nature of the atom. Any rule that might be capable of predicting the allowed energies of a quantum system must also account for the wave-particle duality and implicitly include a wave-like description for particles. Nonetheless, we will attempt a heuristic argument to make the result at least plausible. In classical electromagnetic theory, it follows from Maxwell's equations that each component of the electric and magnetic fields in vacuum is a solution of the 3-D wave equation for electronmagnetic waves: \[\nabla^2 \Psi(x,y,z,t) -\dfrac{1}{c^2}\dfrac{\partial ^2 \Psi(x,y,z,t) }{\partial t^2}=0\label{3.1.1}\] The wave equation in Equation $\ref{3.1.1}$ is the three-dimensional analog to the wave equation presented earlier (Equation 2.1.1) with the velocity fixed to the known speed of light: $c$. Instead of a partial derivative $\dfrac{\partial^2}{\partial t^2}$ in one dimension, the Laplacian (or "del-squared") operator is introduced: \[\nabla^2=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}\label{3.1.2}\] Corresponding, the solution to this 3D equation wave equation is a function of four independent variables: $x$, $y$, $z$, and $t$ and is generally called the wavefunction $\psi$. We will attempt now to create an analogous equation for de Broglie's matter waves. Accordingly, let us consider a only 1-dimensional wave motion propagating in the x-direction. At a given instant of time, the form of a wave might be represented by a function such as \[\Psi(x)=f\left(\dfrac {2\pi x}{ \lambda}\right)\label{3.1.3}\] where $f(\theta)$ represents a sinusoidal function such as $\sin\theta$, $\cos\theta$, $e^{i\theta}$, $e^{-i\theta}$ or some linear combination of these. The most suggestive form will turn out to be the complex exponential, which is related to the sine and cosine by Euler's formula \[e^{\pm i\theta}=\cos\theta \pm i \sin\theta \label{3.1.4}\] Each of the above is a periodic function, its value repeating every time its argument increases by $2\pi$. This happens whenever $x$ increases by one wavelength $\lambda$. At a fixed point in space, the time-dependence of the wave has an analogous structure: \[T(t)=f(2\pi\nu t)\label{3.1.5}\] where $\nu$ gives the number of cycles of the wave per unit time. Taking into account both $x$ and $t$ dependence, we consider a wavefunction of the form \[\Psi(x,t)=exp\left[2\pi i\left(\dfrac{x}{\lambda}-\nu t\right)\right]\label{3.1.6}\] representing waves traveling from left to right. Now we make use of the Planck formula ($E=h\nu$) and de Broglie formulas ($p=\frac{h}{\lambda}$) to replace $\nu$ and $\lambda$ by their particle analogs. This gives \[\Psi(x,t)=\exp \left[\dfrac{i(px-Et)}{\hbar} \right] \label{3.1.7}\] \[\hbar \equiv \dfrac{h}{2\pi}\label{3.1.8}\] Since Planck's constant occurs in most formulas with the denominator $2\pi$, the $\hbar$ symbol was introduced by Peter Dirac. Equation $\ref{3.1.5}$ represents in some way the wavelike nature of a particle with energy $E$ and momentum $p$. The time derivative of Equation $\ref{3.1.7}$ gives \[\dfrac{\partial\Psi}{\partial t} = -\left(\dfrac{iE}{\hbar} \right ) \exp \left[\dfrac{i(px-Et)}{\hbar} \right]\label{3.1.9}\] Thus from a simple comparison of Equations $\ref{3.1.7}$ and $\ref{3.1.9}$ \[i\hbar\dfrac{\partial\Psi}{\partial t} = E\Psi\label{3.1.10}\] or analogously differentiation of Equation $\ref{3.1.9}$ with respect to $x$ \[-i\hbar\dfrac{\partial\Psi}{\partial x} = p\Psi\label{3.1.11}\] and then the second derivative \[-\hbar^2\dfrac{\partial^2\Psi}{\partial x^2} = p^2\Psi\label{3.1.12}\] The energy and momentum for a nonrelativistic free particle (i.e., all energy is kinetic with no potential energy involved) are related by Substituting Equations $\ref{3.1.12}$ and $\ref{3.1.10}$ into Equation $\ref{3.1.13}$ shows that $\Psi(x,t)$ satisfies the following partial differential equation \[i\hbar\dfrac{\partial\Psi}{\partial t}=-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi}{\partial x^2}\label{3.1.14}\] Equation $\ref{3.1.14}$ is the applicable differential equation describing the wavefunction of a free particle that is not bound by any external forces or equivalently not in a region where its potential energy $V(x,t)$ varies. For a particle with a non-zero potential energy $V(x)$, the total energy $E$ is then a sum of kinetics and potential energies we postulate that Equation $\ref{3.1.3}$ for matter waves can be generalized to \[ \underbrace{ i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}=\left[-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}+V(x)\right]\Psi(x,t) }_{\text{time-dependent Schrödinger equation in 1D}}\label{3.1.16}\] For matter waves in three dimensions, Equation $\ref{3.1.6}$ is then expanded \[ \underbrace{ i\hbar\dfrac{\partial}{\partial t}\Psi(\vec{r},t)=\left[-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r})\right]\Psi(\vec{r},t)}_{\text{time-dependent Schrödinger equation in 3D}}\label{3.1.17}\] Here the potential energy and the wavefunctions $\Psi$ depend on the three space coordinates $x$, $y$, $z$, which we write for brevity as $\vec{r}$. Notice that the potential energy is assumed to depend on position only and not time (i.e., particle motion). This is applicable for conservative forces that a potential energy function $V(\vec{r})$ can be formulated. The Laplacian operator The three second derivatives in parentheses together are called the Laplacian operator, or del-squared, \[ \nabla^2 = \left ( \frac {\partial ^2}{\partial x^2} + \dfrac {\partial ^2}{\partial y^2} + \dfrac {\partial ^2}{\partial z^2} \right ) \label {3-20}\] with the del operator, \[\nabla = \left ( \vec {x} \frac {\partial}{\partial x} + \vec {y} \frac {\partial}{\partial y} + \vec {z} \frac {\partial }{\partial z} \right ) \label{3-21}\] also is used in Quantum Mechanics. The symbols with arrows over them are unit vectors. Equation $\ref{3.1.17}$ is the time-dependent Schrödinger equation describing the wavefunction amplitude $\Psi(\vec{r}, t)$ of matter waves associated with the particle within a specified potential $V(\vec{r})$. Its formulation in 1926 represents the start of modern quantum mechanics (Heisenberg in 1925 proposed another version known as matrix mechanics). For conservative systems, the energy is a constant, and the time-dependent factor from Equation $\ref{3.1.7}$ can be separated from the space-only factor (via the Separation of Variables technique discussed in Section 2.2) \[\Psi(\vec{r},t)=\psi(\vec{r})e^{-iEt / \hbar}\label{3.1.18}\] where $\psi(\vec{r})$ is a wavefunction dependent (or time-independent) wavefuction that only depends on space coordinates. Putting Equation $\ref{3.1.18}$ into Equation $\ref{3.1.17}$ and cancelling the exponential factors, we obtain the time-independent Schrödinger equation: \[ \textcolor{red}{ \underbrace{\left[-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r})\right]\psi(\vec{r})=E\psi(\vec{r})} _{\text{time-independent Schrödinger equation}}} \label{3.1.19}\] The overall form of the Equation $\ref{3.1.19}$ is not unusual or unexpected as it uses the principle of the conservation of energy. Most of our applications of quantum mechanics to chemistry will be based on this equation (with the exception of spectroscopy). The terms of the time-independent Schrödinger equation can then be interpreted as total energy of the system, equal to the system kinetic energy plus the system potential energy. In this respect, it is just the same as in classical physics. Time Dependence to the wavefunctions Notice that the wavefunctions used with the time-independent Schrödinger equation (i.e., $\psi(\vec{r})$) do not have explicit $t$ dependences like the wavefunctions of time-dependent analog in Equation $\ref{3.1.17}$ (i.e., $\Psi(\vec{r},t)$). That does not imply that there is no time dependence to the wavefunction. Let's go back to Equation \ref{3.1.18}: The time-dependent (i.e., full spatial and temporal) wavefunction ($\Psi(\vec{r},t)$) differs from from the time-independent (i.e., spatial only) wavefunction $\psi(\vec{r})$ by a "phase factor" of constant magnitude. Using the Euler relationship in Equation \ref{3.1.4}, the total wavefunction above can be expanded \[\Psi(\vec{r},t)=\psi(\vec{r})\left(\cos \dfrac{Et}{\hbar} - i \, \sin \dfrac{Et}{\hbar} \right) \] This means the total wavefunction has a complex behavior with a real part and an imaginary part. Moreover, using the Trigonometry identity $\sin (\theta) = \cos (\theta - \pi/2)$ this can further simplified to \[\Psi(\vec{r},t)=\psi(\vec{r})\cos \left(\dfrac{Et}{\hbar} \right) - i \psi(\vec{r})\cos \left(\dfrac{Et}{\hbar} - \dfrac{\pi}{2} \right) \] Hence, the imaginary part of the total wavefunction oscillates out of phase by $\frac{\pi}{2}$ with respect to the real part. While all wavefunctions have a time-dependence, that dependence may not be manifested in simple quantum problems as the next sections discuss. Before we embark on this, however, let us pause to comment on the validity of quantum mechanics. Despite its weirdness, its abstractness, and its strange view of the universe as a place of randomness and unpredictability, quantum theory has been subject to intense experimental scrutiny. It has been found to agree with experiments to better than $10^{-10}\%$ for all cases studied so far. When the Schrödinger Equation is combined with a quantum description of the electromagnetic field, a theory known as quantum electrodynamics, the result is one of the most accurate theories of matter that has ever been put forth. Keeping this in mind, let us forge ahead in our discussion of the quantum universe and how to apply quantum theory to both model and real situations.
c4cb51955106051c	Quantum Mechanics and the Shape of Graphs Event time: Wednesday, August 14, 2019 - 2:00pm LOM 206 Ivan Contreras Palacios Speaker affiliation: Amherst College Event description: Abstract: Quantum physics has revolutionized the way we understand our world. Since the beginning of the 20th century, beautiful mathematics has been devised and implemented in order to achieve such success. This talk intends to give an overview of a discretized model of quantum mechanics: the Schrödinger equation on graphs. We will use the combinatorial graph Laplacian to describe certain properties of finite graphs such as topological invariants, number of generalized walks and entropy. No prior knowledge of physics or graph theory will be assumed.
756e9eadd40802a8	Physics/Essays/Anonymous/Low energy nuclear matter transformations From Wikiversity Jump to navigation Jump to search Low energy nuclear matter transformations – electromagnetic (or electronic) pulse initialization of the selfamplifying accumulation processes with the intrinsic explosion compression of the target material to the nuclear super density. Here we have almost complete nuclear transformation of the definite primary chemical element (Cu, for example) to different other stable chemical elements (Mg, Fe, Ta, etc.) [1] [2]. The experiments were made during last ten years (start at 1999) in the Electrodynamics Laboratory of Proton-21 company in Kyiv, Ukraine. This work has been carried out within the commercial project called Luch, which is developed on Adamenko initiative (PI) and aims at the creation of new, efficient and environmentally safe nuclear technologies for neutralizing the radioactivity and synthesizing stable isotopes of chemical elements, including superheavy ones. Experimental installation looks like vacuum diode with the niddle anode made for electric field increase. Furthermore, anode was made from the pure technical copper (99.99%), however, other chemical elements could be used, such as argentums, tantalum, plumbum etc. Adamenko experiments used the following electron beam characteristics for atom compression at the anode surface: Electron “coherent” beam energy: J; Electromagnetic pulse duration: s; Electromagnetic pulse power: W; Residual pressure inside camera: Pa. Compressed atom concentration: 1/m^3; «Lattice constant» for compressed atom: m; Atoms number that deal with the “transmutation process”: . Considering that every atom of the target has about 100 atomic mass (), than the total nucleon number (the difference between proton and neutron masses could be neglected for simplicity) which took part in the transformation process will be: One proton compression requires the definite energy: Thus, the input electron beam energy could compress the following proton number of the target: The relationship between the really compressed protons and the protons compressed by input energy is: So, the “energy deficit” will be about fife order of magnitude (it depends of the target material). It has been observed that at the end of experimental compression procedure the target explodes from inside. The explosion results are exploded “volcano” with a tubular crater, and left its traces, some drops, on the surface of one of the “petals” of the exploded tube which formerly was a monolithic target rod. Numerous studies of the element and isotope composition of the exploded target surface and accumulating screen, conducted using various methods, have shown the presence, in different amounts, of all elements of Mendeleev periodic table among the target ejections. Most of chemical elements found in accumulating screens and remnants of the target either were not found in the materials of which targets and screens were initially made, or they were present in those materials in concentrations and amounts several orders lower than in the resulting ones. In addition, for most of the created elements, their isotopic composition has been significantly different from natural conditions. For example, the target №1754 has at one of its part the following composition of different chemical elements, presented at table 1. Table 1: Chemical element percent composition of the explosion results for an arbitrary anod point. n/n Chemical element % 1 O 3.4 2 Al 1.7 3 Si 13.5 4 Ca 3.4 5 Ti 0.3 6 Mn 0.2 7 Fe 0.2 8 Cu 33.7 9 Ta 26.9 The results of the modeling procedures for the target atom compression based on the classical electrodynamics were presented in the number of Adamenko publications [3] [4] [5]. Classical approach to the Adamenko problem[edit] Classical properties of the Adamenko sphere[edit] In the general case the Adamenko sphere has the following classical properties (for copper target): m - Adamenko sphere radius; m^2 - Adamenko sphere surface area; m^3 - Adamenko sphere volume; kg/m^3 – pure copper mass density [6]; kg – copper mass of the Adamenko sphere; - atomic number of the Adamenko sphere; - copper atomic number; - copper electron (charge) number; - total electron number in the Adamenko sphere; - electron compression factor for an arbitrary atom; - proton-electron mass ratio; - fine structure constant. Thus, we should to place the following number of electrons on the Adamenko sphere, to compress the copper atomic electronic shells to the protonic scale: The productivity coefficient of the Adamenko sphere can be defined as: Thus, we can to find out the minimal electronic scale on the Adamenko sphere from the condition: Considering the following limit for “minimal electron radius”: we could to find out the productivity coefficient: where - proton characteristic length. It is evident that we have here the extremely small electron scale, which could be in the very strong electric fields only. In another words, we need additional energy for the external electron compression on the Adamenko sphere. However, we have no any additional energy to do this process real (to say nothing of the Pauli principle!). Therefore, we need another (quantum) mechanism to consider this process properly. Energy balance problem for the Adamenko experiments[edit] Let us consider how much energy used the melting process. It is known that copper melting temperature is C. The heat of melting is defined by the following equation: where J/kg – the heat of melting density for copper [7]. The heat of evaporation is defined by the following equation: where J/kg - the heat of evaporating density for copper. It is evident that melting and evaporation processes need very small energy with compare to the input energy in the Adamenko experiments (J). However, what about the ionization processes? If we use all input energy to the ionization then we obtain the following electron number: where J – ionization energy for Bohr atom. This number is about the total electron number in the Adamenko sphere However this fact does not confirm that all input energy yields to the copper atom ionization. Note that bigger energy part yields to the quantum pumping of the electromagnetic resonator, formed by the Adamenko sphere. Electric field effect in the applied physics[edit] To the contrary of “field theory” the s.c. concept of electric field effect appeared in the technical sphere and therefore it is properly patented. For example, the idea of MOS-transistor appeared at the end of 20-es of the 20-th century and therefore its priority was patented by Lilienfeld in the USA [8], and by Heil in the Great Britain [9]. The construction of these devices was primitive and trivial: metallic and semiconductor plates divided by dielectric. These devices were controlled by the electric field applied to the gate electrode. William Shockley made number attempt to the practical realization of this idea at the end of 30-ies of the 20-th century [10]. He used germanium plate as semiconductor, mica plate as dielectric and metallic plate as gate. Yes, Shockley obtained the conductivity modulation, however the amplification effect was insignificant. Furthermore, these devices were unstable in time, and therefore they did not obtain practical realization in the industry. However at that time the macroscopic theory of the modulation processes was constructed and the dominant role of the “surface states” was discovered by Bardeen [11]. The practical realization of the field effect became possible by using the silicon as semiconductor and after development of the “passivation procedure” of the silicon surface by the Atalla group [12], та Кангом [13]. Thus, from the early 60-es of the 20-th century the field effect MOS-transistors became the leading active devices in the microelectronics industry (up today: the most of the contemporary microprocessors were made on the MOS-technology!). ). It is worth noting that the two dimensional (2D-) layers of the current curriers at the silicon-silicon oxide interface had the rectangular topology, since the device width should be bigger then device length (better amplifying). So, the “comb channel topology” was developed for compactization of the transistors on the silicon surface. However, the “cylindrical topology «for MOS-transistor was used in the physical experiments for currier mobility measurements. The revolutionary discovery by Klaus von Klitzing of the w:Quantum Hall effect (KHE) [14] on the long channel MOS-transistors at the helium temperatures and strong magnetic fields, broadened the field-effect conception on the quantum phenomena range. Thus, at the helium temperatures and strong transverse magnetic field the 2d- quantum electromagnetic resonators of the Hall type arises. At the end of 80-ies of the 20-th century were discovered the quantum galvanomagnetic effects at the silicon interface on the industrial MOS-transistors at the room temperatures and higher [15] [16] [17]. These effects were connected with the surface area and temperature quantization, however to the contrary to KHE, they were observed at the room temperatures. It is worth noting, that here we have the induced 2D-structure with the minimal symmetry and strong transversal electric field. Field effect in the spherical symmetry devices[edit] It is known, that when we place an electric charge on the metallic empty sphere, then the electric field will be outside metallic sphere. However, there is no any electric field (it will be compensated!) inside empty sphere. Furthermore, closed metallic surfaces are used for screening of any electric fields in technique. Other case is the spherical capacitor! We have here electric field inside capacitor only (closed topology). For the case, when an external plate has radius bigger then an internal ones (), then we shall have an amplification of electric field, even in the case when the internal radius has the nuclear radius! Thus, the practical using of the spherical capacitors is perspective for the atomic electron shells compression, placed in the mesoscopic sphere. In another words an accumulation of the excessive elections on en external sphere is equivalent to the increasing of the nuclear charge, which could be used for the compression of the atomic electronic shells! However, the practical realization of such sphere is a very hard problem, but the field effect properties make a good deal in our case. Actually, at the strong electric field it is possible creating the quantum 2D-layers on the limited (quantized) surface area. In the case of flat structure we shall have the flat surface quantum, but in the case of the spherical structures – we shall have the spherical surface quantum. The typical example is the Bohr atom. Note that, the Bohr radius is due to the Schrödinger equation, where it is the standard normalization factor for the length. Let us consider, as an example, the mesoscopic sphere with one charged ion inside. Further, we place the large negative charge on the sphere: Then the electronic atomic shell collapses to the characteristic proton length. In the real case we shall have the nuclear radius due to “uncompressive” properties of the nucleons: Here we considered the nuclear density approximately the same as proton ones: It is evident that compression we should have excessive energy dissipation, which is equal to the electron transportation work from to : We have here the two possibilities for the next events. In the first case, when the energy is taken from the quantum system, we shall have the stable nuclear structure (so called – “neutron matter”) after external electrons are moved out. However, in the second case, when the excessive energy is conservated inside quantum system, we shall have the restoration process to the initial state, when the external electrons are moved from the sphere. Note that, the “restored” element will not be obviously the same as been at the experiment beginning. Quantum approach to the Adamenko problem[edit] In the strong electric field the electronic quantum vortex could be produced [16]. In the general case this effect produces the surface quantization, which is placed perpendicular to the electric field: where is the electron mass. This surface quantum in the case of spherical symmetry by automatically induces the quantum electromagnetic resonator (QER), which has the following parameters: is the capacitance of the QER; is the inductance of the QER; :Om is the characteristic impedance of the QER; : is the resonance angular frequency; is the QER period of oscillations; :Jж is the one photon energy for QER; where m is the electron Compton wave-length, H/m is the magnetic, and F/m is the electric vacuum constants. The external action (macroscopic one) which is applied to the Adamenko system could be presented as: J s, where с is the external pulse duration, and J is the external pulse energy. The microscopic action could be presented as: Note that it is the action quantum here. Thus, the maximal photon number, which the Adamenko sphere produces, will be: Note that this number exceeds the required electron number for the Adamenko sphere which is needed to compress the atoms to the nuclear density inside it. It is worth noting that, electromagnetic oscillations of the QER induce the electric charge: where C is the elementary induced electric charge of the QER. Thus, in the range of QER conception we solve the Pauli principle problem for the external electrons by automatically. Furthermore, the induced electron number is sufficient to compress almost all atoms (ions) in the induced Adamenko sphere. See also[edit] 1. С.В.Адаменко. Концепция искусственно инициируемого коллапса вещества и основные результаты первого этапа ее экспериментальной реализации // Препринт 2004, Киев, Академпериодика, с. 36. Pdf) 2. Controlled Nucleosynthesis. Breakthroughs in Experiment and Theory, Series: Fundamental Theories of Physics , Vol. 156, Adamenko, Stanislav; Selleri, Franco; Merwe, Alwyn van der (Eds.), 780 p. (Springer, 2007). Pdf 3. Adamenko S.V. et al. Effect of auto-focusing of the electron beam in the relativistic vacuum diode. Proceedings of the 1999 Particle Accelerator Conference, New york, 1999. 4. Vysotskii V.I., Adamenko S.V. et al. Creating and using of superdense micrj-beams of relativistic electrons. Nuclear Instruments and Methods in Physics Research. A455 (2000) pp.123-127. 5. Адаменко С.В., Пащенко А.В., Шаповал И.Н. и Новиков В.Е. Процессы с обострением и дробление масштабов в плазменно-полевых структурах. Вопросы атомной науки и техники. 2003, №4,с.171-176. 6. Свойства элементов: Справ. изд./Под ред. Дрица М.Е. М.:Металлургия, 1985, 672с. 7. Кузьмичев В.Е. Законы и формулы физики / Отв.ред. В.К.Тартаковский.- Kиев:Наук.думка,1989.-864с. 8. Lilienfeld J.E. Method and Apparatus for Controlling Electric Currents. US Patent #1745175, 1930, january. 9. Heil O. Impruvements in or Relating to Electric Amplifiers and other Control Arrangements. UK Patent #439457, 1935, December. 10. Shokley W., Pearson G.L. Modulation of Conductance of Thin Films of Semiconductors by Surface Charges. Phys. Rev., 1948, 74, July, p.232-233. 11. Bardeen J., Phys. Rev., 71, 1947, p.717. 12. Atalla M.M., Tannenbaum E., Scheiber E.J. Stabilization of Silicon Surfaces by Thermally Grown Oxides. Bell Syst. Tech. J., 1959, 38, May, p.749-783. 13. Kahng D., Atalla M.M. Silicon— Silicon Dioxide Field Induced Devices. Solid- State Device Research Conference, Pittsburgh, Pa., 1960, June. 14. von Klitzing, K. (1980). "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance". Physical Review Letters 45 (6): 494–497. doi:10.1103/PhysRevLett.45.494. 15. Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu 16. 16.0 16.1 Yakymakha O.L., Kalnibolotskij Y.M., Solid- State Electronics, vol.37, No.10,1994.,pp.1739-1751 Pdf Cite error: Invalid <ref> tag; name "Yakymakha1" defined multiple times with different content 17. Yakymakha O.L., Kalnibolotskij Y.M., Solid- State Electronics, vol.38, No.3,1995.,pp.661-671 pdf External links[edit]
d935fef904ac8292	Skip to main content Chemistry LibreTexts 4.11: Many-Electron Atoms & the Periodic Table • Page ID • Quantum mechanics can account for the periodic structure of the elements, by any measure a major conceptual accomplishment for any theory. Although accurate computations become increasingly more challenging as the number of electrons increases, the general patterns of atomic behavior can be predicted with remarkable accuracy. Figure $\PageIndex{1}$ shows a schematic representation of a helium atom with two electrons whose coordinates are given by the vectors $r_1$ and $r_2$. The electrons are separated by a distance $r_{12} = \|r_1-r_2\|$. The origin of the coordinate system is fixed at the nucleus. As with the hydrogen atom, the nuclei for multi-electron atoms are so much heavier than an electron that the nucleus is assumed to be the center of mass. Fixing the origin of the coordinate system at the nucleus allows us to exclude translational motion of the center of mass from our quantum mechanical treatment. Figure $\PageIndex{1}$: a) The nucleus (++) and electrons (e-) of the helium atom. b) Equivalent reduced particles with the center of mass (approximately located at the nucleus) at the origin of the coordinate system. Note that $μ_1$ and $μ_2 ≈ m_e$. The Schrödinger equation operator for the hydrogen atom serves as a reference point for writing the Schrödinger equation for atoms with more than one electron. Start with the same general form we used for the hydrogen atom Hamiltonian \[ (\hat {T} + \hat {V}) \psi = E \psi \label {9-1}\] Include a kinetic energy term for each electron and a potential energy term for the attraction of each negatively charged electron for the positively charged nucleus and a potential energy term for the mutual repulsion of each pair of negatively charged electrons. The He atom Schrödinger equation is \[ \left( -\dfrac {\hbar ^2}{2m_e} (\nabla ^2_1 + \nabla ^2_2) + V (r_1) + V (r_2) + V (r_{12}) \right) \psi = E \psi \label {9-2}\] \[ V(r_1) = -\dfrac {2e^2}{4 \pi \epsilon _0 r_1} \label {9-3}\] \[ V(r_2) = -\dfrac {2e^2}{4 \pi \epsilon _0 r_2} \label {9-4}\] \[ V(r_{12}) = -\dfrac {e^2}{4 \pi \epsilon _0 r_{12}} \label {9-5}\] Equation $\ref{9-2}$ can be extended to any atom or ion by including terms for the additional electrons and replacing the He nuclear charge +2 with a general charge Z; e.g. \[V(r_1) = -\dfrac {Ze^2}{4 \pi \epsilon _0 r_1} \label {9-6}\] Equation $\ref{9-2}$ then becomes \[ \left( -\dfrac {\hbar ^2}{2m_e} \sum _i \nabla ^2_i + \sum _i V (r_i) + \sum _{i \ne j} V (r_{ij}) \right) \psi = E \psi \label {9-7}\] Given what we have learned from the previous quantum mechanical systems we’ve studied, we predict that exact solutions to the multi-electron Schrödinger equation would consist of a family of multi-electron wavefunctions, each with an associated energy eigenvalue. These wavefunctions and energies would describe the ground and excited states of the multi-electron atom, just as the hydrogen wavefunctions and their associated energies describe the ground and excited states of the hydrogen atom. We would predict quantum numbers to be involved, as well. The fact that electrons interact through their electron-electron repulsion means that an exact wavefunction for a multi-electron system would be a single function that depends simultaneously upon the coordinates of all the electrons; i.e., a multi-electron wavefunction: \[\Psi (r_1, r_2, \cdots r_i) \label{8.3.4}\] Unfortunately, the electron-electron repulsion terms make it impossible to find an exact solution to the Schrödinger equation for many-electron atoms. The most basic ansatz to the exact solutions involve writing a multi-electron wavefunction as a simple product of single-electron wavefunctions \[\psi (r_1, r_2, \cdots , r_i) = \varphi _1 (r_1) \varphi _2 (r_2) \cdots \varphi _i(r_i) \label{8.3.5}\] Obtaining the energy of the atom in the state described by that wavefunction as the sum of the energies of the one-electron components. By writing the multi-electron wavefunction as a product of single-electron functions in Equation $\ref{8.3.5}$, we conceptually transform a multi-electron atom into a collection of individual electrons located in individual orbitals whose spatial characteristics and energies can be separately identified. For atoms these single-electron wavefunctions are called atomic orbitals. For molecules, as we will see in the next chapter, they are called molecular orbitals. While a great deal can be learned from such an analysis, it is important to keep in mind that such a discrete, compartmentalized picture of the electrons is an approximation, albeit a powerful one. Electron Configurations The specific arrangement of electrons in orbitals of an atom determines many of the chemical properties of that atom and is formulated via the Aufbau principle, which means "building-up" in German. Aufbau principles determine the order in which atomic orbitals are filled as the atomic number is increased. For the hydrogen atom, the order of increasing orbital energy is given by 1s < 2s = 2p < 3s = 3p = 3d, etc. The dependence of energy on n alone leads to extensive degeneracy, which is however removed for orbitals in many-electron atoms. Thus 2s lies below 2p, as already observed in helium. Similarly, 3s, 3p and 3d increase energy in that order, and so on. The 4s is lowered sufficiently that it becomes comparable to 3d. The general ordering of atomic orbitals is summarized in the following scheme: \[ 1s < 2s < 2p < 3s < 3p < 4s \sim 3d < 4p < 5s \sim 4d\\< 5p < 6s \sim 5d \sim 4f < 6p < 7s \sim 6d \sim 5f \label{4}\] and illustrated in Figure $\PageIndex{2}$. This provides enough orbitals to fill the ground states of all the atoms in the periodic table. For orbitals designated as comparable in energy, e.g., 4s $\sim$ 3d, the actual order depends which other orbitals are occupied. The energy of atomic orbitals increases as the principal quantum number, $n$, increases. In any atom with two or more electrons, the repulsion between the electrons makes energies of subshells with different values of $l$ differ so that the energy of the orbitals increases within a shell in the order s < p < d < f. Figure $\PageIndex{2}$ depicts how these two trends in increasing energy relate. The 1s orbital at the bottom of the diagram is the orbital with electrons of lowest energy. The energy increases as we move up to the 2s and then 2p, 3s, and 3p orbitals, showing that the increasing n value has more influence on energy than the increasing l value for small atoms. However, this pattern does not hold for larger atoms. The 3d orbital is higher in energy than the 4s orbital. Such overlaps continue to occur frequently as we move up the chart. Figure $\PageIndex{2}$: Generalized energy-level diagram for atomic orbitals in an atom with two or more electrons (not to scale). Electrons in successive atoms on the periodic table tend to fill low-energy orbitals first. The arrangement of electrons in the orbitals of an atom is called the electron configuration of the atom. We describe an electron configuration with a symbol that contains three pieces of information ( Figure $\PageIndex{3}$): 1. The number of the principal quantum shell, n, For example, the notation 2p4 (read "two–p–four") indicates four electrons in a p subshell (l = 1) with a principal quantum number (n) of 2. The notation 3d8 (read "three–d–eight") indicates eight electrons in the d subshell (i.e., l = 2) of the principal shell for which n = 3. Figure $\PageIndex{3}$: The diagram of an electron configuration specifies the subshell (n and l value, with letter symbol) and superscript number of electrons. To determine the electron configuration for any particular atom, we can “build” the structures in the order of atomic numbers. Beginning with hydrogen, and continuing across the periods of the periodic table, we add one proton at a time to the nucleus and one electron to the proper subshell until we have described the electron configurations of all the elements. This procedure is called the Aufbau principle, from the German word Aufbau (“to build up”). Each added electron occupies the subshell of lowest energy available (in the order shown in Figure $\PageIndex{4}$), subject to the limitations imposed by the allowed quantum numbers according to the Pauli exclusion principle. Electrons enter higher-energy subshells only after lower-energy subshells have been filled to capacity. Figure $\PageIndex{3}$ illustrates the traditional way to remember the filling order for atomic orbitals. Figure $\PageIndex{4}$: The arrow leads through each subshell in the appropriate filling order for electron configurations. This chart is straightforward to construct. Simply make a column for all the s orbitals with each n shell on a separate row. Repeat for p, d, and f. Be sure to only include orbitals allowed by the quantum numbers (no 1p or 2d, and so forth). Finally, draw diagonal lines from top to bottom as shown. We will now construct the ground-state electron configuration and orbital diagram for a selection of atoms in the first and second periods of the periodic table. Orbital diagrams are pictorial representations of the electron configuration, showing the individual orbitals and the pairing arrangement of electrons. We start with a single hydrogen atom (atomic number 1), which consists of one proton and one electron. Referring to either Figure $\PageIndex{4}$, we would expect to find the electron in the 1s orbital. By convention, the $m_s=+\dfrac{1}{2}$ value is usually filled first. The electron configuration and the orbital diagram are: Following hydrogen is the noble gas helium, which has an atomic number of 2. The helium atom contains two protons and two electrons. The first electron has the same four quantum numbers as the hydrogen atom electron (n = 1, l = 0, ml = 0, $m_s=+\dfrac{1}{2}$). The second electron also goes into the 1s orbital and fills that orbital. The second electron has the same n, l, and ml quantum numbers, but must have the opposite spin quantum number, $m_s=−\dfrac{1}{2}$. This is in accord with the Pauli exclusion principle: No two electrons in the same atom can have the same set of four quantum numbers. For orbital diagrams, this means two arrows go in each box (representing two electrons in each orbital) and the arrows must point in opposite directions (representing paired spins). The electron configuration and orbital diagram of helium are: In this figure, the element symbol H e is followed by the electron configuration, “1 s superscript 2.” An orbital diagram is provided that consists of a single square. The square is labeled below as “1 s.” It contains a pair of half arrows: one pointing up and the other down. The n = 1 shell is completely filled in a helium atom. The next atom is the alkali metal lithium with an atomic number of 3. The first two electrons in lithium fill the 1s orbital and have the same sets of four quantum numbers as the two electrons in helium. The remaining electron must occupy the orbital of next lowest energy, the 2s orbital (Figure $\PageIndex{4}$ ). Thus, the electron configuration and orbital diagram of lithium are: An atom of the alkaline earth metal beryllium, with an atomic number of 4, contains four protons in the nucleus and four electrons surrounding the nucleus. The fourth electron fills the remaining space in the 2s orbital. In this figure, the element symbol B e is followed by the electron configuration, “1 s superscript 2 2 s superscript 2.” An orbital diagram is provided that consists of two individual squares. The first square is labeled below as, “1 s.” The second square is similarly labeled, “2 s.” Both squares contain a pair of half arrows: one pointing up and the other down. An atom of boron (atomic number 5) contains five electrons. The n = 1 shell is filled with two electrons and three electrons will occupy the n = 2 shell. Because any s subshell can contain only two electrons, the fifth electron must occupy the next energy level, which will be a 2p orbital. There are three degenerate 2p orbitals (ml = −1, 0, +1) and the electron can occupy any one of these p orbitals. When drawing orbital diagrams, we include empty boxes to depict any empty orbitals in the same subshell that we are filling. Carbon (atomic number 6) has six electrons. Four of them fill the 1s and 2s orbitals. The remaining two electrons occupy the 2p subshell. We now have a choice of filling one of the 2p orbitals and pairing the electrons or of leaving the electrons unpaired in two different, but degenerate, p orbitals. The orbitals are filled as described by Hund’s rule: the lowest-energy configuration for an atom with electrons within a set of degenerate orbitals is that having the maximum number of unpaired electrons. Thus, the two electrons in the carbon 2p orbitals have identical n, l, and ms quantum numbers and differ in their ml quantum number (in accord with the Pauli exclusion principle). The electron configuration and orbital diagram for carbon are: In this figure, the element symbol C is followed by the electron configuration, “1 s superscript 2 2 s superscript 2 2 p superscript 2.” The orbital diagram consists of two individual squares followed by 3 connected squares in a single row. The first blue square is labeled below as, “1 s.” The second is similarly labeled, “2 s.” The connected squares are labeled below as, “2 p.” All squares not connected to each other contain a pair of half arrows: one pointing up and the other down. The first two squares in the group of 3 each contain a single upward pointing arrow. Nitrogen (atomic number 7) fills the 1s and 2s subshells and has one electron in each of the three 2p orbitals, in accordance with Hund’s rule. These three electrons have unpaired spins. Oxygen (atomic number 8) has a pair of electrons in any one of the 2p orbitals (the electrons have opposite spins) and a single electron in each of the other two. Fluorine (atomic number 9) has only one 2p orbital containing an unpaired electron. All of the electrons in the noble gas neon (atomic number 10) are paired, and all of the orbitals in the n = 1 and the n = 2 shells are filled. The electron configurations and orbital diagrams of these four elements are: The alkali metal sodium (atomic number 11) has one more electron than the neon atom. This electron must go into the lowest-energy subshell available, the 3s orbital, giving a 1s22s22p63s1 configuration. The electrons occupying the outermost shell orbital(s) (highest value of n) are called valence electrons, and those occupying the inner shell orbitals are called core electrons (Figure $\PageIndex{5}$). Since the core electron shells correspond to noble gas electron configurations, we can abbreviate electron configurations by writing the noble gas that matches the core electron configuration, along with the valence electrons in a condensed format. For our sodium example, the symbol [Ne] represents core electrons, (1s22s22p6) and our abbreviated or condensed configuration is [Ne]3s1. Figure $\PageIndex{5}$: A core-abbreviated electron configuration (right) replaces the core electrons with the noble gas symbol whose configuration matches the core electron configuration of the other element. Similarly, the abbreviated configuration of lithium can be represented as [He]2s1, where [He] represents the configuration of the helium atom, which is identical to that of the filled inner shell of lithium. Writing the configurations in this way emphasizes the similarity of the configurations of lithium and sodium. Both atoms, which are in the alkali metal family, have only one electron in a valence s subshell outside a filled set of inner shells. \[\ce{Li:[He]}\,2s^1\\ \ce{Na:[Ne]}\,3s^1\] The alkaline earth metal magnesium (atomic number 12), with its 12 electrons in a [Ne]3s2 configuration, is analogous to its family member beryllium, [He]2s2. Both atoms have a filled s subshell outside their filled inner shells. Aluminum (atomic number 13), with 13 electrons and the electron configuration [Ne]3s23p1, is analogous to its family member boron, [He]2s22p1. The electron configurations of silicon (14 electrons), phosphorus (15 electrons), sulfur (16 electrons), chlorine (17 electrons), and argon (18 electrons) are analogous in the electron configurations of their outer shells to their corresponding family members carbon, nitrogen, oxygen, fluorine, and neon, respectively, except that the principal quantum number of the outer shell of the heavier elements has increased by one to n = 3. Figure Beginning with the transition metal scandium (atomic number 21), additional electrons are added successively to the 3d subshell. This subshell is filled to its capacity with 10 electrons (remember that for l = 2 [d orbitals], there are 2l + 1 = 5 values of ml, meaning that there are five d orbitals that have a combined capacity of 10 electrons). The 4p subshell fills next. Note that for three series of elements, scandium (Sc) through copper (Cu), yttrium (Y) through silver (Ag), and lutetium (Lu) through gold (Au), a total of 10 d electrons are successively added to the (n – 1) shell next to the n shell to bring that (n – 1) shell from 8 to 18 electrons. For two series, lanthanum (La) through lutetium (Lu) and actinium (Ac) through lawrencium (Lr), 14 f electrons (l = 3, 2l + 1 = 7 ml values; thus, seven orbitals with a combined capacity of 14 electrons) are successively added to the (n – 2) shell to bring that shell from 18 electrons to a total of 32 electrons. Example $\PageIndex{1}$ Quantum Numbers and Electron Configurations What is the electron configuration and orbital diagram for a phosphorus atom? What are the four quantum numbers for the last electron added? The atomic number of phosphorus is 15. Thus, a phosphorus atom contains 15 electrons. The order of filling of the energy levels is 1s, 2s, 2p, 3s, 3p, 4s, . . . The 15 electrons of the phosphorus atom will fill up to the 3p orbital, which will contain three electrons: The last electron added is a 3p electron. Therefore, n = 3 and, for a p-type orbital, l = 1. The ml value could be –1, 0, or +1. The three p orbitals are degenerate, so any of these ml values is correct. For unpaired electrons, convention assigns the value of $+\dfrac{1}{2}$ for the spin quantum number; thus, $m_s=+\dfrac{1}{2}$. Exercise $\PageIndex{1}$ Identify the atoms from the electron configurations given: 1. [Ar]4s23d5 2. [Kr]5s24d105p6 (a) Mn (b) Xe Effective Charge, Shielding and Penetration For an atom or an ion with only a single electron, we can calculate the potential energy by considering only the electrostatic attraction between the positively charged nucleus and the negatively charged electron. When more than one electron is present, however, the total energy of the atom or the ion depends not only on attractive electron-nucleus interactions but also on repulsive electron-electron interactions. When there are two electrons, the repulsive interactions depend on the positions of both electrons at a given instant, but because we cannot specify the exact positions of the electrons, it is impossible to exactly calculate the repulsive interactions. Consequently, we must use approximate methods to deal with the effect of electron-electron repulsions on orbital energies. If an electron is far from the nucleus (i.e., if the distance r between the nucleus and the electron is large), then at any given moment, most of the other electrons will be between that electron and the nucleus. Hence the electrons will cancel a portion of the positive charge of the nucleus and thereby decrease the attractive interaction between it and the electron farther away. As a result, the electron farther away experiences an effective nuclear charge (Zeff) that is less than the actual nuclear charge Z (Figure $\PageIndex{6}$). This effect is called electron shielding. As the distance between an electron and the nucleus approaches infinity, Zeff approaches a value of 1 because all the other (Z − 1) electrons in the neutral atom are, on the average, between it and the nucleus. If, on the other hand, an electron is very close to the nucleus, then at any given moment most of the other electrons are farther from the nucleus and do not shield the nuclear charge. At r ≈ 0, the positive charge experienced by an electron is approximately the full nuclear charge, or ZeffZ. At intermediate values of r, the effective nuclear charge is somewhere between 1 and Z: 1 ≤ ZeffZ. Thus the actual Zeff experienced by an electron in a given orbital depends not only on the spatial distribution of the electron in that orbital but also on the distribution of all the other electrons present. This leads to large differences in Zeff for different elements, as shown in Figure 2.5.1 for the elements of the first three rows of the periodic table. Notice that only for hydrogen does Zeff = Z, and only for helium are Zeff and Z comparable in magnitude. Figure $\PageIndex{6}$: Relationship between the Effective Nuclear Charge Zeff and the Atomic Number Z for the Outer Electrons of the Elements of the First Three Rows of the Periodic Table. Except for hydrogen, Zeff is always less than Z, and Zeff increases from left to right as you go across a row. Because of the effects of shielding and the different radial distributions of orbitals with the same value of n but different values of l, the different subshells are not degenerate in a multielectron atom. For a given value of n, the ns orbital is always lower in energy than the np orbitals, which are lower in energy than the nd orbitals, and so forth. As a result, some subshells with higher principal quantum numbers are actually lower in energy than subshells with a lower value of n; for example, the 4s orbital is lower in energy than the 3d orbitals for most atoms. Except for the single electron containing hydrogen atom, in every other element $Z_{eff}$ is always less than $Z$. Figure $\PageIndex{7}$: Orbital Penetration. A comparison of the radial probability distribution of the 2s and 2p orbitals for various states of the hydrogen atom shows that the 2s orbital penetrates inside the 1s orbital more than the 2p orbital does. Consequently, when an electron is in the small inner lobe of the 2s orbital, it experiences a relatively large value of Zeff, which causes the energy of the 2s orbital to be lower than the energy of the 2p orbital. Ioonization Energy Ionization energy is the energy required to remove an electron from a neutral atom in its gaseous phase. Conceptually, ionization energy is the opposite of electronegativity. The lower this energy is, the more readily the atom becomes a cation. Therefore, the higher this energy is, the more unlikely it is the atom becomes a cation. Generally, elements on the right side of the periodic table have a higher ionization energy because their valence shell is nearly filled. Elements on the left side of the periodic table have low ionization energies because of their willingness to lose electrons and become cations. Thus, ionization energy increases from left to right on the periodic table. Another factor that affects ionization energy is electron shielding. Electron shielding describes the ability of an atom's inner electrons to shield its positively-charged nucleus from its valence electrons. When moving to the right of a period, the number of electrons increases and the strength of shielding increases. As a result, it is easier for valence shell electrons to ionize, and thus the ionization energy decreases down a group. Electron shielding is also known as screening. • The ionization energy of the elements within a period generally increases from left to right. This is due to valence shell stability. • The ionization energy of the elements within a group generally decreases from top to bottom. This is due to electron shielding. • The noble gases possess very high ionization energies because of their full valence shells as indicated in the graph. Note that helium has the highest ionization energy of all the elements. Some elements have several ionization energies; these varying energies are referred to as the first ionization energy, the second ionization energy, third ionization energy, etc. The first ionization energy is the energy requiredto remove the outermost, or highest, energy electron, the second ionization energy is the energy required to remove any subsequent high-energy electron from a gaseous cation, etc. Below are the chemical equations describing the first and second ionization energies: First Ionization Energy: \[ X_{(g)} \rightarrow X^+_{(g)} + e^- \] Second Ionization Energy: \[ X^+_{(g)} \rightarrow X^{2+}_{(g)} + e^- \] Generally, any subsequent ionization energies (2nd, 3rd, etc.) follow the same periodic trend as the first ionization energy. Ionization energies decrease as atomic radii increase. This observation is affected by $n$ (the principal quantum number) and $Z_{eff}$ (based on the atomic number and shows how many protons are seen in the atom) on the ionization energy (I). The relationship is given by the following equation: \[ I = \dfrac{R_H Z^2_{eff}}{n^2} \] • Across a period, $Z_{eff}$ increases and n (principal quantum number) remains the same, so the ionization energy increases. • Down a group, $n$ increases and $Z_{eff}$ increases slightly; the ionization energy decreases. 1st Ionization Energies Figure $\PageIndex{8}$: Periodic trends in ionization energy. The periodic structure of the elements is evident for many physical and chemical properties, including chemical valence, atomic radius, electronegativity, melting point, density, and hardness. The classic prototype for periodic behavior is the variation of the first ionization energy with atomic number, which is plotted in Figure $\PageIndex{8}$. Electron Affinity The electron affinity (EA) of an element E is defined as minus the internal energy change associated with the gain of an electron by a gaseous atom, at 0 K : \[E_{(g)} + e^- → E^-_{(g)}\] Unlike ionization energies, which are always positive for a neutral atom because energy is required to remove an electron, electron affinities can be positive (energy is released when an electron is added), negative (energy must be added to the system to produce an anion), or zero (the process is energetically neutral). Electron Affinities Figure $\PageIndex{9}$: Periodic trends in electron affinities. The periodic trends of electron affinity (Figure $\PageIndex{9}$) shows that chlorine has the most positive electron affinity of any element, which means that more energy is released when an electron is added to a gaseous chlorine atom than to an atom of any other element, EA= 348.6 kJmol-1 and the Group 17 elements have the largest values overall. The addition of a second electron to an element is expected to be much less favored since there will be repulsion between the negatively charged electron and the overall negatively charged anion. For example, for O the values are: \[O_{(g)} + e^ \rightarrow O^-_{(g)} \;\;\;\; EA = +141\; kJmol^{-1}\] \[O^-_{(g)} + e^- → O^{2-}){(g)} \;\;\;\; EA = -798\; kJmol^{-1}\]
1e25e6565e54853d	An extremely brief introduction to computational quantum chemistry In this section, we provide a very brief background for the computational tools to be used in this module, which are based on quantum chemistry. Because quantum theory is not the primary concern of this exercise, important themes in quantum chemistry are addressed in a broad manner. Quantum theory holds that matter has both particle- and wavelike properties. For large particles, the wavelike properties of matter are nearly impossible to detect. However, for small particles—such as electrons—the particle-wave duality must be addressed when writing equations, including energy balance equations. The quantum mechanical energy balance (also known as the Schrödinger equation) for an electron can be written as follows: In this equation, E is the total energy of the electron, psi is the wavefunction for the electron, and H is the Hamiltonian operator that describes the kinetic and potential energy of the electron. In other words, the Schrödinger equation is a simple statement that the total energy is equal to the kinetic plus potential energies for a wavelike particle. The Hamiltonian operator for a single electron can be written as: Where V (x,y,z) = (3) Z is the atomic number, e is the charge associated with an electron, and x,y, and z are spatial components. And is Planck’s constant divided by two Pi, m is the mass of an electron, is the Laplacian operator, and V(x,y,z) is the potential energy term, which depends on the electrostatic interactions of the electron with nuclei (attractive interactions) and other electrons (repulsive interactions) in the system. As we will see from the examples in this module, by finding the total energy of molecular systems by solving Schrödinger equations, we can compute thermodynamic and kinetic quantities that are important in chemical reaction engineering (CRE). We start with a solution for the simplest possible molecule – the hydrogen atom, with a single nucleus and single electron – and then show how the results from this solution can be used to address more complex (and relevant) chemical systems. The Hydrogen-Like Atom The Schrödinger equation can only be solved analytically for a “hydrogen-like” atom, i.e., an atom with one electron and one nucleus. The Schrödinger equation is a special type of equation known as an eigenvalue equation, which some of you may have encountered in higher-level mathematics courses. Eigenvalue problems have a number of special characteristics. One of these is that they have an infinite number of solutions for both the eigenvalues () and the associated eigenfunctions(). The value n (an integer) is known in quantum mechanics as the principle quantum number. The fact that there are an infinite number of solutions for the energy of our molecule may seem confusing at first, but we will soon see what these solutions for different values of n represent. The Schrödinger equation—like many other partial differential equations—can be solved using a solution method known as separation of variables subject to the boundary condition that the energy of the system is zero when the electron and nucleus are infinitely far away from each other. We will not go through the details of the solution method here; that method is described in a number of references (click here for references). Instead, we will discuss the form of the solutions to the Schrödinger equation for the hydrogen-like atom. The lowest energy solution for n=1 describes the state in which the system is most probable to exist; this is also known as the ground state. For n=2 (the next-lowest energy solution) there are four solutions that have equivalent energies; for n=3 (the third-lowest energy solution), the number of energetically equivalent solutions rises to 9. Although the physical meaning of the total energies for the system are relatively easy to understand, the meaning of the wavefunction is not. To gain insights into what the wavefunctions actually are, we will consider the quantity: Which is a Cartesian coordinate triple integral over all of the space that the electron could occupy. It can be shown that this quantity describes the probability that the electron will be located in the volume of space between the endpoints labeled i (initial) and f (final). To make this more clear, one can plot the probability distribution as a function of position for each of our solutions. We see that our eigenfunctions psi actually describe the electronic orbitals of the hydrogen atoms. We can recognize that the lowest energy solution (for n=1) corresponds to the 1s orbital:which is the orbital that we know should be occupied by an electron in the ground state for an H atom. This is the region in space in which the electron is most probable to be present. For n=2, there are four energetically degenerate wavefunctions that represent the 2s and 2px, 2py, and 2pz orbitals; these are orbitals which are less energetically favorable for occupation by the electron, as we know from basic chemistry. Furthermore, the nine energetically degenerate 3s, 3p, and 3d orbitals are still higherin energy. We could continue going to higher energy solutions describing the 4s, p, d, and f orbitals, etc., but this information is usually not of practical interest to us. That’s because we’re interested in the ground state of the molecule, the one that preferentially exists in nature (and in our chemical application). So we’re interested in the lowest-energy solution to the Schrödinger equation, the one in which the wavefunction describes a hydrogen 1s orbital. In the exercises in this module, we will show how to use this ground state energy to evaluate thermodynamic properties. Larger atoms and molecules There is no analytical solution to the Schrödinger equation for systems containing multiple nuclei and electrons. This difficulty is caused by the fact that repulsive interactions with multiple electrons greatly complicate the solution to the Schrödinger equation. Therefore, we must use an approximate, numerical solution. The best way to do this is to assume that the wavefunctions can be based on the same basic shapes identified for the H atom; that is, that they should describe orbitals that look like 1s, 2s, 2p, etc., and (in the case of chemical bonds formed between atoms) linear combinations of these orbitals. We will therefore construct our numerical solution for the eigenvalues and eigenfunctions using a basis set of orbital shapes (i.e., s, p, d, and f wave functions) to describe the wavefunction for each electron: where describes the different functions that approximate electronic orbitals, and are fitting coefficients that are varied in the course of our numerical solution of the Schrödinger equation. In other words, our solution method will involve varying the coefficients until a minimum energy (ground state) solution is found. The energy minimum that is found is not always the absolute global minimum for the molecule, because an molecule can have many local energy minima. A principle known as the variational principle dictates that the lowest energy solution will be closest to the true energy of the molecular system. The larger the basis set size—i.e., the more fitting coefficients we have at our disposal for the solution—the closer to the true ground state energy our solution will be. However, larger basis sets also naturally require longer computational times, because there will be more terms to include in the calculation. An infinite basis set – one that includes innumerable different shapes for the fitting functions – could in principle be used to compute an exact molecular energy. But this calculation would take an infinite amount of time! We typically use the smallest basis set size possible that generates results in reasonably good agreement with experiments. There are a number of ways to use these basis sets in solving the Schrödinger equation; we will deal only with some of the more commonly used methods here. One of the earliest methods for treating many-electron systems is based on the Hartree-Fock (HF) approximation. The discovery of the HF method was central to the field of quantum chemistry, and it forms the heart of many of the more accurate computational methods today. In the HF method, the energy of each electron is solved for individually, using a modified version of the Schrödinger equation called the Hartree-Fock equation: This equation is solved for each electron i, where epsilon () is the energy of the electron analogous to E in the Schrödinger equation, chi () its electronic orbital analogous to , and ( f ) is the so-called Fock operator analogous to H. The Fock operator has the form: Where is the radial distance be tween the ith electron and the Ath nucleus, is the atomic number of the Ath nucleus, and is the average potential felt by the ith electron from the presence of the other electrons present in the system. Solving the complete system of HF equations (as discussed below), one can determine energies for each electron, and sum these together with nuclear nuclear repulsion energies to obtain the total energy of the molecule. The key to the HF approximation is the use of the term , which describes the average electronic charge “felt” by the ith electron; this average is used to compute the effect of electrostatic interactions between electrons on the shapes and energies of the individual orbitals. Taking the average effect of the electrostatic interactions is far less cumbersome to calculate as opposed to calculating the electrostatic effect of the system (the molecule) on each individual electron. Solution of the HF equations is accomplished by an algorithm known as the self-consistent field (SCF) procedure. The computation basically proceeds as follows: 1. An initial guess for the values of the basis set coefficients,, is made. 2. The values of are used to construct the electronic orbitals through a linear combination of single electron orbitals. 3. Utilizing the electronic orbitals constructed in step 2, which tells you on average the location of the electrons, allows one to determine the average electrostatic interaction () between the electron in question and the remaining electrons in the system. This electrostatic interaction is calcuated for each electron and then averaged to more accurately represent the electron-electron interactions in the molecule. 4. Once is known, the Hartree-Fock equations can be solved to determine new electronic orbitals and their associated energies. 5. The electronic energies can be summed to calculated the total energy of the system. If the energy is sufficiently converged with respect to the previous SCF cycle, we’re done. If not, return to step 3 and continue to cycle until the energy value converges to a stable (minimum) value. In many of the programs that are used for these computations, you will be able to watch the progress of the SCF procedure as the calculation progresses. Although the HF method has produced many valuable results, it turns out that significant errors can be induced by treating repulsive forces between electrons in an average sense, rather than treating them individually. We refer to the difference between the HF energy and the true energy of the system (molecule),, as the correlation energy. There are many methods which attempt to improve the accuracy of computed energies using different methods of accounting for the electron correlation. We list several of these below, along with a very brief description of the basis for each method. 1. Density Functional Theory (DFT). In DFT, the correlation energy is treated as a function of the electronic density, which itself is a function of position. In other words, the correlation can be calculated for a given electronic charge distribution in the molecular system. A number of functionals have been devised for the correlation energy; these functionals vary in their effectiveness for given applications. Treating the correlation energy as a functional of the density greatly improves computational accuracy without a significant cost in terms of computer time. In fact, because other terms in the Schrodinger equation can be treated as functionals of the density, DFT calculations sometimes run faster than HF computations. For this reason, DFT is widely used for studying complex systems in heterogeneous catalysis, electronic thin films deposition, etc. 2. Perturbation theories. These theories account for the correlation energy in terms of perturbations to the original eigenfunction and eigenvalue solutions. These perturbations are treated as series expansions of the original eigenfunctions. Two of the most popular perturbation methods are referred to as Moller-Plessett calculations, which are commonly based on 2nd-order or 4th-order expansions and referred to as MP2 and MP4, respectively. MP4 is more accurate, but requires longer computation times. 3. Coupled cluster theories. Coupled cluster theories are a method of electronic structure thoery for predicting molecular properties. The basic ideas of CC theories includes cluster expansion, cluster operators and size extensivity. The theory can be furthur categorized into aa formal theory and a pratical theory. The CC equations are quadratic in the ampitudes and must be solved iteratively. CC is correct to the infinite order for single and double excitations and is correct to the 4th order for quadruple excitations. 4. Configuration interaction. Configuration interaction (CI) methods hypothetically can calculate exactly the correlation energy in the limit of an infinite basis set (i.e., an infinite number of fitting parameters). Meaning that the calculation of the total system energy is exact. The price that is paid for this exactness, of course, is an extremely long computation time – full CI is not yet practical for most systems of chemical engineering interest. In CI, the exact wave function is represented as a linear combination of the HF ground state wave function plus excited state wave functions. Similar to perturbation methods, the more excited states that are included in the calculation, the more accurate (and lengthy) the calculation.
05c73ffba81c55ab	Difference between revisions of "Schrödinger equation" From Encyclopedia of Mathematics Jump to: navigation, search (Importing text file) (No difference) Revision as of 20:54, 24 March 2012 A fundamental equation in quantum mechanics that determines, together with corresponding additional conditions, a wave function characterizing the state of a quantum system. For a non-relativistic system of spin-less particles it was formulated by E. Schrödinger in 1926. It has the form where is the Hamilton operator constructed by the following general rule: in the classical Hamilton function the particle momenta and their coordinates are replaced by operators that have, respectively, the following form in the coordinate representation and in the momentum representation : For charged particles in an electromagnetic field, characterized by a vector potential , the quantity is replaced by . In these representations the Schrödinger equation is a partial differential equation, for example, for particles in the potential field , Discrete representations are possible, in which the function is a multi-component function and the operator has the form of a matrix. If a wave function is defined in the space of occupation numbers, then the operator is represented by some combinations of creation and annihilation operators (the second quantization representation, cf. Annihilation operators; Creation operators). The generalization of the Schrödinger equation to the case of a non-relativistic particle with spin (a two-component function ) is called the Pauli equation (1927); to the case of a relativistic particle with spin (a four-component function ) — the Dirac equation (1928); to the case of a relativistic particle without spin — the Klein–Gordon equation (1926); with spin 1 (the function is a vector) — the Proca equation (1936); etc. The solution of the Schrödinger equation is defined in the class of functions that satisfy the normalization condition for all (the brackets mean integration or summation over all values of ). To find the solution it is necessary to formulate initial and boundary conditions, corresponding to the character of the problem under consideration. The most characteristic among such problems are: 1) The stationary Schrödinger equation and the determination of admissible values of the energy of the system. Assuming that and requiring in conformity with the normalization condition and the condition of absence of flows at infinity that the wave function and its gradients vanish when , one obtains an equation for the eigenvalues and eigenfunctions of the Hamilton operator: Characteristic examples of the exact solution to this problem are: the eigenfunctions and energy levels for a harmonic oscillator, a hydrogen atom, etc. 2) The quantum-mechanical scattering problem. The Schrödinger equation is solved under boundary conditions that correspond at a large distance from the scattering centre (described by the potential ) to the plane waves falling on it and the spherical waves arising from it. Taking into consideration this boundary condition, the Schrödinger equation can be written as an integral equation, the first iteration of which with respect to the term containing corresponds to the so-called Born approximation. This equation is also called the Lippman–Schwinger equation. 3) The case where the Hamiltonian of the system depends on time, , is usually considered in the framework of time-dependent perturbation theory. This is a theory of quantum transition, the determination of the system's reaction to an external perturbation (dynamic susceptibility) and characteristics of relaxation processes. To solve the Schrödinger equation one usually applies approximate methods, regular methods (different types of perturbation theories), variational methods, etc. [1] A. Messiah, "Quantum mechanics" , 1 , North-Holland (1961) [2] L.D. Landau, E.M. Lifshitz, "Quantum mechanics" , Pergamon (1965) (Translated from Russian) [3] L.I. Schiff, "Quantum mechanics" , McGraw-Hill (1955) [a1] R.P. Feynman, R.B. Leighton, M. Sands, "The Feynman lectures on physics" , III , Addison-Wesley (1965) [a2] S. Gasiorowicz, "Quantum physics" , Wiley (1974) [a3] J.M. Lévy-Lehlond, "Quantics-rudiments of quantum physics" , North-Holland (1990) (Translated from French) [a4] F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) How to Cite This Entry: Schrödinger equation. Encyclopedia of Mathematics. URL: This article was adapted from an original article by I.A. Kvasnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
0980fd5b63133ce3	Share this \| Latest Posts A recent opinion in Chemistry World focused on the issues of the practicality of turning ideas into useful technologies. One of the arguments seemed to be that curiosity driven science was giving the world a false sense of what could be achieved, and worse, was taking funding away from where it could be more usefully spent. As usual in such issues, there are several ways of viewing the issue. First, look at the issue of why scientists make some of the outrageous claims. In my view, the answer is simple. It is not because the scientists have lost track of thermodynamics as implied in the article (although I guess some might) and it is not because they are snake-oil merchants. My guess is that the biggest reason is dressing up work to satisfy the providers of funding. Let me confess to one example from my own past. My very first excursion into "the origin of life" issue came in the 1970s. I was supposed to be working on energy research, but funding was extremely tight, energy research needs expensive equipment that we did not have, so there was scope to do experiments that did not cost much. Gerald Smith and I had seen that the theory of the initial atmospheres required it to be carbon dioxide, which was thermodynamically very bad for biogenesis in terms of energy. Carbon dioxide is what life gets rid of at the bottom of the energy chain and is only returned to life by photosynthesis. So, if the geologists were correct, how did carbon biogenic precursors form from such an unpromising start? Our idea was that the carbon dioxide could still be reduced through photochemistry. Water and carbon dioxide attacks olivine, and somewhat more slowly, pyroxenes, to dissolve magnesium ions and ferrous ions, and the concept was, Fe II and light would reduce CO2 to formic acid and thence to formaldehyde, whereupon the magnesium carbonate could help catalyse the Butlerov type reactions. So, we did some photochemistry, and persuaded ourselves that we were reducing CO2. It was then that a thought struck me. The Fe II must end up as Fe III, and what would Fe III do to organic materials? The answer was reasonably obvious: try some and find out. So we irradiated some dilute sugar with Fe III, and the carbohydrates simply fell to pieces, with an action spectrum corresponding to the spectrum of the iron complex. Many other potential biochemical precursors suffered the same fate. So, we wrote up the results, but then came the question, how were we going to justify this work? Well, since energy was the desired activity, we wrote a little comment at the bottom of the paper about the potential of photochemical fuel cells. Did we think this was realistic? No, we did not. Did we think there was any theoretical possibility? Yes, while outrageously unlikely, it remained possible. Did it satisfy the keepers of returns to funding sources? Yes, because they never read past the keywords. You may say there was a little duplicity there, but first, this work cost very little and it did not distract us from doing anything else. We used equipment that otherwise would have been doing nothing, and the only real costs were trivial amounts of chemicals and the time spent writing the paper, because that was a real cost. Was the result meaningful? I leave that to you to decide, BUT for me, it was because it set me off realizing that the standard theory of atmospheric formation cannot be right. The carbon source for life could not have come from carbon dioxide initially, because in getting to reduced carbon from the most available source in the oceans, a much worse agent from the point of view of biogenesis was formed. Had we been able to show how CO2 could be the carbon source for biogenesis, I think that would have been interesting, but just because you fail in the primary objective, that does not mean the time was wasted. The recording of the effects of a failed idea are just as valuable. Posted by Ian Miller on Nov 10, 2013 10:51 PM GMT The first round of results came in from Curiosity at Gale crater, and I found the results to be both comforting but also disappointing. The composition of the rocks, with one exception, and the composition of the dust were very similar to what had been found elsewhere on Mars. We now know the results are more general, but they are not exactly exciting. Dust was heated to 835 oC and a range of volatiles came off, and there was, once again, evidence of some carbonaceous matter, but the products obtained (SO2, CO2, and O2, HCN, H2S, methyl chloride, dichloromethane, chloroform, acetone, acetonitrile, benzene, toluene and a number of others) were almost certainly pyrolysis products. An interesting paper (Nature Geosci. doi:10.1038/ngeo1930) found that when ices similar to those in comets were subjected to high velocity impacts, several aminoacids were produced. However, some were aminoacids such as α-aminoisobutyric acid and isovaline, which are not used for protein, and the question is, why not? One reason may be that our aminoacid resource did not come from such comets. A circumstellar disk was identified around a white dwarf, and the disk was considered to have arisen from a rocky minor planet (Science 342: 218 – 220). There was an excess of oxygen present compared with the metals and silicates, and a lack of carbon, and this is consistent with the parent body having comprised 26% water by mass. This was interpreted as confirming that water-bearing planetesimals exist around A and F-type stars that end their lives as white dwarfs. Of particular interest was the lack of carbon. What sort of body could it have come from? I have seen suggestions that it would be a body like Ceres, in which case my proposed mechanism for the formation of minor planets would not be correct (because of the lack of carbon) but another option might be something that accreted in the Jovian zone, where I argue carbon is not accreted significantly. Finally, Curiosity made a specific search for methane in the Martian atmosphere and put an upper limit of 1.3 ppbv, which suggests that methane seen on Mars did not come from methanogenic microbial activity, but rather from either extraplanetary or geologic sources. The latter fits nicely with my proposed mechanism of formation of Mars. Posted by Ian Miller on Nov 4, 2013 1:35 AM GMT The prize appears to have been given for work that leads to the modeling of how enzymes work. If I follow the information I have seen correctly, the modelling involves three different levels. The very inner site of reactivity involves a quantum mechanical evaluation of the reaction site and the reactivity. Outside this, where the protein strands fold and interact, the situation is simplified with simple (in comparison) classical physics, while outside this there is further simplification by which the situation is considered simply as a dielectric medium. All of that seems eminently sensible, and there is little doubt that even with such simplifications there remains some serious work that has been done. However one thing concerns me: up until this award, I was totally unaware of it. Yes, this might indicate a lack of effort on my part, but in my defence, there is an enormous amount of information available, and for matters outside my immediate research interests, I have to simply rely on more general articles. Which gets me to the point: assuming this work has been successful, it is obviously important, but why has more not been made of it? Again, perhaps this illustrates a fault on my part, but again I feel there is more need to promote important work. I guess the final point I would like to make is, could someone highlight the principles that this modeling work has uncovered? The general chemist has little interest in wading through computations of the various options open to such a complex molecule as an enzyme, but if some general principles are uncovered, could they not be better publicized? After all, they may have more general applicability. Posted by Ian Miller on Oct 28, 2013 5:22 AM GMT There was a recent comment to one of my posts regarding the formation of rocky planets, so I thought I should outline how I think the rocky planets formed, and why. The standard theory involves only physical forces, and is that dust accreted to planetesimals, then these collided, eventually to form embryos (Mars-sized bodies), then these collided to form planets. First, why do I think that is wrong? For me, it is difficult to see how the planetesimals form by simple collision of dust, and it is even harder to see how they stay together. One route might be through melting due to radioactivity, but if that is the case, one would need very recently formed supernova debris to get sufficient radioactivity. Then, as the objects get bigger, collisions will have greater relative velocities, which means much greater kinetic energy in impacts, and because everything is further apart, collisions become less probable and everything takes too long. The models of Moon formation generally lead to the conclusion that such massive impacts lead to a massive loss of material. The difference between the standard theory and mine is that I think chemistry is involved. There are two stages involved for rocky planets. The first is during the accretion of the star, and near the star, temperatures are raised significantly. Once temperatures reach greater than 1200 oC, some silicates become semi-molten and sticky, and this leads to the accretion of lumps. By 1538 oC, iron melts, and hence lumps of iron-bodies form, while around 1500 – 1600 oC. calcium aluminosilicates form separate molten phases, although about 1300 oC a calcium silicate forms a separate phase. (The separation of phases is enhanced by polymerization.) Material at 1 A. U., say, reaches about 1550 - 1600 oC, while near Mars it reaches something like 1300 oC. Of particular relevance are the calcium aluminosilicates, as these form a range of materials that act as hydraulic cements. Also, the closer the material gets to the star, the hotter and more concentrated it gets, so bigger lumps of material form. One possibility is that Mercury is in fact essentially formed from one such accreted lump that scavenged up local lumps. Another important feature is that within this temperature range, significant other chemistry occurred, e.g. the formation of carbides, carbon, nitrides, cyanides, cyanamides, silicides, phosphides, etc. When the disk cooled down, collisions between bodies formed dust, while some bodies would come together. Dust would form preferentially from the more brittle solids, which would tend to be the aluminosilicates, and when such dust accreted onto other bodies, water from the now cool disk would set the cement and make a solid body that would grow by simply accreting more dust and small bodies. Because there is a gradual movement of dust and gas towards the star, there would be a steady supply of such feed, and the bodies would grow at a rate proportional to their cross-section. Eventually, the bodies would be big enough to gravitationally attract other larger bodies, however the important point is that provided initiation is difficult, runaway growth of one body in a zone would predominate. Earth grows to be the biggest because it is in the zone most suitable for forming and setting cement, and because the iron bodies are eminently suitable for forming dust. The atmosphere and biochemical precursors form because the water of accretion reacts within the planet to form a range of chemicals from the nitrides, phosphides, carbides, etc. What is relevant here is high-pressure organic chemistry, which again is somewhat under-studied. Am I right? The more detailed account, including a major literature review, took just under a quarter of a million words in the ebook, and the last chapter contains over 80 predictions, most of which are very difficult to do. Nevertheless, an encouraging sign is that the debris of a minor rocky planet around a white dwarf (what remains of an F or A type star) shows the presence of considerable amounts of water. Such water is (in my opinion) best explained by the water being involved in the initial accretion of the body, because it is extremely unlikely that such an amount of water could arrive on a minor rocky planet by collision of chondrites because the gravity of the minor planet is unlikely to be enough to hold such water. Thus this is strongly supportive of my mechanism, and it is rather difficult to see how this arose through the standard theory. Posted by Ian Miller on Oct 21, 2013 1:57 AM BST Leaving aside the provision of employment for modelers, I am far from convinced that the climate change models are of any use at all. As an example, we often hear the proposition that to fix climate change we should find a way to get carbon dioxide from the atmosphere, or from the gaseous effluent of power stations. This sounds simple. It is reasonably straightforward to absorb carbon dioxide: bubble the gas through a suitable base. Of course, the problem then comes down to, how do you get a suitable base? Calcium oxide is fine, except you broke down a carbonate at quite high temperatures to get it. Amines offer an easier route, but to collect a power station's output, regenerate your amine, and keep the carbon dioxide under control will require up to a third of the power from your power station. Not attractive. The next problem is, what to do with the carbon dioxide? Yes, some can be sunk into wells, preferably wet basaltic ones as this will fix the CO2, and a small amount could be used as a chemical, say to make polycarbonates, but how many power stations do you think will be accounted for by that? The problem for climate change is that we currently burn about 9 Gt of carbon per annum, which means we have to fix/use something like 33 Gt of CO2 per annum just to break even, and breaking even is unlikely to fix this carbon problem. The problem is, CO2 is not a very strong greenhouse gas, but it does stay around in the atmosphere for a considerable time. One point that nobody seems to make in public is that even if we stopped emitting CO2 right now, the additional carbon we have already put in the atmosphere will remain for long enough to do a lot more damage. Everybody seems to behave as if we are in a rapid equilibrium, and that is not so. The Greenland ice sheet is the last relic of the last ice age. If we have created the net warming to melt so much per annum, that will keep going until the ice retreats to a position more resilient, at which point our climate will change significantly because we have a much different albedo over a large area. We cannot "fix" climate change by simply stopping the rate of increase of burning carbon; we have to actively reduce the total integrated amount, and not simply worry about the rate of increased production. I suggest that to fix the climate problem, assuming we see it as a problem, we would be better to put more effort into something with a stronger response than fixing CO2. In the previous post, I attempted (unsuccessfully!) to irritate some people relating to how climate change research is spent. When money becomes available for this, what happens? What I believe happens is that we see numerous proposals for funding to make more accurate measurements of something. My argument is, just supposing we do get more accurate data on, say, the methane output of some swamp, what good does that do? It provides employment for those measuring the output of the swamp, but then what? Certainly it will add more to the literature, but the scientific literature is hardly short of material. Enough such measurements will help models account for what has happened, perhaps, but the one thing I am less confident about is whether such models will be able to answer the question, "Exactly what will happen if we do X?" For example, suppose we decided to try to raise the albedo of the planet by reflecting more light to space, and did this in a region that would lower the temperature of cold fronts coming into Greenland, with the aim of increasing snow deposition over Greenland, how much light would we need to reflect and where should we reflect it? My argument is, until models can give an approximate answer to that sort of question, they are useless. And unless we do something like geo-engineering, we are doomed to have to accommodate the change, because nobody has suggested any alternative that has the capacity to solve the problem. We can wave our hands and "feel virtuous" for claiming that we are doing something, but unless the sum of the somethings solves the problem, it is a complete waste of effort. Worse than that, such acts consume resources that could be better used to accommodate what will come. The only value of a model is to inform us which actions will be sufficient, and so far they cannot do that. Posted by Ian Miller on Oct 14, 2013 10:13 PM BST Currently, NASA is asking for public assistance for their astrobiology program, or they were up until the current government shutdown, and in particular, asking for suggestions as to where their program should be going. I think this is an extremely enlightened view, and I hope they receive plenty of good suggestions and take some of them up. This is a little different from the average way science gets funded, in which academic scientists put in applications for funds to pursue what they think is original. This is supposed to permit the uncovering of "great new advances", and in some areas, perhaps it does, but I rather suspect the most common outcome is to support what Rutherford dismissively called, "stamp collecting". You get a lot of publications, a lot of data, but there is no coherent approach towards answering "big questions". That, I think, is a strength of the NASA approach, and I hope other organizations take this up. For example, if we wish to address climate change, what questions do we really want to have answered? What we tend to get is, "Fund me to set up more data gathering," from those too uninspired to come up with something more incisive. We do not need more data to set the parameters so that current models better represent what we see; we need better models that will represent what will happen if we do or do not do X. So what are the good questions for NASA to address? Obviously there are a very large number of them, but in my view, regarding biogenesis, I think there are some very important ones. Perhaps one of the most important one that has been pursued so far is how do the planets get their water, because if we want life on other planets, they have to have water. The water on the rocky planets is often thought to come from chondrites, as a "late veneer" on the planet. Now, one of the peculiarities of this explanation is that, as I argued in my ebook, Planetary Formation and Biogenesis, this explanation has serious problems. The first is, only a special class of chondrites contains volatiles; the bulk of the bodies from the asteroid belt do not. Further, the isotopes of the heavier elements are different from Earth, the ratios of different volatiles do not correspond to anything we see here or on the other planets, so why is such an explanation persisted with? The short answer is, for most there is no alternative. My alternative is simple: the planets started accreting through chemical processes. Only solids could be accreted in reasonable amounts this close to the star, unless the body got big enough to hold gravitationally gases from the accretion disk. Water can be held as metal and silicon hydroxyl compound, the water subsequently being liberated. This, as far as I know, is the only mechanism by which the various planets can have different atmospheric compositions: different amounts of the various components were formed at different temperatures in the disk. If that is correct, we would have a means of predicting whether alien planets could conceivably contain life. Accordingly, one way to pursue this would be to try to understand the high temperature chemistry of the dusts and volatiles expected to be in the accretion disk. That would involve a lot of work for which chemists alone would be suitable. Now, my question is, how many chemists have shown any interest in this NASA program? Do we always want to complain about insufficient research funds, or are we prepared to go out and do something to collect more? Posted by Ian Miller on Oct 7, 2013 1:10 AM BST Perhaps one of the more interesting questions is where did Earth's volatiles come from? The generally accepted theory is that Earth formed by the catastrophic collisions of planetary embryos (Mars-sized bodies), which effectively turned earth into a giant ball of magma, at which time the iron settled to the core though having a greater density, and took various siderophile elements with it. At this stage, the Earth would have been reasonably anhydrous. Subsequently, Earth got bombarded with chondritic material from the asteroid belt that was dislodged by Jupiter's gravitational field (including, in some models, Jupiter migrating inwards then out again), and it is from here that Earth gets its volatiles and its siderophile elements. This bombardment is often called "the late veneer". In my opinion, there are several reasons why this did not happen, which is where these papers become relevant. What are the reasons? First, while there was obviously a bombardment, to get the volatiles through that, only carbonaceous chondrites will suffice, and if there were sufficient mass to give that to Earth, there should also be a huge mass of silicates from the more normal bodies. There is also the problem of atmospheric composition. While Mars is the closest, it is hit relatively infrequently compared with its cross-section, and hit by moderately wet bodies almost totally deficient in nitrogen. Earth is hit by a large number of bodies with everything, but the Moon is seemingly not hit by wet bodies or carbonaceous bodies. Venus, meanwhile, is hit by more bodies that are very rich in nitrogen, but relatively dry. What does the sorting? The first paper (Nature 501: 208 – 210) notes that if we assume the standard model by which core segregation took place, the iron would have removed about 97% of the Earth's sulphur and transferred it to the core. If so, the Earth's mantle should exhibit fractionated 34S/32S ratio according to the relevant metal-silicate partition coefficients, together with fractionated siderophile metal abundances. However, it is usually thought that Earth's mantle is both homogeneous and chondritic for this sulphur ratio, consistent with the acquisition of sulphur ( and other siderophile elements) from chondrites (the late veneer). An analysis of mantle material from mid-ocean ridge basalts displayed heterogeneous 34S/32S ratios that are compatible with binary mixing between a low 34S/32S ambient mantle ratio and a high 34S/32S recycled component. The depleted end-member cannot reach a chondritic value, even if the most optimistic surface sulphur is added. Accordingly, these results imply that the mantle sulphur is at least partially determined by original accretion, and not all sulphur was deposited by the late veneer. In the second (Geochim. Cosmochim. Acta 121: 67-83), samples from Earth, Moon, Mars, eucrites, carbonaceous chondrites and ordinary chondrites show variation in Si isotopes. Earth and Moon show the heaviest isotopes, and have the same composition, while enstatite chondrites have the lightest. A model of Si partitioning based on continuous planetary formation that takes into account T, P and oxygen fugacity variation during Earth's accretion. If the isotopic difference results solely from Si fractionation during core formation, their model requires at least ~12% by weight Si in the core, which exceeds estimates based on core density or geochemical mass balance calculations. This suggests one of two explanations: Earth's material started with heavier silicon, or (2) there is a further unknown process that leads to fractionation. They suggest vaporization following the Moon forming event, but would not this lead to lighter or different Moon material? One paper (Earth Planet. Sci. Lett. 2013: 88-97) pleased me. My interpretation of the data related to atmospheric formation is that the gaseous elements originally accreted as solids, and were liberated by water as the planet evolved. These authors showed that early early degassing of H2 obtained from reactions of water explains the "high oxygen fugacity" of the Earth's mantle. A loss of only 1/3 of an "ocean" of water from Earth would shift the oxidation state of the upper mantle from the very low oxidation state equivalent to the Moon, and if so, no further processes are required. Hydrogen is an important component of basalts at high pressure and, perforce, low oxygen fugacity. Of particular interest, this process may have been rapid. On the early Earth, over 5 times the amount of heat had to be lost as is lost now, and one proposal (501:501 - 504 ) heat pipe volcanism such as found on Io would manage this, in which case, the evolution of water and volatiles may have also been very rapid. Finally, in (Icarus 226: 1489 -1498), near-infrared spectra show the presence of hydrated poorly crystalline silica with a high silica content on the western rim of Hellas. The surfaces are sporadically exposed over a 650 km section within a limited elevation range. The high abundances and lack of associated aqueous phase material indicate high water to rock ratios were present, but the higher temperatures that would lead to quartz were not present. This latter point is of interest because it is often considered that the water flows on Mars in craters were due to internal heating due to impact, such heat being retained for considerable periods of time. To weather basalt to make silica, there would have to be continuous water of a long time, and if the water was hot and on the surface it would rapidly evaporate, while if it was buried, it would stay super-heated, and presumably some quartz would result. This suggests extensive flows of cold water. Posted by Ian Miller on Sep 30, 2013 3:30 AM BST In a previous post, I questioned whether gold showed relativistic effects in its valence electrons. I also mentioned a paper of mine that proposes that the wave functions of the heavier elements do not correspond exactly to the excited states of hydrogen, but rather are composite functions, some of which have reduced numbers of nodes, and I said that I would provide a figure from the paper once I sorted out the permission issue. That is now sorted, and the following figure comes from my paper. The full paper can be found at and I thank CSIRO for the permission to republish the figure. The lines show the theoretical function, the numbers in brackets are explained in the paper and the squares show the "screening constant" required to get the observed energies. The horizontal axis shows the number of radial nodes, the vertical axis, the "screening constant". The contents of that paper are incompatible with what we use in quantum chemistry because the wave functions do not correspond to the excited states of hydrogen. The theoretical function is obtained by assuming a composite wave in which the quantal system is subdivisible provided discrete quanta of action are associated with any component. The periodic time may involve four "revolutions" to generate the quantum (which is why you see quantum numbers with the quarter quantum). What you may note is that for = 1, gold is not particularly impressive (and there was a shortage of clear data) but for = 0 and = 2 the agreement is not too bad at all, and not particularly worse than that for copper. So, what does this mean? At the time, the relationships were simply put there as propositions, and I did not try to explain their origin. There were two reasons for this. The first was that I thought it better to simply provide the observations and not clutter it up with theory that many would find unacceptable. It is not desirable to make too many uncomfortable points in one paper. I did not even mention "composite waves" clearly. Why not? Because I felt that was against the state vector formalism, and I did not wish to have arguments on that. (That view may not be correct, because you can have "Schrödinger cat states", e.g. as described by Haroche, 2013, Angew. Chem. Int. Ed. 52: 10159 -10178). However, the second reason was perhaps more important. I was developing my own interpretation of quantum mechanics, and I was not there yet. Anyway, I have got about as far as I think is necessary to start thinking about trying to convince others, and yes, it is an alternative. For the motion of a single particle I agree the Schrödinger equation applies (but for ensembles, while a wave equation applies, it is a variation as seen in the graph above.) I also agree the wave function is of the form ψ = A exp(2πiS/h) So, what is the difference? Well, everyone believes the wave function is complex, and here I beg to differ. It is, but not entirely. If you recall Euler's theory of complex numbers, you will recall that exp() = -1, i.e. it is real. That means that twice a period, for the very brief instant that S = h, ψ is real and equals the wave amplitude. No need to multiply by complex conjugates then (which by itself is an interesting concept –where did this conjugate come from? Simple squaring does not eliminate the complex nature!) I then assume the wave only affects the particle when the wave is real, when it forces the particle to behave as the wave requires. To this extent, the interpretation is a little like the pilot wave. If you accept that, and if you accept the interpretation of what the wave function means, then the reason why an electron does not radiate energy and fall into the nucleus becomes apparent, and the Uncertainty Principle and the Exclusion Principle then follow with no further assumptions. I am currently completing a draft of this that I shall self-publish. Why self-publish? That will be the subject of a later blog. Posted by Ian Miller on Sep 23, 2013 3:30 AM BST In the latest Chemistry World, Derek Lowe stated that keeping up with the literature is impossible, and he argued for filtering and prioritizing. I agree with his first statement, but I do not think his second option, while it is necessary right now, is optimal. That leaves open the question, what can be done about it? I think this is important, because the major chemical societies around the world are the only organizations that could conceivably help, and surely this should be of prime importance to them. So, what are the problems? Where to put the information is not a problem because we now seem to have almost unlimited digital storage capacity. Similarly, organizing it is not a problem provided the information is correctly input, in an appropriate format with proper tags. So far, easy! Paying for it? This is more tricky, but it should not necessarily be too costly in terms of cash. The most obvious problem is manpower, but this can also be overcome if all chemists play their part. For example, consider chemical data. The chemist writes a paper, but it would take little extra effort to put the data into some pre-agreed format for entry into the appropriate data base. Some of this is already done with "Supplementary information", but that tends to be attached to papers, which means someone wishing to find the information has to subscribe to the journal. Is there any good reason why data like melting points and spectra cannot be provided free? As an aside, this sort of suggestion would be greatly helped if we could all agree on the formatting requirements, and what tags would be required. This does not solve everything, because there are a lot of other problems too, such as "how to make something". One thing that has always struck me is the enormous wastage of effort in things like biofuels, where very similar work tended to be repeated every crisis. Yes, I know, intellectual property rights tend to get in the way, but surely we can get around this. As an example of this problem, I recall when I was involved in a joint venture with the old ICI empire. For one of the potential products to make, I suggested a polyamide based on a particular diamine that we could, according to me, make. ICINZ took this up, sent it off to the UK, where it was obviously viewed with something approaching indifference, but they let it out to a University for them to devise a way to make said polyamide. After a year, we got back the report, they could not make the diamine, and in any case, my suggested polymer would be useless. I suggested that they rethink that last thought, and got a rude blast back, "What did I know anyway?" So, I gave them the polymer's properties. "How did I know that?" they asked. "Simple," I replied, and showed them the data in an ICI patent, at which point I asked them whether they had simply fabricated the whole thing, or had they really made this diamine? There was one of those embarrassed silences! The institution could not even remember its own work! In principle, how to make something is clearly placed in scientific papers, but again, the problem is, how to find the data, bearing in mind no institute can afford more than a fraction of the available journals. Even worse is the problem of finding something related. "How do you get from one functional group to another in this sort of molecule with these other groups that may interfere?" is a very common problem that in principle could be solved by computer searching, but we need an agreed format for the data, and an agreement that every chemist will do their part to place what they believe to be the best examples of their own synthetic work in it. Could we get that cooperation? Will the learned societies help? Posted by Ian Miller on Sep 16, 2013 8:07 PM BST One concern I have as a scientist, and one I have alluded to previously, lies in the question of computations. The problem is, we have now entered an age where computers permit modeling of a complexity unknown to previous generations. Accordingly, we can tackle problems that were never possible before, and that should be good. The problem for me is, the reports of the computations tell almost nothing about how they were done, and they are so opaque that one might even question whether the people making them fully understand the underlying code. The reason is, of course, that the code is never written by one person, but by rather a team. The code is then validated by using the computations for a sequence of known examples, and during this time, certain constants of integration that are required by the process are fixed. My problem with this follows a comment that I understand was attributed to Fermi: give me five constants and I will fit any data to an elephant. Since there is a constant associated with every integration, it is only too easy to get agreement with observation. An example that particularly irritated me was a paper that tried "evolved" programs on molecules from which they evolved (Moran et al. 2006. J. Am Chem Soc. 128: 9342-9343). What they did was to apply a number of readily available and popular molecular orbital programs to compounds that had been the strong point of molecular orbital theory, such as benzene and other arenes. What they found was that these programs "predicted" benzene to be non-planar with quite erroneous spectral signals. That such problems occur is, I suppose, inevitable, but what I found of concern is that nowhere that I know was the reason for the deviations identified, and how such propensity to error can be corrected, and once such corrections are made, what do they do to the subsequent computations that allegedly gave outputs that agreed well with observation. If the values of various constants are changed, presumably the previous agreement would disappear. There are several reasons why I get a little grumpy over this. One example is this question of planetary formation. Computations up to about 1995 indicated that Earth would take about 100 My to accrete from planetary embryos, however, because of the problem of Moon formation, subsequent computations have reduced this to about 30 My, and assertions are made that computations reduce the formation of gas giants to a few My. My question is, what changed? There is no question that someone can make a mistake, and subsequently correct it, but surely it should be announced what the correction was. An even worse problem, from my point of view, was what followed from my PhD project, which involved, do cyclopropane electrons delocalize into adjacent unsaturation? Computations said yes, which is hardly surprising because molecular orbital theory starts by assuming it, and subsequently tries to show why bonds should be localized. If it is going to make a mistake, it will favour delocalization. The trouble was, my results, which involved varying substituents at another ring carbon and looking at Hammett relationships, said it does not. Subsequent computational theory said that cyclopropane conjugates with adjacent unsaturation, BUT it does not transmit it, while giving no clues as to how it came to this conclusion, apart from the desire to be in agreement with the growing list of observations. Now, if theory says that conjugation involves a common wave function over the region, then the energy at all parts of that wave must be equal. (The electrons can redistribute themselves to accommodate this, but a stationary solution to the Schrödinger equation can have only one frequency.) Now, if A has a common energy with B, and B has a common energy with C, why does A not have a common energy with C? Nobody has ever answered that satisfactorily. What further irritates me is that the statement that persists in current textbooks employed the same computational programs that "proved" the existence of polywater. That was hardly a highlight, so why are we so convinced the other results are valid? So, what would I like to see? In computations, the underpinning physics, the assumptions made, and how the constants of integration were set should be clearly stated. I am quite happy to concede that computers will not make mistakes in addition, etc, but that does not mean that the instructions for the computer cannot be questioned. Posted by Ian Miller on Sep 9, 2013 4:31 AM BST
59286dcc021f330c	Google+ Followers Sunday, November 26, 2017 Ionization Energy of Hydrogen - Experiment vs. Theory This paper explains one of the typical ways to extrapolate the ionization energy from hydrogen line measurements. Consequently, if the increase in frequency is plotted against the actual frequency, the curve can be extrapolated to the point at which the increase becomes zero, the frequency of the series limit. This extrapolation is required to be able to extract the frequency for the fully ionized $n={\infty}$ case for the ionization energy. This is where the electron is pushed from the $n=1$ energy state (ground state or fundamental state) of hydrogen to the $n={\infty}$ (free state) and when the electron returns to the $n=1$ state, it gives off a photon of the frequency equal to the ionization energy (or frequency). Going back to the original equation for the energy of the electron (derived from the full two body proton-electron hydrogen atom Schrödinger equation): And, using $E=hf$, the ionization energy of hydrogen is: Note the reduced mass term, $m_r$. Here is where a potential misapplication of this equation can happen in concert with experimental data that can lead to possible misinterpretations of theory or experiment: 1. Step 1, since $m_r\equiv{m_pm_e\over m_p+m_e}$ and ${m_pm_e\over m_p+m_e}\approx m_e$, let $m_r=m_e$ <~~~reduced mass approximation 2. Since $m_r\approx m_e$, just assume the equation is valid for the electron, reducing the two body problem into a one body problem. 3. Assume a quantum number/function, do experiment, and get excited how closely data matches theory! Let's analyze the results of this experiment using the actual complete reduced mass term and compare it to using the reduced mass approximation: From the linked experiment, the experimentally numerically/graphically extrapolated hydrogen ionization frequency is $3.28\times10^{15}Hertz$. From this, the ionization energy can be calculated using: $E=h\times3.28\times10^{15}Hertz =$ 13.564990eV <~~~ Experimental data $E_{ionization}(m_r)={m_re^4\over8ce_0^2h^2}=$ 13.598287eV <~~~ Using reduced mass matches data better!!! $E_{ionization}(m_e)={m_ee^4\over8ce_0^2h^2}=$ 13.605693eV <~~~ Using electron mass theory does not match data as well Note using the reduced mass is a better match to the experimental data for the results of this one experiment. Due to possible experimental error and the huge implications of this, it would be wise to check more than one experiment, however, I am confident all experiments of this type the data of the experiment will match the theory more closely if reduced mass is used rather than the electron mass. (I will check more and get back to ya if I find otherwise and retract this) The purpose of this post's investigation is to determine if there is any experimental data to support our viewpoint about properly using the equations from theory, and first indications are there is support for our view. Specifically, the handling of the proton to electron mass ratio and all the theoretical work around an atomic viewpoint of the proton and electron that is just slightly different than mainstream. Actually, it may simply be turning into an interpretation question. And, if correct, an approach that gives us all the masses and constants of physics via numerical algorithm. This is the reason for pushing forward - to see if the algorithm is true (TOP-PCG2). Also, this is favorable info for all the proton radius, mass ratio, and magnetic moment work. Nothing conclusive, yet it is favorable. While this changes little about mainstream science efforts and experiments, it should, perhaps, start opening minds to the idea that maybe, just maybe, something was missed, or passed over, or covered up, or just plain hidden. The implications are great to science and humanity. Forgot to mention, the other thing that may possibly come out of this, what I promised, a while back, a calculation - an equation for things such as muon mass, tau mass, these are likely, or the Higgs, or something, is in this error term (mixed in with the quantum number factor): $m_{err} = m_e-m_r$ $m_{err} = m_e-{m_pm_e\over m_p+m_e}$ $m_{err} = m_e{m_p+m_e\over m_p+m_e}-{m_pm_e\over m_p+m_e}$ $m_{err} = {m_e^2\over m_p+m_e}$ that, combined with $\left({1\over l^2}+{1\over n^2}\right)$ might give some interesting masses and energies to find with this "goggle/viewpoint" of the mainstream, perhaps explaining some of the confusion on particle zoo follies over the years. There are other terms for the quantum numbers in the parenthesis if things such as spin are considered in more detail which have been experimentally played with - key thing is it is all playing around unless there is a clear theoretical and experimental verification plan, and that's what all these years of hammering on the collider have been, looking at more and more advanced terms for the things in parenthesis while missing a blaring 4-5 decimal point error (1 in 2000? approx?) (and swept all errors, theoretical and measurement or otherwise, into a 4% proton radius error, and very, very likely, the same 4% error in the proton magnetic moment, and this overall 1/1836 or so error with this reduced mass assumption blunder, in all other constants, coefficients, and masses...) The Surfer, OM-IV
4011f6d2e38b260d	Sometimes, Things Are Really Simple But We Insist on Making Them Complicated Teaching quantum mechanics to students who have not seen the subject before can be extremely challenging. Take, for example, one of its postulates (A postulate is a statement that is assumed true without proof.): This is the time-dependent Schrödinger equation, which describes how a system evolves in time when acted upon by a force or energy. For a situation like this, I try to remind my students of the time they were in kindergarten and the teacher taught them that 1 + 1 = 2. This is no different. Being able to accept nature the way it is can be very difficult especially when our mind has been conditioned to rationalize all the time. There are building blocks which we must assume as starting material. How we connect or assemble these blocks to create something is indeed a skill, but we must not confuse skills with fundamentals. In General Chemistry, there are likewise fundamental concepts. An example is the Law of Definite Proportions: "A chemical compound always contains exactly the same proportion of elements by mass". One can illustrate this with the following hydrocarbons: methane and octane. By mass, methane is about 75% carbon and 25% hydrogen. Octane, on the other hand, is about 84% carbon and 16% hydrogen. In order to grasp these percentages, a student needs to be able to work with atomic and molecular masses. These are among the basic principles in chemistry. Without providing students the opportunity to work with these rudimentary, and sometimes referred to as "rote learning" practices, it is inappropriate, for example, to expect to students to tackle the problem below: Why is compressed natural gas (mostly methane, CH4) advertised as friendlier to the earth’s climate than gasoline (take octane, C8H18, for example)? The heats of combustion of methane and octane are 800 and 5000 kJ mol-1, respectively. The question does require a higher level of thinking, but it is inappropriate to ask when students do not even know how to evaluate percent compositions of compounds. It is important to identify the fundamentals. In arithmetic, addition of numbers is as rudimentary as counting. Students nowadays in American grade schools unfortunately not only have to contend with the challenge of learning to add, but also using strategies recommended by curriculum writers. Take, for example, the following strategies: • Make a 10: For example, to solve 5 + 7, a students is advised to add 5 to 5 to make 10, then subtract 5 from 7, that is 5 + 7 = (5 + 5) + (7 - 5) = 10 + 2 = 12. • Make doubles: In this strategy, it is assumed that a student can do doubles easier, to solve 5 + 7, one breaks 7 first into 5 + 2, thus, 5 + 7 = 5 + 5 + 2. The above can be extended to adding several numbers. An example is shown below (from Thorn County Primary School): There are children who can in fact add 7 + 8 + 3, without doing the steps recommended or in this case, "commanded", in the above exercise. And it can confuse students. Of course, in later years, being able to distribute terms in a sequence is necessary. For example, in algebra, it is useful to understand that 7x + 3 + 2x + 4 = 9x + 7. For arithmetic, we need to be more careful. Oftentimes, strategies are designed by people who already understand the process. This does not necessarily mean that it is helpful for beginners. These strategies require that particular attention is given to each student individually. These are in fact prescriptions. Some may be appropriate, but definitely, these are not generally applicable. Ill-advised approaches also proliferate in reading. One of the fundamentals of reading is vocabulary. Yet, primary school children are likewise bombarded with strategies and interventions. First, these interventions come with their corresponding benchmark assessments. Without administering properly and correctly the assessments, these interventions may in fact harm not help a struggling student. Unfortunately, some teachers are excited to embrace seemingly fashionable strategies such as, "sound it out", "relate the story from beginning to end", without realizing that these interventions are sometimes very specific. These are interventions, no different from medicines or procedures prescribed by a physician. The wrong medicine can make things worse when prescribed and administered incorrectly or inappropriately. Popular posts from this blog The National Achievement Test in the Philippines K to 12 Program ng Gobyerno ng Pilipinas Absenteeism and Student Performance
2bddf32f946c047f	Module code: PHY3046 Module Overview The module addresses the advanced physics and technology of photonic nanostructures, where photons and/or electrons are spatially confined to dimensions comparable to or smaller than their wavelength. The propagation of the light and its interaction with matter are determined by factors such as length scales, periodicity, and dimensionality, and lead to phenomena not observed in nature. This is a rapidly developing field where fundamental science and technological advance hand-in-hand, and the module aims to demonstrate how new science drives new technologies that have a significant impact on society, for example through energy production, communications, and healthcare. Module provider Module Leader ALLAM J Prof (Physics) Number of Credits: 15 ECTS Credits: 7.5 Framework: FHEQ Level 6 JACs code: F390 Module cap (Maximum number of students): N/A Module Availability Semester 2 Prerequisites / Co-requisites Module content Indicative content includes: A. Introduction and Review 1. Introduction What is photonics? What is nanotechnology? Description of module: organisation, teaching methods, assessment A look ahead: nanophotonics and the quantum playground 2. Brief review of physics of photons and electrons These review lectures briefly summarize the minimum background in Electromagnetism (EM) and Quantum Mechanics (QM) required for this module. They also introduce concepts, methodology and nomenclature to be followed in the module. Wave equations: propagation, dispersion, velocities, impedance Interfaces, barriers and tunneling Confinement: total internal reflection, standing waves, waveguides and resonant cavities Materials: dielectrics, metals, and semiconductors EM waves: Maxwell’s equations and EM wave equation, in dielectrics and metals Electron waves and Schrödinger equation Semi-classical interaction of light and atoms Light emission and lasers 3. Introduction to optical resonators and microcavities Light emission and lasers Losses and Quality (Q) factor of a resonator Finesse, free-spectral range, and mode volume Fabry-Perot resonators 'Whispering gallery' micro cavities (disks, rings, spheres, tori) 4. Waves in periodic media Electromagnetic waves in periodic media: Floquet (Bloch) theorem and band structure Analytical solution of wave equation in periodic medium Distributed Bragg Reflectors, band gaps and mini-bands Overview of computational methods: Fourier methods, Transfer Matrix, FDTD Electron waves in semiconductors, heterostructures and superlattices B. Light in Nanostructures 5. Photonic crystals Photonics crystals and photonic bandgaps (PBG) in periodic, quasiperiodic and disordered dielectric structures Dispersion of 1D photonic crystal Natural and man-made photonic crystals; from butterfly wings to “holey fibres” 2D and 3D PBGs Defects, cavities and photonic crystal resonators PBGs for functional photonic components Dispersion control and ‘slow light’ 6. Meta materials and negative refraction Conditions for negative refraction Consequences for refraction and Doppler shift Materials and structures for negative refraction Scaling of operation frequency with size Meta materials and negative refraction An application (selected from: superlens, invisibility cloak, trapped light) 7. Plasmonics Bulk and surface plasmons: derivation of dispersion relation Plasmons in nanoparticles: resonance and field enhancement Plasmonic waveguides and cavities Applications: from solar cells to cancer therapy C. Electrons in Nanostructures 8. Low-dimensional semiconductors Review of density of states and dimensionality Excitons in low dimensions Quantum dots as ‘artificial atoms’ Introduction to exciton-polaritons and quantum optics D. Applications of Nanophotonics 9. Survey of Applications Critical evaluation of advantages from use of nanophotonic structures Impact of manufacturability, tolerances and cost on their adoption 10. Computional simulation of nanophotonic structure or device use of provided MATLAB simulations to understand behaviour and optimise performance Assessment pattern Assessment type Unit of assessment Weighting Alternative Assessment Assessment Strategy (1) technical knowledge and understanding of the core principles of nanophotonics, and (2) the application to simple nanophotonic devices and systems, and (3) skills in group working and technical reporting. Thus, the summative assessment for this module consists of: final exam on core principles of nanophotonics (1.5 hours) course work on MATLAB simulations (3 sets of exercises on photonic crystals, quantum wells and superlattice, and plasmonic nanoparticles) Formative assessment and feedback Problem sheets on the material delivered in lectures will be available, with follow-up tutorials, which allow the students to test their understanding of course material. Model answers and verbal feedback are provided to allow the students to assess their progress. The coursework will be preceded by 3 introductory sessions which will include preliminary exercises on which feedback will be given. Module aims • provide students with an overview of photonics and nanotechnology, sufficient to enter technical employment or pursue further research in these fields. • expose students to examples of latest developments in a fast-moving field. • provide practice in the application of known physical concepts and mathematical techniques to new situations. • provide an experience of computational simulation and performance optimization Learning outcomes Attributes Developed 001 Recognize the main optical and electrical properties of metals, dielectrics and semiconductors that determine their use in nanophotonics K 002 Identify similarities and differences between the propagation of light and electron waves in materials with reference to Maxwell's and Schrodinger’s equations KC 003 Describe how photon and electron confinement is achieved in nanostructured materials . K 004 Explain the origin of five principal classes of nanophotonic phenomena and structures K 005 1. photonic bandgaps in photonic crystals, 006 1. plasmons in metals, at metal-dielectric interfaces and in nano-particles, 007 1. quantum confinement and excitons in low-dimensional semiconductors, 008 1. polaritons in an optical cavity, and 009 1. negative refraction in metamaterials. 010 Analyse the influence of size, dimensionality, inhomogeneity, periodicity and anisotropy in these phenomena C 011 Recognise graphs of the dispersion relations associated with nanophotonic phenomena and identify the main features C 012 Evaluate the dispersion in specified examples of nanophotonic structures including use of appropriate approximations C 013 Examine the application of nanophotonics in devices for the manipulation of light KC 014 Perform computer simulations of a nanophotonic structure CP Attributes Developed C - Cognitive/analytical K - Subject knowledge T - Transferable skills P - Professional/Practical skills Overall student workload Independent Study Hours: 117 Lecture Hours: 22 Tutorial Hours: 15 Methods of Teaching / Learning The learning and teaching strategy is designed to: deliver core material in a familiar format of traditional lectures, supported by occcasional tutorials and students’ reading; incorporate a synoptic element: integrating understanding gained in compulsory modules on electromagnetism, quantum mechanics and solid-state physics, and refreshing some key physical concepts in preparation for employment or further studies after graduation; The learning and teaching methods include: 3 hours lectures / tutorials per week, including 3 hours introductory sessions to computational simulations Reading list Programmes this module appears in Programme Semester Classification Qualifying conditions Liberal Arts and Sciences BA (Hons)/BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module Physics with Nuclear Astrophysics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module Physics with Astronomy BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module Physics with Quantum Technologies BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
d179019e6823ed19	a repository of mathematical know-how The tensor power trick Quick description If one wants to prove an inequality X \leq Y for some non-negative quantities X, Y, but can only see how to prove a quasi-inequality X \leq CY that loses a multiplicative constant C, then try to replace all objects involved in the problem by "tensor powers" of themselves and apply the quasi-inequality to those powers. If all goes well, one can show that X^M \leq C Y^M for all M \geq 1, with a constant C which is independent of M, which implies that X \leq Y as desired by taking M^{th} roots and then taking limits as M \to \infty. General discussion This trick works best when using techniques which are "dimension-independent" (or depend only weakly (e.g. polynomially) on the ambient dimension M). On the other hand, the constant C is allowed to depend on other parameters than the dimension M. In particular, even if the eventual inequality X \leq Y one wishes to prove is supposed to be uniform over all choices of some auxiliary parameter n, it is possible to establish this estimate by first establishing a non-uniform estimate X \leq C_n Y, so long as this estimate "tensorises" to X^M \leq C_n Y^M, where n does not grow with M. It is of course essential that the problem "commutes" or otherwise "plays well" with the tensor power operation in order for the trick to be effective. In order for this to occur, one often has to phrase the problem in a sufficiently abstract setting, for instance generalizing a one-dimensional problem to one in arbitrary dimensions. Advanced undergraduate analysis (real, complex, and Fourier). The user of the trick also has to be very comfortable with working in high-dimensional spaces such as {\Bbb R}^M, or more generally G^M for some mathematical object G (e.g. a set, a group, a measure space, etc.) The later examples are more advanced but are only given as sketches. Example 1: convexity of L^p norms Let {\Bbb R} \to {\Bbb C} be a measurable function such that \int_{\Bbb R} \|f(x)\|^p\ dx \leq 1 and \int_{\Bbb R} \|f(x)\|^q\ dx \leq 1 for some 0 < p < q < \infty. Show that \int_{\Bbb R} \|f(x)\|^r\ dx \leq 1 for all p < r < q. As a first attempt to solve this problem, observe that \|f(x)\|^r is less than \|f(x)\|^q when \|f(x)\| \geq 1, and is less than \|f(x)\|^p when \|f(x)\| \leq 1. Thus the inequality \|f(x)\|^r \leq \|f(x)\|^p + \|f(x)\|^q holds for all x, regardless of whether \|f(x)\| is larger or smaller than 1. This gives us the bound \int_{\Bbb R} \|f(x)\|^r\ dx \leq 2, which is off by a factor of 2 from what we need. But we can eliminate this loss of 2 by the tensor power trick. We pick a large integer M \ge 1, and define the tensor power {\Bbb R}^M \to {\Bbb C} of f by the formula f^{\otimes M}(x_1,\ldots,x_M) = f(x_1) \ldots f(x_M). From the Fubini–Tonelli theorem we see that \int_{{\Bbb R}^M} \|f^{\otimes M}\|^p\ dx = (\int_{\Bbb R} \|f\|^p\ dx)^M \leq 1 and similarly \int_{{\Bbb R}^M} \|f^{\otimes M}\|^q\ dx = (\int_{\Bbb R} \|f\|^q\ dx)^M \leq 1. If we thus apply our previous arguments with f and {\Bbb R} replaced by f^{\otimes M} and {\Bbb R}^M respectively, we conclude that \int_{{\Bbb R}^M} \|f^{\otimes M}\|^r\ dx \leq 2; applying Fubini–Tonelli again we conclude that (\int_{\Bbb R} \|f\|^r\ dx)^M \leq 2. Taking M^{th} roots and then taking limits as M \to \infty we obtain the claim. More generally, show that if 0 < p < r < q < \infty, (X,\mu) is a measure space, and X \to {\Bbb C} is measurable, then we have the inequality \\| f\\|_{L^r(X,\mu)} \leq \\|f\\|_{L^p(X,\mu)}^{1-\theta} \\|f\\|_{L^q(X,\mu)}^\theta whenever the right-hand side is finite, where 0 < \theta < 1 is such that \frac{1}{r} = (1-\theta) \frac{1}{p} + \theta \frac{1}{q}. (Hint: by multiplying f and \mu by appropriate constants one can normalize \\|f\\|_{L^p(X,\mu)} = \\|f\\|_{L^q(X,\mu)}=1.) Use the previous exercise (and a clever choice of f, r, \theta and \mu - there is more than one choice available) to prove Hölder's inequality. Example 2: the maximum principle This example is due to Landau. Let \gamma be a simple closed curve in the complex plane that bounds a domain D, and let \overline{D} \to {\Bbb C} be a function which is complex analytic in the closure of this domain, and which obeys a bound \|f(z)\| \leq A on the boundary \gamma. The maximum principle for such functions asserts that one also has \|f(z)\|\leq A on the interior D as well. One way to prove this is by using the Cauchy integral formula f(z) =\frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z}\ dw for z \in D (assuming that \gamma is oriented anti-clockwise). Taking absolute values and using the triangle inequality, we obtain the crude bound \|f(z)\| \leq \frac{1}{2\pi} \frac{\|\gamma\|}{\hbox{dist}(z,\gamma)} A where \|\gamma\| is the length of \gamma. This bound is off by a factor of \frac{1}{2\pi} \frac{\|\gamma\|}{\hbox{dist}(z,\gamma)}. This loss depends on the point z and the curve \gamma, but it is crucial to observe that it does not depend on f. In particular, one can apply it with f replaced by f^M (and A replaced by A^M) for any positive integer M, noting that f^M is of course also complex analytic. We conclude that \|f(z)\|^M \leq \frac{1}{2\pi} \frac{\|\gamma\|}{\hbox{dist}(z,\gamma)} A^M and on taking M^{th} roots and then taking limits as M \to \infty we obtain the maximum principle. Example 3: new Strichartz estimates from old This observation is due to Jon Bennet. Technically, it is not an application of the tensor power trick as we will not let the dimension go off to infinity, but it is certainly in a similar spirit. Strichartz estimates are an important tool in the theory of linear and nonlinear dispersive equations. Here is a typical such estimate: if {\Bbb R} \times {\Bbb R}^2 \to {\Bbb C} solves the two-dimensional Schrödinger equation i u_t + \Delta u = 0, then one has the inequality \\| u \\|_{L^4_t L^4_x( {\Bbb R} \times {\Bbb R}^2 )} \leq C \\| u(0) \\|_{L^2( {\Bbb R}^2 )} for some absolute constant C. (In this case, the best value of C is known to equal 1/\sqrt{2}, a result of Foschi and of Hundertmark-Zharnitsky.) It is possible to use this two-dimensional Strichartz estimate and a version of the tensor power trick to deduce a one-dimensional Strichartz estimate. Specifically, if {\Bbb R} \times {\Bbb R} \to {\Bbb C} solves the one-dimensional Schrödinger equation iu_t + u_{xx} = 0, then observe that the tensor square = u(t,x) u(t,y) solves the two-dimensional Schrödinger equation. (This tensor product symmetry of the Schrödinger equation is fundamental in quantum physics; it allows one to model many-particle systems and single-particle systems by essentially the same equation.) Applying the above Strichartz inequality to this product we conclude that \\| u \\|_{L^8_t L^4_x( {\Bbb R} \times {\Bbb R} )} \leq C^{1/2} \\| u(0) \\|_{L^2( {\Bbb R} )}. A similar trick allows us to deduce "interaction" or "many-particle" Morawetz estimates for the Schrödinger equation from their more traditional "single-particle" counterparts; see for instance Chapter 3.5 of Terence Tao's book. Example 4: the Hausdorff-Young inequality Let G be a finite abelian group, and let G \to {\Bbb C} be a function. We let \hat G be the group of characters G \to S^1 of G, and define the Fourier transform \hat G \to {\Bbb C} by the formula = \frac{1}{\|G\|} \sum_{x \in G} f(x) \overline{\chi(x)}. From the triangle inequality we have \sup_{\chi \in \hat G} \|\hat f(\chi)\| \leq \frac{1}{\|G\|} \sum_{x \in G} \|f(x)\| (1) while from Plancherel's theorem we have (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^2)^{1/2} = (\frac{1}{\|G\|} \sum_{x \in G} \|f(x)\|^2)^{1/2} (2) By applying the Riesz-Thorin interpolation theorem, we can then conclude the Hausdorff-Young inequality (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^q)^{1/q} \leq (\frac{1}{\|G\|} \sum_{x \in G} \|f(x)\|^p)^{1/p} (3) whenever 1 < p < 2 and \frac{1}{p}+\frac{1}{q}=1. However, it is also possible to deduce (3) from (2) and (1) by a more elementary method based on the tensor power trick. First suppose that f is supported on a set A \subset G and that \|f\| takes values between (say) 2^m and 2^{m+1} on A. Then from (1) and (2) we have \displaystyle \sup_{\chi \in \hat G} \|\hat f(\chi)\| \leq \frac{\|A\|}{\|G\|} 2^{m+1} (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^2)^{1/2} \leq (\frac{\|A\|}{\|G\|})^{1/2} 2^{m+1} from which we establish a "restricted weak-type" version of Hausdorff-Young, namely (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^q)^{1/q} \leq (\frac{\|A\|}{\|G\|})^{1/p} 2^{m+1}, and thus (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^q)^{1/q} \leq 2 (\frac{1}{\|G\|} \sum_{x \in G} \|f(x)\|^p)^{1/p}. This inequality is restricted to those f whose non-zero magnitudes \|f(x)\| range inside a dyadic interval {}[2^m, 2^{m+1}], but by the technique of dyadic decomposition one can remove this restriction at the cost of an additional factor of O(1 + \log \|G\| ), thus obtaining the estimate (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^q)^{1/q} \leq C (1 + \log \|G\|) (\frac{1}{\|G\|} \sum_{x \in G} \|f(x)\|^p)^{1/p} (4) for general f. (There are of course an infinite number of dyadic scales {}[2^m, 2^{m+1}] in the world, but if one normalizes (\frac{1}{\|G\|} \sum_{x \in G} \|f(x)\|^p)^{1/p} = 1, then it is not hard to see that any scale above, say, \|G\|^{100} or below \|G\|^{-100} is very easy to deal with, leaving only O( 1 + \log \|G\| ) dyadic scales to consider.) The estimate (4) is off by a factor of O( 1 + \log \|G\| ) from the true Hausdorff-Young inequality, but if one replaces G with a Cartesian power G^M and f with its tensor power f^{\otimes M}, and makes the crucial observation that the Fourier transform of a tensor power is the tensor power of the Fourier transform, we see from applying (4) to the tensor power that (\sum_{\chi \in \hat G} \|\hat f(\chi)\|^q)^{M/q} \leq C (1 + \log \|G\|^M) (\frac{1}{\|G\|} \sum_{x\in G} \|f(x)\|^p)^{M/p}. Taking M^{th} roots and sending M \to \infty we obtain the desired inequality. Note that the logarithmic dependence on \|G\| in the constants turned out not to be a problem, because M^{1/M} \to 1. Thus the tensor power trick is able to handle a certain amount of dependence on the dimension in the constants, as long as the loss does not grow too rapidly in that dimension. Establish Young's inequality \\|fg\\|_{l^r(G)} \leq \\|f\\|_{l^p(G)} \\|g\\|_{l^q(G)} for 1 \leq p,q,r < \infty and \frac{1}{r}+1=\frac{1}{p}+\frac{1}{q} and finite abelian groups G, where = \sum_G f(y) g(x-y), by the same method. Prove the Riesz-Thorin interpolation theorem by this method, thus avoiding all use of complex analysis. (Note the similarity here with Example 1 and Example 2.) Example 5: an example from additive combinatorics, due to Imre Ruzsa An important inequality of Plünnecke asserts, among other things, that for finite non-empty sets A, B of an additive group G, and any positive integer k, the iterated sumset kB = B +\ldots + B, which is defined as the set of all possible sums of k not necessarily distinct elements of B, obeys the bound \|kB\| \leq \frac{\|A+B\|^k}{\|A\|^{k-1}}. (5) (This inequality, incidentally, is itself proved using a version of the tensor power trick, in conjunction with Hall's marriage theorem, but never mind that here.) This inequality can be amplified to the more general inequality \|B_1 + \ldots + B_k\| \leq \frac{\|A+B_1\| \ldots \|A+B_k\|}{\|A\|^{k-1}} via the tensor power trick as follows. Applying (5) with = B_1 \cup \ldots \cup B_k, we obtain \|B_1 + \ldots + B_k\| \leq \frac{(\|A+B_1\| + \ldots + \|A+B_k\|)^k}{\|A\|^{k-1}}. The right-hand side looks a bit too big, but we can resolve this problem with a Cartesian product trick (which can be viewed as a cousin of the tensor product trick). If we replace G with the larger group G \times {\Bbb Z}^k and replace each set B_i with the larger set B_i \times \{ e_i, 2e_i, \ldots, N_i e_i \}, where e_1,\ldots,e_k is the standard basis for {\Bbb Z}^k and N_i are arbitrary positive integers (and replacing A with A \times \{0\}), we obtain N_1 \ldots N_k \|B_1 + \ldots + B_k\| \leq \frac{(N_1 \|A+B_1\| + \ldots + N_k \|A+B_k\|)^k}{\|A\|^{k-1}}. Optimizing this in N_1,\ldots,N_k (basically, by making the N_i \|A+B_i\| close to constant; this is a general rule in optimization, namely that to optimize X+Y it makes sense to make X and Y comparable in magnitude – someone should think of some examples and write a Tricki article on this) we obtain the amplified estimate \|B_1 + \ldots + B_k\| \leq C_k \frac{\|A+B_1\| \ldots \|A+B_k\|}{\|A\|^{k-1}}. for some constant C_k; but then if one replaces A, B_1, \ldots, B_k with their Cartesian powers A^M, B_1^M, \ldots, B_k^M, takes M^{th} roots, and then sends M to infinity, one can delete the constant C_k and recover the inequality. Example 6: the Cotlar-Knapp-Stein lemma This example is not exactly a use of the tensor power trick, but is very much in the same spirit. Let H \to H be bounded linear operators on a Hilbert space; for simplicity of discussion let us say that they are self-adjoint, though one can certainly generalize the discussion here to more general operators. If we assume that the operators are uniformly bounded in the operator norm, say \\| T_i \\|_{op} \leq A (6) for all i and some fixed A, then this is not enough to ensure that the sum \sum_{i=1}^N T_i is bounded uniformly in N; indeed, the operator norm can be as large as AN. If however one has the stronger almost orthogonality estimate \sum_{j=1}^N \\| T_i T_j \\|_{op}^{1/2} \leq A (7) for all i, then the very useful Cotlar-Knapp-Stein lemma asserts that \sum_{i=1}^N T_i is now bounded uniformly in N, with \\| \sum_{i=1}^N T_i \\|_{op} \leq A. To prove this, we first recall that a direct application of the triangle inequality using (6) (which is a consequence of (7)) only gave the inferior bound of AN. To improve this, first observe from self-adjointness that \\| \sum_{i=1}^N T_i \\|_{op} = \\|(\sum_{i=1}^N T_i)^2 \\|_{op}^{1/2}. Expanding the right-hand side out using the triangle inequality, we obtain \\| \sum_{i=1}^N T_i \\|_{op} \leq (\sum_{i=1}^N \sum_{j=1}^N \\| T_i T_j \\|_{op})^{1/2}; using (7) one soon ends up with a bound of A N^{1/2}. This is better than before as far as the dependence on N is concerned, but one can do better by using higher powers. In particular, if we raise \sum_{i=1}^N to the 2M^{th} power for some M, and repeat the previous arguments, we end up with \\| \sum_{i=1}^N T_i \\|_{op} \leq(\sum_{1 \leq i_1,\ldots,i_{2M} \leq N} \\| T_{i_1} \ldots T_{i_{2M}} \\|_{op})^{1/2M}. Now, one can estimate the operator norm \\| T_{i_1} \ldots T_{i_{2M}} \\|_{op} either by \\| T_{i_1} T_{i_2} \\|_{op} \ldots \\| T_{i_{2M-1}} T_{i_{2M}} \\|_{op}, or (using (6)) by A^2 \\| T_{i_2} T_{i_3} \\|_{op} \ldots \\| T_{i_{2M-2}} T_{i_{2M-1}} \\|_{op}. We take the geometric mean of these upper bounds to obtain A \\| T_{i_1} T_{i_2} \\|_{op}^{1/2} \\|T_{i_2} T_{i_3} \\|_{op}^{1/2} \ldots \\| T_{i_{2M-1}} T_{i_{2M}} \\|_{op}^{1/2}. Summing this in i_{2M}, then i_{2M-1}, and so forth down to i_1 using (7) repeatedly, one eventually establishes the bound \\| \sum_{i=1}^N T_i \\|_{op} \leq (N A^{2M})^{1/2M}. Sending M \to \infty one eliminates the dependence on N and obtains the claim. Show that the hypothesis of self-adjointness can be dropped if one replaces (7) with the two conditions \sum_{j=1}^N \\| T_i T_j^ \\|_{op}^{1/2} \leq A \sum_{j=1}^N \\| T_i^* T_j \\|_{op}^{1/2} \leq A for all i. Example 7: entropy estimates Suppose that X is a random variable taking finitely many values. The Shannon entropy H(X) of this random variable is defined by the formula = - \sum_x {\Bbb P}(X=x) \log {\Bbb P}(X=x), (8) where x runs over all possible values of X. For instance, if X is uniformly distributed over N values, then H(X)=\log N. (9) If X is not uniformly distributed, then the formula is not quite as simple. However, the entropy formula (8) does simplify to the uniform distribution formula (9) after using the tensor power trick. More precisely, let X^{\otimes M} = (X_1,\ldots,X_M) be the random variable formed by taking M independent and identicaly distributed samples of X; thus for instance, if X was uniformly distributed on N values, then X^{\otimes M} is uniformly distributed on N^M values. For more general X, it is not hard to verify the formula which is of course consistent with the uniformly distributed case thanks to (9). A key observation is that as M \to \infty, the probability distribution of X^{\otimes M} becomes "asymptotically uniform" in a certain sense. Indeed, the law of large numbers tells us that with very high probability, each possible value x of X will be attained by ({\Bbb P}(X=x)+o(1)) M of the M trials X_1,\ldots,X_M. The number of possible configurations of X^{\otimes M} = (X_1,\ldots,X_M) which are consistent with this distribution can be computed (using Stirling's formula) to be e^{M (H(X)+o(1))}, and each such configuration appears with probability e^{-M (H(X)+o(1))} (again by Stirling's formula). Thus, at a heuristic level at least, X^{\otimes M} behaves like a uniform distribution on e^{M (H(X)+o(1))} possible values; note that this is consistent with (9) and (10), and can in fact be taken as a definition of Shannon entropy. One can use this "microstate" or "statistical mechanics" interpretation of entropy in conjunction with the tensor power trick to give short (heuristic) proofs of various fundamental entropy inequalities, such as the subadditivity inequality H(X,Y) \leq H(X) + H(Y) whenever X, Y are discrete random variables which are not necessarily independent. Indeed, since X^{\otimes M} and Y^{\otimes M} (heuristically) take only e^{M (H(X)+o(1))} and e^{M (H(Y)+o(1))} values respectively, then (X,Y)^{\otimes M} \equiv (X^{\otimes M}, Y^{\otimes M}) will (mostly) take on at most e^{M (H(X)+o(1))} e^{M (H(Y)+o(1))} values. On the other hand, this random variable is supposed to behave like a uniformly distributed random variable over e^{M (H(X,Y)+o(1))} values. These facts are only compatible if e^{M (H(X,Y)+o(1))} \leq e^{M(H(X)+o(1))} e^{M (H(Y)+o(1))}; taking M^{th} roots and then sending M \to \infty we obtain the claim. Make the above arguments rigorous. (The final proof will be significantly longer than the standard proof of subadditivity based on Jensen's inequality, but it may be clearer conceptually. One may also compare the arguments here with those in Example 1.) Example 8: the monotonicity of Perelman's reduced volume One of the key ingredients of Perelman's proof of the Poincaré conjecture is a monotonicity formula for Ricci flows, which establishes that a certain geometric quantity, now known as the Perelman reduced volume, increases as time goes to negative infinity. Perelman gave both a rigorous proof and a heuristic proof of this formula. The heuristic proof is much shorter, and proceeds by first (formally) applying the Bishop-Gromov inequality for Ricci-flat metrics (which shows that another geometric quantity - the Bishop-Gromov reduced volume, increases as the radius goes to infinity) not to the Ricci flow itself, but to a high-dimensional version of this Ricci flow formed by adjoining an M-dimensional sphere to the flow in a certain way, and then taking (formal) limits as M \to \infty. This is not precisely the tensor power trick, but is certainly in a similar spirit. For further discussion see "285G Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman's reduced volume." Example 9: the Riemann hypothesis for function fields For background, see the Wikipedia entry on the Riemann hypothesis for function fields. This example is much deeper than the previous ones, and I am not qualified to explain it in its entirety, but I can at least describe the one piece of this argument that uses the tensor power trick. For simplicity let us restrict attention to the Riemann hypothesis for curves C over a finite field F_{p^M} of prime power order. Using the Riemann-Roch theorem and some other tools from arithmetic geometry, one can show that the number \|C(F_{p^M})\| of points in C taking values (projectively) in F_{p^r} is given by the formula where g is the genus of the curve, and \omega_1,\ldots,\omega_{2g} are complex numbers that depend on p but not on M. Weil showed that all of these complex numbers had magnitude exactly p^{1/2} (the elliptic curve case g=1 being done earlier by Hasse, and the g=0 case morally going all the way back to Diophantus!), which is the analogue of the Riemann hypotheses for such curves. There is also an analogue of the functional equation for these curves, which asserts that the 2g numbers \omega_1,\ldots,\omega_{2g} come in pairs \omega, p/\omega. As a corollary, we see that \|C(F_{p^M})\| stays quite close to p^M: But because of the formula (11) and the functional equation, one can in fact deduce the Riemann hypothesis from the apparently weaker statement where the implied constants can depend on the curve C, the prime p, and the genus g, but must be independent of the "dimension" M. Indeed, from (13) and (12) one can see that the largest magnitude of the \omega_1,\ldots,\omega_{2g} (which can be viewed as a sort of "spectral radius" for an underlying operator) is at most p^{1/2}; combining this with the functional equation, one obtains the Riemann hypothesis. [This is of course only a small part of the story; the proof of (13) is by far the hardest part of the whole proof, and beyond the scope of this article.] Further reading The blog post "Amplification, arbitrage, and the tensor power trick" discusses the tensor product trick as an example of a more general "amplification trick". Minor correction to 'The tensor power trick' Before (1), the equation for the Fourier transform has \chi when it should have \xi. For consistency I've now decided to use \chi throughout that section. Examples in complexity theory 1. (Approximate counting of \mathsf{\# P} problems in {\mathsf{BPP}}^{\mathsf{NP}}) Let N(\phi) be the number of satisfying assignments of a Boolean formula \phi. Using the leftover hash lemma, there exists a polynomial time probabilistic algorithm taking oracle for SAT that given a formula \phi outputs with high probability a number a such that \frac{1}{2} N(\phi) \leq a \leq 2N(\phi) i.e. approximates within factor of 2. By replacing \phi with "tensor power" = \phi_1 \wedge \dots \wedge \phi_k, the number of assignments rises to N(\phi^k) = N(\phi)^k. Running the algorithm on this formula, \frac{1}{2^{1/k}} N(\phi) \leq a^{1/k} \leq 2^{1/k} N(\phi) and putting k=O(\frac{1}{\epsilon}) we get approximation within factor of \epsilon in time polynomial in \frac{1}{\epsilon} and length of \phi. The algorithm is by Stockmeyer. 2. (Independent set cannot be approximated to a constant factor) For any constant \delta \in (0, 1), \delta-approximating maximal independent set in a graph is \mathsf{NP}-hard. (Taking complement, the same argument goes to clique) Proof idea: The PCP theorem implies that there is a constant \delta' < 1 such that MAX-3SAT cannot be approximated within \delta'. The usual reduction from 3SAT to independent set takes a formula with m clauses and gives a graph with 7m vertices, where each vertex corresponds to one of 7 ways a clause can be satisfied, and edges correspond to conflicts. Therefore the approximation ratio is preserved, and independent set cannot be approximated within some constant factor. Applying "tensor square" G^2 allows to lower the constant arbitrarily.
34934019c734593f	onsdag 2 februari 2011 Judy Curry and "BackRadiation" In the comment Febr 1 12:28 to the thread on Slaying the Sky Dragon on Judy Curry's blog, Judy asks me: • Do you dispute that if you put an infrared radiometer on the surface of the earth and point it upwards, that it will measure an IR radiance or irradiance (depending on how the instrument is configured)? Go to http://www.arm.gov for decades worth of such measurements. And that this infrared radiation comes from IR emission by gases such as CO2 and H2O and also clouds? If you say yes, well this is what people are calling back radiation (a term that I don’t use myself). If you say no, then I will call you a crank – all your manipulations of Maxwell’s equation will not make this downwelling IR flux from the atmosphere go away. I address this question in Section 7.4 of my Sky Dragon article Computational Blackbody Radiation, and Judy's question indicates that she has not read my article. I explain there that an IR camera (infrared radiometer) directed to the sky measures the frequency of incoming light and computes by Wien's displacement law the temperature T of the emitter, and then by Stefan-Boltzmann's law Q = sigma T^4 associates a "downwelling IR-flux from the atmosphere" of size Q. The IR camera thus measures frequency/temperature which by SB is translated to "downwelling IR-flux" or "backradiation". So everything hinges on this translation. Is it Is it correct to use SB in the form Q = sigma T^4? No, because this law gives the radiated energy from a blackbody into an environment of 0 K. But the Earth surface is not at 0 K, but even warmer than the atmospheric emitter. The translation Q = sigma T^4 is thus incorrect in the sense that it indicates a fictitious "downwelling IR flux from the atmosphere" obtained by an erronous translation. Judy calls me a "crank" because I say "no to downwelling IR flux from the atmosphere". Let me then remind Judy that just saying "crank" does not mean that I am a crank in reality, and just saying "downwelling IR flux from the atmosphere" does not mean that in reality there is anything like that. Right Judy? 22 kommentarer: 1. Claes its good that you brought up the question of instrumentation. The calibration of the instrument is most important. Also the use that the instrument is put to. For instance if an instrument is properly calibrated to measure black body radiation by placing it in a cavity. Then measuring its response at cavity temperatures from +50C to -50C. Such an elaborate process is almost never carried out. However lets say we have such a properly calibrated instrument and point it to the night sky. The emissions from CO2 and H2O are line spectra not continuous like black body spectra. Do they care if they get a "reading" that the reading means anything. When IPCC advocates lose the theoretical argument their last resort is to say "well I have an meter that proves my point". Scientologists also have a meter called an E-meter to support their bogus claims. 2. I agree with you completely that Stefan-Boltzmann is being routinely, even addictively misused by climate scientists, so your answer here is right to the point: That climate scientists are fundamentally deluded in trying to apply only radiative transfer theory to the atmosphere, at the expense of the real, complex physics. Amazingly, they short-circuit the entire atmosphere when they presume the Earth's surface to be a blackbody. But I see little chance that Curry will heed your words, much less give them any respect. She and all the other "97% of climate scientists" will have to jerk their thoughts around 180 degrees to do that. As for why "physicists say nothing", what is there to say to a wayward, headstrong climatologist beyond, "you are using Stefan-Boltzmann wrong here, and here, and here, and...." And if they can't say that, it's better they say nothing. Anyway, the wayward Curry is plainly telling you she is not disposed to listen, because she has chosen her "side" and the ballgame is in play. It is such a pain to change your mind, when your side is driving the ball downfield, or even over a cliff. 3. It would help if physicists said something, but I guess they are fully occupied with string theory, quantum loop theory and multiverses, and have little to say about Maxwell's equations and radiation. 4. This is a revolutionary time, when the consensus is incompetent, and it is an amazing fact that, so far, very few are capable of saying, "the emperor has no clothes." I once briefly presented my own work to a physics department head, at the end of which she said, "I just try to keep my head down." At bottom, it is due to the reigning belief paradigm of Darwin, "survival of the fittest." The truth is, only the truth survives, but that requires a long view (indeed, a spiritual view). We can only engage the world as it is, and hopefully be the agents for the correction that must come to science, and seems long overdue. I agree with you also about string theory and quantum theory, but at present all one can point to is their barrenness in the struggle to bring forth more true knowledge, as proof of their uselessness. 5. Would like to challenge backradiation believers to construct a cooling bag for cooling beer that is based on the phenomenon backradiation! I volunteer myself to test it during next summer :>) 6. johnosullivan 7. In other words, Maxwell's equations are dead. These equations can't explain the frequency spectrum of blackbody radiation. They can't explain why electrons accelerating under the influence of nearby protons don't emit radiation. The don't explain the existence of lasers and masers. Most of all, they aren't consistent with the particulate nature of light. You can, of course, find numerous situations where Maxwell's equations and quantum mechanics make indistinguishable predictions. However, when these predictions disagree, Maxwell's equations turn out to be wrong. You may have discovered a modification of Maxwell's equations that allows the modified equation to properly predict the spectrum of blackbody radiation. TO BE USEFUL, your modification needs to give the correct answer in all other situations where Maxwell's equations disagree with QED and observation, especially situations where light appears to behave like a particle. Until then, it is senseless and irresponsible to speculate about interactions between radiation and the atmosphere using an theory that was proven to be incorrect a century ago. Even better, you need to identify some situations where your modification of Maxwell's equations and QED give different answers and then see which theory gives the correct answer. In theory, you have already identified such a situation - radiation emitted downward from the atmosphere. Which theory is correct? You can see some examples of IR spectral data for downward radiation at http://scienceofdoom.com/2010/07/24/the-amazing-case-of-back-radiation-part-two/. SOD also has references showing the agreement between observed and theoretical DLR calculated using overhead temperature and humidity data obtained from a radiosonde. I'm skeptical of what the IPCC is telling us about AGW, but abandoning well-established scientific theories about radiation for new untested hypotheses isn't going help and hurts the credibility of all skeptics until there is substantial evidence showing the new hypotheses are more relevant than established theory. 8. Maxwell is dead but not his equations which describe macro physics beyond the realm of quantum mech. 9. No, quantum mechanisms describes all macro and micro physics that can be explained by Maxwell and cannot be explained by Maxwell. To supplant QM, you need to explain everything known that is consistent with QM and at least one phenomenon that is not consistent. [A theory that gives the same results as QM (some sort of unification theory, for example), would be valuable, but wouldn't change QM's predictions for atmospheric phenomena. (Some of my previous post was lost by the blog software.) Frank. 10. Above you mentioned lack of input from physicists. I suggest the introduction to QED (Quantum Electrodynamics), a series of popular lectures by Feynman. p15: "I want emphasize that light comes in this form - particles. It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I'm telling you how it does behave - like particles." "If you put a whole lot of photomultipliers around and let some very dim light shine in various directions, the light goes into one multiplier or another and makes a click of full intensity. It is all or nothing: if one photomultiplier goes off at a given moment, none of the others goes off at the same moment... There is no splitting of light into "half-particles" that go different places." "Every instrument that has been designed to be sensitive enough to detect weak light has always ended up discovering the same thing: light behaves like particles." p14: "If we ... could see ten times more sensitively, we wouldn't need to have this discussion - we would have all seen very dim light of one color as a series of intermittent little flashes of equal intensity." Phenomena of this type killed Maxwell's equations. If wave equations can't reproduce all aspects of this "particulate" behavior, they are worthless. Frank 11. A little more of Feynman's wisdom on understanding physics from QED (p10) : "The next reason that you might think you do not understand what I am telling you is, while I am describing to you HOW Nature works, you won't understand WHY Nature works that way. But you see, nobody understand that. I can't explain why Nature behaves in this peculiar way. Finally, ... I'm going to describe to you how Nature is - and if you don't like it, that's going to get in the way of your understanding it. It's a problem that physicists have learned to deal with: They have learned to realize that whether they like a theory or they don't like a theory is NOT the essential question. Rather, it is whether or not the theory agrees with experiment. It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the point of view of common sense. The theory of QED describes Nature as absurd from the point of view of common sense. And it agrees fully with experiment. So I hope you can accept Nature as She is - absurd." Unlike Feynman, Planck and Einstein didn't have the benefit of seeing an older generation of physicists rendered obsolete because they insisted looking for a common sense theory that explained why things happen, rather than a pragmatic theory that explained what actually happens. "Experiments have Dirac's number at 1.00115965221 +/- 0.00000000004; theory puts it at 1.00115965246 +/- 0.00000000020 .... By the way, I have chosen only one number to show you. There are other things in QED that have been measured with comparable accuracy, which also agree very well. Things have been checked at distance scales that range from one hundred times the size of the earth down to one-hundredth the size of an atomic nucleus. These number as meant to intimidate you into believing that the theory is probably not too far off!" p7 Frank 12. No, QM is useless for macrophysics because the wave function is viewed to depend on 3N variables for N electrons. 13. Finally, we come to Feynman's infamous "Cargo Cult Science", which is available on the web at: http://calteches.library.caltech.edu/51/2/CargoCult.pdf Since that is short and freely available online, I don't have to quote from it. None of this has been meant as an "appeal to authority", just a measure of the size of the mountain that needs to be climbed Frank. 14. "No, QM is useless for macrophysics because the wave function is viewed to depend on 3N variables for N electrons. " That is a meaningless statement. Depending on a large number of variables does not mean that nothing at all can be predicted. As an example, a simple average of N quantities can be evaluated for huge values of N. Often there are additional symmetries which help us evaluate even quite complicated functions on a macro scale. Maxwell's equations which you want to modify and use have same type of dependencies since they depend on the precise detailed shape of the domain boundary. 15. OK, let me then ask you to predict the propagation of electromagnetic waves in the visible spectrum say, using QM. Can you do that or anybody else? 16. Your question is ill defined since you do not say what you mean by "propagation of electromagnetic waves in the visible spectrum" The emission rate of photons can be calculated. The expected absorption rate of photons in a gas can be calculated. But more to tho point. You asked a counter question instead of saying why your own modified Maxwell's equations do not have exactly the same problem which you accuse QM of having. So I'll do the same. Why does not your equations have the same problem? 17. Let me then ask you for the QM mathematical model showning the propagation of a photon? My model is a deterministic macroscopic continuum wave model subject to finite precision computation. QM is (in the Copenhagen interpretation) a microscopic statistical particle model. The difference between the models is immense. 18. That "model" is given directly by the Schroedinger equation, which give the time evolution of the probability amplitude for the photon. The differences between the models are indeed immense. The main difference being that QM has passed every experimental test it has been put to, and your model is a truncated model incapable of describing anything but the single thing you have tried to make it model. 19. Can you describe which version of the Schrödinger equation has the propagation of a photon as its solution? 20. Since you refuse to explain why your model is not "useless for macrophysics" as you say that QM is I will not humor you by repating the basic course in QM here. You say "QM is useless for macrophysics because the wave function is viewed to depend on 3N variables for N electrons." Why does your model not have this problem itself when it depends on the precise details of the physical environment in which the waves travel, the exact shapes of boundaries and exact local densities of the gases? 21. The Schwartzschild equation contains the predictions of quantum mechanics for radiation traveling through a gas. For light of a given wavelength): dI = -nkI.ds + nkB(T).ds The incremental change in the intensity of the radiation (dI) as radiation of a given wavelength passes an incremental distance (ds) through a gas has two components: 1) An absorption term that is proportional to the number of absorbing/emitting/GHG molecules (n) and the intensity of the radiation (I). 2) An emission term that is proportional to the number of absorbing/emitting/GHG molecules (n) and Planck's function, B(T), for that wavelength and the temperature of the gas. The constant of proportionality, k, is called the absorption or emission coefficient and is the same in both terms because emission is the same as absorption with time running backwards. For total intensity, one integrates over all wavelengths. One presumably doesn't need the Schrodinger eqn to derive the Schwartzschild eqn. Derivations of Planck's Law use an enclosure with a "photon gas". If one places a gas capable of absorbing and emitting radiation inside the same enclosure (Planck's oscillators, if you like the analogy), the intensity I used the absorption term will contain B(T) and the emission term needed to produce equilibrium will also contain the same factor. The Schwartzschild equation is consequence of the Planck's work, not later QM 22. Anonym 200211 08:44 You seem to be mixing up waves and photons. Claes has made his analyses around threshold energy levels which can be received by a body. This make some sense from engineering measurements. Another explanation could be that higher intensity outgoing waves from a source cancel lower intensity incoming waves from lower temperature sources and the opposite effect at receivers where high intensity waves from higher temperature sources cancel any outgoing waves. This results in a net flow only from higher temperature to lower temperature. Your explanation of absorption in terms of number of molecules is basically what Prof Hoyt Hottel found by measurements in heat exchangers. The absorptivity is a function of the partial pressure of absorbing gases. The absorptivity is also a function of the wavelengths that the receiver can absorb (ie its total emissivity) and the temperature and emissivity of the source. Now considering the trace amount of CO2 in the atmosphere it can be calculated that the absorptivity of CO2 (from the two factors very low receiver emissivity and very low quantity present) is insignificant in the radiant loss from the earth surface (which is also a variable unknown amount due to other heat transfer). This article from Dr Van Andel http://climategate.nl/wp-content/uploads/2011/02/CO2_and_climate_v7.pdf makes the point that measurements by radiosondes and satellites show that CO2 has no effect in the lower atmosphere but has a cooling effect at the TOA. Miskolczi stated in his 2010 article the physics of the atmosphere needs to be reconsidered. Van Andel supports the findings of Miskolczi and indicates something of the many complex issues such as Cosmic Rays which need to be considered. keep well cementafriend
86ebf8ede4329f66	• Tunnel Visions • Statistical Methods for Data Analysis in Particle Physics • Path Integrals for Pedestrians • Books received Tunnel Visions By M Riordan, L Hoddeson and A W Kolb University of Chicago Press Also available at the CERN bookshop The Superconducting Super Collider (SSC), a huge accelerator to be built in Texas in the US, was expected by the physicists who supported it to be the place where the Higgs boson would be discovered. Instead, the remnants of the SSC facilities at Waxahachie are now property of the chemical company Magnablend, Inc. What happened in between? What did go wrong? What are the lessons to be learnt? Tunnel Visions responds to these historical questions in a very precise and exhaustive way. Contrary to my expectations, it is not a doom and gloom narration but a down to earth story of the national pride, good physics and bad economics of one of the biggest collider projects in history. The book depicts the political panorama during the 10 years (~1983–1993) of life of the SSC project. It started during the Reaganomics, hand in hand with the International Space Station (ISS), and concluded during the first Clinton presidency after the 1990s recession and the end of the Cold War. The ISS survived, possibly because political justifications for space adventure are easier to find, but most probably because from the beginning it was an international project. The book explains the management intricacies of such a large project, the partisan support and disregard, until the final SSC demise in the US congress. For the particle-physics community this is a well-known tale, but the historical details are welcome. However, the book is more than that, because it also sheds light on the lessons learnt. The final woes of the SSC signed the definitive opening of the US particle-physics community to full international collaboration. For 50 years, without doubt, the US had been the place to go for any particle physicist. Fermilab, SLAC and Brookhaven were, and still are, great stars in the physics firmament. Even if the SSC project had not been cut, those three had to keep working in order to maintain the progress in the field. But that was too much for essentially a zero-sum budget game. The show must go on, so Fermilab got the main injector, SLAC the BaBar factory, and Brookhaven the RHIC collider. Thanks to these upgrades, the three laboratories made important progress in particle physics: top quark discovery; W and Z boson precision measurements; Higgs boson mass hunt narrowing between 113 and 170 GeV; detection of possible discrepancies in the Standard Model associated with b-meson decay; and the discovery of the liquid-like quark–gluon plasma. Why did the SSC project collapse? The authors explain the real reasons, not related to technical problems but to poor management in the first years and the clash of cultures between the US particle-physics community and the US military-industrial system. But there are also reasons of opportunity. The SSC was several steps beyond its time. To put it into context: during the years of the SSC project, at CERN the conversion of the SPS into a collider took place, along with the whole LEP programme and the beginning of the LHC project. That effort prevented any possible European contribution to the SSC. The last-ditch attempt to internationalize the SSC into a trans-Pacific partnership with Japan was also unsuccessful. The lessons from history, the authors conclude, are that at the beginning of the 1990s the costs of frontier experimental particle physics had grown too much, even for a country like the US. Multilateral international collaboration was the only way out, as the ISS showed. The Higgs boson discovery was possible at CERN. The book avoids any “hare and tortoise” comparison here, however, since in the dawning of the new century, the US became a CERN observer state with a very important in-kind contribution. In my opinion, this is where the book grows in interest because it explains how the US particle-physics community took part in the LHC programme, becoming decisive. In particular, the US technological effort in developing superconducting magnets was not wasted. The book also talks about the suspense around the Higgs search when the Tevatron was the only one still in the game during the LHC shutdown after the infamous incident in September 2008. Useful appendices providing notes, a bibliography and even a short explanation of the Standard Model complete the text. • Rogelio Palomo, University of Sevilla, Spain. Statistical Methods for Data Analysis in Particle Physics By Luca Lista Also available at the CERN bookshop Particle-physics experiments are very expensive, not only in terms of the cost of building accelerators and detectors, but also due to the time spent by physicists and engineers in designing, building and running them. With the statistical analysis of the resulting data being relatively inexpensive, it is worth trying to use it optimally to extract the maximum information about the topic of interest, whilst avoiding claiming more than is justified. Thus, lectures on statistics have become regular in graduate courses, and workshops have been devoted to statistical issues in high-energy physics analysis. This also explains the number of books written by particle physicists on the practical applications of statistics to their field. This latest book by Lista is based on the lectures that he has given at his home university in Naples, and elsewhere. As part of the Springer series of “Lecture Notes in Particle Physics”, it has the attractive feature of being short – a mere 172 pages. The disadvantage of this is that some of the explanations of statistical concepts would have benefited from a somewhat fuller treatment. The range of topics covered is remarkably wide. The book starts with definitions of probability, while the final chapter is about discovery criteria and upper limits in searches for new phenomena, and benefits from Lista’s direct involvement in one of the large experiments at CERN’s LHC. It mentions such topics as the Feldman–Cousins method for confidence intervals, the CLs approach for upper limits, and the “look elsewhere effect”, which is relevant for discovery claims. However, there seems to be no mention of the fact that a motivation for the Feldman–Cousins method was to avoid empty intervals; the CLs method was introduced to protect against the possibility of excluding the signal plus background hypothesis when the analysis had little or no sensitivity to the presence or absence of the signal. The book has no index, nor problems for readers to solve. The latter is unfortunate. In common with learning to swim, play the violin and many other activities, it is virtually impossible to become proficient at statistics by merely reading about it: some practical exercise is also required. However, many worked examples are included. There are several minor typos that the editorial system failed to notice; and in addition, figure 2.17, in which the uncertainty region for a pair of parameters is compared to the uncertainties in each of them separately, is confusing. There are places where I disagree with Lista’s emphasis (although statistics is a subject that often does produce interesting discussions). For example, Lista claims it is counter-intuitive that, for a given observed number of events, an experiment that has a larger than expected number of background events (b) provides a tighter upper limit than one with a smaller background (i.e. a better experiment). However, if there are 10 observed events, it is reasonable that the upper limit on any possible signal is better if b = 10 than if b = 0. What is true is that the expected limit is better for the experiment with smaller backgrounds. Finally, the last three chapters could be useful to graduate students and postdocs entering the exciting field of searching for signs of new physics in high energy or non-accelerator experiments, provided that they have other resources to expand on some of Lista’s shorter explanations. • Louis Lyons, University of Oxford, UK. Path Integrals for Pedestrians By E Gozzi, E Cattaruzza and C Pagani World Scientific The path integral formulation of quantum mechanics is one the basic tools used to construct quantum field theories, especially gauge-invariant theories. It is the bread and butter of modern field theory. Feynman’s original formulation developed and extended some of the work of Dirac in the early 1930s, and provided an elegant and insightful solution to a generic Schrödinger equation. This short book provides a clear, pedagogical and insightful presentation of the subject. The derivations of the basic results are crystal clear, and the applications worked out to be rather original. It includes a nice presentation of the WKB approximation within this context, including the Van Vleck and functional determinant, the connections formulae and the semiclassical propagator. An interesting innovation in this book is that the authors provide a clear presentation of the path integral formulation of the Wigner functions, which are fundamental in the study of quantum statistical mechanics; and, for the first time in an elementary book, the work of Koopman and von Neumann on classical and statistical mechanics. The book closes with a well selected set of appendices, where some further technical details and clarifications are presented. Some of the more mathematical details in the basic derivations can be found there, as well as aspects of operator ordering as seen from the path integral point formulation, the formulation in momentum space, and the use of Grassmann variables, etc. It will be difficult to find a better and more compact introduction to this fundamental subject. • Luis Álvarez-Gaumé, CERN. Books received Bananaworld: Quantum Mechanics for Primates By Jeffrey Bub Oxford University Press This is not another “quantum mechanics for dummies” book, as the author himself states. Nevertheless, it is a text that talks about quantum mechanics but is not meant for experts in the field. It explains complex concepts of theoretical physics almost without bringing up formulas, and makes no reference to a specialist background. The book focuses on an intriguing issue of present-day physics: nonlocality and the associated phenomenon of entanglement. Thinking in macroscopic terms, we know that what happens here affects only the surrounding environment. But going down to the microscopic level where quantum mechanics applies, we see that things work in a different way. Scientists discovered that in this case, besides the local effects, there are less evident effects that reveal themselves in strange correlations that occur instantaneously between remote locations. Even stronger nonlocal correlations, still consistent with relativity, have been theoretically supposed, but have not been observed up to now. This complex subject is treated by the author using a particular metaphor, which is actually more than just that: he draws a metaphoric world made of magic bananas, and simple actions that can be performed on them. Thanks to this, he is able to explain nonlocality and other difficult physics concepts in a relatively easy and comprehensive way. Even if it requires some general knowledge of mathematics and familiarity with science, this book will be accessible and interesting to a wide range of readers, as well as being an entertaining read. Particles and the Universe: From the Ionian School to the Higgs Boson and Beyond By Stephan Narison World Scientific This book aims to present the history of particle physics, from the introduction of the concept of particles by Greek philosophers, to the discovery of the last tile of the Standard Model, the Higgs boson particle, which took place at CERN in 2012. Chronologically following the development of this field of science, the author gives an overview of the most important notions and theories of particle physics. The text is divided into seven sections. The first part provides the basics concepts and a summary of the history of physics, arriving at the modern theory of forces, which are the subject of the second part. It carries on with the Higgs boson discovery and the description of some of the experimental apparatus used to study particles (from the LHC at CERN to cosmic rays and neutrino experiments). The author also provides a brief treatment of general relativity, the Big Bang model and the evolution of the universe, and discusses the future developments of particle physics. In the main body of the book, the topics are presented in a non-technical fashion, in order to be accessible to non-experts. Nevertheless, a rich appendix provides demonstrations and further details for advanced readers. The text is accompanied by plenty of images, including paintings and photographs of many of the protagonists of particle physics. Beyond the Galaxy: How Humanity Looked Beyond our Milky Way and Discovered the Entire Universe By Ethan Siegel World Scientific This book provides an introduction to astrophysics and cosmology for absolute beginners, as well as for any reader looking for a general overview of the subject and an account of its latest developments. Besides presenting what we know about the history of the universe and the marvellous objects that populate it, the author is interested in explaining how we came to such knowledge. He traces a trajectory through the various theories and the discoveries that defined what we know about our universe, as well as the boundary of what is still to be understood. The first six chapters deal with the state-of-the-art of our knowledge about the structure of the universe, its origin and evolution, general relativity and the life of stars. The following five address the most important open problems, such as: why there is more matter than antimatter, what dark matter and dark energy are, what there was before the Big Bang, and what the fate of the universe is. Written in plain English, without formulas and equations, and characterized by a clear and fluid prose, this book is suitable for a wide range of readers. Modern Physics Letters A: Special Issue on Hadrontherapy By Saverio Braccini (ed.) World Scientific The applications of nuclear and particle physics to medicine have seen extraordinary development since the discovery of X-rays by Röntgen at the end of the 19th century. Medical imaging and oncologic therapy with photons and charged particles (specifically hadrons) are currently hot research topics. This special issue of Modern Physics Letters is dedicated to hadron therapy, which is the frontier of cancer radiation therapy, and aims at filling a gap in the current literature on medical physics. Through 10 invited review papers, the volume presents the basics of hadron therapy, along with the most recent scientific and technological developments in the field. The first part covers topics such as the history of hadron therapy, radiation biophysics, particle accelerators, dose-delivery systems and treatment planning. In the second part, more specific topics are treated, including dose and beam monitoring, proton computer tomography, innoacustics and microdosimetry. This volume will be very useful to students, researchers approaching medical physics, and scientists interested in this interdisciplinary and fast-moving field. By R Wagner and A Briggs Oxford University Press Wagner, Briggs This book uses an original perspective to trace the history of the human quest for making sense of the world we live in. Written in collaboration by a painter specialising in religious subjects and a physical scientist who is a professor in the UK and also the director of a centre for research in quantum information processing, it starts from the assumption that both religion and science are manifestations of human curiosity. Science and its methods, based on reproducible experiments and evidence-based conclusions, are able to find answers to the “how” questions, to explain how nature works. This is what the authors call the “penultimate curiosity”. But the “ultimate curiosity” is “why” the world is like it is. Science doesn’t necessarily have the answer to such a question. Religions were born to try and give an answer to this. In the book, science and religion are not placed in opposition to one another. On the contrary, it is shown how they can live in a mutually enriching relationship. The authors sweep human history from caveman times to the present day, explaining the nature and evolution of the entanglement between the two. The text is also accompanied by many beautiful illustrations that are an integral part of the argument. Entropy Demystified: The Second Law Reduced to Plain Common Sense (2nd edition) By Arieh Ben-Naim World Scientific In this book, the author explains entropy and the second law of thermodynamics in a clear and easy way, and with the help of many examples. He intends, in particular, to show that these physics laws are not intrinsically incomprehensible, as they appear at first. The fact that entropy, which is defined in terms of heat and temperature, can be also expressed in terms of order and disorder, which are intangible concepts, together with the evidence that entropy (or, in other words, disorder) increases perpetually, can puzzle students. Some mystery seems to be inevitably associated with these concepts. The author asserts that, looking at the second law from the molecular point of view, everything clears up. What a student needs to know is the atomistic formulation of entropy, which comes from statistical mechanics. The aim of the book is to clarify these concepts to readers who haven’t studied statistical mechanics. Many dice games and examples from everyday life are used to make readers familiar with the subject. They are guided along a path that allows them to discover by themselves what entropy is, how it changes, and why it always changes in one direction in a spontaneous process. In this second edition, seven simulated games are also included, so that the reader can experiment with and appreciate the joy of understanding the second law of thermodynamics. About the author Compiled by Virginia Greco, CERN.
838d04bbdcc44018	About this Journal Submit a Manuscript Table of Contents Advances in Mathematical Physics Volume 2014 (2014), Article ID 795730, 14 pages Research Article On the Use of Lie Group Homomorphisms for Treating Similarity Transformations in Nonadiabatic Photochemistry CTMM, Institut Charles Gerhardt Montpellier, CNRS/Université Montpellier 2, CC 15001, Place Eugène Bataillon, 34095 Montpellier, France Received 25 March 2014; Accepted 22 May 2014; Published 15 July 2014 Academic Editor: Fabien Gatti A formulation based on Lie group homomorphisms is presented for simplifying the treatment of unitary similarity transformations of Hamiltonian matrices in nonadiabatic photochemistry. A general derivation is provided whereby it is shown that a similarity transformation acting on a traceless, Hermitian matrix through a unitary matrix of is equivalent to the product of a single matrix of by a real vector. We recall how Pauli matrices are the adequate tool when and show how the same is achieved for with Gell-Mann matrices. 1. Introduction The construction of quasidiabatic states capable of reproducing the properties of a limited set of strongly interacting adiabatic states (block or group Born-Oppenheimer approximation) is a central problem in nonadiabatic photochemistry. As discussed in [1, 2], this is closely related to the concept of an effective Hamiltonian matrix (quasidiabatic) obtained as the similarity transform of a real diagonal matrix (adiabatic). The reciprocal problem corresponds to the diagonalisation (or block-diagonalisation) of a Hermitian matrix through an invertible transformation. Few-state cases can be parameterised explicitly in terms of rotation angles whereby the operational formulae are made more tractable upon reformulating the transformation within a vector space spanned by a basis set of matrices through a Lie group homomorphism, following the original suggestion of Mead [3], further explored by Yarkony and coworkers [47]. The objective of the present work is to provide the noninitiate theoretical chemists with some basic aspects of the required mathematical background underlying this formulation and to lay the foundations of a general treatment of three-state problems where a few helpful tricks are highlighted to make the operational formulae as compact as possible. The approach proposed by Yarkony and coworkers is generalised in terms of Gell-Mann matrices. Their results based on Euler angles are confirmed and an alternative parameterisation based on Cardan angles is proposed. Pauli matrices [8], originally introduced for treating two-level spin systems in quantum mechanics and further extended to isospin symmetry and quantum electrodynamics, are well-established mathematical tools for treating two-state problems in theoretical chemistry, for example, when applied to nonadiabatic photochemistry involving conical intersections between two electronic states (see, e.g., [9, 10]). This formulation ultimately relies on a Lie group homomorphism from to , where the former is a double cover of the latter [11]. In this, traceless, Hermitian matrices are isomorphic to vectors and treated as such, and unitary similarity transformations act on them as rotation matrices act on the isomorphic vectors. Although elegant and compact, this formulation does not provide much more insight than directly separating the trace from the traceless part of a Hamiltonian matrix when considering a similarity transformation (e.g., when diagonalising) and noticing that a rotation of the two states through an angle implies a rotation through twice this angle of the half-difference and coupling entries. However, it can really make a difference to treat problems with three states or more, as it yields relationships that are much more compact and easier to manipulate, as exemplified by the seminal papers of Yarkony and coworkers on conical intersections with more than two electronic states [47]. Here, we reanalysed their approach for three-state problems upon examining the properties of Gell-Mann matrices [12], first introduced for describing the colour charge of quarks and gluons in quantum chromodynamics. We propose a trick based on a threefold equivalence to simplify the derivation of the relevant matrices of . This paper is written from a theoretical-chemistry perspective and is aimed at nonexperts in the formalism of group theory and linear algebra. It is purposely pedestrian and nonexhaustive, as its objective is to provide operational tools to facilitate the treatment of similarity transforms of Hamiltonian matrices in nonadiabatic photochemistry, where a finite set of electronic states must be considered as coupled. For details on the underlying mathematical foundations, the reader is referred to textbooks on Lie groups and algebras such as, for example, [13]. We first recall some general properties of traceless, Hermitian matrices in the context of Lie group homomorphisms for treating similarity transformations and illustrate this with Pauli matrices. Then, we focus on Gell-Mann matrices and provide some practical examples showing how this formulation can prove useful when dealing with three-state Hamiltonian matrices. 2. Lie Group Homomorphisms for Similarity Transformations Let . Any complex matrix can be uniquely expanded as where is the identity matrix of rank and is the traceless part of . As is a complex vector space (a vector space over the scalar field of complex numbers, ) with respect to matrix addition and scalar multiplication, it is possible to define a complete and linearly independent set of basis matrices, , such that where the entries of and the entries of are related through an isomorphism that depends on the particular choice made for the basis set. In what follows, bold scripts such as will be used to denote the corresponding column-vectors (and when is excluded). Note that, hereafter, we will deliberately identify the -tuple vectors of to the isomorphic -line column-vectors of for notation simplicity. The complex Frobenius (also known as Hilbert-Schmidt) inner product, defines a Hermitian metric, with respect to which the basis set can be chosen orthogonal, where is the Kronecker symbol. This implies that the -matrices are traceless, since . In addition, we choose the Hermitian, that is, . Note that they are not normalised and would be so if multiplied by a factor (and the identity by a factor ). The weight factor 2 is conventional and chosen according to the definition of Pauli and Gell-Mann matrices (see below). Any other homogeneous scaling factor would work just as well. The corresponding closure relationship reads such that the entries of satisfy It is also true that is isomorphic to . Let us now consider any complex Hermitian matrix , where The previous properties hold, except that all entries of are now real. This defines an isomorphism between and and a similar isomorphism between and , where , the set of traceless, complex Hermitian matrices, is thus an -dimensional real vector space (a vector space over the scalar field of real numbers, ) with respect to matrix addition and scalar multiplication, is a complete and orthogonal basis set of and is a Euclidean space with respect to the, now real, Frobenius inner product, which, upon expanding over and halving, canonically identifies the standard Euclidean dot product of , In addition, is related through matrix exponentiation to the -fold special unitary Lie group , for which the basis set of traceless, skew-Hermitian matrices, , is a possible representation of the infinitesimal generators belonging to the corresponding Lie algebra, . For and 3 we will consider as the three Pauli matrices, , and the eight Gell-Mann matrices, , respectively, (see next sections). Now, let us turn to unitary similarity transformations of Hermitian matrices. Through any unitary matrix , that is, , and for any , the similarity transform, , is Hermitian and breaks into where ; that is, . The trace is preserved, and the only practical difficulty from an operational perspective lies in transforming the traceless part. As both and are Hermitian and traceless too, the similarity transformation defines a linear map of the real vector space , such that, for and its isomorphic column-vector , there exists a unique, real matrix that satisfies that is, This is the first central idea of the formulation exposed here and in the papers of Yarkony and coworkers [47]. The product of three complex matrices required to evaluate the similarity transform is readily expressed as an invariant trace augmented with the product of an real matrix by an real vector. Although this may not be more efficient computationally in all cases, the corresponding expressions are less entangled. Let us now examine the properties of . The similarity transformation preserves the trace through matrix product, using the cyclicity of the trace. In other words, it preserves the inner product of , and is thus an isometry with respect to the metric of this vector space. From the isomorphism between and , we get that is, and is thus an orthogonal matrix; that is, and . The explicit expression of can be obtained from the transformation of , as any is isomorphic to the corresponding canonical basis vector of , . Hence, that is, In other words, each column of , obtained as , is made of the components of with respect to . The map corresponding to the unitary similarity transformation, is a group homomorphism. The image of in is in , as The image of the product is the product of the images, since where One, thus, deduces that the image of the inverse is the inverse of the image, This is the second central idea of this formulation: if is decomposed as a product of elementary unitary matrices, the corresponding orthogonal matrix will simply be the product of their images. An operational parameterisation of such a transformation in terms of a set of angles is thus better-factorised, as each angle appears only once (in a single matrix factor contained in ) rather than twice (in a two matrix factors contained both in and ). This group homomorphism is not an isomorphism. Indeed, for any , since , then and . The “useful” set of unitary matrices can obviously be restricted to by getting rid of the complex phases of their determinants, which is of no consequence on the similarity transform. However, there still are distinct matrices of sharing the same image. Indeed, In the following examples with and 3, we will further consider restrictions of complex matrices of to real matrices of when similarity transformations are limited to rotations applied to real Hermitian matrices. 3. Pauli Matrices for Two-State Hamiltonian Matrices We now recall some properties of the well-known Pauli matrices that can prove elegant, if not useful, in the context of two-electronic-state problems in nonadiabatic photochemistry. The three Pauli matrices are traceless, Hermitian, and defined as Thus, the corresponding isomorphism is such that For a complex Hermitian matrix , we get four real parameters, as required: and Pauli matrices satisfy, for any , where is the threefold Levi-Civita symbol. This is summarised in Table 1. Table 1: Multiplication table of Pauli matrices (left times top). As a consequence, and, for any , In the case , it is easy to demonstrate that, for any given such that , the determinant of is one and is a rotation matrix. First, let us notice that Then, As , then . The aforementioned group homomorphism can thus be restricted to and, in practice, to . The latter is of kernel such that is known as a double cover of . This result formally reflects the invariance of half-unit spins through -rotation and the Berry geometrical phase (also known as molecular Bohm-Aharonov effect) in two-state systems [3, 9, 10, 1416]. Let us now consider the particular example of a direct rotation through an angle , represented with the following orthogonal matrix: It can be expressed from matrix exponentiation of as and is such that . The corresponding similarity transformation satisfies for each basis matrix that is, The second entry of is unaffected and is rarely required in practical applications to nonadiabatic photochemistry, as the electronic states are often chosen real-valued. If so, the Hamiltonian matrix, , is a real symmetric matrix that depends only on three real parameters: and the two-entry column-vector which leads to the well-known expansion [9, 10]: The corresponding restriction of reads The similarity transform, , is thus obtained from which yields the well-known relationships used in nonadiabatic photochemistry for a two-state problem, For real symmetric matrices and rotations , the group homomorphism reduces to where . For obvious reasons, we also get , which is a manifestation of the double-valuedness [3, 9, 10, 1416] issue for such systems. These relationships can be used, for example, when deriving the condition to be fulfilled by to diagonalise , where the adiabatic Hamiltonian matrix reads with by convention. We define such that the columns of give the adiabatic states (eigenstates) in terms of the original states, where the Schrödinger equation, for , reads Hence, and rearranging the last two equations yields Alternatively, these can be used to generate an effective Hamiltonian matrix, , from the adiabatic Hamiltonian matrix and a predefined angle. The inverse transformation, , yields rotated states that span the same Hilbert subspace, and the corresponding effective Hamiltonian matrix reads Hence, This formulation is elegant and compact but is not required as such in order to derive the same relationships directly. However, it can become useful in situations where three states or more are coupled, as shown in the next section. 4. Gell-Mann Matrices for Three-State Hamiltonian Matrices The Gell-Mann matrices are the analogue of Pauli matrices for . They are traceless, Hermitian, and defined as The corresponding isomorphism is such that As required, for a complex Hermitian matrix , we get nine real parameters: and The definition of the Gell-Mann matrices and seems to imply an arbitrary choice upon which the first two labels are not treated on the same footing as the third (by labels we mean the line and column indices of , e.g., the red, green, and blue colour charges of quarks in quantum chromodynamics). In fact, this apparent distinction hides a threefold equivalence where and form a degenerate irreducible representation of -type in the threefold-rotation point group (e.g., and are the analogue of -orbitals in H3). We thus propose to define conveniently four alternative linear combinations, which are not linearly independent from and , We now have three equivalent basis sets, , , and (where for ). We further introduce such that and are the corresponding threefold rotation matrices used to particularise labels 1 and 2 instead of 3, respectively. Indeed, where for and We will later show how back and forth transformations among the three basis sets can be used as a trick that simplifies the treatment of similarity transformations for a three-state problem compared to the direct approach discussed by Yarkony and coworkers [57]. Gell-Mann matrices satisfy, for any , where the nonzero structure constants are given by They are antisymmetric under the permutation of any pair of indices; for example, . The nonzero elements of the -coefficients are In contrast, these are symmetric under the permutation of any pair of indices. As a consequence, and, for any , The corresponding multiplication table seems more complicated than in the case of . However, the threefold equivalence between the three basis sets corresponds to three underlying subgroups embedded within , based on isomorphisms between Cartan subalgebras: . Note that occurs rather than in this mapping because this preserves “direct orientation” through circular permutations of the labels 1, 2, and 3. We also introduce the -restricted identity matrices, For obvious reasons, acts as with respect to and thus commutes with them. Similar considerations apply to , , and , as well as , , and . These relationships are summarised in Tables 2, 3, 4, 5, 6, and 7. Table 2: Multiplication table of Gell-Mann matrices (left times top). Table 3: Multiplication table of Gell-Mann matrices (left times top). Table 4: Multiplication table of Gell-Mann matrices (left times top). Table 5: Multiplication table of Gell-Mann matrices (left times top). Table 6: Multiplication table of Gell-Mann matrices (left times top). Table 7: Multiplication table of Gell-Mann matrices (left times top). We now consider the three elementary, direct rotations through the angles , represented with the three following orthogonal matrices: They can be expressed as and are such that for . The corresponding similarity transformations for each basis matrix are given in Table 8 in the most adequate basis set. Only one is required, as the remaining two can be derived from it upon noticing the structural threefold equivalence between the three reordered basis sets, . Table 8: Similarity transforms of Gell-Mann matrices through elementary rotations. This yields three rotation matrices: The latter two can be transformed back into the original Gell-Mann basis set, , according to which will prove to be a useful trick from an operational point of view. All involved matrices (including ) are rotation matrices, so that any composition of the three basic rotations of yields a rotation matrix, the image of which in is a rotation matrix. In other words, the aforementioned group homomorphism can thus be restricted to (when is chosen as a rotation matrix). Let us now focus on the similarity transformation of a real symmetric matrix through a rotation. As already pointed out in the previous section, electronic states are often chosen real-valued in practice. Gell-Mann matrices can be partitioned into the set , used to define rotations through matrix exponentiation (see above), and , a basis set for real symmetric matrices. Any real symmetric matrix is now given in terms of six real parameters: and the five-entry column-vector (restricted to the basis set of real Gell-Mann matrices), which leads to the expansion Note the isomorphism between and on the one hand and and on the other hand. The similarity transform, , is thus obtained from where is the restriction of to real Gell-Mann matrices. At this stage, there are several possibilities for representing as a given rotation of . We will consider two examples, one based on Euler angles, the other one based on Cardan angles. As a first example, the rotation matrix is defined from three Euler angles as follows: The corresponding matrix satisfies where The resulting entries of are given in Table 9. They are in agreement with the entries of the matrix given in [5], except that the authors used different labels and scaling factors for their basis matrices, Let us now consider Cardan angles such that The corresponding matrix satisfies where Note that the last occurrence of is irrelevant, as, in practice, one can consider a simpler transformation acting on , such that where The resulting entries
84938dd0074d39eb	Sturm–Liouville theory From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the form \frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right]+q(x)y=-\lambda w(x)y, where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation ('1') at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function. The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of ('1') satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S–L) problem. Such values of λ, when they exist, are called the eigenvalues of the boundary value problem defined by ('1') and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equations that are separable. A Sturm–Liouville (S–L) problem is said to be regular if p(x), w(x) > 0, and p(x), p'(x), q(x), and w(x) are continuous functions over the finite interval [ab], and have separated boundary conditions of the form Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that: • The eigenvalues λ1, λ2, λ3, ... of the regular Sturm–Liouville problem ('1')-('2')-('3') are real and can be ordered such that \lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n < \cdots \to \infty; • Corresponding to each eigenvalue λn is a unique (up to a normalization constant) eigenfunction yn(x) which has exactly n − 1 zeros in (ab). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regular Sturm–Liouville problem ('1')-('2')-('3'). \int_a^b y_n(x)y_m(x)w(x)\,\mathrm{d}x = \delta_{mn}, in the Hilbert space L2([ab], w(x)dx). Here δmn is a Kronecker delta. Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense. Sturm–Liouville form[edit] The differential equation ('1') is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear ordinary differential equations can be recast in the form on the left-hand side of ('1') by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector.) The Bessel equation[edit] x^2y''+xy'+\left (x^2-\nu^2 \right )y=0 which can be written in Sturm–Liouville form as (xy')'+ \left (x-\frac{\nu^2}{x}\right )y=0. The Legendre equation[edit] which can easily be put into Sturm–Liouville form, since D(1 − x2) = −2x, so, the Legendre equation is equivalent to An example using an integrating factor[edit] Divide throughout by x3: Multiplying throughout by an integrating factor of \mu(x) =e^{\int -\frac{1}{x^2}\, \mathrm{d}x}=e^{\frac{1}{x}}, e^{\frac{1}{x}}y''-\frac{e^{\frac{1}{x}}}{x^2} y'+ \frac{2 e^{\frac{1}{x}}}{x^3} y = 0 which can be easily put into Sturm–Liouville form since D e^{\frac{1}{x}} = -\frac{e^{\frac{1}{x}}}{x^2} so the differential equation is equivalent to (e^{\frac{1}{x}}y')'+\frac{2 e^{\frac{1}{x}}}{x^3} y =0. The integrating factor for a general second order differential equation[edit] multiplying through by the integrating factor \mu(x) = \frac{1}{P(x)} e^{\int \frac{Q(x)}{P(x)} \mathrm{d}x}, and then collecting gives the Sturm–Liouville form: \frac{d}{dx} (\mu(x)P(x)y')+\mu(x)R(x)y=0 or, explicitly, \frac{d}{dx} \left (e^{\int \frac{Q(x)}{P(x)} \mathrm{d}x}y' \right )+\frac{R(x)}{P(x)} e^{\int \frac{Q(x)}{P(x)}\,\mathrm{d}x} y = 0 Sturm–Liouville equations as self-adjoint differential operators[edit] The map Lu = -\frac{1}{w(x)} \left(\frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}u}{\mathrm{d}x}\right]+q(x)u \right) can be viewed as a linear operator mapping a function u to another function Lu. One may study this linear operator in the context of functional analysis. In fact, equation ('1') can be written as L u = \lambda u. This is precisely the eigenvalue problem; that is, one is trying to find the eigenvalues λ1, λ2, λ3, ... and the corresponding eigenvectors u1, u2, u3, ... of the L operator. The proper setting for this problem is the Hilbert space L2([a, b], w(x) dx) with scalar product \langle f, g\rangle = \int_{a}^{b} \overline{f(x)} g(x)w(x)\,\mathrm{d}x. In this space L is defined on sufficiently smooth functions which satisfy the above boundary conditions. Moreover, L gives rise to a self-adjoint operator. This can be seen formally by using integration by parts twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem, one looks at the resolvent (L - z)^{-1}, \qquad z \in\mathbb{C}, where z is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous equation, which can be done using the variation of parameters formula. This shows that the resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem). As a consequence of the Arzelà–Ascoli theorem, this integral operator is compact and existence of a sequence of eigenvalues αn which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that (L-z)^{-1} u = \alpha u, \qquad L u = \left (z+\alpha^{-1} \right ) u, are equivalent. If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case, the spectrum no longer consists of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in quantum mechanics, since the one-dimensional time-independent Schrödinger equation is a special case of a S–L equation. We wish to find a function u(x) which solves the following Sturm–Liouville problem: L u = -\frac{\mathrm{d}^2u}{\mathrm{d}x^2} = \lambda u where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example u(0) = u(\pi) = 0. Observe that if k is any integer, then the function u(x) = \sin kx is a solution with eigenvalue λ = k2. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors. Given the preceding, let us now solve the inhomogeneous problem L u =x, \qquad x\in(0,\pi) with the same boundary conditions. In this case, we must write f(x) = x in a Fourier series. The reader may check, either by integrating ∫exp(ikx)x dx or by consulting a table of Fourier transforms, that we thus obtain L u =\sum_{k=1}^{\infty}-2\frac{(-1)^k}{k}\sin kx. This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in L2 which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier's series converges at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series). Therefore, by using formula ('4'), we obtain that the solution is u=\sum_{k=1}^{\infty}2\frac{(-1)^k}{k^3}\sin kx. In this case, we could have found the answer using anti-differentiation. This technique yields u= \tfrac{1}{6} \left (x^3 -\pi^2 x \right), whose Fourier series agrees with the solution we found. The anti-differentiation technique is no longer useful in most cases when the differential equation is in many variables. Application to normal modes[edit] Certain partial differential equations can be solved with the help of S–L theory. Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L1, 0 ≤ y ≤ L2. The equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation: \frac{\partial^2W}{\partial x^2}+\frac{\partial^2W}{\partial y^2} = \frac{1}{c^2}\frac{\partial^2W}{\partial t^2}. The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c2)T"/T. Since the three terms of this equation are functions of x,y,t separately, they must be constants. For example, the first term gives X" = λX for a constant λ. The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L1 or y = 0, L2 and define the simplest possible S–L eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence, W_{mn}(x,y,t) = A_{mn}\sin\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi y}{L_2}\right)\cos\left(\omega_{mn}t\right) where m and n are non-zero integers, Amn are arbitrary constants, and \omega^2_{mn} = c^2 \left(\frac{m^2\pi^2}{L_1^2}+\frac{n^2\pi^2}{L_2^2}\right). The functions Wmn form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies \omega_{mn}. This representation may require a convergent infinite sum. Representation of solutions and numerical calculation[edit] The Sturm–Liouville differential equation (1) with boundary conditions may be solved in practice by a variety of numerical methods. In difficult cases, one may need to carry out the intermediate calculations to several hundred decimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places. 1. Shooting methods.[1][2] These methods proceed by guessing a value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [ab], comparing the value this solution takes at the other endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues. 2. Finite difference method. 3. The Spectral Parameter Power Series (SPPS) method[3] makes use of a generalization of the following fact about second order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then the function y(x) \int_a^x \frac{\mathrm{d}t}{p(t)y(t)^2} is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ0* (often λ0* = 0; it does not need to be an eigenvalue) and any solution y0 of (1) with λ = λ0* which does not vanish on [ab]. (Discussion below of ways to find appropriate y0 and λ0.) Two sequences of functions X(n)(t), X~(n)(t) on [ab], referred to as iterated integrals, are defined recursively as follows. First when n = 0, they are taken to be identically equal to 1 on [ab]. To obtain the next functions they are multiplied alternately by 1/(py02) and wy02 and integrated, specifically X^{(n)}(t) = \begin{cases} - \int_a^x X^{(n-1)}(t) p(t)^{-1} y_0(t)^{-2}\,\mathrm{d}t & n \text{ odd}, \\ \int_a^x X^{(n-1)}(t)y_0(t)^{2} w(t) \,\mathrm{d}t & n \text{ even} \end{cases} \widetilde X^{(n)}(t) = \begin{cases} \int_a^x \widetilde X^{(n-1)}(t)y_0(t)^{2} w(t)\,\mathrm{d}t &n \text{ odd}, \\ when n > 0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ: u_0 = y_0 \sum_{k=0}^\infty \left (\lambda-\lambda_0^ \right )^k \widetilde X^{(2k)}, u_1 = y_0 \sum_{k=0}^\infty \left (\lambda-\lambda_0^* \right )^k X^{(2k+1)}. Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.) Next one chooses coefficients c0, c1 so that the combination y = c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a) = 0 and X~(n)(a) = 0, for n > 0. The values of X(n)(b) and X~(n)(b) provide the values of u0(b) and u1(b) and the derivatives u0'(b) and u1'(b), so the second boundary condition (3) becomes an equation in a power series in λ. For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the sought-after eigenvalues. When λ = λ0, this reduces to the original construction described above for a solution linearly independent to a given one. The representations ('5'),('6') also have theoretical applications in Sturm–Liouville theory.[3] Construction of a nonvanishing solution[edit] The SPPS method can, itself, be used to find a starting solution y0. Consider the equation (py')' = μqy; i.e., q, w, and λ are replaced in (1) by 0, −q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ0 = 0. While there is no guarantee that u0 or u1 will not vanish, the complex function y0 = u0 + iu1 will never vanish because two linearly independent solutions of a regular S–L equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y0 of (1) for the value λ0 = 0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded. Application to PDEs[edit] For a linear second order in one spatial dimension and first order in time of the form: f(x) \frac{\partial^2 u}{\partial x^2} + g(x) \frac{\partial u}{\partial x}+h(x) u= \frac{\partial u}{\partial t}+k(t) u Let us apply separation of variables, which in doing we must impose that: u(x,t) =X(x) T(t) Then our above PDE may be written as: \frac{\hat{L} X(x)}{X(x)} = \frac{\hat{M} T(t)}{T(t)} \hat{L}=f(x) \frac{\mathrm{d}^2}{\mathrm{d} x^2}+g(x) \frac{\mathrm{d}}{\mathrm{d}x}+h(x), \qquad \hat{M}=\frac{\mathrm{d}}{\mathrm{d}t} +k(t) Since, by definition, \hat{L} and X(x) are independent of time t and \hat{M} and T(t) are independent of position x, then both sides of the above equation must be equal to a constant: \hat{L} X(x) =\lambda X(x) X(a)=X(b)=0 \, \hat{M} T(t) =\lambda T(t) \, The first of these equations must be solved as a Sturm–Liouville problem. Since there is no general analytic (exact) solution to Sturm–Liouville problems, we can assume we already have the solution to this problem, that is, we have the eigenfunctions X_n (x) and eigenvalues \lambda_n . The second of these equations can be analytically solved once the eigenvalues are known. \frac{\mathrm{d}}{\mathrm{d}t} T_n (t)= (\lambda_n -k(t)) T_n (t) T_n (t) = a_n e^{-\left(\lambda_n t -\int_0^t k(\tau) \mathrm{d}\tau\right)} u(x,t) =\sum_n a_n X_n (x) e^{-\left(\lambda_n t -\int_0^t k(\tau) \mathrm{d}\tau\right)} a_n =\frac{\langle X_n (x), s(x)\rangle}{\langle X_n(x),X_n (x)\rangle} \langle y(x),z(x)\rangle = \int_a^b y(x) z(x) w(x) \mathrm{d}x w(x)= \frac{e^{\int \frac{g(x)}{f(x)} \mathrm{d}x}}{f(x)} See also[edit] 1. ^ J. D. Pryce, Numerical Solution of Sturm–Liouville Problems, Clarendon Press, Oxford, 1993. 2. ^ V. Ledoux, M. Van Daele, G. Vanden Berghe, "Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics," Comput. Phys. Comm. 180, 2009, 532–554. 3. ^ a b V. V. Kravchenko, R. M. Porter, "Spectral parameter power series for Sturm–Liouville problems," Mathematical Methods in the Applied Sciences (MMAS) 33, 2010, 459–468 Further reading[edit]
ad3caf9c4c15e3d1	Take the 2-minute tour × While investigating the EPR Paradox, it seems like only two options are given, when there could be a third that is not mentioned - Heisenberg's Uncertainty Principle being given up. The setup is this (in the wikipedia article): given two entangled particles, separated by a large distance, if one is measured then some additional information is known about the other; the example is that Alice measures the z-axis and Bob measures the x-axis position, but to preserve the uncertainty principle it's thought that either information is transmitted instantaneously (faster than light, violating the special theory of relativity) or information is pre-determined in hidden variables, which looks to be not the case. What I'm wondering is why the HUP is not questioned? Why don't we investigate whether a situation like this does indeed violate it, instead of no mention of its possibility? Has the HUP been verified experimentally to the point where it is foolish to question it (like gravity, perhaps)? It seems that all the answers are not addressing my question, but addressing waveforms/commutative relations/fourier transforms. I am not arguing against commutative relations or fourier transforms. Is not QM the theory that particles can be represented as these fourier transforms/commutative relations? What I'm asking this: is it conceivable that QM is wrong about this in certain instances, for example a zero state energy, or at absolute zero, or in some area of the universe or under certain conditions we haven't explored? As in: Is the claim then that if momentum and position of a particle were ever to be known somehow under any circumstance, Quantum Mechanics would have to be completely tossed out? Or could we say QM doesn't represent particles at {absolute zero or some other bizarre condition} the same way we say Newtonian Physics is pretty close but doesn't represent objects moving at a decent fraction of the speed of light? EPR Paradox: "It considered two entangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance." "According to EPR there were two possible explanations. Either there was some interaction between the particles, even though they were separated, or the information about the outcome of all possible measurements was already present in both particles." These are from the wikipedia article on the EPR Paradox. This seems to me to be a false dichotomy; the third option being: we could measure the momentum of one entangled particle, the position of the other simultaneously, and just know both momentum and position and beat the HUP. However, this is just 'not an option,' apparently. I'm not disputing that two quantities that are fourier transforms of each other are commutative / both can be known simultaneously, as a mathematical construct. Nor am I arguing that the HUP is indeed false. I'm looking for justification not just that subatomic particles can be models at waveforms under certain conditions (Earth like ones, notably), but that a waveform is the only thing that can possibly represent them, and any other representation is wrong. You van verify the positive all day long, that still doesn't disprove the negative. It is POSSIBLE that waveforms do not correctly model particles in all cases at all times. This wouldn't automatically mean all of QM is false, either - just that QM isn't the best model under certain conditions. Why is this not discussed? share\|improve this question I +1d to get rid of the downvote you had. It's the last line that did it for me. – Olly Price Aug 13 '12 at 22:37 Anyone who is downvoting care to elaborate on where my question is unclear, unuseful or shows no effort? I'd be glad to improve it if I can. – Ehryk Aug 13 '12 at 23:40 Try Bohmian mechanics. – MBN Sep 6 '12 at 11:02 @Ehryk: Not my downvote, but this question is a waste of time. You misunderstood what EPR is all about. The EPR effects have nothing to do with HUP, and you can show that they are inconsistent with local variables determining experimental outcomes without doing quantum mechanics, just from the experimental outcomes themselves. This means the weirdness is not due to the formalism, but really there in nature. – Ron Maimon Sep 11 '12 at 6:15 So in a universe without the commutative relation/HUP, where the commutative relation was sometimes zero / position and momentum could both be known, where's the paradox with EPR? You could just determine the values of both entangled particles, no paradox necessary. – Ehryk Sep 11 '12 at 7:24 12 Answers 12 up vote 5 down vote accepted In precise terms, the Heisenberg uncertainty relation states that the product of the expected uncertainties in position and in momentum of the same object is bounded away from zero. Your entanglement example at the end of your edit does not fit this, as you measure only once, hence have no means to evaluate expectations. You may claim to know something but you have no way to check it. In other entanglement experiments, you can compare statistics on both sides, and see that they conform to the predictions of QM. In your case, there is nothing to compare, so the alleged knowledge is void. The reason why the Heisenberg uncertainty relation is undoubted is that it is a simple algebraic consequence of the formalism of quantum mechanics and the fundamental relation $[x,p]=i\hbar$ that stood at the beginning of an immensely successful development. Its invalidity would therefore imply the invalidity of most of current physics. Bell inequalities are also a simple algebraic consequence of the formalism of quantum mechanics but already in a more complex set-up. They were tested experimentally mainly because they shed light on the problem of hidden variables, not because they are believed to be violated. The Heisenberg uncertainty relation is mainly checked for consistency using Gedanken experiments, which show that it is very difficult to come up with a feasible way of defeating it. In the past, there have been numerous Gedanken experiments along various lines, including intuitive and less intuitive settings, and none could even come close to establishing a potential violation of the HUP. Edit: One reaches experimental limitations long before the HUP requires it. Nobody has found a Gedankenexperiment for how to do defeat the HUP, even in principle. We don't know of any mechanism to stop an electron, thereby bringing it to rest. It is not enough to pretend such a mechanism exists; one must show a way how to achieve it in principle. For example, electron traps only confine an electron to a small region a few atoms wide, where it will roam with a large and unpredictable momentum, due to the confinement. Thus until QM is proven false, the HUP is considered true. Any invalidation of the foundations of QM (and this includes the HUP) would shake the world of physicists, and nobody expects it to happen. share\|improve this answer Why wouldn't it just invalidate it under certain conditions? For example: by some means, we completely arrest an electron. Position = center of device, momentum = 0. Both known simultaneously. Couldn't we just say QM is 'not a valid model for arrested particles but works for moving ones' without invalidating most of current physics? – Ehryk Sep 7 '12 at 21:43 Any invalidation of the foundations of QM would shake the world of physicists. - But the center of a device is usually poorly definable, and an electron cannot be arrested completely, neither in position nor in momentum. One reaches experimental limitations long before the HUP requires it. - In the past, there have been numerous Gedanken experiments along similar and many other lines, and none could even come close to establishing a violation of the HUP. – Arnold Neumaier Sep 9 '12 at 13:12 In practice, I get this. I'm not claiming that we can make such a machine now, or soon. But the HUP seems to say such a machine cannot exist, now or ever, with more advanced races or technologies or anything. Are you saying an electron cannot be arrested completely - anywhere, ever, inside a black hole, at absolute zero - under no conditions ever? – Ehryk Sep 9 '12 at 19:40 until QM is proven false, the HUP is true. – Arnold Neumaier Sep 10 '12 at 11:53 @Ehryk: here's why you're seeming nonsensical to everyone here: at small length scales, an electron looks very, very much like a wave. You get interference patterns and everything. Now, you want to 'stop' it. Well, a "slower" electron has a longer wavelength than a "faster" one, but this longer wavelength is going to spread it out farther. By the time you get to your limit of a 'stopped' electron, the electron will be spread out over all of space. – Jerry Schirmer Sep 14 '12 at 23:13 In quantum mechanics, two observables that cannot be simultaneously determined are said to be non-commuting. This means that if you write down the commutation relation for them, it turns out to be non-zero. A commutation relation for any two operators $A$ and $B$ is just the following $$[A, B] = AB - BA$$ If they commute, it's equal to zero. For position and momentum, it is easy to calculate the commutation relation for the position and momentum operators. It turns out to be $$[\hat x ,\hat p] = \hat x \hat p - \hat p \hat x = i \hbar$$ As mentioned, it will always be some non-zero number for non-commuting observables. So, what does that mean physically? It means that no state can exist that has both a perfectly defined momentum and a perfectly defined position (since $ \|\psi \rangle$ would be both a right eigenstate of momentum and of position, so the commutator would become zero. And we see that it isn't.). So, if the uncertainty principle was false, so would the commutation relations. And therefore the rest of quantum mechanics. Considering the mountains of evidence for quantum mechanics, this isn't a possibility. I think I should clarify the difference between the HUP and the classical observer effect. In classical physics, you also can't determine the position and momentum of a particle. Firstly, knowing the position to perfect accuracy would require you to use a light of infinite frequency (I said wavelength in my comment, that's a mistake), which is impossible. See Heisenberg's microscope. Also, determining the position of a particle to better accuracy requires you use higher frequencies, which means higher energy photons. These will disturb the velocity of the particle. So, knowing the position better means knowing the momentum less. The uncertainty principle is different than this. Not only does it say you can't determine both, but that the particle literally doesn't have a well defined momentum to be measured if you know the position to a high accuracy. This is a part of the more general fact in quantum mechanics that it is meaningless to speak of the physical properties of a particle before you take measurements on them. So, the EPR paradox is as follows - if the particles don't have well-defined properties (such as spin in the case of EPR), then observing them will 'collapse' the wavefunction to a more precise value. Since the two particles are entangled, this would seem to transfer information FTL, violating special relativity. However, it certainly doesn't. Even if you now know the state of the other particle, you need to use slower than light transfer of information to do anything with it. Also, Bell's theorem, and Aspect's tests based off of it, show that quantum mechanics is correct, not local realism. share\|improve this answer So how do we know that all particles have a non-commuting relationship, always and forever, under all conditions, even the ones we aren't able to measure or with technology or knowledge we don't yet possess? – Ehryk Sep 6 '12 at 10:47 What if you define position and momentum as the two real numbers that you measure at time t from experiment? (That's what most people consider "position" and "momentum" to be anyway.) What is this "new" definition of position and momentum? – Nick Sep 8 '12 at 23:11 Let me add this: I've taken QM and done those calculations for the commutation plenty of times to figure out what sets of compatible observables there are. But I could give someone a random formula for some random integral and divide by 6.3 and say "look, this always comes out to a real value -- thus position and momentum can't be simultaneously well-defined!" and that makes no sense whatsoever. Yeah, I know the whole spiel about eigenvalues and eigenstates and identical preparations of quantum systems, but what kind of physical experiment demonstrates this limit? – Nick Sep 8 '12 at 23:15 Noncommutativity of operators nicely explains emission spectra, which I believe were the subject of Heisenberg's (?) initial ponderings. There's a nice bit of this history explained at page 40 of this book by Alain Connes alainconnes.org/docs/book94bigpdf.pdf (there is probably a more focused reference for this history, but I don't know of one) – Ryan Thorngren Sep 9 '12 at 0:51 The Heisenberg's relation is not tied to quantum mechanics. It is a relation between the width of a function and the width of its fourier transform. The only way to get rid of it is to say that x and p are not a pair of fourier transform: ie to get rid of QM. share\|improve this answer So if by any means at all (entanglement, future machines, or divine powers) one could measure both position and momentum simultaneously, then all of quantum mechanics is false? There could be no QM in a universe in which this is possible? – Ehryk Aug 14 '12 at 9:35 You necessarily need to change the relationship between position and momemtum. It is mathematically impossible if they just form a fourier transform pair. But considering the huge amount of datas validating QM, one can try to extend QM by adding a small term in the pair or by using a fractional commutator (with fractional derivative) for instance. – Shaktyai Aug 14 '12 at 9:43 How about saying x and p are a pair of fourier transforms USUALLY, but not in certain circumstances such as {inside a black hole, at absolute zero, under certain entanglement experiments, in a zero rest energy universe, etc.} How do we know that because QM is right USUALLY or from what we can observe, that it is right ALWAYS and FOREVER? – Ehryk Sep 6 '12 at 10:43 That is to say: QM as we know it is not valid in these cases. There is no possible objections to such a statement, but for it to get accepted by the physicists, you need to prove that you can explain things in a simpler way and that you can predict something measurable. – Shaktyai Sep 6 '12 at 10:46 Because there is no proof whatsoever that QM fails. The day it fails we shall reconsider the question. However, there are many theorists working on alternative theories, so you have your chances. – Shaktyai Sep 6 '12 at 15:09 The wave formulation has in its seed the uncertainty relation. Let me be precise what is meant by the wave formulation: the amplitude over space points will give information about localization on space, while amplitude over momenta will give information about localization in momentum space. But for a function, the amplitude over momenta is nothing else but the Fourier transform of the space amplitude. The following is jut a mathematical fact, not up to physical discussion: the standard deviation, or the spread of the space amplitude, multiplied by the spread of the momenta amplitude (given by the Fourier transform of the former) will be bounded from below by one. So, it should be pretty clear that, as long as we stick to a wave formulation for matter fields, we are bound mathematically by the uncertainty relation. No work around over that. Why we stick to a wave formulation? because it works pretty nicely. The only way someone is going to seriously doubt that is the right description is to either: 1) find an alternate description that at least explains everything that the wave formulation describes, and hopefully some extra phenomena not predicted by wave formulation alone. 2) find an inconsistency in the wave formulation. In fact, if someone ever manages to measure both momenta and position for some electron below the Planck factor, it would be definitely an inconsistency in the wave formulation. It would mean we would have to tweak the De Broglie ansatz or something equally fundamental about it. Needless to say, nothing like that has happened share\|improve this answer It's a mathematical fact IF the particle can indeed be wholly represented by that specific function, right? So in the entanglement experiment, perhaps that function does not represent the state of TWO entangled particles? Maybe we have entanglement wrong, or maybe that function does not represent particles in certain conditions? Why are these possibilities not even discussed? – Ehryk Sep 6 '12 at 18:05 @Ehryk, because scientists, as all humans, tend to do the least amount of effort that will get the job done, it really does not make economical sense to do otherwise. As i said, there would be something to discuss if something in the experiment would not turn out as expected, but it does. If you want to do your life's mission to prove false the wave representation, then you need to build an experiment that will either confirm it or disprove it. then, people will likely start seriously discussing other possibilities. – lurscher Sep 6 '12 at 18:13 We can't prove Zeus doesn't exist, yet we don't accept his existence because of this. An idea shouldn't have to be 'debunked' to have a healthy amount of doubt in it, yet the wave formulation representing all particles, everywhere, at all times and locations seems to be presented 'beyond doubt' - so why is it stated with such certainty about unknowability and when challenged, the opposition gives in without so much as a mention? – Ehryk Sep 6 '12 at 18:26 (I'm not trying to prove it wrong, or stating that it is, I'm asking if it can be false and if so, why it's not treated as such) – Ehryk Sep 6 '12 at 18:28 @Ehryk, suppose someone starts asking why physicists assume that we only have one time dimension, and why we don't try to debunk that. We would reply the same thing; we have no reason to devote resources to debunk something that seems to fit so nicely with existing phenomena, so the ball is in the court of the person that insist that, say, two-dimensional time makes great deal of sense for X or Y experiment. Then, if the experiment sounds like something that has not been tested, and is under budget to implement, maybe some experimentalists will try to do it. That is how science works – lurscher Sep 6 '12 at 18:30 If we want the position and the momentum to be well-defined at each moment of time, the particle has to be classical. We inherited these notions from classical mechanics, where they apply successfully. Also they apply at macroscopic level. So, it is a natural question to ask if we can keep their good behavior in QM. Frankly, there is nothing to stop us to do this. We can conceive a world in which the particles are pointlike all the time, and move along definite trajectories, and this will "beat HUP". This was the first solution to be looked for. Einstein and de Broglie tried it, and not only them. Even Bohr, in his model, envisioned electrons as moving along definite trajectories in the atom (before QM). David Bohm was able to develop a model which has at a sub-quantum level this property, and in the meantime behaves like QM at quantum level. The price to be paid is to allow interactions which "beat the speed of light", and to adjust the model whenever something incompatible with QM was found. IMHO, this process of adjustments still continues today, and this looks very much like adding epicycles in the heliocentric model. But I don't want to be unfair with Bohm and others: it is possible to emulate QM like this, and if we learn QM facts which contradict it, it will always be possible to find such a model which behaves like QM, but also has a subquantum level which consists of classical-like point particles with definite positions and momenta. At this time, these examples prove that what you want is possible. One may argue that they are unaesthetic, because they are indeed more complicated than QM. But this doesn't mean that they are not true. Also, at this time they don't offer anything testable which QM can't offer. So, while QM describes what we observe, the additional features of hidden variable theories are not observable, more complicated, and violate special relativity. Or, if they don't violate special relativity, they contradict what QM predicts and we observed in experiments of entanglement like that of Alan Aspect. If EPR presents us with two alternatives, (1) spooky action at distance, (2) QM is incomplete, and that you propose, (3) HUP is false, let's not forget that Aspect's experiment and many others confirmed the alternative (1). Now, it would be much better for such models if they would stop adjusting themselves to mimic QM, and predict something new, like a violation of HUP. This would really be something. In conclusion, yes, you are right and in principle it is possible to beat HUP. The reason why most physicists don't care too much about this, is that the known ways to beat HUP are ugly, have hidden elements, violate other principles. But others consider them beautiful and useful, and if you are interested, start with Bohm's theory and the more recent developments of this. Synopsis: The Certainty of Uncertainty Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements (arXiv link) share\|improve this answer This was rather helpful, so I appreciate it. I'm still just having difficulty wrestling with unknowability in relation to this; for example if we ever found a way to arrest a particle completely; we'd know it's position and momentum (0) both at the same time, and while it violated HUP, it could just be said 'this particle cannot be represented by a wavefunction.' The reach of the HUP seems to include this though, with no provisions, and just be accepted so OBVIOUSLY you can't stop a particle. Would we just say the particle is classical in that instance? – Ehryk Sep 6 '12 at 18:20 @Cristi I see (and generally have no objections to) your argument, but that conclusion seems misleading. Yes, it's possible to beat HUP (by discarding quantum mechanics) in the same sort of sense that it's possible to create a macroscopic stable wormhole: not strictly ruled out, but there is no evidence to support it. So I think it's misleading to be saying that this is possible. – David Z Sep 6 '12 at 18:45 @David Zaslavsky: Thanks. To make clear my conclusion, and less misleading, I wrote the first, rather lengthy, paragraph. This contains for instance the statement "while QM describes what we observe, the additional features of hidden variable theories are not observable, more complicated, and violate special relativity." Anyway, I considered it would be more misleading to claim that one knows HUP can't be violated no matter what. – Cristi Stoica Sep 6 '12 at 19:31 @Ehryk: "What happened to particle-wave duality?". Particles are represented as wavefunctions. They are defined on the full space, but may have a small support (bump functions). At limit, when concentrated at a point, bump becomes Dirac's $\delta$ function. Then, it has definite position $x$, but indefinite wave vector, so it spreads immediately (this corresponds to HUP). Its "dual" is a pure wave (with definite wave vector $k_x$, hence momentum $p_x$). The "particle-wave" duality refers to these two extreme cases. But most of the times the "wavicles" are somewhere between these two extremes. – Cristi Stoica Sep 6 '12 at 20:41 @Ehryk: "How does a wave have mass?" They have momentum and energy: multiply wave 4-vector with $\hbar$ and obtain $4$-momentum, so yes, they have mass. Interesting thing: the rest mass $m_0$ is the same, even though in general the wave 4-vector is undetermined. By "undetermined" you can understand that the wavefunction is a superposition of a large (usually infinite) number of pure wavefunctions. Pure wavefunctions have definite wave vector (hence momentum), but totally undetermined position. – Cristi Stoica Sep 6 '12 at 20:50 You are asking if a more complete theory might show that HUP is wrong and that position and momentum do exist simultaneously. But a more complete theory has to explain all the observations that QM already explains, and those observations already show that position and momentum cannot have definite values simultaneously. This is known because when particles such as photon, electrons, or even molecules are sent through a pair of slits one at a time, an interference pattern on the detector plate appears that shows that the probability of the measured location and time follows a specific mathematical relationship. The fact that certain regions have zero probability shows that before measurement, the particles exist in a superposition of possible states, such that the wave function for those states can cancel out with other states resulting in areas of low probability of observation. The observed relationships through increasingly complex experiments rules out possibilities other than what is described by QM. The only way that QM could be superseded by a new theory is for new observations to be made that violate QM, but the new theory would still result in the same predictions as QM in the circumstances that QM has already been tested. Since HUP results directly from QM, HUP would also follow from a new theory with the only possible exception in conditions such as super high energy conditions such when a single particle is nearly a black hole. Basically you have to get used to the idea that particles are really quantized fluctuations in a field and that the field exists in a superposition of states. Any better theory will simply provide additional details about why the field behaves in that way. share\|improve this answer "Accept it as true until it's debunked" is not scientific. "When a particle can be perfectly represented by a waveform and ONLY a waveform, then it cannot have definite momentum and position" is acceptable. Asserting the "When" is "Always and Forever" is not. – Ehryk Sep 15 '12 at 0:05 if can help Open timelike curves violate Heisenberg's uncertainty principle ...and show that the Heisenberg uncertainty principle between canonical variables, such as position and momentum, can be violated in the presence of interaction-free CTCs.... Foundations of Physics, March 2012, Volume 42, Issue 3, pp 341-361 ...considering that a D-CTC-assisted quantum computer can violate both the uncertainty principle... Phys Rev Lett, 102(21):210402, May 2009. arxiv 0811.1209 ...show how a party with access to CTCs, or a "CTC-assisted" party, can perfectly distin- guish among a set of non-orthogonal quantum states.... Phys. Rev. A 82, 062330 2010. arxiv 1003.1987v2 ...and can be interacted with in the way described by this simple model, our results conﬁrm those of Brun et al that non-orthogonal states can be discriminated... ...Our work supports the conclusions of Brun et al that an observer can use interactions with a CTC to allow them to discriminate unknown, non-orthogonal quantum states – in contradiction of the uncertainty principle... share\|improve this answer The only way to make Heisenberg's principle irrelevant is to measure the speed and the position (to make it simple) of a fundamental particle. In other words, you would have to observe a particle, without having it collide with a photon or reacting to a magnetic force, or without interacting with it. There might be an other way, which would be to find a very general law (but not statistical) which describes the characteristics (spin, speed, position etc) of an elemental particle in an absolute way.... share\|improve this answer I think that's just the observer effect, described in another answer, and I can beat that by hypothesyzing a future race that has developed a gravitational particle-position-and-momentum sensor machine, which does not use photons or interact with the particle in any way that would change the position or momentum (a read only sensor). Even in this case, the HUP says they CANNOT be known simultaneously. – Ehryk Sep 6 '12 at 10:56 I want to know what evidence there is to support this, even in the case of such a hypothetical machine. – Ehryk Sep 6 '12 at 10:57 In this case you interact using the gravitationnal interaction, so that's almost the same. – Yves Sep 6 '12 at 11:21 Not really. Bombarding it with photons are distinct events; surrounding it by a machine that is sensitive to the gravitation inside of it would only exert the same gravity that any other matter around it would, and if done as stated in my hypothetical, would not alter the position or momentum in any way once the particle has settled inside the machine. – Ehryk Sep 6 '12 at 11:24 Very interesting, and it would be possible if such a machine existed (my first point). But how would you measure something else than a change in the surronding gravitationnal field (which would imply an interaction with the particule) and how would you measure a spin ? It sounds like your method is equivalent to trying to measure an absolute quantity of energy, or to "forcing" the position or momentum of your particle, a case which doesn't fall under Heisenberg's principle. This reasoning might end up as a Ouroboros.. – Yves Sep 6 '12 at 11:33 "Heisenberg uncertainty principle" is a school term that is used in popular literature. It simply does not matter. What matters is the wavefunction and Schroedinger equation. The EPR paradox experiment never used any explicit "uncertainty principle" in the proof. share\|improve this answer As @MarkM pointed out above, what I meant but wasn't able to espouse was a 'non-commutation' property (a term I've not heard of in this context), or the claim that the exact position and momentum of a particle cannot be known simultaneously. I thought this was semantically equivalent to the Heisenberg Uncertainty Principle, which I guess it is not. – Ehryk Aug 13 '12 at 23:30 Also, from wikipedia: "The uncertainty principle is a fundamental concept in quantum physics." (from the disambiguation page, main article here: en.wikipedia.org/wiki/Uncertainty_principle ). Could you explain or give sources for it 'not matter'ing? Further, the wiki article on the EPR Paradox explicitly uses the Heisenberg Uncertainty Principle - I'm not claiming WP is any authority, but it would be the source of my confusion. – Ehryk Aug 13 '12 at 23:38 @Annix This isn't true. Firstly, Heisenberg's matrix mechanics is an equally valid formulation of QM as wave mechanics, see Zettlli page 3. Second, the uncertainty principle is a part of wave mechanics. As you say, you can easily derive it from the Schrodinger equation. I find it odd that you say that this somehow makes the uncertainty principle irrelevant. You can't simultaneously know position and momentum to perfect accuracy, since localizing the position of the particle involves adding plane waves, which then makes the momentum uncertain. – Mark M Aug 13 '12 at 23:58 @Anixx If you claim that you may derive the HUP from the Schrödinger equation, you should show it. I actually think it is not possible, but I'm curious. One usually derive the HUP from the commutation relations and later one shows it is preserved by the unitary evolution. The Schr. equation tells us how the states evolve in time, while the HUP must be verified even in the initial state so I'm very skeptical about your derivation. In any case, the HUP is at least as fundamental as the Schr. equation and it is a term very often used in technical papers and seminars. – drake Aug 14 '12 at 0:28 @drake You can't derive it from the SE, but from the wave mechanics formulation (which is what I guess Annix means). See the'Proof of Kennard Inequality using Wave Mechanics' sub-section here: en.wikipedia.org/wiki/… However, I agree with you that the HUP is fundamental (see my above post.). – Mark M Aug 14 '12 at 1:32 Without gravity: The uncertainty principle is not really a principle because it is a derivable statement, it is not postulated. It is derivable and proven mathematically. Once you prove something you cannot unprove it. That means it cannot turn out to be false. For experimental verifications, see for example this article by Zeilinger et al and the references inside. Zeilinger is a world expert on quantum phenomena and it is expected that he will get Nobel prize in the future. With gravity, (and that matters only at extremely high energy, as high as the Planck scale): Intuitively you can use the uncertainty principle to give an estimate about the energy needed to resolve tiny region of space. For sufficiently small region in space you will create a black hole. So there is a limit on the spacial resolution one can achieve, because of gravity. If you try to use higher energy you will create a bigger black hole. Bottom line is, uncertainty principle does not make sense in this case because space loses its meaning and it cannot be defined operationally. share\|improve this answer Things can be unproven if one of the axioms or postulates they are based on is proven false. HUP may be true if <x, y and z> are true, but it certainly is based on foundations (waveforms representing matter, for one) that are not infallible. – Ehryk Aug 14 '12 at 11:37 @Ehryk You cannot unprove something by changing the postulates, because then you are talking about totally different problem. You can compare only 2 situations giving the same postulates/axioms. The axioms are true and not false in the sense that the coherent structure coming out of those postulates leads to predictions that are consistent with experimental observations. The world is quantum mechanical. – Revo Aug 14 '12 at 16:03 You cannot unprove it as a model of how things could work, no, but you could show that it is just not the most accurate model of the world we live it - just like we can theorize about hyperbolic geometry as a model, though it's unlikely to be the model of reality. Is it the case that you could not have a variant of something like QM that produces similar results while in some instances allowing precise position and momentum values, in the same way newton's laws were 'good enough' for the values we had measured at non relativistic speeds up until that point? – Ehryk Aug 15 '12 at 1:43 @Ehryk No. You could not have had something similar to Newtonian meachanics that underlies Quantum Mechanics. What you are thinking of has been thought of for long time ago, it is unknown as hidden variables theories. It has been proven experimentally that something like Newtonian mechanics or any deterministic theory cannot be the basis of Quantum Mechanics. May be you should also keep in mind the following main point: QM is more general than CM, hence it is more fundamental. Since QM is more general than CM, one should understand how CM emerges from QM, not the other way around. – Revo Aug 15 '12 at 1:50 @Ehryk One should understand CM in terms of QM not QM in terms of CM. – Revo Aug 15 '12 at 1:52 The way I see it, HUP cannot be disproven "at absolute zero", because absolute zero cannot be physically reached, er... due to HUP... is circular reasoning good enough? Let's try something else. Maybe try to imagine what would happen if HUP was to be violated? For one, I guess the proton - electron charges would cause one or two electrons to fall down into the nucleus, as HUP normally prevents that (if the electron fell down on nucleus we'd know it's position with great precision, requiring it to have indeterminate but large momentum, so it kind of settles for orbiting around nucleus). If you know more about the stuff than I do, try to imagine what else would happen, and how likely is that effect. For example, if HUP violation would imply violation of 2nd law of thermodynamics, this would render HUP violation pretty unlikely. That much from a layman. share\|improve this answer But then why can't we just say 'HUP is only for particles not at absolute zero'? It seems like violating it is 'not an option', even as above - so an electron falls into the nucleus. It has a measurable position and momentum. Why does HUP have to hold so strongly that we instead are comfortable with 'that particle must always have energy'? – Ehryk Sep 6 '12 at 18:31 The way I see it "absolute zero" is purely theoretical concept. Look up Bose-Einstein condensate, get a feeling for what happens at extremely low temperatures and then try to project that further to zero. Doesn't click. So saying "HUP is only for particles at absolute zero" is like saying "HUP is for all particles", for absolute zero can't be reached. – pafau k. Sep 6 '12 at 18:54 Do you have evidence or citations that nothing can be absolute zero? Or are you just asserting it? Note that saying 'we can't get to absolute zero' is different than 'no particle anywhere, at any time, can be at absolute zero.' – Ehryk Sep 6 '12 at 19:10 Let me quote the beginning of Wikipedia entry on absolute zero :) "Absolute zero is the theoretical temperature at which entropy reaches its minimum value", note the word theoretical. Temperature always flows from + to -, so the simple explanation is: you'd have to have something below absolute zero to cool something else to absolute zero. (this would violate laws of thermodynamics). – pafau k. Sep 6 '12 at 19:59 Transfer heat from hot to hotter? Decrease the volume of the container. Cool matter? Increase the volume of the container. In both cases, heat is not 'transferred', but temperature (average kinetic energy) has been changed without the interaction of other matter, either hotter or colder. – Ehryk Sep 10 '12 at 11:18 The Heisenberg uncertainty principle forms one of the most important pillars in physics. It can't be proven wrong because too many experimentally determined phenomena are a result of the uncertainty principle. However, something may be discovered in the future that can make a modification to the uncertainty principle - in a similar way that Newton's laws were modified by Einstein's special relativity. Saying that the uncertainty principle is wrong is like saying that Newton's law is wrong. In reply to the comments, I'm not saying that it can be falsified. It can't. In a classical sense, it will always be correct, in a similar way that Newton's law will always be correct.However, it can be modified. Until the day that all the open questions in physics have been resolved, how can you claim that the uncertainty principle can't be modified further? Do we know everything about extra dimensions? Do we know everything about string theory and physics at the Planck scale? By the way, it has already been modified. Please check this link. The uncertainty principle will always be correct. However, it can and has been modified. In its current formalism and interpretation, it could represent a special case of a larger underlying theory. The claim that the current formalism and limitations to the uncertainty principle are absolute and can never be modified under any circumstance in the universe, is a claim that does not obey the uncertainty principle itself. share\|improve this answer The uncertainty principle is a lot closer to uncertainty law than your answer lets on. It's not really about measurement so much as it's about a Fourier Transform. – Brandon Enright Jan 26 '14 at 23:47 The Heisenberg Uncertainty Principle is an unfalsifiable claim? All of (good) science is falsifiable. See the first paragraph: en.wikipedia.org/wiki/Falsifiability – Ehryk Jan 28 '14 at 6:12 protected by Qmechanic Jan 26 '14 at 23:37 Would you like to answer one of these unanswered questions instead?
1c46c983998c8105	Does quantum theory have to be interpreted? by aotell Witnessing the ongoing discussion about how quantum theory should be interpreted, and the strong opinions and sometimes even dogmatic arguments, I decided to write a series of blog posts that will try to discuss the issue of interpretation as objectively as I possibly can. I will not specifically try to compare the different mainstream interpretation with each other, but rather explore the requirement of an interpretation at all and the possibility of answering the same fundamental questions using strong scientific rigor instead. A scientific theory is usually defined as consisting of a mathematical apparatus that allows to perform calculations of predictive nature, and a layer of interpretational glue that connects the resulting numbers with measurements that we can actually perform. The separation of measurement and prediction works very well for all classical theories, where observer and experiment can be regarded as entirely separate entities. Quantum theory however makes a clean cut between observer and the observed experiment impossible, because after the experiment the two subsystems are interwoven in a very fundamental and complicated way, even if spatially separated. The nonlocal entanglement of the quantum state space does not allow us to use the approximation of objectivity anymore. Understanding this problem, there are two main approaches of dealing with it. The older one insists on the classical separation and is willing to live with the necessary consequences. The Copenhagen interpretation introduces the Heisenberg cut between quantum and classical domains to recover the notion of an objective observer that can make classical statements about the measurement outcome. And with that cut we also get the interpretational glue back that relates mathematics with measurement results. This happens in the form of the well known measurement postulate which includes the Born rule describing the statistical outcome of a measurement. The approach has several drawbacks however. Firstly, the location of the Heisenberg cut is more or less arbitrary as long as the observer and the system are well distinguishable, but becomes impossible as soon as this is not the case anymore. Often this does not pose a problem, but it is still a shortcoming as it keeps us from understanding certain realizable situations. Secondly, the Copenhagen and related interpretations leave us entirely in the dark as to what precisely happens during a measurement. Still, the Copenhagen interpretation is fundamentally scientific, as it focuses on measurements and predictions only, and does not take into account what is not observable. The other main approach to the problem of observation takes the alternative route. Instead of introducing a cut, everything is taken into account. Experiment and measurement device become one system, which is itself a part of the largest system, the universe. It is then only consequential to assume the time evolution of undisturbed quantum systems as formulated in the Copenhagen interpretation, the Schrödinger equation, as the evolution law for the universe. Within this approach, all predictions and results must emerge only from the properties of the evolving system, as there is no external observer that can measure anything, and no classical measurement device either. The time evolution would also be fully deterministic and the randomness of the measurement outcome could also be regarded as an emergent property. So when Hugh Everett III came up with his many worlds or relative state interpretation, he did really not at all want to create an interpretation in the sense of the Copenhagen interpretation, namely as a layer of translation between math and measurement. Rather, he wanted to create a scientific theory of emergence, where all results are derived as inherent properties of the system itself. And he was willing to accept all the consequences it brought, because the approach was rigorously scientific and only the logical consequence of avoiding the artificial Heisenberg cut. Unfortunately, not everything worked out as well as this approach had been promising. Of course, the most famous consequence is the existence of arbitrarily many worlds containing observers that have seen any possible experimental outcome. While this is philosophically hard to accept for some, it surely is only an acceptable consequence if the other results work out correctly. And these results ought to be the precise statements of the measurement postulate of the Copenhagen interpretation, because those are experimentally verified. However, while the many worlds theory gives a reasonably good explanation for the state collapse, it fails to give the right statistics. There has been some criticism regarding the collapse too, but more importantly it is generally agreed that the Born rule does not come out of the relative state theory unless extra postulates are added. Decoherence theory, which incorporates the environment to move coherence away from the experiment, or more recent attempts to use psychologically founded counting mechanisms for calculating the relative outcome probabilities, have not been generally convincing enough to consider the issues of the theory be solved. And adding postulates of course spoils the initial idea of having an actual theory of emergence. So where does this leave us? We have a practical approach that works most of the time, but hides some possibly important features and mechanisms from us. And we have a holistic approach that stands on a beautiful theoretical idea, but fails to deliver the right results and comes with some curious side effects. The question that I will explore in the following articles is what Everett’s approach has to do with the relationship between simulation and reality, and whether something that he and others have potentially overlooked could lead to a new theory with better results. And I promise, I’ll have a few surprises for you! About these ads
2a6e870383aeca12	Take the 2-minute tour × For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the explanation. However I'm not so stupid to think I can understand this landmark without some years of physics. I'll just be happy to be able to read the equation and recognize it in its various forms. To be more precise, here are my questions. Hyperphysics tell me that Shrodinger's equation "is a wave equation in terms of the wavefunction". 1. Where is the wave equation in the most general form of the equation? $$\mathrm{i}\hbar\frac{\partial}{\partial t}\Psi=H\Psi$$ I thought wave equation should be of the type It's the difference in order of of derivation that is bugging me. From Wikipedia "The equation is derived by partially differentiating the standard wave equation and substituting the relation between the momentum of the particle and the wavelength of the wave associated with the particle in De Broglie's hypothesis." 2. Can somebody show me the passages in a simple (or better general) case? 3. I think this questions is the most difficult to answer to a newbie. What is the Hamiltonian of a state? How much, generally speaking, does the Hamiltonian have to do do with the energy of a state? 4. What assumptions did Schrödinger make about the wave function of a state, to be able to write the equation? Or what are the important things I should note in a wave function that are basilar to proof the equation? With both questions I mean, what are the passages between de Broglie (yes there are these waves) and Schrödinger (the wave function is characterized by)? 5. It's often said "The equation helps finds the form of the wave function" as often as "The equation helps us predict the evolution of a wave function" Which of the two? When one, when the other? share\|improve this question Philisophically I always find requests to explain an equation for the laymen to be a little strange. The point of writing it in math is to have a precise, and complete representation of the theory... – dmckee Dec 15 '12 at 16:13 You're right. That's why I tried to make it clear I'm not asking an explanation of the "equation" as you mean it, rather the meaning of the "symbols in it". In particulart question number 1 is the most important for me now. – Temitope.A Dec 15 '12 at 17:04 For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein. – Qmechanic Dec 15 '12 at 18:21 3 Answers 3 up vote 10 down vote accepted You should not think of the Schrödinger equation as a true wave equation. In electricity and magnetism, the wave equation is typically written as with two temporal and two spatial derivatives. In particular, it puts time and space on 'equal footing', in other words, the equation is invariant under the Lorentz transformations of special relativity. The one-dimensional time-dependent Schrödinger equation for a free particle is $$ \mathrm{i} \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}$$ which has one temporal derivative but two spatial derivatives, and so it is not Lorentz invariant (but it is Galilean invariant). For a conservative potential, we usually add $V(x) \psi$ to the right hand side. Now, you can solve the Schrödinger equation is various situations, with potentials and boundary conditions, just like any other differential equation. You in general will solve for a complex (analytic) solution $\psi(\vec r)$: quantum mechanics demands complex functions, whereas in the (classical, E&M) wave equation complex solutions are simply shorthand for real ones. Moreover, due to the probabilistic interpretation of $\psi(\vec r)$, we make the demand that all solutions must be normalized such that $\int \|\psi(\vec r)\|^2 dr = 1$. We're allowed to do that because it's linear (think 'linear' as in linear algebra), it just restricts the number of solutions you can have. This requirements, plus linearity, gives you the following properties: 1. You can put any $\psi(\vec r)$ into Schrödinger's equation (as long as it is normalized and 'nice'), and the time-dependence in the equation will predict how that state evolves. 2. If $\psi$ is a solution to a linear equation, $a \psi$ is also a solution for some (complex) $a$. However, we say all such states are 'the same', and anyway we only accept normalized solutions ($\int \|a\psi(\vec r)\|^2 dr = 1$). We say that solutions like $-\psi$, and more generally $e^{i\theta}\psi$, represent the same physical state. 3. Some special solutions $\psi_E$ are eigenstates of the right-hand-side of the time-dependent Schrödinger equation, and therefore they can be written as $$-\frac{\hbar^2}{2m} \frac{\partial^2 \psi_E}{\partial x^2} = E \psi_E$$ and it can be shown that these solutions have the particular time dependence $\psi_E(\vec r, t) = \psi_E(\vec r) e^{-i E t/\hbar}$. As you may know from linear algebra, the eigenstates decomposition is very useful. Physically, these solutions are 'energy eigenstates' and represent states of constant energy. 4. If $\psi$ and $\phi$ are solutions, so is $a \psi + b \phi$, as long as $\|a\|^2 + \|b\|^2 = 1$ to keep the solution normalized. This is what we call a 'superposition'. A very important component here is that there are many ways to 'add' two solutions with equal weights: $\frac{1}{\sqrt 2}(\psi + e^{i \theta} \phi)$ are solutions for all angles $\theta$, hence we can combine states with plus or minus signs. This turns out to be critical in many quantum phenomena, especially interference phenomena such as Rabi and Ramsey oscillations that you'll surely learn about in a quantum computing class. Now, the connection to physics. 1. If $\psi(\vec r, t)$ is a solution to the Schrödinger's equation at position $\vec r$ and time $t$, then the probability of finding the particle in a specific region can be found by integrating $\|\psi^2\|$ around that region. For that reason, we identify $\|\psi\|^2$ as the probability solution for the particle. • We expect the probability of finding a particle somewhere at any particular time $t$. The Schrödinger equation has the (essential) property that if $\int \|\psi(\vec r, t)\|^2 dr = 1$ at a given time, then the property holds at all times. In other words, the Schrödinger equation conserves probability. This implies that there exists a continuity equation. 2. If you want to know the mean value of an observable $A$ at a given time just integrate $$ <A> = \int \psi(\vec r, t)^* \hat A \psi(\vec r, t) d\vec r$$ where $\hat A$ is the linear operator associated to the observable. In the position representation, the position operator is $\hat A = x$, and the momentum operator, $\hat p = - i\hbar \partial / \partial x$, which is a differential operator. The connection to de Broglie is best thought of as historical. It's related to how Schrödinger figured out the equation, but don't look for a rigorous connection. As for the Hamiltonian, that's a very useful concept from classical mechanics. In this case, the Hamiltonian is a measure of the total energy of the system and is defined classically as $H = \frac{p^2}{2m} + V(\vec r)$. In many classical systems it's a conserved quantity. $H$ also lets you calculate classical equations of motion in terms of position and momentum. One big jump to quantum mechanics is that position and momentum are linked, so knowing 'everything' about the position (the wavefunction $\psi(\vec r))$ at one point in time tells you 'everything' about momentum and evolution. In classical mechanics, that's not enough information, you must know both a particle's position and momentum to predict its future motion. share\|improve this answer Thank you! One last question. How do somebody relate the measurment principle to the equations, that an act of measurment will cause the state to collapse to an eigenstate? Or is time a concept indipendent of the equation? – Temitope.A Dec 16 '12 at 11:37 Can states of entanglement be seen in the equation to? – Temitope.A Dec 16 '12 at 11:47 Note that user10347 talks of a potential added to the differential equation. To get real world solutions that predict the result of a measurement one has to apply the boundary conditions of the problem. The "collapse" vocabulary is misleading. A measurement has a specific probability of existing in the space coordinates or with the fourvectors measured. The measurement itself disturbs the potential and the boundary conditions change, so that after the measurement different solutions/psi functions will apply. – anna v Dec 16 '12 at 13:23 One type of measurement is strong measurement, where we the experimentalists, measure some differential operator $A$, and find some particular (real) number $a_i$, which is one of the eigenvalues of $A$. (Important detail: for $A$ to be measureable, it must have all real eigenvalues.) Then, we know the wavefunction "suddenly" turns into $\psi_i$, which is the eigenfunction of $A$ whose eigenvalue was that number $a_i$ we measured. The system has lost of knowledge of the original wavefunction $\psi$. The probability of measuring $a_i$ is $\|<\psi_i \| \psi>\|^2$. – emarti Dec 18 '12 at 7:12 @Temitope.A: Entanglement isn't obvious in anything here because I've only written single-particle wavefunctions. A two-particle wavefunction $\Psi(\vec r_1, \vec r_2)$ gives a probability $\int_{V_1}\int_{V_2}\|\Psi\|^2 d \vec r_1 d \vec r_2$ of detecting one particle in a region $V_1$ and a second particle in a region $V_2$. A simple solution for distinguishable particles is $\Psi(\vec r_1, \vec r_2) = \psi_1(\vec r_1) \psi_2(\vec r_2)$, and it can be shown that this satisfies all our conditions. An entangled state cannot be written so simply. (Indistinguishable particles take more care.) – emarti Dec 18 '12 at 9:32 If you take the wave equation $$\nabla^2\phi = \frac{1}{u^2}\frac{d^2\phi}{dt^2}\text{,}$$ and consider a single frequency component of a wave while taking out its time dependence, $\phi = \psi e^{-i\omega t}$, then: $$\nabla^2 \phi = -\frac{4\pi^2}{\lambda^2}\phi\text{,}$$ but that means the wave amplitude should satisfy an equation of the same form: $$\nabla^2 \psi = -\frac{4\pi^2}{\lambda^2}\psi\text{,}$$ and if you know the de Broglie relation $\lambda = h/p$, where for a particle of energy $E$ in a potential $V$ has momentum $p = \sqrt{2m(E-V)}$, so that: $$\underbrace{-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi}_{\hat{H}\psi} = E\psi\text{,}$$ Therefore, the time-independent Schrödinger equation has a connection to the wave equation. The full Schrödinger equation can be recovered by putting time-dependence back in, $\Psi = \psi e^{-i\omega t}$ while respecting the de Broglie $E = \hbar\omega$: $$\hat{H}\Psi = (\hat{H}\psi)e^{-i\omega t} = \hbar\omega \psi e^{-i\omega t} = i\hbar\frac{\partial\Psi}{\partial t}\text{,}$$ and then applying the principle of superposition for the general case. However, in this process the repeated application of the de Broglie relations takes us away from either classical waves or classical particles; to what extent the resulting "wave function" should be considered a wave is mostly a semantic issue, but it's definitely not at all a classical wave. As other answers have delved into, the proper interpretation for this new "wave function" $\Psi$ is inherently probabilistic, with its modulus-squared representing a probability density and the gradient of the complex phase being the probability current (scaled by some constants and the probability density). As for the de Broglie relations themselves, it's possible to "guess" them from making an analogy from waves to particles. Writing $u = c/n$ and looking for solutions close to plane wave in form, $\phi = e^{A+ik_0(S-ct)}$, the wave equation gives: $$\begin{eqnarray} \nabla^2A + (\nabla A)^2 &=& k_0^2[(\nabla S)^2 - n^2]\text{,}\\ \nabla^2 S +2\nabla A\cdot\nabla S &=& 0\text{.} \end{eqnarray}$$ Under the assumption that the index of refraction $n$ changes slowly over distances on the order of the wavelength, then $A$ does not vary extremely, the wavelength is small, and so $k_0^2 \propto \lambda^{-2}$ is large. Therefore the term in the square brackets should be small, and we can make the approximation: $$(\nabla S)^2 = n^2\text{,}$$ which is the eikonal equation that links the wave equation with geometrical optics, in which motion of light of small wavelengths in a medium of well-behaved refractive index can be treated as rays, i.e., as if described by paths of particles/corpuscles. For the particle analogy to work, the eikonal function $S$ must take the role of Hamilton's characteristic function $W$ formed by separation of variables from the classical Hamilton-Jacobi equation into $W - Et$, which forces the latter to be proportional to the total phase of the wave, giving $E = h\nu$ for some unknown constant of proportionality $h$ (physically Planck's constant). The index of refraction $n$ corresponds to $\sqrt{2m(E-V)}$. This is discussed in, e.g., Goldstein's Classical Mechanics, if you're interested in details. share\|improve this answer Your first equation is a wave equation, only if you substitute the total time derivatives by partial ones. Moreover, you introduce a $\Psi = \psi e^{-i\omega t} = \phi$, but the wavefunction $\Psi$ does not satisfy the first equation for a wave. – juanrga Dec 18 '12 at 11:21 What you write is the time-dependent Schrödinger equation. This is not the equation of a true wave. He postulated the equation using a heuristic approach and some ideas/analogies from optics, and he believed on the existence of a true wave. However, the correct interpretation of $\Psi$ was given by Born: $\Psi$ is an unobservable function, whose complex square $\|\Psi\|^2$ gives probabilities. In older literature $\Psi$ is still named the wavefunction, In modern literature the term state function is preferred. The terms "wave equation" and "wave formulation" are legacy terms. In fact, part of the confusion that had Schrödinger, when he believed that his equation described a physical wave, is due to the fact he worked with single particles. In that case $\Psi$ is defined in an abstract space which is isomorphic to the tri-dimensional space. However, when you consider a second particle and write $\Psi$ for a two-body system, the isomorphism is broken and the superficial analogy with a physical wave is completely lost. A good discussion of this is given in Ballentine textbook on quantum mechanics (section 4.2). The Schrödinger equation cannot be derived from wave theory. This is why the equation is postulated in quantum mechanics. There is no Hamiltonian for one state; the Hamiltonian is characteristic of a given system with independence of its state. Energy is a possible physical property of a system, one of the possible observables of a system; it is more correct to say that the Hamiltonian gives the energy of a system in the cases when the system is in a certain state. A quantum system always has a Hamiltonian, but not always has a defined energy. Only certain states $\Psi_E$ that satisfy the time-independent Schrödinger equation $H\Psi_E = E \Psi_E$ are associated to a value $E$ of energy. The quantum system can be in a superposition of the $\Psi_E$ states or can be in more general states for which energy is not defined. Wavefunctions $\Psi$ have to satisfy a number of basic requirements such as continuity, differentiability, finiteness, normalization... Some texts emphasize that the wavefunctions would be single-valued, but I already take this in the definition of function. The Schrödinger equation gives both "the form of the wave function" and "the evolution of a wave function". If you know $\Psi$ at some initial time and integrate the time-dependent Schrödinger equation you obtain the form of the wavefunction to some other instant: e.g. the integration is direct and gives $\Psi(t) = \mathrm{Texp}(-\mathrm{i}/\hbar \int_0^t H(t') dt') \Psi(0)$, where $\mathrm{Texp}$ denotes a time-ordered exponential. This equation also gives the evolution of the initial wavefunction $\Psi(0)$. When the Hamiltonian is time-independent, the solution simplifies to $\Psi(t) = \exp(-\mathrm{i}Ht/\hbar) \Psi(0)$. For stationary states, the time-dependent Schrödinger equation that you write reduces to the time-independent Schrödinger equation $H\Psi_E = E \Psi_E$; the demonstration is given in any textbook. For stationary states there is no evolution of the wavefunction, $\Psi_E$ does not depend on time, and solving the equation only gives the form of the wavefunction. share\|improve this answer Good answer. I would only add that regarding the last point, I think the confusion comes from references to the "time-independent" Schrodinger eigenvalue equation $H\psi_E = E\psi_E$ being conflated with the "time-dependent" evolution equation $\mathrm{i}\hbar \dot{\psi} = H\psi$, when of course the two are entirely different beasts. – Chris White Dec 15 '12 at 21:07 @ChrisWhite Good point. Made. – juanrga Dec 16 '12 at 2:33 6 paragraph: maybe you should add that the equation only holds if H is time-independent. – ungerade Dec 16 '12 at 12:19 @ungerade Another good point! Added evolution when H is time-dependent. – juanrga Dec 16 '12 at 12:49 Your Answer
ce0c6a3cdcc58c22	Reactivity (chemistry) From Wikipedia, the free encyclopedia Jump to: navigation, search Reactivity in chemistry refers to • the chemical reactions of a single substance, • the chemical reactions of two or more substances that interact with each other, • the systematic study of sets of reactions of these two kinds, • methodology that applies to the study of reactivity of chemicals of all kinds, • experimental methods that are used to observe these processes, • theories to predict and to account for these processes. The chemical reactivity of a single substance (reactant) covers its behaviour in which it: • Decomposes • Forms new substances by addition of atoms from another reactant or reactants • Interacts with two or more other reactants to form two or more products The chemical reactivity of a substance can refer to the variety of circumstances (conditions that include temperature, pressure, presence of catalysts) in which it reacts, in combination with the: • Variety of substances with which it reacts, • Equilibrium point of the reaction (i.e., the extent to which all of it reacts) • Rate of the reaction The term reactivity is related to the concepts of chemical stability and chemical compatibility. An alternative point of view[edit] Reactivity is a somewhat vague concept in chemistry. It appears to embody both thermodynamic factors and kinetic factors—i.e., whether or not a substance reacts and how fast it reacts. Both factors are actually distinct, and both commonly depend on temperature. For example, it is commonly asserted that the reactivity of group one metals (Na, K, etc.) increases down the group in the periodic table, or that hydrogen's reactivity is evidenced by its reaction with oxygen. In fact, the rate of reaction of alkali metals (as evidenced by their reaction with water for example) is a function not only of position within the group but particle size. Hydrogen does not react with oxygen—even though the equilibrium constant is very large—unless a flame initiates the radical reaction, which leads to an explosion. Restriction of the term to refer to reaction rates leads to a more consistent view. Reactivity then refers to the rate at which a chemical substance tends to undergo a chemical reaction in time. In pure compounds, reactivity is regulated by the physical properties of the sample. For instance, grinding a sample to a higher specific surface area increases its reactivity. In impure compounds, the reactivity is also affected by the inclusion of contaminants. In crystalline compounds, the crystalline form can also affect reactivity. However in all cases, reactivity is primarily due to the sub-atomic properties of the compound. Although it is commonplace to make statements that substance 'X is reactive', all substances react with some reagents and not others. For example, in making the statement that 'sodium metal is reactive', we are alluding to the fact that sodium reacts with many common reagents (including pure oxygen, chlorine, hydrochloric acid, water) and/or that it reacts rapidly with such materials at either room temperature or using a bunsen flame. 'Stability' should not be confused with reactivity. For example, an isolated molecule of an electronically state of the oxygen molecule spontaneously emits light after a statistically defined period. The half-life of such a species is another manifestation of its stability, but its reactivity can only be ascertained via its reactions with other species. Causes of reactivity[edit] The second meaning of 'reactivity', that of whether or not a substance reacts, can be rationalised at the atomic and molecular level using older and simpler valence bond theory and also atomic and molecular orbital theory. Thermodynamically, a chemical reaction occurs because the products (taken as a group) are at a lower free energy than the reactants; the lower energy state is referred to as the 'more stable state'. Quantum chemistry provides the most in-depth and exact understanding of the reason this occurs. Generally, electrons exist in orbitals that are the result of solving the Schrödinger equation for specific situations. All things (values of the n and ml quantum numbers) being equal, the order of stability of electrons in a system from least to greatest is unpaired with no other electrons in similar orbitals, unpaired with all degenerate orbitals half filled and the most stable is a filled set of orbitals. To achieve one of these orders of stability, an atom reacts with another atom to stabilize both. For example, a lone hydrogen atom has a single electron in its 1s orbital. It becomes significantly more stable (as much as 100 kilocalories per mole, or 420 kilojoules per mole) when reacting to form H2. It is for this same reason that carbon almost always forms four bonds. Its ground state valence configuration is 2s2 2p2, half filled. However, the activation energy to go from half filled to fully filled p orbitals is so small it is negligible, and as such carbon forms them almost instantaneously. Meanwhile the process releases a significant amount of energy (exothermic). This four equal bond configuration is called sp3 hybridization. The above three paragraphs rationalise, albeit very generally, the reactions of some common species, particularly atoms, but chemists have so far been unable to jump from such general considerations to quantitative models of reactivity. == The rate of any given reaction, Reactants → Products is governed by the rate law: where the rate is the change in the molar concentration in one second in the rate-determining step of the reaction (the slowest step), [A] is the product of the molar concentration of all the reactants raised to the correct order, known as the reaction order, and k is the reaction constant, which is constant for one given set of circumstances (generally temperature and pressure) and independent of concentration. The greater the reactivity of a compound the higher the value of k and the higher the rate. For instance, if, A+B → C+D where n is the reaction order of A, m is the reaction order of B, n+m is the reaction order of the full reaction, and k is the reaction constant. See also[edit]
b5cea60b7fb5912d	Higgs Standard Model The fundamental types of particles in the Universe, now complete. 5 sigma annoucement So, the Higgs boson has been discovered! That’s good news. You may have also heard that the Higgs gives mass to everything in the Universe, and that it’s a field. The odd thing is that all of these things are true, if not intuitive. There are some attempts to explain it simply, but as you can see, even the top ones are not very clear. So let’s give you something to sink your teeth into: How do fundamental particles, including the Higgs boson, get their mass? Cow Moose in a Rain Storm Image credit: Highway Man of WhiteBlaze.net. The Higgs field is like rain, and there is no place you can go to keep dry. Just like there’s no way to shield yourself from gravitation, there’s no way to hide from the rain that is the Higgs field. If there were no Higgs field, all the fundamental particles would be like dried-out sponges. Massless, dried-out sponges. Dried-out sponges You have to use your imagination, if only slightly, for the massless part. But you can’t keep these sponges out of the rain, and when you can’t stop them from getting wet, they carry that water with them. Some sponges can only carry a little bit of water, while others can expand to many times their original size, carrying very large amounts of water with them once they’re fully expanded. Compressed Sponge Image credit: GNI Phoenix International, via DIYTrade.com. The most massive fundamental particles are the ones that couple most strongly to the Higgs field, and are like the sponges that expand the most and hold the most water in the rain. Of all the particles I’ve shown you, atop, there are just two that are truly massless, and hence don’t couple to the Higgs at all: the photon and the gluon. They can be represented by massless sponges, too, except they are water repellent. Water Repellant Image credit: CETEX Water Repellent, from Waltar Enterprises; photo by © Gregory Alan Dunbar. So, the Higgs field is rain, all the particles are like various types of sponges (with various absorbancies), and then… then there’s the Higgs Boson. How can the field — the rain — be a particle, too? deflated balloons Image credit: stockmedia.cc / stockarch.com. If it weren’t raining — if there were no source of water — your intended water balloon would be a sad failure. If there were no Higgs field, there wouldn’t be a Higgs boson; at least, not one of any interest, and not one with any mass. But the water comes from the Higgs field, and it also fills the balloon that is the Higgs boson: the Higgs field gives mass to all the particles that couple to the Higgs field, including the Higgs boson itself! Image credit: Laura Williams from SheKnows.com. Without the water, the balloons and the sponges would be far less interesting, and without the Higgs field, the Higgs boson and all the other fundamental particles would have no intrinsic mass to them. It's only kind of like the Higgs boson "I've found the Higgs boson! And I'm very, very wet!" So now you not only know that we’ve found the Higgs Boson, but how the Higgs field gives mass to all the particles in the Universe, including the newly-discovered boson itself. Just like water can seep its way into almost anything, making it heavier, the Higgs field couples to almost all types of fundamental particles — some more than others — giving them mass. And the great new find? We’ve been able to create and detect enough Higgs Bosons at the Large Hadron Collider to confidently announce — for the first time — that we’ve discovered it, that we’ve determined its mass (around 133 times the mass of a proton), and that it agrees perfectly with what our understanding of the Universe currently is. Higgs Event Image credit: A Higgs creation, decay and detection event, courtesy of CERN. Like I told you yesterday, keep up with the latest Particle Physics news here, and if you want to see/hear me on TV talking about the discovery of the Higgs in all its glory, you get to, tonight! I’ll be talking about the discovery of the Higgs Boson at CERN later today, July 4th, at 7PM (Pacific Time) live on Portland, OR’s own KGW NewsChannel 8 on The Square: Live @ 7! If you missed my last appearance on the show, talking about the Higgs, you can watch it anytime. But if you want to catch tonight’s show? Tune in to channel 8 if you’re in Portland, otherwise you can watch the live stream from anywhere in the world at 7PM Pacific at this link. See you then, and enjoy your Higgs-Discovery/Independence Day! 1. #1 CharlieG July 4, 2012 2. #2 Gethyn Jones July 4, 2012 I love the rain analogy! Would it make sense to think of the Higgs boson as the raindrop and the Higgs field as the rain? 3. #3 wow July 4, 2012 I would say rather than perfectly agrees that it supports the validity. After all, the median expected value for the mass of the Higgs particle was less than the figure we have, but within the range of what concords with the rest of the standard model outcomes. How this value sets other values that are to some extent free variables in the standard model will be interesting to me (and comprehensible to me too). Interesting times. 4. #4 Peido Velho Rio de Janeiro July 4, 2012 If we build a pizza collider, each resulting fragment will contain less pasta, right? So, why does a proton smasher unveil particles whose mass is 125-126 times that of the whole enchilada? This reminds me of the miracle of multiplication of loaves and fishes, but don’t let’s change the subject. 5. #5 JesseS July 4, 2012 @Peido Velho. I’m not a physicist so I might be wrong here but; - In physics mass and matter are different. In your anology you are talking about the total matter of the pizza, each fragment, added back together, must contain the same amount of matter as the two pizzas. - Since mass is equivallent to energy (the famous E=mc2) I would imagine that the mass of the created particles goes up significantly because the acceleration goes waaaay down when they smuck into each other. You have two particles, moving at somewhere around 99% of the speed of light, going nearly equal speeds, suddenly slamming into each other. Since they’re going in opposite directions they cancel each other out but all that energy has to go SOMEWHERE, so it gets converted to mass. I think… 6. #6 Michael Kelsey SLAC National Accelerator Lab July 4, 2012 @Peido Velho: The energy of the protons involved in the collision also contributes to producing the outgoing particles. With proton-proton collisions, only a fraction of the total energy actually goes toward new particle masses; the rest goes into their kinetic energy. The CERN accelerator is currently running with 4 TeV proton beams, so there is a total of 8 TeV potentially available to create new particles. 7. #7 Adithya The Netherlands July 4, 2012 If I understand correctly, what they have managed to do is create extremely high energies which causes a disturbance in the higgs field. This disturbance is manifested as the higgs boson but since it is unstable it decays. Now what would happen if during the very limited existence of the higgs boson it is exposed to another field like an electron field? Is there any possibility of interaction between the higgs boson and the field it is exposed to? Considering the early universe had high enough energy to create higgs bosons and perhaps other fields for it to interact with. Could an interaction like this be the basis for dark matter and dark energy? 8. #8 PhysicsDummy July 4, 2012 If photons do not couple with the higgs field, why is their path affected by gravity? 9. #9 david July 4, 2012 Also not a physicist, so would appreciate setting me straight if I get this wrong: I thought that the Higgs mechanism for mass involved a 4-component spinor field. 3 of the components couple to other particles, producing mass, and the 4th component is free to do whatever it wants. The 4th component is the “scalar” Higgs boson. But that means that the Higgs boson itself isn’t what produces mass. The mass of other particles comes from interaction with the other 3 parts of the spinor. 10. #10 bob July 4, 2012 i think this analogy has a couple of major flaws: 1) It doesn’t seem to have anything to do with symmetry breaking 2) Your Higgs as the water just gets absorbed to the sponges, so the sponges are then “made out of” higgses. But really, the SM particles are not made out of higgses, they just “bump into it”, which is a different idea. 11. #11 Kenny July 4, 2012 I assume people mean Inertial mass when they say it ‘gives mass’ as Gravity is and has never been part of standard model. 12. #12 Cleon Teunissen July 4, 2012 It is my understanding that non-zero masses of the neutrino, muon neutrino and tau neutrino do not involve the Higgs mechanism. Generally, the neutrino particle type is quite a source of surprises. For years physicists were convinced the neutrino’s had to be massless. As we know, today a non-zero mass is attributed to neutrino’s. As I recall, the known case of parity violation involves the neutrino. 13. #13 Cam July 4, 2012 The best analogy I’ve read Ethan, thank you. I have a question – if the higgs particle is nothing without the higgs field, what part does the particle play, i.e. in what way are the particle and field connected? Or was the particle just useful in confirming the existence of the field? Maybe there’s a way to shoehorn it into your analogy… 14. #14 bob July 4, 2012 The discussion of the Higgs field giving mass to itself is not right. If the Higgs field were zero, i.e., if the vacuum expectation value were zero, then the Higgs would still have a mass. In fact, the Higgs is the only field in the SM with a fundamental mass. 15. #15 Torbjörn Larsson, OM July 4, 2012 Hippity Higgs, Hurrah! Some early reflections: - They did really well as mentioned here, better than expected. - What they didn’t handle well was the press release. Apparently they put up press videos leaking the result yesterday and press releases before the talks were finished, as well as collaboration members leaking. - The production rates and the different combinations of observed particles produced by the Higgs, the “channels”, are still somewhat rickety statistics. But they are all consistent with a standard Higgs. What is interesting is that a standard 125-126 GeV Higgs, if that is what it is, immediately points to new physics. For example, as I understand it several analysis including this update find that there should be supersymmetry at the weak scale, which is where LHC works. And the vacuum should be quasistable, with a lot of indication of an underlying dynamical process (multiverses). @ david: 4 components, yes, that is what particle physicist Matt Strassler’s notes on his blog Of Paricular Significance. They are all from the Higgs field, they are all “higgs” including the Higgs. 3 of them goes into Z&Ws which has mass, one is massless, _the field_ is the mechanism giving mass. (By virtual particles, same as how EM fields give potentials with virtual photons.) Oh, and while Higgs field gives fundamental particle’s mass proportionally to energy, it doesn’t do proportionality for its own particles (so it ain’t gravity). Something else is required, precisely as neutrinos are SM particles (I think, sort of, it’s a kludge) but they get mass elsewhere. 16. #16 Torbjörn Larsson, OM July 4, 2012 Oh, I see bob was already there regarding that Higgs’s masses are different. And I fumbled the “massless”, its the massive Higgs natch. Here is a description. 17. #17 P Velho Rio de Janeiro July 4, 2012 Given that physical reality is awfully non-intuitive, people complicate matters even further by mishandling the instrument of language. ‘God particle’ is obviously just a bad slogan. But ‘hadron collider’, ‘atom smasher’, and the like, when used to refer to the discovery of ‘elementary’ particles, are expressions that induce innocents like me to believe we’re talking about proton debris. The same language problem arises when we say that not even light escapes from a black hole, as if massless photons were newtonian apples. The effect of gravity on space-time is often illustrated by some sort of bowl into which things ‘fall’. And so on. Wittgenstein, we have a problem. 18. #18 Andrew Foland July 4, 2012 Depending on exactly what you mean by “mass”, most of the mass of the universe is either Dark Energy or Dark Matter. The former with near certainty does not get its mass from the Higgs, and the latter may or may not, depending on what it is. As for “everyday” baryonic matter in the universe, the Higgs contribution to baryonic mass is very small, on the order of a percent or less. Most of the universe’s bayronic mass is from the confinement energy of the gluon fields inside the nucleon. Frank Wilczek wrote a nice article on all this recently. 19. #19 Bob July 4, 2012 Good summary Foland. You are right, there is a lot of misimformation presented here in Ethan’s blogs. 20. #20 Hannay July 5, 2012 So, if some particles acquire their mass through interaction with the Higgs field, then where does the Higgs boson fit in? Why is the Higgs boson needed? 21. #21 hannay July 5, 2012 So, if most particles acquire their mass through interactions with the Higgs field, then why is the Higgs boson important? what does the Higgs boson do here? 22. #22 wow July 5, 2012 Andrew until we know what dark matter and dark energy actually are, your statement is unsupported. Wrong even. It’s like saying invisible pink unicorns are not affected by electric fields (which is why you can’t touch them either). 23. #23 wow July 5, 2012 Bob the highs field particles are virtual particles. This means they have no mass (to within the limits of the uncertainty principle, which gives at least one limit to the mass of a free higgs) 24. #24 wow July 5, 2012 Physicsdummy, that is one of the ways we know we don’t yet have all the answers. Higgs gives everything inertial mass. But it doesn’t give gravitational mass. And one of the huge questions is “why are they the same value?” 25. #25 Michel July 5, 2012 Now I wait for homeopathic light nanowater that sucks up, by means of quantum mechanics, those bosons that make you heavy 26. #26 Slawek July 5, 2012 But I still don’t understand few things. About density of Higgs Field. Is it constans? I mean if some bosoms wet the sponge they will be absent in place without sponge. How about space between bosons? Do bosoms multiply to fill the emptyness? Is the field thinner or fatter? Does ideal vacuum exist or not? 27. #27 Wow July 5, 2012 You just rule-34′d homeopathy, Michael. 28. #28 Sinisa Lazarek July 5, 2012 Have to say that I’m deeply disappointed by this latest post Ethan :( Was expecting some real explanations on the Higg’s field and how it interacts with particles. Yet you have said nothing on the subject. Water and sponges.. come on. While basically wrong as someone pointed out here (as given an impression that particles somehow “absorb” the field) I am really sad that you made no attempts to even try to explain it in physics terms. How does work? How do particles interact with it? What’s the difference in proton’s interaction with it in comparison with photons… etc. etc? Not some kindergarden grade explanation (which makes it even more confusing) about some water baloons and whatnot, but a genuine explanation as we have them today. If we as a society don’t know something still, ok, then say.. we don’t know how this or that works. But at least try to explain. Your post is “how the higgs gives mass to the universe”, and yet there is nothing physics about it :(( Would you please try another go at it, for all of us curious to know more without high-end mathematics. I loved your post about quarks and chromodinamics, I was really hoping that this post about the Higg’s would be along those lines, but it’s not :( 29. #29 Wow July 5, 2012 Since we can’t see them directly and our monkey-brain doesn’t do thinking on the subatomic scale too easy, we have ot use analogies. You’re merely whining that you don’t like the analogy. Touch noogies. 30. #30 Sinisa Lazarek July 5, 2012 I’m whining because there is nothing scientific or physics related in the post. How does Higgs’ field interact with other particles in real terms? Is it through strong or weak interaction or some other “new” force? I.e. we have proton moving through Higgs field… how does it interact with it? What is that “drag” (not rain) that happens to it. What are the forces in play? Is there any emittions, absorptions etc.. If yes, what are day. Then in contrast, what happens to the photon i.e. or some “less” massive particle. There is no explanation here about the mechanism nor even a hint to it. That’s what I commented about. I don’t care if we use fish in the sea or ping-pong balls sitting on a bed of sugar, or whatever other way popular press is trying to describe it. From Ethan I came to expect real physics and science in his posts. This particular one fell really short for me, and I just commented on that. If someone now has a better understanding about Higgs mechanism by reading about different spunges abosrbing different ammounts of water, great fro you. It brings me no closer to understanding what Higg’s bosson is really about and how Higgs field really works. 31. #31 chelle July 5, 2012 I agree here with ‘Sinisa Lazarek’ although I do not want to point the finger at Ethan specifically. We have amazing Visual FX technology that can create imagery of just anything imaginable, and what we get here is a sponge and some balloons to explain what’s going on. It can’t get any more amateurish for “the discovery of the century for physics”, knowing that the Standard Model and the Higgs mechanism is nothing new, it is already more than 30 years old. Why can’t CERN and all those genius physicists take a more serious approach at educating the general public to explain how this all works. This is some very poor communication. 32. #32 P Velho Rio de Janeiro July 5, 2012 I have a far better analogy: the Higgs boson is like The Girl from Ipanema permeating all the elevators of the world, so that neither elevators nor the world would fall apart. You just can’t get rid of them, I mean, that godforsaken boson and the unstoppable song. The end of the Universe shall consist of a lukewarm soup of Higgs bosons with The Girl from Ipanema as background radiation. I hope I have clarified the matter once and for all. 33. #33 OKThen Planet Earth July 5, 2012 How does the Higgs work? Sponges are nice, thank you Ethan; But this 3 minute video explains the Higgs without sponges or need of a PhD Just clinck on the video when you get to the New Scientist site. 34. #34 Michel July 5, 2012 And BTW Ethan, now have watched you on TV, I would love it to see put little speeches on all kind a things here too. Would be great to have a little seminar once a month or so. Not that I´m a lazy reader (far from it) but to see and hear adds so much. 35. #35 Sinisa Lazarek July 5, 2012 thanx for the video link to newscientist. Is ok, but nowhere informative enough. I mean not to my apetite :) I really want to know what happens to particles in the higgs field and how it “gives” mass to particles. Guess I’ll digg deep into wiki and other resources to find out what really happens and how. 36. #36 Wow July 5, 2012 Higgs boson as water and everything else as sponges rather happily explains why some things are heavy and other things light, SL. The sponge doesn’t get bigger, it gets more filled, meaning heavier. And a sponge that is water repellent will not contain water and remain “sponge only” and light. This explains how the higgs field can make things heavier or lighter by binding to the material that we see as “massive particle”. This neatly explains this aspect of the Higgs field. There are other aspects that are not covered by this analogy and therefore this analogy for those aspects is invalid. HOWEVER, this isn’t trying to explain those features. If you want to explain those features, you do it. But don’t complain about an analogy to explain one feature doesn’t explain another, because it was never meant to. Make your own analogy. With hookers and blackjack if you want, but you do the damn work if you’re so damn cheesed off. 37. #37 Michel July 5, 2012 I bet that if Nethan was a gorgious girl who wrote about sponges and balloons they would be more than happy. 38. #38 Sinisa Lazarek July 5, 2012 @ Wow “Higgs boson as water and everything else as sponges rather happily explains why some things are heavy and other things light,” – my issue was with this in the first place. Why use water and sponges or big fish and small fish etc.. in the first place. Why not talk about the higgs field and particles in the first place?? Why the unnecesary metaphore? “The sponge doesn’t get bigger, it gets more filled, meaning heavier.” – ok.. now let’s get back to particles please. What happens to the particles in the higgs field? Do they absorb the field somehow? If so, how, by what process? Does it “suck” the energy from the higgs field and therefore increases it’s own energy? Do higgs bossons get somehow coupled to particles? By what process, what energy? What is a carrier of that coupling? Those are my questions, among others. “And a sponge that is water repellent will not contain water and remain “sponge only” and light.” – so this is in reference to photons (or EM fields) not interacting with Higgs field, while other quanta do. Again, how? “How” was never touched in real physical sense and yet it’s the first word of the title. How does that interaction take place, not as a metaphore but as a physical process? “This explains how the higgs field can make things heavier or lighter by binding to the material that we see as “massive particle”. – well, no it doesn’t. It explains in a metaphore WHAT happens, but doesn’t explain HOW it happens. “If you want to explain those features, you do it.” – I don’t want to explain anything, I want to know first. “Make your own analogy.” – one first needs to know what happens in order to make analogies. If you know what happens, I’m glad for you. If you know how it happens, even better. But we who are not physicists don’t know. But some of us would like to know. I just don’t understand why it can’t be written as is and needs balls, and guests and fish and whatnot. Why not use words like field, potential, charge, vector, scalar, tensor, operator, particle, quanta, etc etc etc….? Why can’t it be explain in plain physics language… why these analogies that confuse? 39. #39 Michel July 5, 2012 So you just found Ethan´s explanation too simple and wanted more. There are more sources than Ethan alone. Go search and expand your mind. But now you just sound ungratefull towards someone who does his best to explain something hard to a wider public. Or maby you just get angry quickly. FYI they are working right now, as we speak, to make the afformentioned homeopathic light nanowater that sucks up, by means of quantum mechanics, those bosons that make you heavy thoughts. 40. #40 JM Hanes July 5, 2012 Your Particle Physics TrapIt page is great. I loved the headline on one of the news articles you’re collecting there: “God Discovers the Elusive ‘Physicist Particle.’” LOL! 41. #41 Sinisa Lazarek July 5, 2012 Ok think I understand now. Did some wiki digging, and reading and think I have the essence of it. And without any math :)) yeeey. Please correct me if I’m wrong. this is from wiki and think it gives the best summary possible: “According to the Standard Model, the W and Z bosons gain mass via the Higgs mechanism. In the Higgs mechanism, the four gauge bosons (of SU(2)×U(1) symmetry) of the unified electroweak interaction couple to a Higgs field. This field undergoes spontaneous symmetry breaking due to the shape of its interaction potential. As a result, the universe is permeated by a nonzero Higgs vacuum expectation value (VEV). This VEV couples to three of the electroweak gauge bosons (the Ws and Z), giving them mass; the remaining gauge boson remains massless (the photon). This theory also predicts the existence of a scalar Higgs boson, which has just been observed[4].” So it’s basically an interaction of one type of field with the other at a fundametal interaction level (W and Z bosons being the carriers of weak interaction, the interactions between quarks i.e. ) those fundamental force carriers interact with a Higgs field which then breaks and gives masss/energy to those very bosons, while others remain intact. So no mysterious fishes and ping pong balls in sugar :) 42. #42 Sinisa Lazarek July 5, 2012 p.s. another interesting thing that I didn’t know before is that symmetry breaking occurred right after the big bang (the energies involved to have em and weak field unifing). So W and Z bosons “got their mass” at that instant. Everything from then on as far as mass goes is just an effect. It’s not like we are “swimming” now in the “higgs field” sort of fluid that resists our movement. It;s the mass of W and Z bosons that gives mass to everything else. Is this correct? 43. #43 bob July 5, 2012 Sinisa, no, the Higgs gives mass to the other particles, not the W and Z bosons. Did you really think that every single person in the world were all saying it wrong? 44. #44 Sinisa Lazarek July 6, 2012 Don’t want to argue, since it’s not my field, but from everything I read, it’s the W and Z bosons that are first to get directly “modified” by the interaction with the higgs field. Quarks and leptons are thought to interact via Yukawa mechanism with the higgs, but the whole point of the field being non zero is because of the initial interaction with the unified field which cuased it’s symmetry to be broken. I do not think that every single person is wrong, not did I say that. But would like if you could explain how higgs gives mass since you say it’s not the W and Z bosons. 45. #45 Michel July 6, 2012 46. #46 Sinisa Lazarek July 6, 2012 Wow Michael… hilarious. What’s even more funny is none of you even bother to answer. 47. #47 Wow July 6, 2012 “I do not think that every single person is wrong, not did I say that.” Then why are you continually complaining about everyone else? 48. #48 Sinisa Lazarek July 6, 2012 “why are you continually complaining about everyone…?” what? everyone who? don’t put words in my mouth which i never said or ment 49. #49 Wow July 6, 2012 Because you’re weaselling out of your comments against everyone by using the pedantic “absolutely everyone” meaning rather than the colloquial “everyone”. And you’re whinging about everyone else, SL. 50. #50 Gethyn Jones July 6, 2012 I really like Ethan’s rain analogy, but I am confused about the relationship between the Higgs field (“rain”) and Higgs boson. I am going to try and torture the analogy a little further using the idea that a gauge boson is the minimum-sized “ripple” in a quantum field e.g. a single photon is the smallest enrgy “ripple” in an electromagnetic field The “sponges” (particles with non-zero rest mass) absorb the “rain” which gives them mass… OK…so what happens when you bang two sponges together? Nothing – these are incredibly absorbent sponges we have here. In fact, some Sponge-physicists suggested that it wasn’t actually raining at all! Nevertheless, physicists in Sponge-world went on to build a Large Sponge Collider in order to bang them together really, really hard to see if they were really absorbing water. And when they did so, the minimum mass of the water droplet released was about 126 GeV. Sponge-physicists now triumphantly concluded that it really is raining…. (Apologies – I know an analogy is only an analogy but just trying to get my non-expert head around the ideas….) 51. #51 Wow July 6, 2012 “but I am confused about the relationship between the Higgs field (“rain”) and Higgs boson.” Well, it’s not a good bit of the analogy. But mostly because we don’t have 100% rain all the time everywhere, even indoors. Since the higgs field is everywhere (even indoors), for the rain to be like it, it has to be everywhere. Ethan does try to get this across, but if you’re spending too much time trying to find the faults, you can easily miss it: Ethan: “The Higgs field is like rain, and there is no place you can go to keep dry.” But here is another attempt to find fault rather than look for enlightenment from you: Gethan: “so what happens when you bang two sponges together? Nothing – these are incredibly absorbent sponges we have here.” And when you bang two items that can tough each other (bind together), you lose mass. 52. #52 hikaye July 6, 2012 Why is the Higgs boson needed? 53. #53 Gethyn Jones July 6, 2012 Actually, far from finding fault with Ethan’s analogy, I was attempting to extend it. I was puzzled about why high energies are required to detect the HB, and I was playing with the analogy to see if it could help me picture the relationship between field and boson. 54. #54 wow July 6, 2012 Well, one way to look at it is the be broglie wavelenght. Higher energies mean you see smaller structures. I.e. dimensions that are wrapped up smaller. Dimensions that the higgs field sits in. Another way is harmonics on a very short tight string. To excite that string you need a certain energy before you het a standing wave that will last. The string theory view. You can look at it like pair production: you neded at least enough energy to create the mass of the particle, and the higher the energy, the more you’ll make and the liklier you get to see one. None of these views work as water drops because the analogy isn’t explaining that bit and attempting to stretch it that far tears it. Like I said earlier, an analogy is not the thing it analogises, therefore you’ll always find a way it doesn’t work. Picky pedants who like to pick holes in things for pleasure love analogies from other people for this reason. 55. #55 bob July 6, 2012 @ Sinisa, what are you talking about? Its not the W and Z bosons that gives mass. Its the Higgs. As you said, the Higgs has Yukawa couplings to the fermions and its this interaction that endows the fermions with a mass. What is it you don’t get? 56. #56 Sinisa Lazarek July 7, 2012 From the research I did in the past few days, this is what I have in summary. And seems that we are diverging in something, and would like to understand what it is. So here it is: “Actually, there’s a significant caveat to “the Higgs field gives all particles mass.” Many strongly interacting particles, such as the proton and neutron, would still be massive even if all quarks had zero mass. In fact most of the mass of the proton and neutron comes from strong interaction effects and not the Higgs-produced quark masses. For instance the proton weighs almost 1 GeV, and only a small fraction of this comes from the three up and down quarks that compose it, which weigh only around 5 MeV each. If that 5 MeV was reduced to 0 the proton mass wouldn’t change very much.” and this… “An example of energy contributing to mass occurs in the most familiar kind of matter in the universe–the protons and neutrons that make up atomic nuclei in stars, planets, people and all that we see. These particles amount to 4 to 5 percent of the mass-energy of the universe. The Standard Model tells us that protons and neutrons are composed of elementary particles called quarks that are bound together by massless particles called gluons. Although the constituents are whirling around inside each proton, from outside we see a proton as a coherent object with an intrinsic mass, which is given by adding up the masses and energies of its constituents. The Standard Model lets us calculate that nearly all the mass of protons and neutrons is from the kinetic energy of their constituent quarks and gluons (the remainder is from the quarks’ rest mass). Thus, about 4 to 5 percent of the entire universe–almost all the familiar matter around us–comes from the energy of motion of quarks and gluons in protons and neutrons. ” So yes, compound particles also get a small portion of their mass from Higg’s field, but only a small part. The main mass is already there by the process’ we already know and understand. What we didn’t understand is why some bosons have mass (w and z) while others (photon and gluon) are massless. And this is where higgs mechanism really shows itself. It gives all the mass to those bosons. Or in other words, the interactions of higgs field and boson’s gives the terms in lagrangian that corresponds to mass value of those bosons. If I’m mistaken, please correct me. But please do give some examples and explanations instead of just saying yes or no. I want to learn more, and just saying “this isn;t so” without a follow up isn’t helping :) 57. #57 Dai July 7, 2012 Way out of my depth here, but in case it helps you Sinisa; IIRC kinetic energy is dependent on mass, so if the quarks had no rest mass I assume they would also have no kinetic energy. Of course it could be my school-level physics is not relevant at this scale, not sure.. ;) 58. #58 Gethyn Jones July 7, 2012 Helpful comments – thank you. I agree the analogy as originally presented by Ethan isn’t intended to illustrate the relationship between boson and field, and that I’m probably overextending it…but what the heck so here goes nothing Ethans rain analogy cleverly explains why hadrons and leptons and some bosons have mass: they are “spongy” and absorb “water”. OK but the “rain” is the Higgs field, not the Higgs boson. So can the HB be represented? One possible way would be to picture a boson as the minimum energy wave in its associated field. I guess for a e-m field this would be a low energy photon, perhaps in the radio frequency region. For a Higgs field, this is a high energy Higgs boson. Using the analogy, the Higgs field would be a fine mist of rain droplets (what some people call mizzle) while the HB would be a more substantial drop. If the “sponges” were very, very absorbent then you’d have to squeeze them pretty damn hard to get even the tiniest drop of water…which is one way of picturing why the HB can only be detected at high energies… However, an analogy is only an analogy as you rightfully point out – but they can be a lot of fun too. 59. #59 wow July 7, 2012 I eould have put it that the higgs field iis the fact that it’s rsining and a raindrop is the particle. It’s not about being low energy, it’s about being a virtual photon or higgs boson. Electric fields have the forces transferred by whatever energy photon it needs to do the job, but they’re virtual photons, not real ones. 60. #60 Sinisa Lazarek July 7, 2012 photons have no rest mass yet they have kinetic energy, actually all of it’s energy is kinetic. 61. #61 wow July 7, 2012 Actually we don’t know that. Kinetic energy = mass times velocity squared divided by two. Mass zero, kinetic energy zero. Photons do have momentum, though. Or at least can impart momentum or sosk it up. Whether thst’s momentum as you get in matter is a little unclear. But photons could have no kinetic energy, but only energy from existing (at the speed of light), as the equicalent of things at resthaving mass (=energy) 62. #62 wow July 7, 2012 Those infinities are hard to deal with in a language developed to tell other apes whete the bannanas were. Classical mechanics terms don’t really do much better. 63. #63 bob July 7, 2012 sinisa, it is true that the strong interaction provides most of the mass of the proton and neutron. However, the point of the Higgs is to give mass to the “elementary particles”. The proton and neutron are NOT elementary, they are made out of the elementary quarks. The quark’s masses and the lepton’s masses (including the electron), as well as the W and Z bosons, are acquired from interaction with the Higgs field. 64. #64 bob July 7, 2012 Dai and wow, you are both wrong. Even if a particle’s mass is zero, the particle can carry kinetic energy. This is the difference between Einsteinian theory and Newtonian theory. 65. #65 Sinisa Lazarek July 7, 2012 Bob, thanx for the reply. with this, we are in total agreement. 66. #66 Bob July 7, 2012 Sinisa, the reason for your confusion was that Ethan claimed that the Higgs gives mass to everything in the universe, when in fatc this is competeky wrong. Almost all the mass in the universe cmes from the dark sector and nuclei, whose mass does not come from the Huggs. Instead only a very tiny percentage, less than 0.002% such as electrons, comes from the Higgs. 67. #67 wow July 8, 2012 Bob, kinetic energy is for a photon its energy in and of itself. Try to remobe some snd the photon is reduced in itself. Red shifted. Something different is going on here. And note I merely maintained “we don’t know that for sure” If you’re goung to say “wrong”, you,re saying we ARE sure. 68. #68 wow July 8, 2012 Using a tablet sucks. 69. #69 Gary D Houston, TX July 9, 2012 Theoritical physicists get the best dope. That’s crazy man. 70. #70 Dirk July 9, 2012 Go Sinisa Lazarek! I’m with you. Though there is a place for providing ‘real world’ analogies to roughly explain a phenomenon, indulging in the analogy does more harm than good especially where it gives the impression that it has explained anything. Funny how the posts of those who accuse Sinisa as being ‘cheesed off’ (Wow) and ‘ungrateful’ (not ungratefull btw) (Michel) are the ones that sound most agressive – Sinisa is just stating his thoughts in a decent and polite way. 71. #71 wow July 9, 2012 Why do you say that this analogy has explained nothing, dink? Making it up, yes? 72. #72 CB July 9, 2012 Jeeze. This is like the time someone complained about an analogy to red and black marbles in closed bags to explain why quantum entanglement cannot be used to send information, because the marbles represent hidden variables and the outcome of the experiment would not match the statistical distribution of actual quantum entanglement. Even though neither of those things are relevant to explaining why you can’t send information with entanglement. The only analogy that correctly explains all aspects of a phenomenon is not an analogy, it’s the actual phenomenon in question. That doesn’t make analogies useless. If you understood that summary on Wikipedia, then congratulations you’re more informed on the subject than the vast majority of people with science degrees. You don’t need an analogy. Most do, and this is a good one for explaining what it does. 73. #73 CB July 9, 2012 On a non-analogy note, does this really create a problem with inertial vs gravitational mass? The intrinsic (and hypothetically inertial-only) mass granted by the Higgs is the result of a particle’s potential with respect to the Higgs Field. That potential is a form of energy. Energy creates gravity. So is it really any more surprising that the gravity exactly matches the potential energy of the Higgs than it is that it also exactly matches the binding energy of a proton, or water molecule? 74. #74 Sinisa Lazarek July 11, 2012 I guess I should say “thank you”. But I think you went a bit too far with the “science degrees”. If in biology, then ok. But as far as physics goes, there isn’t much not to understand. All the terminology is from high-school grade physics (relativity, qm and some math terms). I learned in high-school what leptons and quarks are, what the fundamental forces are, how mass equals energy, what symmetry and symmetry breaking is in math and physics. So it’s all there. Just needs some “dot connecting” and perhaps some cross referencing, nothing more. My strong belief is that anyone with a general notion of relativity and qm can understand that quote I took from wiki. If in fact it’s not so, especially for science majors, then something is terribly wrong with the educational system. :) 75. #75 bob July 11, 2012 Sinisa, most of the particles of the Standard Model have an interaction with the Higgs field – it is a new kind of force, a “higgs force” if you like. (Technically, for the fermions it is a type of Yukawa interaction, and for the W bosons it is a gauge interaction). The Higgs field takes on a non-zero value, even in the vacuum. So the interaction is always present. It leads to an effective mass for those particles. What more do you want to know? Did you try opening a book and finding out for yourself? 76. #76 Sinisa Lazarek July 11, 2012 don’t know what this last post of yours to me was about. Couple of days back i posted to you that I agree completely with what you posted then and that the statement that higgs gives mass to everything and anything is not correct. after that I haven’t posted any questions about the higgs. my post to which you now comment was to CB who said that that paragraph from wiki which I quoted is above the understanding of most science majors, which I find hard to believe. It’s wasn’t in any way connected to anything dealing with higgs directly. Am sorry if I am hard to understand sometimes. English is not my native language, so something might get lost on the way. “Did you try opening a book and finding out for yourself?” … of course.. that’s how you I and started discussing higgs. But again I don’t know why this last post from you? And in such a way? Wasn’t about higgs or questions about it. Was about understanding the wiki quote 77. #77 wow July 12, 2012 SL who said that higgs gives mass to everything? Strawman. 78. #78 Sinisa Lazarek July 12, 2012 @ Wow what’s the title of this post? 79. #79 wow July 12, 2012 And you only read that??? You did notice there were more words below that, right? 80. [...] For those interested, this is the best article for the layman I have found on the Higgs boson. There is plenty of stuff on http://www.youtube.com as well: http://scienceblogs.com/startswithabang/2012/07/04/how-the-higgs-gives-mass-to-the-universe/. [...] 81. #81 Nathan July 14, 2012 One of the comments above “Higgs, Higgs ,Hurrah” is real. By the way I just posted this in another article in this blog. Explains very nicely… 82. #82 Van den Bogaert Joannes Belgium B2970 July 17, 2012 Mass is an inherent property of elementary particles. The mass of the proton has been calculated from spin, charge and particle radius on pages 3-4 of Belgian patent BE1002781; see e-Cat Site “Belgian LANR Patents”. For the electron mass a similar formula has been used. 83. #83 Wow July 17, 2012 Science discoveries are not patentable. 84. #84 chelle July 17, 2012 “Science discoveries are not patentable.” Depends on what you call a discovery. And I suppose that in theory one could see the production/creation of a Higgs-boson by the LHC as a patentable thing, no? 85. #85 wow July 17, 2012 Nope depends on the definition of discovery by patent offices. And these discoveries are not patentable by ANY patent office. 86. #86 wow July 17, 2012 You csn patent the design of the macine to make the measurement. But maths and the discoveries of science in nature are not. 87. #87 wow July 17, 2012 In ansewr to your question- no. 88. #88 chelle July 17, 2012 I was looking here at the broad sense of science and the controversial gray zone of gene patenting. But with “these discoveries” you surely mean in the field of physics, here I’m not going to argue with you. Regarding the Higgs-boson, there are two parts, the collider making them, and the detectors measuring them. I think that you could patent almost everything that CERN makes, and perhaps lots of the parts being used are already patented? So you either can scoop them Higgs for free coming out of a cosmic ray collision, or probably having to pay for an artificially created one. 89. #89 wow July 18, 2012 There’s no grey area here, chelle, thankfully enough. Discovering the electron charge value is not patentable. Inventing a machine to measure the electron charge is, but I can’t think of any scientist who does that because there’s no market for the singular purpose machine, and they’d rather get on with research. They’ll use patented tools. Like hammers. But they don’t purchase a licence to the patent on them any more than you do. Bogart there was claiming a patent on the theory of how to calculate masses. As maths, this is not patentable. 90. #90 chelle July 18, 2012 “There’s no grey area here, chelle, thankfully enough.” You might want to read ‘The Immortal Life of Henrietta Lacks’ by Rebecca Skloot, or follow up on some other patent cases. “Each nation has its own patent law, and what is patentable in some countries is not allowed to be patented in others.” It’s all about politics and company’s lobbying. Anyway, the way you keep on ignoring facts just amazes me. 91. #91 Wow July 18, 2012 Nope, I won’t. Guess why? Because discoveries in science and maths are not patentable. It’s not about politics, by the way, it’s about money and the capitalist system that equates power with money and allows it to accumulate freely. Maybe you want to read up on an Aus patent on swinging on a swing. 92. #92 Wow July 18, 2012 PS Irony: Chelle saying “the way you keep on ignoring facts just amazes me.” ROFL indeed… 93. #93 Van den Bogaert Joannes Belgium B-2970 Schilde July 26, 2012 To Mr. Wow, The patent BE1002781 does not relate to a method of calculating the rest mass of the proton, it relates to a kind of “cold fusion” based on Coulomb explosion of charged deuterated electroconductive particles. Read the patent text in English published on the “e-Cat Site” under the title “Belgian LANR Patents” and have a look to BE1003296 published on the same site under the title “LANR by Coulomb explosion retarded from publication for 2 years by the Belgian Government of Defense. The calculation of said rest mass is dimensionally correct and does not infringes quantumphysics or mathematics. The rest mass of the proton is intrinsic linked to spin angular momentum, electric charge and particle radius. The product of mass and spin radius is constant and charge is an inverse function of the root value of mass and said radius. the proton is composed of “spinning quarks” . Two of them spinning in the same sense , one having opposite charge spinning in opposite sense being quenched between the the two others attracting each other by the current law of Ampère. Electric charge comes out of the formula as dualistic, positive and negative, that is correct! 94. #94 Wow July 26, 2012 Then your post earlier was lying. 95. #95 Van den Bogaert Joannes Belgium, B-2970 August 2, 2012 To Wow, Have you read already BE1002781 through the e-Cat Site and what do you think of the equation for the rest mass of the proton on pages3-4 of that BE patent relating to lattice assisted nuclear fusion (LANR) by Coulomb explosion? 96. #96 Wow August 2, 2012 I can’t even work out what that patent is trying to patent. Patents are pretty pointless now. They’re nothing but lawsuit fodder. However, in this case it looks more likely that the patent is patenting rubbish, hiding the result in obtuse verbiage and using the PTO as a proxy for publishing in a journal to lend unearned authority to the idea. That, however, is a conclusion based on likely utility. This patent may be genuinely intended as a patent, in which case, you wasted your money, but hey, who cares? 97. #97 Van den Bogaert Joannes August 2, 2012 To Wow, I still not have comments to the equation on pages 3-4 of BE1002781. I do not like your vocabulary “rubbish”. Blogs are developed to have worthful discussions, certainly when it concerns science. Cheers! 98. #98 Wow August 2, 2012 I don’t really see why your dislike is my problem. 99. #99 Bernard August 23, 2012 Does an understanding of the Higgs field provide any hints (perhaps vague hints) about why General Relativity’s equivalence of inertial and gravitation mass should be expected? 100. #100 marino January 20, 2013 this will give mass it’s matter. 101. #101 marino January 20, 2013 E=mc2 gives an explosion E/m=c2 gives you fusion A.E.I.O.U (Absolute Energy equals Input, Output Utilization) 102. #102 Wow January 21, 2013 I don’t think so, Bernard. It could do if, for example, Higgs tied to Higgs in short range interactions. Then again, we don’t know WHY vacuum has a permitivity or permeability either. Well, not since I last looked. Not why an electron has one electron’s charge (though it may have more: the excess hidden by charged virtual particles hiding some of the electrons’ “true” charge). It may be that these figures are self-correcting to some “most stable local value” and gravitational mass does the same thing. All this, however, is well beyond my pay grade… 103. #103 Martin Burger Vancouver BC October 15, 2013 Earth science discovery is exciting work but if your new data goes against the accepted models it can take time for the community to incorporate new data. Carl Sagan wrote, A sad truth indeed from someone who carried the burdens of innovation. This brings us to the big bang dogma today that all elements were created in a singularity event out of nothing, when new data reveals that cosmological processes are creating new elements continuously and on a massive scale. ie, Navy drill cores from ocean rifting, covering massive planet surface areas are only from a few years old to 180 million years old. Those still wearing the big bang blinders can not appreciate that we indeed have a growing earth with a changing radius (continental mass is growing and ocean bottom surface is growing) from new elements being created at the core and not from space dust accretions. Maverick scientists at Blue Eagle have now confirmed using LENR Interferometry Microsmelt Technology Processes (basically mimicking nature’s elemental bloom conditions) and are now making new precious elements. Not the wispy Hadron atomic scale elements but visible gold beads measured on a gram scale. Our team of credentialed scientists and entrepreneurial engineers have accomplished more science in the last 18 months than the legions of those labouring over bosons in billion dollars budgets. For their efforts they are labelled as Crackpots when they should be recipients of the highest awards for progressing science. To see a video of a modern day alchemist makng real gold in an LENR Interferometry Microsmelt low budget lab go to: http://www.kickstarter.com/projects/56975959/2129178196?token=7173274e 104. #104 Wow October 16, 2013 Well, of course. For a start you’d need to find a new model if it is going to be refuted by the new data and that takes time UNLESS you’ve gone looking for data to fit a preconceived model. Which may be correct (e.g. looking for how the photoelectric effect disproves the wave theory of light and proves the quantisation of same). Or completely anti-science (e.g. looking at the time of diagnosis of autism and the similar time you can first be immunised against deadly childhood diseases so you can push your own “miracle cure” and rubbish the vaccine). Either case does DEMAND you state a priori what model you did your measurements to fit so that others can check for confirmation bias. Much as Carl Sagan constantly exhorts guarding against, but almost never quoted by cranks and quacks. 105. #105 Sean T October 16, 2013 Martin Burger, Please educate yourself about what the big bang theory actually says before you try to criticize it. The big bang theory does NOT say that all the elements were created “in a singularity event”. In fact, in the earliest moments (ie fractions of a second) after the BB, there was nothing that could conceivably be called an element. The universe was too energetic for atomic nuclei to remain intact. Nuclei only formed later as the universe cooled. Furthermore, not all elements formed at this time. The BB theory does quite well at predicting the abundance of elements formed at this time, and it consisted mainly of hydrogen, with a smaller amount of helium formed and trace amounts of other light nuclei such as lithium. Heavier elements (up to iron) formed via nuclear fusion in stars. Elements heavier than iron formed in supernovae. Short story: formation of new elements today in no way invalidates the big bang theory. 106. #106 CB October 16, 2013 I love when people call the Big Bang “dogma”, ignorant of the fact that the Big Bang suffered all the resistance one could imagine but eventually won everyone over by its overwhelming predictive success. In the same way, even if scientists are obstinately opposed, if you can do as you say and produce gold from silicon dioxide, then they will be forced to accept the evidence. It’s the E-Cat all over again: - If the goal is to convince science of the new theory behind this invention, it would be easy to produce the necessary evidence. But how much do you want to be that a proper test that controls for any possible source of fraud will NEVER be done. Maybe sham “demonstrations”, but never the kind of test you would design if you really wanted to prove the device worked. Just the kind you would design if you wanted to sucker in gullible investors. - Screw those dogmatic scientists! You have a device that makes GOLD from SAND. Much like a simple cold fusion reactor, this is a project that, if real, would have zero problem funding itself. Once deployed at industrial scale it would drop the bottom out of the gold market, but in the meantime you’d be raking in the cash. In fact to prevent speculation, you’d probably keep really quiet, just slowly selling enough gold on the market to keep going (and getting rich) until one day you open your factory and reveal you’re now the world’s gold supplier. Instead, you have a kickstarter page. 107. #107 dean October 16, 2013 there is probably a good reason for your “scientists” to be called crackpots. You website indicates that. Cb, it isn’t quite a kick starter page, given this disclaimer at the top This is not a live project. This is a draft shared by martin burger for feedback. 108. #108 Sinisa Lazarek October 17, 2013 @ Martin Burger hahha… OMG… hahaha… 109. #109 CB October 17, 2013 Dean: Wouldn’t the purpose of the draft be to eventually set up a proper kickstarter based on the feedback? Seems like otherwise there’s no reason for it to be on kickstarter (with sponsorship prizes and all); it could just be a facebook post. Or am I missing something? Sinisa: Once again you find a way to summarize my own thoughts in far fewer words. 110. #110 dean October 17, 2013 “Wouldn’t the purpose of the draft be to eventually set up a proper kickstarter based on the feedback? ” It seems that this was a feeler to get a sense of interest – my take away that whoever put it there hasn’t done the hoop jumping to get it okayed to get to the point of taking money. It seems that there has not been any interest in it at all. That could be because * there is little interest for certain science items, or * people skim over it because of what this particular item is Or, I could be missing a bigger piece of the puzzle. My wife tells me that happens quite often. 111. #111 CB October 17, 2013 I’m just trying to infer the intention to eventually, should interest be sufficient etc. etc., fund the miraculous alchemical gold-making machine (which supposedly already works and can make significant amounts of gold!) using kickstarter. Because that’s hilarious to me. 112. #112 Joannes Van den Bogaert March 5, 2014 Gravity waves are the result of the product of mass of an elementary particle (fermion) and its spin radius. Said product is constant but results in zitterbewegung. Longitudinal waves are produced by the trembling motion of the particles. The spin radius is fluctuating inversionally proportional to the value of the mass. Mass fluctuations are gravity fluctuations. Interference of gravity waves inbetween massive objects is at the origin of attraction (pushing from the other side). Photons have no rest mass but are composed of matter and antimatter particles in equal strength with curvature infinite. Their traject curvature (bending) is influenced by the permittivity and permeability of the vacuum in the neighbourhood of massive objects such as the sun. 113. #113 Michael Kelsey SLAC National Accelerator Laboratory March 5, 2014 @Joannes #112: [citation needed] 114. #114 Van den Bogaert Joannes March 13, 2014 In connection with the preceding blog of mine have look at the equation for the mass of the proton in the Belgian patent BE1002781 available in English on the blog site ; “e-Cat Site” in the article “Belgian LANR Patents” Have a look also at the article of Rockenbauer concerning the cause of mass formation through spin of the elementary particle. 115. #115 Michael Kelsey SLAC National Acclerator Laboratory March 13, 2014 @Joannes #114: Thanks. So no published journal papers, then. Just blog posts, patents (which are neither reviewed for, nor required to meet, conditions of reality), and vanity-press papers. 116. #116 Van den Bogaert Joannes March 18, 2014 To Mr. Michael Kelsey Dear Sir, You are right about the non-existence of publications of mine in journal papers. Being a self-teached person in quantum physics I read some books about it, e.g. “101 Quantum Questions” from Kenneth Ford and was impressed by the statement that nowone knows the real nature of “electric charge” (je ne sais quoi) statement in the book. I like to draw your attention to the Bohr-atom formulae of the electron (Essentials of Physics from Borowitz-Beiser wherein you will find how to calculate electric charge in function of the product mass and (spin)radius of a fermiparticle such as an electron. See also BE1002781 PAGES 3 AND 4 for the proton restmass and its connection with electric charge. Further I like to draw your attention to my Belgian patent BE904719 (in Dutch) for calculating the spin radius of the electron using a time independent Schrödinger equation for a “Standing wave” and have a look at the BE-patent referenced therein (Fig. 2 and 3). It has been a pleasure to me to hear from you. Have a look at my article “Cold Fusion Catalyst” on the E-Cat Site and have a comment thereto if possible . Thanks! 117. #117 Van den Bogaert Joannes March 20, 2014 The Figures 2 and 3 are in the Belgian patent BE895572 (abstract in English available through ESPACENET. My e-mail address is : jan.van.den.bogaert@hotmail.com. Do not hesitate to ask questions about my patents(12) no longer in force. 118. #118 Van den Bogaert Joannes March 20, 2014 The frequency of the gravity waves emitted by the trembling in the ground state of the electron “f” is 0.000008717 cycles/sec. This value has been obtained starting with the pendulum equation of Huygens (Dutch scientist) . In that equation L has been put equal to the spin radius calculated according to my Belgian patent BE904719 viz. 2.64X10^-11 m. The period (T) is consequently 114715.2798 seconds and the energy E being h.f = 5.7758842×10^39 joule. For calculating the acceleration factor (a) in the Newton formula of gravity force( (F) I had to divide through the restmass of the electron being 9.108×10^31 kg. For calculating the electromagnetic trembling being origin of electromagnetic attraction or repulsion I started with the Coulomb formula for electrostatic force giving (a) by dividing through the restmass of the electron . The force relationship of 10^42 of electromagnetic force to gravity force comes out. Comments are welcome. 119. #119 Van den Bogaert Joannes March 22, 2014 The rest maas of the electron is 9.108×10^-31 kg and the outcome of h.f = 5.7758842×10^-39 joule . Sorry for the typist error. 120. #120 S Kennnedy July 8, 2014 When an electron in an atom goes from a higher state to lower state the mass of the atom decreases.This is explained by electromagnetism and quantum mechanics. The “Higgs Field” is not needed. The Higgs theory is incomplete and the predicted mass of the “Higgs particle” kept changing to higher and higher values used to justify to the European politicians to fund the Hadron Collider. Peoples jobs depended on the Hadron Collider finding the “Higgs boson”. I tried to ask Steven Weinberg about this and he wouldn’t look me in the eyes, I suspect there is something very incomplete about this even to the people who created it. 121. #121 lakshminarayana gopalan August 9, 2014 Different levels of sponginess,indeed a good anology. But the concept can be deemed to be fully explained only after establishing why and different levels of sponginess? If this picture too is clear to the dedicated scintists it may merit a similar clarification and explanations . 122. #122 lakshminarayana gopalan August 9, 2014 Similar to Higgs Particle and its field can there be say “Kiggs”particle and field for force fields?-
f72925ba2746bd46	Sign up Here's how it works: 1. Anybody can ask a question 2. Anybody can answer Is time simply the rate of change? If this is the case and time was created during the big bang would it be the case that the closer you get to the start of the big bang the "slower" things change until you essentially approach a static, unchanging entity at the beginning of creation? Also, to put this definition in relation to Einstein's conclusions that "observers in motion relative to one another will measure different elapsed times for the same event." : Wouldn't it be the case that saying the difference in elapsed time is the same as saying the difference in the rate of change. With this definition there is no point in describing the "flow" of time or the "direction" of time because time doesn't move forward but rather things simply change according to the laws of physics. Edit: Adding clarification based on @neil's comments: The beginning of the big bang would be very busy, but if time was then created if you go back to the very beginning it seems there is no time and there is only a static environment. So it seems to me that saying time has a direction makes no sense. There is no direction in which time flows. There is no time; unless time is defined as change. So we have our three dimensional objects: and then we have those objects interact. The interaction is what we experience as time. Is this correct or is time more complicated than this? share\|cite\|improve this question If you're concerned with the rate at which things change, shouldn't things go faster as you approach the Big Bang? The first hour of the universe was an extremely busy time. – Niel de Beaudrap Oct 4 '11 at 15:36 More generally and to the point: how do you determine "the rate of change" without a fixed standard for time, anyhow? Fast processes still happen now; just perhaps less frequently than before. That, and we're often more interested in glacially slow processes, such as human behaviour, and well, the movements of glaciers. It makes the most sense to establish a collection of commensurable standards of time reaching back to the Big Bang; but commensurability pretty much prevents any process of "time inflation" --- at least in how we measure time. – Niel de Beaudrap Oct 4 '11 at 15:38 Things may happen "faster" compared to things happening on earth now but wouldn't you eventually reach the beginning where nothing is happening and you reach a static/stable environment – coder Oct 4 '11 at 16:02 It depends on how you're trying to define a changing scale of time! If the "activity" (very vague) of the universe is getting slower with time in an exponential decay, then going backwards in time would look like watching a computer which performs one instruction in 1Gyr, a second instruction in .5Gyr, a third in .25Gyr, getting faster with time. If you "rescale time" so that each instruction takes one "operational time unit", what you find is not that things come to a rest but that you can squeeze in an infinite regress of activity immediately after the Big Bang. Very speculative of course! – Niel de Beaudrap Oct 4 '11 at 16:10 I admit how time apparently "flows" is a difficult problem and one of the most mysterious in physics. But reading one comment above I remember one of the famous quotes "Any intelligent fool can make things bigger and more complex... It takes a touch of genius - and a lot of courage to move in the opposite direction. " – user1355 Oct 7 '11 at 17:05 11 Answers 11 up vote 1 down vote accepted Since for some reason this question has resurfaced, I would like to point to a similar one posed later than this. Observation of change is important to defining a concept of time. If there are no changes, no time can be defined. But it is also true that if space were not changing, no contours, we would not have a concept of space either. A total three dimensional uniformity would not register. Our scientific time definition uses the concept of entropy to codify change in space, and entropy tells us that there exists an arrow of time. In special relativity and general relativity time is defined as a fourth coordinate on par with the three space directions, with an extension to imaginary numbers for the mathematical transformations involved. The successful description of nature, particularly by special relativity, confirms the use of time as a coordinate on par with the space coordinates. It is the arrow of time that distinguishes it in behavior from the other coordinates as far as the theoretical description of nature goes. share\|cite\|improve this answer it could be also added that entropy as an indicator of time is a relatively modern concept and not the only one. Surely the earliest thinkers compared cyclic processes with non-cyclic ones and found that some other non entropy-rising systems would behave as only one directional processes. I think of aging for example, compared to the tides or moon cycle. – rmhleo Aug 20 '15 at 9:42 This question ("Is time simply the rate of change?") is too ambiguous to have any meaningful answer. I can think of interpretations in which the question is vacuous (begging the question: "what is meant by 'rate of change'?"), tautological ("rate of change" == d/dt), or in which the answer is 'no' (GR). You might find the answer you seek in this book: share\|cite\|improve this answer to rephrase: is time a thing in itself or is time simply things changing? This is probably a hard question to articulate. Add that to my lack of understanding of physics :-) – coder Oct 10 '11 at 14:25 @Jeremy: most questions that are hard to articulate in this way are not meaningful, they are only philosophical words that make the brain go in circles. The questions about time which are meaningful are those that can be answered by observations. – Ron Maimon Dec 8 '11 at 5:47 Time is what is measured by clocks. But how is time modelled in physical theories ? In the Schrödinger equation time enters as an external parameter. How does this parameter correspond to the time measured by clocks ? The following reference might be a good introduction to this and related questions concerning time and quantum mechanics : share\|cite\|improve this answer There's is no such notion as "time" in isolation from space. Since time is a measure of entropy of space, then time wouldn't exist if the space is absolutely static. Imagine that one will somehow manage to 'rollback' the matter & energy to a state in which it was yesterday. Would this be a time travel? I don't see reasons why it wouldn't. There are things not affected by time - say, physical laws and regularities. Since we assume that they are the innate property of the universe, we also assume that they exist out of the scope of time and space. That is, time didn't exist before the BigBang, but the laws did. Edit: it's rather difficult for me, though, to imagine a physical law existing in isolation from things that it governs. share\|cite\|improve this answer Physics Law is the description of the thing that it governs. – Prathyush Oct 14 '12 at 14:45 Certainly time is intriguing, but there are two different things going on here: (1) there is (classically) the manifold, (2) and the zeroth component of the momentum 4-vector. To start, the temporal part of the gravitational potential does have some weird geometry that we aren't used to in everyday life and this certainly plays a role in some of the strangeness surrounding "time", but a decomposition of the EFE demonstrates that actually $g_{00}$ and $g_{0i}$ don't have time derivatives. The temporal parts of the space-time manifold, are static, only the spatial parts, $g_{ij}$ are dynamic. So where is this notion of "flow" coming from? Instead, think of the manifold as a landscape, with something like a "temporal" direction. Our movement through that direction, is determined by the zeroth component of the momentum 4-vector, energy, temporal momentum. Why are almost all things in everyday life moving in the same "direction" of time? its not because we are all in the same river, its because we are all made of the same stuff. If you want to relate "time" with a rate of change, a place to start looking is at the momentum 4-vector, not the spacetime manifold. share\|cite\|improve this answer A clear understanding of time in my opinion Still eludes us. Within scope of classical Concepts there is a Perfectly valid practical definition of time. Which essentially is the correlations between the periodic behaviour of systems. For instance the behaviour of a pendulum is correlated with the behaviour of the motion of the sun around the planet as N number of periods pendulum corresponds to 1 Period of earth orbit around the sun. The property of periodicity in classical systems is essential in the definition of a clock. The Question about arrow of time in my opinion Boils down to our inability to prepare system in precise initial conditions, Which only allows the possibility of predicting its behaviour on in a statistical Sense. We are also limited to measure only certain properties of a system, and we cannot acquire complete information. This is a limitation we must accept on our ability to perform experiments. In this sense, if we use the clock we defined only using Classical Concepts, then this implies that flows have a preferred direction ie, a the direction of incerasing entropy. Question of time In my opinion will completely resolve Itself if one understands what a memory impression is. Memory being impression being permanent contains a record of the passing of time. I think it is very closely connected to the foundational Issues that plague Quantum measurement. Saying this coming to your question on Time running Slower closer to the big-bang, In one sense One can say that there is no structure to measure the movement of time. But really to answer this question we have to wait for the discovery of Quantum Gravity. share\|cite\|improve this answer The 7 fundamental quantities are hard to define. • Time • Displacement/position • Mass • Temperature • Current • Amount of substance (e.i. the mole) • Luminous intensity More here: Time is just one of them. If someone asked me, I would say something like "Time is how long something lasts" or "Time is the duration of something". But that's circular; it's the same as saying "Time is how much time has passed". So it really doesn't say anything. Any other physical quantity can be explained in terms of these 7 fundamental ones. E.i., velocity is how far something moves per time unit. But how can you explain/define the fundamental ones themselves, then? My best answer is: You can't! You can only define it empirically or with examples. Time is what a clock shows, I heard someone say once. You do though say that we can define displacement and current. But how? How would you do that without ending up in a similar looping circular explanation? share\|cite\|improve this answer I go with Time is the separation between distinct events that happen in the same place. which is very general and not quantitative at all, but covers the basics. Given three distinct events that happen at the same place we can determine which happened between the other from just the values of the three separations. And it agrees with the notion that "time is what a clock measures". From the perspective of relativity this definition is the proper time. share\|cite\|improve this answer The concept of time is intimately related with the concept of causality. If we don't have the notion that something can cause some other thing then there is no objective meaning to the word "time". It's causality which enables us to decide and describe which event is past and which is present. In relativity, as we know, space and time are intimately related to each other. What is just space to some observer may be a combination of space and time for another. It is therefore helpful to think of a $4$ dimensional space called spacetime whose points represent events. An event therefore needs $4$ independent numbers to be uniquely specified. Out of this $4$ numbers one is a little special. If you draw a light cone at any point in this space then all except one of the axes will be outside the light cone. This special axis is the direction of time and the numbers it represent is "time". share\|cite\|improve this answer Thanks for not commenting on the reason for the down vote. That's a huge relief ;) – user1355 Oct 7 '11 at 17:08 +1 because this definitely doesn't deserve a -1 :-). You essentially rephrase my question in the first part. I could have rephrased as asking, "is it the case that time is simply causality?" If this is the case then it seems the notion of a "flow" of time only exists because we have a memory of the past events; when in fact there is no past, there is no future, there is only stuff which interacts. So the words "time", "causality" and "interaction" are interchangeable leaving us with only stuff that changes. – coder Oct 7 '11 at 17:44 Not my downvote, but the concept of time does not require a concept of causality, which is notoriously hard to pin down in the microrealm and probably doesn't make any sense. – Ron Maimon Oct 10 '11 at 5:30 I disagree with you Ron. First it is dangerous to say that causality doesn't make any sense in the micro realm. If it were so, then how can you ever trust QFT which is fully consistent with S.R. and which requires causality to hold strictly. Secondly, if two events are space like separated in the micro realm how do you decide which has taken place earlier? You can't, unless you seriously modify the existing theories or unless you are talking about some as yet unknown QG theory. – user1355 Oct 11 '11 at 16:02 @RonMaimon: However, it is true that in the microscopic world there may be processes which may not have any intrinsic "arrow of time". But that's an altogether different issue, right? – user1355 Oct 11 '11 at 16:09 well the way time should be conceived is the same way you should look at motion or any type of energy kinetic or potential, ergo it should be treated as such. example, when a object falls from a table the time it takes to travel through the air co insists with the space around it ("space-time to be precise which the fine gent below me is proclaiming). So your question comes up which the answer would most likely be yes. but not to forget time is also a unit of measurement such as length width and depth and we use it as such. it simply being the rate of change is plausible under certain theoretical works in the past which many have been trying to prove fact in the present. share\|cite\|improve this answer you seem to think time started with the big bang, how long was the matter there before the big bang? we are still only talking about half an equation. two points are still needed to justify either of our perspectives share\|cite\|improve this answer Your Answer
d52cdfab28fe9a92	Mauro Murzi's pages on Philosophy of Science - Quantum mechanics prev Index Features of Schrödinger quantum mechanics next 1. Introduction. The main goal of this article is to provide a mathematical introduction to Schrödinger quantum mechanics suitable for people interested in its philosophical implications. A brief explanation of complex functions, including derivatives and partial derivatives, is given. First and second Schrödinger equations are formulated and some of their physical consequences are analysed, particularly the derivation of Bohr energy levels, the forecast of the tunnel effect and an explanation of alpha radioactivity. These examples are chosen in order to show real physical applications of Schrödinger equations. The exposition of Heisenberg indeterminacy principle begins with an analysis of the properties of commutative and non commutative operators, continues with a brief explanation of mean values and ends with some physical applications. Schrödinger quantum theory is formulated in an axiomatic fashion. No historical analysis is developed to justify the formulation of the two Schrödinger equation: Their only justification derives from their success in explaining physical facts. The philosophical background I use in this article is due to logical positivism and its analysis of the structure of a scientific theory. In this perspective, Schrödinger equations are the theoretical axioms of the theory; the probabilistic interpretation of Schrödinger equations plays the roles of the rules of correspondence, establishing a correlation between real objects and the abstract concepts of the theory; the observational part of the theory describes observation about radioactivity, spectral wavelengths and similar events. prev Index Features of Schrödinger quantum mechanics next
205d4df01391cb98	Complex potential model for low-energy neutron scattering Fiedeldey H. ; Frahn W.E. (1961) The optical model for low-energy neutron scattering is treated explicitly by means of a new form of complex potential which permits an exact solution of the S-wave Schrödinger equation. This potential is everywhere continuously differentiable and its imaginary part consists of both a volume and a surface absorption term which is in close agreement with recent theoretical calculations of the spatial distribution of the imaginary potential. Closed-form expressions are obtained for the logarithmic derivative of the wave function, and hence for the S-wave strength function and scattering length, from which their dependence on all potential parameters can be studied explicitly. In particular, it is shown that concentrating the absorption in the nuclear surface can serve as a remedy for a well-known discrepancy, by lowering the minima of the strength function to more realistic values. © 1961. This item appears in the following collections:
2e2e626b3b2ac012	Take the 2-minute tour × I am reading the wikipedia article on the Laplacian matrix: I don't understand what is the particular use of it; having the diagonals as the degree and why the negative adjacency elements off the diagonal? What use would this have? Then on reading about its norm, first of all what does a norm really mean? And what is the norm for the Laplacian matrix delivering? This norm does not result in a matrix whose terms cancel out or sum to one. Or that the determinant is equal to any consistent value. Any insight? share\|improve this question 2 Answers 2 up vote 12 down vote accepted The Laplacian is a discrete analogue of the Laplacian $\sum \frac{\partial^2 f}{\partial x_i^2}$ in multivariable calculus, and it serves a similar purpose: it measures to what extent a function differs at a point from its values at nearby points. The Laplacian appears in the analysis of random walks and electrical networks on a graph (the standard reference here being Doyle and Snell), and so it is not surprising that it encodes some of its structural properties: as I described in this blog post, it can be used to set up three differential equations on a graph (the wave equation, the heat equation, and the Schrödinger equation). (To be totally clear, when you're using this interpretation you should think of the Laplacian, not as a matrix, but as an operator acting on functions $f : V \to \mathbb{R}$. In this setting there is a discrete notion of gradient (which sends a function $f$ to a function $\text{grad } f : E \to \mathbb{R}$) and a discrete notion of divergence (which sends a function $g : E \to \mathbb{R}$ to a function $\text{div } g : V \to \mathbb{R}$), and the divergence of the gradient is the Laplacian - just like in the infinitary case. So the Laplacian defines a certain analogy between graphs and Riemannian manifolds.) The quadratic form defined by the Laplacian appears, for example, as the power running through a circuit with given voltages at each point and unit resistances on each edge. It is the discrete analogue of the Dirichlet energy. The Laplacian appears in the matrix-tree theorem: the determinant of the Laplacian (with a bit removed) counts the number of spanning trees. This is related to its appearance in the study of electrical networks and is still totally mysterious to me. The group $\mathbb{Z}^n / L$ where $L$ is the Laplacian has rank $1$, and its torsion subgroup is the critical group of the graph, which has size the number of spanning trees. The critical group appears in the description of chip-firing games on the graph (another name for this is the abelian sandpile model), and is an interesting invariant of graphs. There is some evidence that finite graphs are analogous to curves over finite fields, and in this analogy the critical group appears to be analogous to the ideal class group (that is, the Jacobian). Its size even appears in a class number formula for graphs coming from the Ihara zeta function (the analogue of the Dedekind zeta function). Again, all of this is totally mysterious to me. Here is a nice survey paper by Mohar on what graph theorists actually use the Laplacian for. In the literature there are several different normalizations; they correspond to either using a different preferred basis for the space of functions $f : V \to \mathbb{R}$ or varying physical properties of the graph (e.g. changing resistances, adding a potential). share\|improve this answer When I say "the group $\mathbb{Z}^n/L$ where $L$ is the Laplacian" I think I mean "where $L$ is the image of the Laplacian." – Qiaochu Yuan Oct 10 '13 at 7:15 Laplacian matrices are important objects in the field of Spectral Graph Theory. Many properties of the graph can be read off easily from the properties of the corresponding Laplacian Matrix. For instance: • The multiplicity of the eigenvalue zero gives the number of connected components of the graph. • The largest eigenvalue is 2 if and only if a connected component of the graph is a non-trivial bipartite graph. There are plenty more results like these. I would recommend you try the book by Fan Chung (Ronald Graham's wife, I believe) conveniently titled Spectral Graph Theory. Here is a link to the book page: http://www.math.ucsd.edu/~fan/research/revised.html An explanation of the possible motivation for using the Laplacian (and it's normalized form, which is what you seem to be talking about, when you say norm) appears in the first chapter of that book: http://www.math.ucsd.edu/~fan/research/cb/ch1.pdf Also search the web for a linear algebra proof of Friendship theorem, which uses matrices. Even though Laplacian matrices are not used, I would still recommend you read that, as it will give you an idea about the power of spectral methods in graph theory. Hope that helps. share\|improve this answer Your Answer
0f7679ebb880e9a9	quantum numbers Quantum mechanics Quantum mechanics is the study of mechanical systems whose dimensions are close to the atomic scale, such as molecules, atoms, electrons, protons and other subatomic particles. Quantum mechanics is a fundamental branch of physics with wide applications. Quantum theory generalizes classical mechanics to provide accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits. The effects of quantum mechanics become evident at the atomic and subatomic level, and they are typically not observable on macroscopic scales. Superfluidity is one of the known exceptions to this rule. E = h nu = hbar omega, where h is Planck's Action Constant. Although Planck insisted that this was simply an aspect of the absorption and radiation of energy and had nothing to do with the physical reality of the energy itself, in 1905, to explain the photoelectric effect (1839), i.e. that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, as based on Planck’s quantum hypothesis, that light itself consists of individual quanta, which later came to be called photons (1926). From Einstein's simple postulation was borne a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics. Relativity and quantum mechanics The modern world of physics is founded on two tested and demonstrably sound theories of general relativity and quantum mechanics —theories which appear to contradict one another. The defining postulates of both Einstein's theory of relativity and quantum theory are indisputably supported by rigorous and repeated empirical evidence. However, while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model. Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly inventive in this field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time. He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics has unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that Einstein's assumption that quantum mechanics is correct, but has to be completed by hidden variables, was based on wrong philosophical assumptions: according to the paper of J. Bell and the Copenhagen interpretation (the common interpretation of quantum mechanics by physicists for decades), and contrary to Einstein's ideas, quantum mechanics is • nor a local theory (essentially not, because the state vector scriptstyle \|psirangle determines simultaneously the probability amplitudes at all sites, \|psirangletopsi(mathbf r), forall mathbf r). The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens. These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically scriptstyle le 1000 km, being on the market since 2004. There do exist quantum theories which incorporate special relativity—for example, quantum electrodynamics (QED), which is currently the most accurately tested physical theory —and these lie at the very heart of modern particle physics. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology. Attempts at a unified theory Inconsistencies arise when one tries to join the quantum laws with general relativity, a more elaborate description of spacetime which incorporates gravitation. Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics. Many prominent physicists, including Stephen Hawking, have labored in the attempt to discover a "Grand Unification Theory" that combines not only different models of subatomic physics, but also derives the universe's four forces—the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon. Quantum mechanics and classical physics This is in accordance with the following observations: Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" can be roughly translated from German as inherent or as a characteristic). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time; rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for (a) the state of something having an uncertainty relation and (b) a state that has a definite value. The latter is called the "eigenstate" of the property being measured. For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wave function. The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. If one performs a position measurement on such a wavefunction, the result x will be obtained with almost 100% probability. In other words, the position of the free particle will almost be known. This is called an eigenstate of position (mathematically more precise: a generalized eigenstate (eigendistribution) ). If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. It can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate. If the particle is in an eigenstate of momentum then its position is completely blurred out. Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse. It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x. Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (HAtomOrbitals.png). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric). Mathematical formulation Interactions with other scientific theories The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical -frac{e^2}{4 pi epsilon_0 } frac{1}{r} Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. Derivation of quantization - frac {hbar ^2}{2m} frac {d ^2 psi}{dx^2} = E psi. The general solutions are: psi = A e^{ikx} + B e ^{-ikx} ;;;;;; E = frac{k^2 hbar^2}{2m} psi = C sin kx + D cos kx ; (exponential rewrite) The presence of the walls of the box restricts the acceptable solutions of the wavefunction. At each wall : psi = 0 ; mathrm{at} ;; x = 0,; x = L Consider x = 0 • sin 0 = 0, cos 0 = 1. To satisfy scriptstyle psi = 0 ; the cos term has to be removed. Hence D = 0 Now consider: scriptstyle psi = C sin kx; • at x = L, scriptstyle psi = C sin kL =0; • If C = 0 then scriptstyle psi =0 ; for all x. This would conflict with the Born interpretation • therefore sin kL = 0 must be satisfied, yielding the condition kL = n pi ;;;; n = 1,2,3,4,5,... ; In this situation, n must be an integer showing the quantization of the energy levels. Philosophical consequences Main article: Interpretation of quantum mechanics The writer C. S. Lewis viewed quantum mechanics as incomplete. Lewis, a professor of English, was of the opinion that the Heisenberg uncertainty principle was more of an epistemic limitation than an indication of ontological indeterminacy, and in this respect believed similarly to many advocates of hidden variables theories. The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view. See also • Quantum electrochemistry • Quantum electronics • Quantum field theory • Quantum information • Quantum mind • Quantum optics • Quantum thermodynamics • Quasi-set theory • Theoretical and experimental justification for the Schrödinger equation • Theoretical chemistry • Notes • P. A. M. Dirac, The Principles of Quantum Mechanics (1930) -- the beginning chapters provide a very clear and comprehensible introduction • Richard P. Feynman, Robert B. Leighton and Matthew Sands (1965). The Feynman Lectures on Physics, Addison-Wesley. • Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of Modern Physics vol 29, (1957) pp 454-462. • Richard P. Feynman, QED: The Strange Theory of Light and Matter -- a popular science book about quantum mechanics and quantum field theory that contains many enlightening insights that are interesting for the expert as well • Marvin Chester, Primer of Quantum Mechanics, 1987, John Wiley, N.Y. ISBN 0-486-42878-8 • Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3th edition, World Scientific (Singapore, 2004) (drafts of a forthcoming fourth edition available online here) • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. -- a standard undergraduate level text • H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications 1950. • Max Jammer, "The Conceptual Development of Quantum Mechanics" (McGraw Hill Book Co., 1966) • Gunther Ludwig, "Wave Mechanics" (Pergamon Press, 1968) ISBN 0-08-203204-1 • Albert Messiah, Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer, fourth printing 1966, North Holland, John Wiley & Sons. • Eric R. Scerri, The Periodic Table: Its Story and Its Significance, Oxford University Press, 2006. Considers the extent to which chemistry and especially the periodic system has been reduced to quantum mechanics. ISBN 0-19-530573-6 External links Search another word or see quantum numberson Dictionary \| Thesaurus \|Spanish Copyright © 2015, LLC. All rights reserved. • Please Login or Sign Up to use the Recent Searches feature
dd618484b0a30082	Skip to main content Chemistry LibreTexts 1: The Basics of Quantum Mechanics • Page ID • In this portion of the text, most of the topics that are appropriate to an undergraduate reader are covered. Many of these subjects are subsequently discussed again in Chapter 5, where a broad perspective of what theoretical chemistry is about is offered. They are treated again in greater detail in Chapters 6-8 where the three main disciplines of theory, electronic structure, chemical dynamics, and statistical mechanics, are covered in depth appropriate to a graduate-student reader. In this Chapter, you should have learned about the following things: 1. Why quantum mechanics is needed; that is, what things classical mechanics does not describe correctly. How quantum and classical descriptions can sometimes agree and when they will not. How certain questions can only be asked when classical mechanics applies, not when quantum mechanics is needed. 2. The Schrödinger equation, operators, wave functions, eigenvalues and eigenfunctions and their relations to experimental observations. 3. Time propagation of wave functions. 4. Free particle motion and corresponding eigenfunctions in one, two, and three dimensions and the associated energy levels, and the relevance of these models to various chemistry issues. 5. Action quantization and the resulting semi-classical wave functions and how this point of view offers connections between classical and quantum perspectives.
7dc8d25810188a3f	We gratefully acknowledge support from the Simons Foundation and member institutions. New submissions [ total of 227 entries: 1-227 ] New submissions for Mon, 27 Jan 20 [1] arXiv:2001.08763 [pdf, other] Title: The classification of multiplicity-free plethysms of Schur functions Subjects: Representation Theory (math.RT); Combinatorics (math.CO) We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near maximal in the dominance ordering and those of small Durfee size. [2] arXiv:2001.08771 [pdf, ps, other] Title: The Dehn twist on a sum of two K3 surfaces Comments: 15 pages Ruberman gave the first examples of self-diffeomorphisms of four-manifolds that are isotopic to the identity in the topological category but not smoothly so. We give another example of this phenomenon, using the Dehn twist along a 3-sphere in the connected sum of two K3 surfaces. [3] arXiv:2001.08775 [pdf, ps, other] Title: Atomic decompositions for noncommutative martingales Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA); Probability (math.PR) We prove an atomic type decomposition for the noncommutative martingale Hardy space $\h_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of $\h_p$ for all $0< p < 1,$ and provide a constructive proof of the atomic decomposition for $p=1$. We also study $(p,\8)_c$-atoms, and show that every $(p,2)_c$-atom can be decomposed into a sum of $(p,\8)_c$-atoms; consequently, for every $0<p\le 1$, the $(p,q)_c$-atoms lead to the same atomic space for all $2\le q\le\8$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space $\h_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to proving some sharp martingale inequalities. [4] arXiv:2001.08789 [pdf, ps, other] Title: Short-time heat content asymptotics via the wave and eikonal equations Subjects: Analysis of PDEs (math.AP) In this short paper, we derive an alternative proof for some known [Van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation} \int_{S} \exp(t\Delta) f \mathbb 1_S\, \mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}{\pi}} \int_{\partial S} f \,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation} and (partially new) explicit expressions for similar expansions involving arbitrary powers of $\sqrt t$. By the same method, we also obtain short-time asymptotics of $\int_S \exp(t^m\Delta^m)f \mathbb 1_S\, \mathrm{d}x$, $m \in \mathbb N$, and more generally for one-parameter families of operators $t \mapsto k(\sqrt{-t\Delta})$ defined by an even Schwartz function $k$. [5] arXiv:2001.08795 [pdf, ps, other] Title: Grothendieck duality and Greenlees-May duality on graded rings Authors: Wai-Kit Yeung Comments: 29 pages; originally part of arXiv:1907.06190, now split into three papers Subjects: Algebraic Geometry (math.AG) We formulate and prove Serre's equivalence for $\mathbb{Z}$-graded rings. When restricted to the usual case of $\mathbb{N}$-graded rings, our version of Serre's equivalence also sharpens the usual one by replacing the condition that $A$ be generated by $A_1$ over $A_0$ by a more natural condition, which we call the Cartier condition. For $\mathbb{Z}$-graded rings coming from flips and flops, this Cartier condition relates more naturally to the geometry of the flip/flop in question. We also interpret Grothendieck duality as an instance of Greenlees-May duality for graded rings. These form the basic setting for a homological study of flips and flops in [Yeu20a, Yeu20b]. [6] arXiv:2001.08796 [pdf, ps, other] Title: Approximation by sampling-type operators in $L_p$-spaces Subjects: Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA) Approximation properties of the sampling-type quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$ for functions $f$ from anisotropic Besov spaces are studied. Error estimates in $L_p$-norm are obtained for a large class of tempered distributions $\widetilde{\varphi}$ and a large class of functions $\varphi$ under the assumptions that $\varphi$ has enough decay, satisfies the Strang-Fix conditions and a compatibility condition with $\widetilde{\varphi}$. The estimates are given in terms of moduli of smoothness and best approximations. [7] arXiv:2001.08797 [pdf, ps, other] Title: Specker Algebras: A Survey Subjects: Rings and Algebras (math.RA) For a commutative ring $R$ with identity, a Specker $R$-algebra is a commutative unital $R$-algebra generated by a Boolean algebra of idempotents, each nonzero element of which is faithful. Such algebras have arisen in the study of $\ell$-groups, idempotent-generated rings, Boolean powers of commutative rings, Pierce duality, and rings of continuous real-valued functions. We trace the origin of this notion from early studies of subgroups of bounded integer-valued functions to a variety of current contexts involving ring-theoretic, topological, and homological aspects of idempotent-generated algebras. [8] arXiv:2001.08799 [pdf, ps, other] Title: Characterizations of the Borel triangle and Borel polynomials Authors: Paul Barry Comments: 24 pages Subjects: Combinatorics (math.CO) We use Riordan array theory to give characterizations of the Borel triangle and its associated polynomial sequence. We show that the Borel polynomials are the moment sequence for a family of orthogonal polynomials whose coefficient array is a Riordan array. The role of the Catalan matrix in defining the Borel triangle is examined. We generalize the Borel triangle to a family of two parameter triangles. Generating functions are expressed as Jacobi continued fractions, as well as the zeros of appropriate quadratic expressions. The Borel triangle is exhibited as a Hadamard product of matrices. We investigate the reversions of the triangles studied. We introduce the notion of Fuss-Borel triangles and Fuss-Catalan triangles. We end with some remarks on the Catalan triangle. [9] arXiv:2001.08800 [pdf, ps, other] Title: A new approach to the Katětov-Tong theorem Subjects: General Topology (math.GN) We give a new proof of the Kat\v{e}tov-Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces. The key ingredient is how the ring of bounded continuous real-valued functions embeds in the ring of all bounded real-valued functions. In the compact case this embedding can be described by an appropriate statement, which we prove implies both the Kat\v{e}tov-Tong theorem and a version of the Stone-Weierstrass theorem. We then extend the Kat\v{e}tov-Tong theorem to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone-\v Cech compactification so that the less than or equal relation between the functions is preserved. [10] arXiv:2001.08801 [pdf, ps, other] Title: A Construction of Uniquely Colourable Graphs with Equal Colour Class Sizes Authors: Samuel Mohr Subjects: Combinatorics (math.CO) A uniquely $k$-colourable graph is a graph with exactly one partition of the vertex set into at most $k$ colour classes. Here, we investigate some constructions of uniquely $k$-colourable graphs and give a construction of $K_k$-free uniquely $k$-colourable graphs with equal colour class sizes. [11] arXiv:2001.08813 [pdf, ps, other] Title: Maximizing the Bregman divergence from a Bregman family Comments: 11 pages, 5 theorems, no figure Subjects: Information Theory (cs.IT) The problem to maximize the information divergence from an exponential family is generalized to the setting of Bregman divergences and suitably defined Bregman families. [12] arXiv:2001.08820 [pdf, ps, other] Title: The metric theory of the pair correlation function of real-valued lacunary sequences Comments: Comments welcome Subjects: Number Theory (math.NT) Let $\{ a(x) \}_{x=1}^{\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a(x)$ is Poissonian for Lebesgue almost every $\alpha\in \mathbb{R}$. By using harmonic analysis, our result - irrespective of the choice of the real-valued sequence $\{ a(x) \}_{x=1}^{\infty}$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$ \vert n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) \vert < 1 $$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N]^6$ with the ``excluded diagonals'', that is $$x_1\neq y_1, \quad x_2 \neq y_2, \quad (n_1,n_2)\neq (0,0),$$ is at most $N^{4-\delta}$ for some fixed $\delta>0$, for all sufficiently large $N$. [13] arXiv:2001.08822 [pdf, other] Title: Counting linear extensions of posets with determinants of hook lengths Comments: 26 pages, 26 figures Subjects: Combinatorics (math.CO) We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also give $q$-analogues of this determinantal formula in terms of the major index and inversion statistics. As applications, we give families of tree posets whose numbers of linear extensions are given by generalizations of Euler numbers, we draw relations to Naruse-Okada's positive formulas for the number of linear extensions of skew $d$-complete posets, and we give polynomiality results analogous to those of descent polynomials by Billey-Burdzy-Sagan and Diaz-L\'opez et. al. [14] arXiv:2001.08824 [pdf, other] Title: Goal-oriented a posteriori estimation of numerical errors in the solution of multiphysics systems Comments: 25 pages, 7 figures Subjects: Numerical Analysis (math.NA) This paper develops a general methodology for a posteriori error estimation in time-dependent multiphysics numerical simulations. The methodology builds upon the generalized-structure additive Runge--Kutta (GARK) approach to time integration. GARK provides a unified formulation of multimethods that simulate complex systems by applying different discretization formulas and/or different time steps to individual components of the system. We derive discrete GARK adjoints and analyze their time accuracy. Based on the adjoint method, we establish computable a posteriori identities for the impacts of both temporal and spatial discretization errors on a given goal function. Numerical examples with reaction-diffusion systems illustrate the accuracy of the derived error measures. Local error decompositions are used to illustrate the power of this framework in adaptive refinements of both temporal and spatial meshes. [15] arXiv:2001.08825 [pdf, other] Title: Numerical Approximation of the Fractional Laplacian on $\mathbb R$ Using Orthogonal Families Comments: 20 pages, 5 figures Subjects: Numerical Analysis (math.NA) In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the ${}_2F_1$ Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the Higgins functions, the Christov functions, and their sine-like and cosine-like versions. After discussing the numerical difficulties in the implementation of the proposed formulas, we develop a method using variable precision arithmetic that gives accurate results. [16] arXiv:2001.08826 [pdf, other] Title: An $O(s^r)$-Resolution ODE Framework for Discrete-Time Optimization Algorithms and Applications to Convex-Concave Saddle-Point Problems Authors: Haihao Lu There has been a long history of using Ordinary Differential Equations (ODEs) to understand the dynamic of discrete-time optimization algorithms. However, one major difficulty of applying this approach is that there can be multiple ODEs that correspond to the same discrete-time algorithm, depending on how to take the continuous limit, which makes it unclear how to obtain the suitable ODE from a discrete-time optimization algorithm. Inspired by the recent paper \cite{shi2018understanding}, we propose the $r$-th degree ODE expansion of a discrete-time optimization algorithm, which provides a principal approach to construct the unique $O(s^r)$-resolution ODE systems for a given discrete-time algorithm, where $s$ is the step-size of the algorithm. We utilize this machinery to study three classic algorithms -- gradient method (GM), proximal point method (PPM) and extra-gradient method (EGM) -- for finding the solution to the unconstrained convex-concave saddle-point problem $\min_{x\in\RR^n} \max_{y\in \RR^m} L(x,y)$, which explains their puzzling convergent/divergent behaviors when $L(x,y)$ is a bilinear function. Moreover, their $O(s)$-resolution ODEs inspire us to define the $O(s)$-linear-convergence condition on $L(x,y)$, under which PPM and EGM exhabit linear convergence. This condition not only unifies the known linear convergence rate of PPM and EGM, but also showcases that these two algorithms exhibit linear convergence in broader contexts. [17] arXiv:2001.08829 [pdf, other] Title: High dimensional expansion using zig-zag product Comments: 19 Pages. Some of the results(Conlon's case) were presented in HUJI Combinatorics seminar on June 10th, 2019 Subjects: Combinatorics (math.CO) We wish to renew the discussion over recent combinatorial structures that are 3-uniform hypergraph expanders, viewing them in a more general perspective, shedding light on a previously unknown relation to the zig-zag product. We do so by introducing a new structure called triplet structure, that maintains the same local environment around each vertex. The structure is expected to yield, in some cases, a bounded family of hypergraph expanders whose 2-dimensional random walk converges. We have applied the results obtained here to several known constructions, obtaining a better expansion rate than previously known. Namely, we did so in the case of Conlon's construction and the $S=[1,1,0]$ construction by Chapman, Linal and Peled. [18] arXiv:2001.08848 [pdf, ps, other] Title: Strong solutions of semilinear SPDEs with unbounded diffusion Authors: Florian Bechtold Subjects: Probability (math.PR) We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions. [19] arXiv:2001.08857 [pdf, ps, other] Title: Arcsine laws for random walks generated from random permutations with applications to genomics Subjects: Probability (math.PR); Statistics Theory (math.ST) A classical result for the simple symmetric random walk with $2n$ steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows($q$) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step $2n$ for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Stein's method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows$(q)$ permutation which is of independent interest. [20] arXiv:2001.08859 [pdf, ps, other] Title: Convergence of a finite element method for degenerate two-phase flow in porous media Subjects: Numerical Analysis (math.NA) A finite element method with mass-lumping and flux upwinding, is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Theoretical convergence is proved via a compactness argument. The proof is convoluted because of the degeneracy of the phase mobilities and the unboundedness of the capillary pressure. [21] arXiv:2001.08860 [pdf, ps, other] Title: Notes on Graph Product Structure Theory Comments: 19 pages, 0 figures It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometrically defined graph classes, and present several open problems. [22] arXiv:2001.08862 [pdf, ps, other] Title: A flow method for the dual Orlicz-Minkowski problem Authors: YanNan Liu, Jian Lu Comments: Accepted by Transactions of the American Mathematical Society [23] arXiv:2001.08872 [pdf, ps, other] Title: Irreducible cone spherical metrics and stable extensions of two line bundles Comments: This manuscript supersedes arXiv:1808.04106. 34 pages Subjects: Algebraic Geometry (math.AG) A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${\rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X \geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 \pi \mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X \geq 2$. As an application, we prove the following two results about irreducible metrics: $\bullet$ as $g_X \geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $\geq 2(d+3-3g_X)$ in ${\rm Sym}^d(X)$; $\bullet$ as $g_X \geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$. [24] arXiv:2001.08874 [pdf, other] Title: Goal-Oriented Adaptive THB-Spline Schemes for PDE-Based Planar Parameterization Subjects: Numerical Analysis (math.NA) This paper presents a PDE-based planar parameterization framework with support for Truncated Hierarchical B-Splines (THB-splines). For this, we adopt the a posteriori refinement strategy of Dual Weighted Residual and present several adaptive numerical schemes for the purpose of approximating an inversely harmonic geometry parameterization. Hereby, the combination of goal-oriented a posteriori strategies and THB-enabled local refinement avoids over-refinement, in particular in geometries with complex boundaries. To control the parametric properties of the outcome, we introduce the concept of domain refinement. Hereby, the properties of the domain into which the mapping maps inversely harmonically, are optimized in order to fine-tune the parametric properties of the recomputed geometry parameterization. [25] arXiv:2001.08875 [pdf, ps, other] Title: Weight distributions and weight hierarchies of two classes of binary linear codes Authors: Fei Li Comments: 17 First, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Second, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes. [26] arXiv:2001.08876 [pdf, other] Title: From Nesterov's Estimate Sequence to Riemannian Acceleration Comments: 30 pages We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the key difficulty due to non-Euclidean structure into a certain ``metric distortion.'' We control this distortion by developing a novel geometric inequality, which permits us to propose and analyze a Riemannian counterpart to Nesterov's accelerated gradient method. [27] arXiv:2001.08877 [pdf, other] Title: Distributed Gaussian Mean Estimation under Communication Constraints: Optimal Rates and Communication-Efficient Algorithms Subjects: Statistics Theory (math.ST); Distributed, Parallel, and Cluster Computing (cs.DC); Information Theory (cs.IT); Machine Learning (cs.LG) We study distributed estimation of a Gaussian mean under communication constraints in a decision theoretical framework. Minimax rates of convergence, which characterize the tradeoff between the communication costs and statistical accuracy, are established in both the univariate and multivariate settings. Communication-efficient and statistically optimal procedures are developed. In the univariate case, the optimal rate depends only on the total communication budget, so long as each local machine has at least one bit. However, in the multivariate case, the minimax rate depends on the specific allocations of the communication budgets among the local machines. Although optimal estimation of a Gaussian mean is relatively simple in the conventional setting, it is quite involved under the communication constraints, both in terms of the optimal procedure design and lower bound argument. The techniques developed in this paper can be of independent interest. An essential step is the decomposition of the minimax estimation problem into two stages, localization and refinement. This critical decomposition provides a framework for both the lower bound analysis and optimal procedure design. [28] arXiv:2001.08881 [pdf, ps, other] Title: A note on purely imaginary independence roots Subjects: Combinatorics (math.CO) The independence polynomial of a graph is the generating polynomial for the number of independent sets of each cardinality and its roots are called independence roots. We investigate here purely imaginary independence roots. We show that there are infinitely many connected graphs with purely imaginary independence roots and that every graph is a subgraph of such a graph. We also classify every rational purely imaginary number that is an independence root. [29] arXiv:2001.08894 [pdf, other] Title: Families of Multidimensional Arrays with Good Autocorrelation and Asymptotically Optimal Cross-correlation Authors: Sam Blake Subjects: Information Theory (cs.IT) We introduce a construction for families of 2n-dimensional arrays with asymptotically optimal pairwise cross-correlation. These arrays are constructed using a circulant array of n-dimensional Legendre arrays. We also introduce an application of these higher-dimensional arrays to high-capacity digital watermarking of images. [30] arXiv:2001.08902 [pdf, ps, other] Title: Distance problems for dissipative Hamiltonian systems and related matrix polynomials Subjects: Numerical Analysis (math.NA) We study the characterization of several distance problems for linear differential-algebraic systems with dissipative Hamiltonian structure. Since all models are only approximations of reality and data are always inaccurate, it is an important question whether a given model is close to a 'bad' model that could be considered as ill-posed or singular. This is usually done by computing a distance to the nearest model with such properties. We will discuss the distance to singularity and the distance to the nearest high index problem for dissipative Hamiltonian systems. While for general unstructured differential-algebraic systems the characterization of these distances are partially open problems, we will show that for dissipative Hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically. [31] arXiv:2001.08905 [pdf, ps, other] Title: Every finite abelian group is a subgroup of the additive group of a finite simple left brace Comments: 9 pages Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,\cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,\cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces. [32] arXiv:2001.08907 [pdf, ps, other] Title: Weights of Semiregular Nilpotents in Simple Lie Algebras of D Type Authors: Yassir Dinar Subjects: Representation Theory (math.RT) We compute the weights of the adjoint action of semiregular $sl_2$-triples in simple Lie algebras of type $D_n$ using mathematical induction. [33] arXiv:2001.08909 [pdf, ps, other] Title: Intelligent Reflecting Surface-Assisted Multiple Access with User Pairing: NOMA or OMA? Comments: This work has been accepted in IEEE Communications Letters (5 Pages, 4 Figures). In this work, we pursue a theoretical performance comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) in the IRS-assisted downlink communication The integration of intelligent reflecting surface (IRS) to multiple access networks is a cost-effective solution for boosting spectrum/energy efficiency and enlarging network coverage/connections. However, due to the new capability of IRS in reconfiguring the wireless propagation channels, it is fundamentally unknown which multiple access scheme is superior in the IRS-assisted wireless network. In this letter, we pursue a theoretical performance comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) in the IRS-assisted downlink communication, for which the transmit power minimization problems are formulated under the discrete unit-modulus reflection constraint on each IRS element. We analyze the minimum transmit powers required by different multiple access schemes and compare them numerically, which turn out to not fully comply with the stereotyped superiority of NOMA over OMA in conventional systems without IRS. Moreover, to avoid the exponential complexity of the brute-force search for the optimal discrete IRS phase shifts, we propose a low-complexity solution to achieve near-optimal performance. [34] arXiv:2001.08912 [pdf, other] Title: Flexible models for overdispersed and underdispersed count data Subjects: Probability (math.PR) Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to a real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD's) to allow both overdispersion and underdispersion. Similar to Kemp's generalized hypergeometric probability distribution, based on hypergeometric functions, we introduce a novel WPD case where the weight function is chosen as a suitable ratio of three-parameter Mittag--Leffler functions. The proposed family includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature. [35] arXiv:2001.08915 [pdf, ps, other] Title: Permutations of N generated by left-right filling algorithms Authors: F. M. Dekking Subjects: Combinatorics (math.CO) We give an in depth analysis of an algorithm that generates permutations of the natural numbers introduced by Clark Kimberling in the On Line Encyclopedia of Integer Sequences. It turns out that the examples of such permutations in OEIS are completely determined by 3-automatic sequences. [36] arXiv:2001.08916 [pdf, other] Title: Normal-normal resonances in a double Hopf bifurcation Comments: 22 pages, 6 figures Subjects: Dynamical Systems (math.DS) We investigate the stability loss of invariant n-dimensional quasi-periodic tori during a double Hopf bifurcation, where at bifurcation the two normal frequencies are in normal-normal resonance. Invariants are used to analyse the normal form approximations in a unified manner. The corresponding dynamics form a skeleton for the dynamics of the original system. Here both normal hyperbolicity and KAM theory are being used. [37] arXiv:2001.08919 [pdf, ps, other] Title: Homogenization of ferromagnetic energies on Poisson random sets in the plane We prove that by scaling nearest-neighbour ferromagnetic energies defined on Poisson random sets in the plane we obtain an isotropic perimeter energy with a surface tension characterised by an asymptotic formula. The result relies on proving that cells with `very long' or `very short' edges of the corresponding Voronoi tessellation can be neglected. In this way we may apply Geometry Measure Theory tools to define a compact convergence, and a characterisation of metric properties of clusters of Voronoi cells using limit theorems for subadditive processes. [38] arXiv:2001.08923 [pdf, ps, other] Title: On accumulation points of $F$-pure thresholds on regular local rings Authors: Kenta Sato Comments: 18pages Blickle, Musta\c{t}\u{a} and Smith proposed two conjectures on the limits of $F$-pure thresholds. One conjecture asks whether or not the limit of a sequence of $F$-pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an $F$-pure thresholds in lower dimension. Another conjecture predicts that any $F$-pure threshold of a formal power series can be written as the $F$-pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture. [39] arXiv:2001.08931 [pdf, ps, other] Title: Distribution of missing differences in diffsets Comments: Version 1.0, 17 pages, 3 figures Subjects: Combinatorics (math.CO) Lazarev, Miller and O'Bryant investigated the distribution of $\|S+S\|$ for $S$ chosen uniformly at random from $\{0, 1, \dots, n-1\}$, and proved the existence of a divot at missing 7 sums (the probability of missing exactly 7 sums is less than missing 6 or missing 8 sums). We study related questions for $\|S-S\|$, and shows some divots from one end of the probability distribution, $P(\|S-S\|=k)$, as well as a peak at $k=4$ from the other end, $P(2n-1-\|S-S\|=k)$. A corollary of our results is an asymptotic bound for the number of complete rulers of length $n$. [40] arXiv:2001.08932 [pdf, ps, other] Title: On the enhanced power graph of a group Subjects: Group Theory (math.GR); Combinatorics (math.CO) The enhanced power graph $\mathcal{P}_e(G)$ of a group $G$ is a graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number and matching number of enhanced power graphs of finite groups. We first study these graph invariants for $\mathcal{P}_e(G)$ when $G$ is any finite group, and then determine them when $G$ is a finite abelian $p$-group, $U_{6n} = \langle a, b : a^{2n} = b^3 = e, ba =ab^{-1} \rangle$, the dihedral group $D_{2n}$, or the semidihedral group $SD_{8n}$. If $G$ is any of these groups, we prove that $\mathcal{P}_e(G)$ is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of $\mathcal{P}_e(G)$ for any finite abelian group $G$. These results along with certain known equalities yield the edge connectivity, vertex covering number and edge covering number of enhanced power graphs of the respective groups as well. [41] arXiv:2001.08936 [pdf, other] Title: Clustering Methods Assessment for Investment in Zero Emission Neighborhoods Energy System Authors: Dimitri Pinel Comments: 12 pages, 19 figures, 7 tables, 1 Appendix, This paper investigates the use of clustering in the context of designing the energy system of Zero Emission Neighborhoods (ZEN). ZENs are neighborhoods who aim to have net zero emissions during their lifetime. While previous work has used and studied clustering for designing the energy system of neighborhoods, no article dealt with neighborhoods such as ZEN, which have high requirements for the solar irradiance time series, include a CO2 factor time series and have a zero emission balance limiting the possibilities. To this end several methods are used and their results compared. The results are on the one hand the performances of the clustering itself and on the other hand, the performances of each method in the optimization model where the data is used. Various aspects related to the clustering methods are tested. The different aspects studied are: the goal (clustering to obtain days or hours), the algorithm (k-means or k-medoids), the normalization method (based on the standard deviation or range of values) and the use of heuristic. The results highlight that k-means offers better results than k-medoids and that k-means was systematically underestimating the objective value while k-medoids was constantly overestimating it. When the choice between clustering days and hours is possible, it appears that clustering days offers the best precision and solving time. The choice depends on the formulation used for the optimization model and the need to model seasonal storage. The choice of the normalization method has the least impact, but the range of values method show some advantages in terms of solving time. When a good representation of the solar irradiance time series is needed, a higher number of days or using hours is necessary. The choice depends on what solving time is acceptable. [42] arXiv:2001.08937 [pdf, other] Title: Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors Comments: 12 pages, 7 figures. Submitted In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure. [43] arXiv:2001.08955 [pdf, ps, other] Title: The model structure for chain complexes Authors: Neil Strickland Subjects: Algebraic Topology (math.AT) Let $\text{Ch}$ be the category of (possibly unbounded) chain complexes of abelian groups. In this note we construct the standard Quillen model structure on $\text{Ch}$, by a method that is somewhat different from the standard one. Essentially, we use a functorial two-stage projective resolution for abelian groups, and build everything directly from that. This has the advantage of being very concrete, explicit and functorial. It does not rely on the small object argument, or make any explicit use of transfinite induction. On the other hand, it is not so conceptual, and it does use the fact that subgroups of free abelian groups are free, so it does not generalise to many rings other than $\mathbb{Z}$. We do not claim any great technical benefit for this approach, but it seems like an interesting alternative, and may be pedagogically useful. [44] arXiv:2001.08956 [pdf, ps, other] Title: Online Resource Procurement and Allocation in a Hybrid Edge-Cloud Computing System By acquiring cloud-like capacities at the edge of a network, edge computing is expected to significantly improve user experience. In this paper, we formulate a hybrid edge-cloud computing system where an edge device with limited local resources can rent more from a cloud node and perform resource allocation to serve its users. The resource procurement and allocation decisions depend not only on the cloud's multiple rental options but also on the edge's local processing cost and capacity. We first propose an offline algorithm whose decisions are made with full information of future demand. Then, an online algorithm is proposed where the edge node makes irrevocable decisions in each timeslot without future information of demand. We show that both algorithms have constant performance bounds from the offline optimum. Numerical results acquired with Google cluster-usage traces indicate that the cost of the edge node can be substantially reduced by using the proposed algorithms, up to $80\%$ in comparison with baseline algorithms. We also observe how the cloud's pricing structure and edge's local cost influence the procurement decisions. [45] arXiv:2001.08959 [pdf, other] Title: Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane Subjects: Probability (math.PR) In this work, we focus on the stationary analysis of a specific class of continuous time Markov modulated reflected random walk in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. Using a queueing network example, we numerically validated the theoretical findings. [46] arXiv:2001.08963 [pdf, other] Title: Intelligent Reflecting Surface Assisted Secure Wireless Communications with Multiple-Transmit and Multiple-Receive Antennas Comments: 26 pages, 5 figures Subjects: Information Theory (cs.IT) In this paper, we propose intelligent reflecting surfaces (IRS) assisted secure wireless communications with multi-input and multi-output antennas (IRS-MIMOME). The considered scenario is an access point (AP) equipped with multiple antennas communicates with a multi-antenna enabled legitimate user in the downlink at the present of an eavesdropper configured with multiple antennas. Particularly, the joint optimization of the transmit covariance matrix at the AP and the reflecting coefficients at the IRS to maximize the secrecy rate for the IRS-MIMOME system is investigated, with two different assumptions on the phase shifting capabilities at the IRS, i.e., the IRS has the continuous reflecting coefficients and the IRS has the discrete reflecting coefficients. For the former case, due to the non-convexity of the formulated problem, an alternating optimization (AO)-based algorithm is proposed, i.e., for given the reflecting coefficients at the IRS, the successive convex approximation (SCA)-based algorithm is used to solve the transmit covariance matrix optimization, while given the transmit covariance matrix at the AP, alternative optimization is used again in individually optimizing of each reflecting coefficient at the IRS with other fixed reflecting coefficients. For the individual reflecting coefficient optimization, the close-form or an interval of the optimal solution is provided. Then, the proposed algorithm is extended to the discrete reflecting coefficient model at the IRS. Finally, some numerical simulations have been done to demonstrate that the proposed algorithm outperforms other benchmark schemes. [47] arXiv:2001.08970 [pdf, ps, other] Title: Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models Subjects: Analysis of PDEs (math.AP) We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell-Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic--hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short--times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier-Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity. [48] arXiv:2001.08973 [pdf, ps, other] Title: A continuum limit for the PageRank algorithm Subjects: Analysis of PDEs (math.AP); Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR) Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological, or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm, and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalized graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion, and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit PDE. We give applications to proving stability and asymptotic regularity of the PageRank vector. [49] arXiv:2001.08978 [pdf, other] Title: Symplectic hats Comments: 45 pages, 5 figures; comments welcome! We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes. We apply the construction to give constraints on the algebraic topology of fillings of double covers of the 3-sphere branched over certain transverse quasipositive knots. [50] arXiv:2001.08982 [pdf, ps, other] Title: Circuit-Difference Matroids Comments: 11 pages Subjects: Combinatorics (math.CO) One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class. [51] arXiv:2001.08984 [pdf, ps, other] Title: Smoothing and growth bound of periodic generalized Korteweg-de Vries equation Subjects: Analysis of PDEs (math.AP) For generalized KdV models with polynomial nonlinearity, we establish nonlinear smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing effect persists globally, provided that the $H^1$ norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains $\min(2s-1,1)-$ derivatives for $s>\frac{1}{2}$. Following a new simple method, which is of independent interest, we establish that, for $s>1$, $H^s$ norm of a solution grows at most by $\langle t\rangle^{s-1+}$ if $H^1$ norm is a priori controlled. [52] arXiv:2001.08989 [pdf, ps, other] Title: Generalized Realizability and Basic Logic Comments: 25 pages Subjects: Logic (math.LO) Let V be a set of number-theoretical functions. We define a notion of absolute V-realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this paper we prove that Basic Logic is sound with respect to the semantics of absolute V-realizability if V satisfies some natural conditions. [53] arXiv:2001.09000 [pdf, ps, other] Title: Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs Comments: arXiv admin note: text overlap with arXiv:1809.03227 Subjects: Numerical Analysis (math.NA); Functional Analysis (math.FA) In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the scheme toward the mild solution of the PDE under polynomial growth condition of the nonlinearity. Our convergence rate are obtain for smooth and non-smooth initial data and is similar to that of the autonomous case. Our convergence result for smooth initial data is very important in numerical analysis. For instance, it is one step forward in approximating non-autonomous stochastic partial differential equations by the finite element method. In addition, we provide realistic conditions on the nonlinearity, appropriated to achieve optimal convergence rate without logarithmic reduction by exploiting the smooth properties of the two parameters evolution operator. [54] arXiv:2001.09002 [pdf, ps, other] Title: Stochastic homogenization of multicontinuum heterogeneous flows Comments: arXiv admin note: text overlap with arXiv:1810.07534, arXiv:1504.04845 We consider a multicontinuum model in porous media applications, which is described as a system of coupled flow equations. The coupling between different continua depends on many factors and its modeling is important for porous media applications. The coefficients depend on particle deposition that is described in term of a stochastic process solution of an SDE. The stochastic process is considered to be faster than the flow motion and we introduce time-space scales to model the problem. Our goal is to pass to the limit in time and space and to find an associated averaged system. This is an averaging-homogenization problem, where the averages are computed in terms of the invariant measure associated to the fast motion and the spatial variable. We use the techniques developed in our previous paper to model the interactions between the continua and derive the averaged model problem that can be used in many applications. [55] arXiv:2001.09004 [pdf, ps, other] Title: New unitals in projective planes of order 16 Authors: Mustafa Gezek Comments: arXiv admin note: text overlap with arXiv:1702.06909 by other authors Subjects: Combinatorics (math.CO) In this study we performed a computer search for unitals in planes of order 16. Some new unitals were found and we show that some unitals can be embedded in two or more different planes. [56] arXiv:2001.09009 [pdf, ps, other] Title: Numerator polynomials of the Riordan matrices Authors: E. Burlachenko Subjects: Number Theory (math.NT) Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials ${{g}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( m \right)$ is the polynomial of degree $\le n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials ${{h}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the exponential Riordan matrices. In paper, the properties of these two types of numerator polynomials and the constructive relationships between them are considered. Separate attention is paid to the numerator polynomials of Riordan matrices associated with the family of series $_{\left( \beta \right)}a\left( x \right)=a\left( x{}_{\left( \beta \right)}{{a}^{\beta }}\left( x \right) \right)$. [57] arXiv:2001.09010 [pdf, ps, other] Title: The impact on mathematics of the paper ''Oscillation and Chaos in Physiological Control Systems'' by Mackey and Glass in Science, 1977 Comments: 7 pages Subjects: Dynamical Systems (math.DS) The note describes the role of the Mackey-Glass equation and of a similar equation due to Lasota in the study of nonlinear delay differential equations between 1977 and 2012, as far as rigorous mathematical results are concerned, and from the very personal point of view of the author's involvement. [58] arXiv:2001.09012 [pdf, ps, other] Title: Proximity in Triangulations and Quadrangulations Comments: arXiv admin note: text overlap with arXiv:1905.06753 Subjects: Combinatorics (math.CO) Let $ G $ be a connected graph. If $\bar{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the proximity, $\pi(G)$, of $G$ is defined as the smallest value of $\bar{\sigma}(v)$ over all vertices $v$ of $G$. We give upper bounds for the proximity of simple triangulations and quadrangulations of given order and connectivity. We also construct simple triangulations and quadrangulations of given order and connectivity that match the upper bounds asymptotically and are likely optimal. [59] arXiv:2001.09013 [pdf, ps, other] Title: Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model Comments: arXiv admin note: text overlap with arXiv:1902.00990 Subjects: Optimization and Control (math.OC) In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities. This paper is an extended and updated version of [arXiv:1902.00990]. In particular, we add an extension of relative strong convexity for optimization and variational inequalities. [60] arXiv:2001.09014 [pdf, ps, other] Title: The identification problem for BSDEs driven by possibly non quasi-left-continuous random measures Comments: arXiv admin note: text overlap with arXiv:1512.06234 Subjects: Probability (math.PR) In this paper we focus on the so called identification problem for a backward SDE driven by a continuous local martingale and a possibly non quasi-left-continuous random measure. Supposing that a solution (Y, Z, U) of a backward SDE is such that $Y(t) = v(t, X(t))$ where X is an underlying process and v is a deterministic function, solving the identification problem consists in determining Z and U in term of v. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when X is a non-semimartingale jump process solution of an SDE with singular coefficients. [61] arXiv:2001.09017 [pdf] Title: Discrete graphical models -- an optimization perspective Comments: 270 pages Journal-ref: Foundations and Trends in Computer Graphics and Vision: Vol. 11: No. 3-4, pp 160-429 (2019) Subjects: Optimization and Control (math.OC); Computer Vision and Pattern Recognition (cs.CV); Discrete Mathematics (cs.DM) This monograph is about discrete energy minimization for discrete graphical models. It considers graphical models, or, more precisely, maximum a posteriori inference for graphical models, purely as a combinatorial optimization problem. Modeling, applications, probabilistic interpretations and many other aspects are either ignored here or find their place in examples and remarks only. It covers the integer linear programming formulation of the problem as well as its linear programming, Lagrange and Lagrange decomposition-based relaxations. In particular, it provides a detailed analysis of the polynomially solvable acyclic and submodular problems, along with the corresponding exact optimization methods. Major approximate methods, such as message passing and graph cut techniques are also described and analyzed comprehensively. The monograph can be useful for undergraduate and graduate students studying optimization or graphical models, as well as for experts in optimization who want to have a look into graphical models. To make the monograph suitable for both categories of readers we explicitly separate the mathematical optimization background chapters from those specific to graphical models. [62] arXiv:2001.09022 [pdf, ps, other] Title: How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings Subjects: Numerical Analysis (math.NA) We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert-Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension $d$. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into $L_2$ and $H^1$. In other words, we investigate the worst-case approximation error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for $n$. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Wo{\'z}niakowski. [63] arXiv:2001.09024 [pdf, ps, other] Title: Deterministic equivalence for noisy perturbations Subjects: Spectral Theory (math.SP); Probability (math.PR) We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is, up to a small error, determined by the singular values of the unperturbed matrix which are larger than some small $N$-dependent cut-off parameter. [64] arXiv:2001.09029 [pdf, ps, other] Title: Rewriting Structured Cospans Authors: Daniel Cicala Subjects: Category Theory (math.CT); Formal Languages and Automata Theory (cs.FL); Social and Information Networks (cs.SI) To foster the study of networks on an abstract level, we further study the formalism of structured cospans. We define a topos of structured cospans and establish its theory of rewriting. For the rewrite relation, we propose a double categorical semantics to encode the compositionality of the structure cospans. For an application, we generalize the inductive viewpoint of graph rewriting to rewriting in a wider class of topoi. [65] arXiv:2001.09030 [pdf, ps, other] Title: Bounds for the capacity error function for unidirectional channels with noiseless feedback Comments: 8 pages, short version submitted to ISIT 2020 Subjects: Information Theory (cs.IT) In digital systems such as fiber optical communications the ratio between probability of errors of type $1\to 0$ and $0 \to 1$ can be large. Practically, one can assume that only one type of errors can occur. These errors are called asymmetric. Unidirectional errors differ from asymmetric type of errors, here both $1 \to 0$ and $0 \to 1$ type of errors are possible, but in any submitted codeword all the errors are of the same type. We consider $q$-ary unidirectional channels with feedback and give bounds for the capacity error function. It turns out that the bounds depend on the parity of the alphabet $q$. Furthermore we show that for feedback the capacity error function for the binary asymmetric channel is different from the symmetric channel. This is in contrast to the behavior of that function without feedback. [66] arXiv:2001.09032 [pdf, ps, other] Title: Limits on Gradient Compression for Stochastic Optimization Subjects: Information Theory (cs.IT) We consider stochastic optimization over $\ell_p$ spaces using access to a first-order oracle. We ask: {What is the minimum precision required for oracle outputs to retain the unrestricted convergence rates?} We characterize this precision for every $p\geq 1$ by deriving information theoretic lower bounds and by providing quantizers that (almost) achieve these lower bounds. Our quantizers are new and easy to implement. In particular, our results are exact for $p=2$ and $p=\infty$, showing the minimum precision needed in these settings are $\Theta(d)$ and $\Theta(\log d)$, respectively. The latter result is surprising since recovering the gradient vector will require $\Omega(d)$ bits. [67] arXiv:2001.09041 [pdf, other] Title: A moduli space for supersingular Enriques surfaces Authors: Kai Behrens Comments: 27 pages Subjects: Algebraic Geometry (math.AG) We construct a moduli space of adequately marked Enriques surfaces that have a supersingular K3 cover over fields of characteristic $p \geq 3$. We show that this moduli space exists as a quasi-separated algebraic space locally of finite type over $\mathbb{F}_p$. Moreover, there exists a period map from this moduli space to a period scheme and we obtain a Torelli theorem for supersingular Enriques surfaces. [68] arXiv:2001.09042 [pdf, ps, other] Title: Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime Subjects: Probability (math.PR) We investigate the characteristic polynomials $\varphi_N$ of the Gaussian $\beta$--ensemble for general $\beta>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $\varphi_N$ in terms of a Gaussian log--correlated field in order to ultimately deduce some of its fine asymptotic properties. We distinguish between different types of transfer matrices and analyze completely the hyperbolic regime of the recurrence. As a result, we obtain a new coupling between $\varphi_N(z)$ and a Gaussian analytic function with an error which is uniform for $z \in \mathbb{C}$ separated from the support of the semicircle law. This also constitutes the first step in obtaining analogous strong approximations for the characteristic polynomials inside of the bulk of the semicircle law. Our analysis relies on moderate deviation estimates for the product of transfer matrices and this approach might also be useful in different contexts. [69] arXiv:2001.09047 [pdf, ps, other] Title: Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher Comments: Almost orthogonal decompositions from arXiv:1905.01490 [math.AP] are adapted to the higher dimensional setting Subjects: Analysis of PDEs (math.AP) The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(\mathbb{R}^3)$ and, under a smallness condition, in $H^1(\mathbb{R}^4)$, follow. [70] arXiv:2001.09048 [pdf, ps, other] Title: Cooperative versus decentralized strategies in three-pursuer single-evader games Comments: Preliminary version submitted to ECC 2020 Subjects: Optimization and Control (math.OC); Robotics (cs.RO) The value of cooperation in pursuit-evasion games is investigated. The considered setting is that of three pursuers chasing one evader in a planar environment. The optimal evader trajectory for a well-known decentralized pursuer strategy is characterized. This result is instrumental to derive upper and lower bounds to the game length, in the case in which the pursuers cooperate in the chasing strategy. It is shown that the cooperation cannot reduce the capture time by more than one half with respect to the decentralized case, and that such bound is tight. [71] arXiv:2001.09049 [pdf, other] Title: Increasing the Raw Key Rate in Energy-Time Entanglement Based Quantum Key Distribution Comments: 14 pages; 3 figures Subjects: Information Theory (cs.IT) A Quantum Key Distribution (QKD) protocol describes how two remote parties can establish a secret key by communicating over a quantum and a public classical channel that both can be accessed by an eavesdropper. QKD protocols using energy-time entangled photon pairs are of growing practical interest because of their potential to provide a higher secure key rate over long distances by carrying multiple bits per entangled photon pair. We consider a system where information can be extracted by measuring random times of a sequence of entangled photon arrivals. Our goal is to maximize the utility of each such pair. We propose a discrete time model for the photon arrival process, and establish a theoretical bound on the number of raw bits that can be generated under this model. We first analyse a well known simple binning encoding scheme, and show that it generates significantly lower information rate than what is theoretically possible. We then propose three adaptive schemes that increase the number of raw bits generated per photon, and compute and compare the information rates they offer. Moreover, the effect of public channel communication on the secret key rates of the proposed schemes is investigated. [72] arXiv:2001.09064 [pdf, ps, other] Title: Five-Linear Singular Integral Estimates of Brascamp-Lieb Type Subjects: Classical Analysis and ODEs (math.CA) We prove the full range of estimates for a five-linear singular integral of Brascamp-Lieb type. The study is methodology-oriented with the goal to develop a sufficiently general technique to estimate singular integral variants of Brascamp-Lieb inequalities that are not of H\"older type. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule which captures nonlinear interactions of waves from transversal directions. [73] arXiv:2001.09068 [pdf, ps, other] Title: On the subring of special cycles Authors: Stephen Kudla For a totally real field F of degree d>1 and a quadratic space V of signature (m,2)^{d_+} x (m+2,0)^{d-d_+} with associated Shimura variety Sh(V), we consider the subring of cohomology generated by the classes of weighted special cycles. We assume that d_+<d. We take the quotient SC(V) of this ring by the radical of the restriction of the intersection pairing to it. We show that the inner products of classes in SC(V) are determined by Fourier coefficients of pullbacks of Hilbert-Siegel Eisenstein series of genus m to products of smaller Siegel spaces and that the products of classes in SC(V) are determined by Fourier coefficients of pullbacks to triple products of smaller Siegel spaces. As a consequence, we show that, for quadratic spaces V and V' over F that are isomorphic at all finite places, but with no restriction on d_+(V) and d_+(V') other than the necessary condition that they have the same parity, the special cycles rings SC(V) and SC(V') are isometrically isomorphic. This is a consequence of the Siegel-Weil formula and the matching principle. Finally, we give a combinatorial construction of a ring SC(V_+) associated to a totally positive definite quadratic space V_+ of dimension m+2 over F and show that the comparison isomorphism extends to this case. [74] arXiv:2001.09075 [pdf, ps, other] Title: A topos-theoretic view of difference algebra Authors: Ivan Tomasic We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference algebraic geometry, and cohomology theory of difference schemes. [75] arXiv:2001.09076 [pdf, ps, other] Title: ECM factorization with QRT maps Authors: Andrew N.W. Hone Subjects: Number Theory (math.NT); Exactly Solvable and Integrable Systems (nlin.SI) Quispel-Roberts-Thompson (QRT) maps are a family of birational maps of the plane which provide the simplest discrete analogue of an integrable Hamiltonian system, and are associated with elliptic fibrations in terms of biquadratic curves. Each generic orbit of a QRT map corresponds to a sequence of points on an elliptic curve. In this preliminary study, we explore versions of the elliptic curve method (ECM) for integer factorization based on iterating three different QRT maps with particular initial data. [76] arXiv:2001.09080 [pdf, ps, other] Title: Cylindrical martingale-valued measures, stochastic integration and stochastic PDEs in Hilbert space Subjects: Probability (math.PR) We introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed using a novel technique that utilizes the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem. We apply the developed theory of stochastic integration to establish existence and uniqueness of weak and mild solutions for stochastic evolution equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Our theory covers the study of integration and of SPDEs driven by Hilbert space valued L\'{e}vy noise (which is not required to satisfy any moment condition), cylindrical L\'{e}vy noise with (weak) second moments and L\'{e}vy-valued random martingale measures with finite second moment. [77] arXiv:2001.09091 [pdf, ps, other] Title: Quantum computation and measurements from an exotic space-time R4 Comments: 16 pages, 8 figires, 2 tables Subjects: Geometric Topology (math.GT); Quantum Physics (quant-ph) The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group $G$ with relations. A valid subgroup $H$ of index $d$ in $G$ leads to a 'magic' state $\left\|\psi\right\rangle$ in $d$-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite 'contextual' geometry. In the present work, we choose $G$ as the fundamental group $\pi_1(V)$ of an exotic $4$-manifold $V$, more precisely a 'small exotic' (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $\mathbb{R}^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence of standard contextual geometries such as the Fano plane (at index $7$), Mermin's pentagram (at index $10$), the two-qubit commutation picture $GQ(2,2)$ (at index $15$) as well as the combinatorial Grassmannian Gr$(2,8)$ (at index $28$) , (ii) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$'s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of 'quantum gravity'. [78] arXiv:2001.09092 [pdf, other] Title: Learning nonlocal regularization operators Comments: 29 pages, 2 figures Subjects: Optimization and Control (math.OC) A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided. [79] arXiv:2001.09098 [pdf, other] Title: Geometric Braid Groups Comments: 16 pages, 8 figures Building on the presentation of the fundamental group of discriminant complements, we associate a geometric braid group with each positive braid word and prove that it gives rise to an invariant of braid isotopy classes of positive braid links. [80] arXiv:2001.09102 [pdf, other] Title: Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements Subjects: Numerical Analysis (math.NA) The well-known Prager-Synge identity is valid in $H^1(\Omega)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise $H^1(\Omega)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; \Omega)$ through a local element-wise scheme and that recovers a gradient in $H(curl;\Omega)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution. [81] arXiv:2001.09103 [pdf, ps, other] Title: Block-avoiding point sequencings Comments: 22 pages Subjects: Combinatorics (math.CO) Recent papers by Kreher, Stinson and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system, directed triple system or Mendelsohn triple system) so that no block occurs in a short segment of consecutive entries (so the ordering is locally block-avoiding). The paper describes a greedy algorithm which shows that such an ordering exists, provided the number of points is sufficiently large. This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (for example, orderings that avoid the blocks of a block design). Similar results for a cyclic variant of this situation are also established. The results above were originally inspired by results of Alspach, Kreher and Pastine, who (motivated by zero-sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach et al. show that, when the system contains at most $k$ pairwise disjoint blocks, an ordering exists when the number of points is more than $15k-5$. By making use of a greedy approach, the paper improves this bound to $9k+O(k^{2/3})$. [82] arXiv:2001.09104 [pdf, ps, other] Title: Spectral maps associated to semialgebraic branched coverings Subjects: Algebraic Geometry (math.AG) In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial result to prove this is the characterization of the prime ideals whose fiber under the previous spectral map is a singleton. [83] arXiv:2001.09106 [pdf, ps, other] Title: On the basin of attraction of McKean-Vlasov paths Authors: Kaveh Bashiri Comments: 12 pages In this paper we provide short proofs and mild extensions of some statements about the ergodicity and the basins of attraction of the McKean-Vlasov evolution. The proofs are based on the representation of these as Wasserstein gradient flows. [84] arXiv:2001.09112 [pdf, ps, other] Title: Context-free languages and associative algebras with algebraic Hilbert series Comments: 11 pages Subjects: Rings and Algebras (math.RA) In this paper, homological methods together with the theory of formal languages of theoretical computer science are proved to be effective tools to determine the growth and the Hilbert series of an associative algebra. Namely, we construct a class of finitely presented associative algebras related to a family of context-free languages. This allows us to connect the Hilbert series of these algebras with the generating functions of such languages. In particular, we obtain a class of finitely presented graded algebras with non-rational algebraic Hilbert series. [85] arXiv:2001.09115 [pdf, ps, other] Title: Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities Comments: 46 pages; comments welcome The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we establish a novel tool for this task by deriving a quantitative lower bound on the Lyapunov exponent in terms of a matrix sum which is efficiently computable in ergodic situations. Our approach combines two deep results from matrix analysis --- the $n$-matrix extension of the Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds to the Lyapunov exponents of Schr\"odinger cocycles with certain ergodic potentials of polymer type and arbitrary correlation structure. We also derive related quantitative stability results for the Lyapunov exponent near aligned diagonal matrices and a bound for almost-commuting matrices. [86] arXiv:2001.09119 [pdf, ps, other] Title: Global Regularity of the 2D HVBK equations Subjects: Analysis of PDEs (math.AP); Fluid Dynamics (physics.flu-dyn) The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations are a macroscopic model of superfluidity at non-zero temperatures. For smooth, compactly supported data, we prove the global well-posedness of strong solutions to these equations in $\mathbb{R}^2$, in the incompressible and isothermal case. The proof utilises a contraction mapping argument to establish local well-posedness for high-regularity data, following which we demonstrate global regularity using an analogue of the Beale-Kato-Majda criterion in this context. In the appendix, we address the sufficient conditions on a 2D vorticity field, in order to have a finite kinetic energy. [87] arXiv:2001.09120 [pdf, ps, other] Title: Graded Morita theory over a G-graded G-acted algebra Subjects: Representation Theory (math.RT); Rings and Algebras (math.RA) We develop a group graded Morita theory over a G-graded G-acted algebra, where G is a finite group. [88] arXiv:2001.09121 [pdf, ps, other] Title: A Geometric View of the Service Rates of Codes Problem and its Application to the Service Rate of the First Order Reed-Muller Codes Subjects: Information Theory (cs.IT) Service rate is an important, recently introduced, performance metric associated with distributed coded storage systems. Among other interpretations, it measures the number of users that can be simultaneously served by the storage system. We introduce a geometric approach to address this problem. One of the most significant advantages of this approach over the existing approaches is that it allows one to derive bounds on the service rate of a code without explicitly knowing the list of all possible recovery sets. To illustrate the power of our geometric approach, we derive upper bounds on the service rates of the first order Reed-Muller codes and simplex codes. Then, we show how these upper bounds can be achieved. Furthermore, utilizing the proposed geometric technique, we show that given the service rate region of a code, a lower bound on the minimum distance of the code can be obtained. [89] arXiv:2001.09123 [pdf, ps, other] Title: What fraction of an $S_n$-orbit can lie on a hyperplane? Comments: 16 pages Subjects: Combinatorics (math.CO) Consider the $S_n$-action on $\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\max_{v,H} \|H\cap S_nv\|$ as $H\subset\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\in\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\leq n$, then $\max_{v,H} \|H\cap S_nv\|\leq \frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime. [90] arXiv:2001.09126 [pdf, ps, other] Title: A Sharp Convergence Rate for the Asynchronous Stochastic Gradient Descent We give a sharp convergence rate for the asynchronous stochastic gradient descent (ASGD) algorithms when the loss function is a perturbed quadratic function based on the stochastic modified equations introduced in [An et al. Stochastic modified equations for the asynchronous stochastic gradient descent, arXiv:1805.08244]. We prove that when the number of local workers is larger than the expected staleness, then ASGD is more efficient than stochastic gradient descent. Our theoretical result also suggests that longer delays result in slower convergence rate. Besides, the learning rate cannot be smaller than a threshold inversely proportional to the expected staleness. [91] arXiv:2001.09137 [pdf, ps, other] Title: On the boundary local time measure of super-Brownian motion Authors: Jieliang Hong Subjects: Probability (math.PR) If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-\lambda L^x} dx$ as $\lambda\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mathcal{L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal{L}$ in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial\mathcal{R}$ in \cite{HMP18}. The construction requires a refinement of the $L^2$ upper bounds in \cite{MP17} and \cite{HMP18} to exact $L^2$ asymptotics. The methods also refine the left tail bounds for $L^x$ in \cite{MP17} to exact asymptotics. We conjecture that the Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant. [92] arXiv:2001.09139 [pdf, other] Title: Characteristic classes and stability conditions for projective Kleinian orbisurfaces Comments: 26 pages, comments are welcome! Subjects: Algebraic Geometry (math.AG) We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the To\"en-Hirzebruch-Riemann-Roch theorem. [93] arXiv:2001.09143 [pdf, ps, other] Title: On infinite variants of De Morgan law in locale theory Authors: Igor Arrieta Subjects: General Topology (math.GN); Category Theory (math.CT); Logic (math.LO) A locale, being a complete Heyting algebra, satisfies De Morgan law $(a\vee b)^=a^\wedge b^$ for pseudocomplements. The dual De Morgan law $(a\wedge b)^={a^* \vee b^}$ (here referred to as the second De Morgan law) is equivalent to, among other conditions, $(a\vee b)^{} =a^{}\vee b^{}$, and characterizes the class of extremally disconnected locales. This paper presents a study of the subclasses of extremally disconnected locales determined by the infinite versions of the second De Morgan law and its equivalents. [94] arXiv:2001.09145 [pdf, other] Title: The geometric Burge correspondence and the partition function of polymer replicas Comments: 39 pages, 1 figure Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO); Representation Theory (math.RT) We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We prove a fundamental link with the geometric Robinson-Schensted-Knuth correspondence and with the geometric Sch\"utzenberger involution. We also show a number of properties of the geometric Burge correspondence, also specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O'Connell, Sepp\"al\"ainen, and Zygouras (2014). [95] arXiv:2001.09146 [pdf, ps, other] Title: A Combinatorial View of the Service Rates of Codes Problem, its Equivalence to Fractional Matching and its Connection with Batch Codes Subjects: Information Theory (cs.IT) We propose a novel technique for constructing a graph representation of a code through which we establish a significant connection between the service rate problem and the well-known fractional matching problem. Using this connection, we show that the service capacity of a coded storage system equals the fractional matching number in the graph representation of the code, and thus is lower bounded and upper bounded by the matching number and the vertex cover number, respectively. This is of great interest because if the graph representation of a code is bipartite, then the derived upper and lower bounds are equal, and we obtain the capacity. Leveraging this result, we characterize the service capacity of the binary simplex code whose graph representation, as we show, is bipartite. Moreover, we show that the service rate problem can be viewed as a generalization of the multiset primitive batch codes problem. [96] arXiv:2001.09147 [pdf, ps, other] Title: On $σ$-arithmetic graphs of finite groups Subjects: Group Theory (math.GR) Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$ is an integer, we write $\sigma(n)=\{\sigma_i \mid \sigma_{i}\cap \pi (n)\ne \emptyset \}$ and $\sigma (G)=\sigma (\|G\|)$. We call a graph $\Gamma$ with the set of all vertices $V(\Gamma)=\sigma (G)$ ($G\ne 1$) a $\sigma$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $\sigma$-arithmetic graphs: (1) the $\sigma$-Hawkes graph $\Gamma_{H\sigma }(G)$ of $G$ is a $\sigma$-arithmetic graph of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{H\sigma }(G))$ if $\sigma_j\in \sigma (G/F_{\{\sigma_i\}}(G))$; (2) the $\sigma$-Hall graph $\Gamma_{\sigma Hal}(G)$ of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{\sigma Hal}(G))$ if for some Hall $\sigma_i$-subgroup $H$ of $G$ we have $\sigma_j\in \sigma (N_{G}(H)/HC_{G}(H))$; (3) the $\sigma$-Vasil'ev-Murashko graph $\Gamma_{{\mathfrak{N}_\sigma }}(G)$ of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{{\mathfrak{N}_\sigma}}(G))$ if for some ${\mathfrak{N}_{\sigma }}$-critical subgroup $H$ of $G$ we have $\sigma_i \in \sigma (H)$ and $\sigma_j\in \sigma (H/F_{\{\sigma_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$. Cross-lists for Mon, 27 Jan 20 [97] arXiv:2001.08655 (cross-list from cs.LG) [pdf, other] Title: Best Arm Identification for Cascading Bandits in the Fixed Confidence Setting Comments: 38 pages, 25 figures We design and analyze CascadeBAI, an algorithm for finding the best set of $K$ items, also called an arm, within the framework of cascading bandits. An upper bound on the time complexity of CascadeBAI is derived by overcoming a crucial analytical challenge, namely, that of probabilistically estimating the amount of available feedback at each step. To do so, we define a new class of random variables (r.v.'s) which we term as left-sided sub-Gaussian r.v.'s; these are r.v.'s whose cumulant generating functions (CGFs) can be bounded by a quadratic only for non-positive arguments of the CGFs. This enables the application of a sufficiently tight Bernstein-type concentration inequality. We show, through the derivation of a lower bound on the time complexity, that the performance of CascadeBAI is optimal in some practical regimes. Finally, extensive numerical simulations corroborate the efficacy of CascadeBAI as well as the tightness of our upper bound on its time complexity. [98] arXiv:2001.08759 (cross-list from gr-qc) [pdf, ps, other] Title: The gauge symmetries of f(R) gravity with torsion in the Cartan formalism Comments: It contains a detailed derivation of the generalization of 3-dimensional "local translations" for 4-dimensional first-order general relativity Journal-ref: Class. Quantum Grav. 37, 045008 (2020) First-order general relativity in $n$ dimensions ($n \geq 3$) has an internal gauge symmetry that is the higher-dimensional generalization of three-dimensional local translations. We report the extension of this symmetry for $n$-dimensional f(R) gravity with torsion in the Cartan formalism. The new symmetry arises from the direct application of the converse of Noether's second theorem to the action principle of f(R) gravity with torsion. We show that infinitesimal diffeomorphisms can be written as a linear combination of the new internal gauge symmetry, local Lorentz transformations, and terms proportional to the variational derivatives of the f(R) action. It means that the new internal symmetry together with local Lorentz transformations can be used to describe the full gauge symmetry of f(R) gravity with torsion, and thus diffeomorphisms become a derived symmetry in this setting. [99] arXiv:2001.08766 (cross-list from quant-ph) [pdf, other] Title: Extremal elements of a sublattice of the majorization lattice and approximate majorization Comments: 26 pages, 1 figure Given a probability vector $x$ with its components sorted in non-increasing order, we consider the closed ball ${\mathcal{B}}^p_\epsilon(x)$ with $p \geq 1$ formed by the probability vectors whose $\ell^p$-norm distance to the center $x$ is less than or equal to a radius $\epsilon$. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case $p=1$ previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls ${\mathcal{B}}^p_\epsilon(x)$ with $1<p<\infty$. On the other hand, we show that ${\mathcal{B}}^\infty_\epsilon(x)$ is a complete sublattice of the majorization lattice. As a consequence, this ball has also extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the $\ell^1$-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity. [100] arXiv:2001.08769 (cross-list from physics.flu-dyn) [pdf, other] Title: A gradient-based framework for maximizing mixing in binary fluids Comments: 23 pages, 55 figures Journal-ref: Journal of Computational Physics 368, 131 - 153 (2018) Subjects: Fluid Dynamics (physics.flu-dyn); Optimization and Control (math.OC); Computational Physics (physics.comp-ph) A computational framework based on nonlinear direct-adjoint looping is presented for optimizing mixing strategies for binary fluid systems. The governing equations are the nonlinear Navier-Stokes equations, augmented by an evolution equation for a passive scalar, which are solved by a spectral Fourier-based method. The stirrers are embedded in the computational domain by a Brinkman-penalization technique, and shape and path gradients for the stirrers are computed from the adjoint solution. Four cases of increasing complexity are considered, which demonstrate the efficiency and effectiveness of the computational approach and algorithm. Significant improvements in mixing efficiency, within the externally imposed bounds, are achieved in all cases. [101] arXiv:2001.08778 (cross-list from hep-th) [pdf, other] Title: Distributions in CFT I. Cross-Ratio Space Comments: 24 pages + appendices We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory. [102] arXiv:2001.08941 (cross-list from eess.SY) [pdf, other] Title: Symmetries and periodic orbits in simple hybrid Routhian systems Comments: Nonlinear Analysis: Hybrid Systems 36, 100857, 2020 Journal-ref: Nonlinear Analysis: Hybrid Systems, Vol 36, May 2020, 100857 Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics are determined by a Lagrangian or Hamiltonian function. For hybrid systems which possess a continuous-time dynamics determined by a Lagrangian function, with a cyclic variable, the degrees of freedom for the corresponding hybrid Lagrangian system can be reduced by means of a method known as \textit{hybrid Routhian reduction}. In this paper we study sufficient conditions for the existence of periodic orbits in hybrid Routhian systems which also exhibit time-reversal symmetry. Likewise, we explore some stability aspects of such orbits through the characterization of the eigenvalues for the corresponding linearized Poincar\'e map. Finally, we apply the results to find periodic solutions in underactuated hybrid Routhian control systems. [103] arXiv:2001.09036 (cross-list from stat.ME) [pdf, ps, other] Title: Optimal Design for Probit Choice Models with Dependent Utilities Subjects: Methodology (stat.ME); Statistics Theory (math.ST) In this paper we derive locally D-optimal designs for discrete choice experiments based on multinomial probit models. These models include several discrete explanatory variables as well as a quantitative one. The commonly used multinomial logit model assumes independent utilities for different choice options. Thus, D-optimal optimal designs for such multinomial logit models may comprise choice sets, e.g., consisting of alternatives which are identical in all discrete attributes but different in the quantitative variable. Obviously such designs are not appropriate for many empirical choice experiments. It will be shown that locally D-optimal designs for multinomial probit models supposing independent utilities consist of counterintuitive choice sets as well. However, locally D-optimal designs for multinomial probit models allowing for dependent utilities turn out to be reasonable for analyzing decisions using discrete choice studies. [104] arXiv:2001.09040 (cross-list from cs.LG) [pdf, other] Title: Estimation for Compositional Data using Measurements from Nonlinear Systems using Artificial Neural Networks Authors: Se Un Park Comments: 43 pages, 20 figures Subjects: Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Optimization and Control (math.OC); Statistics Theory (math.ST); Machine Learning (stat.ML) Our objective is to estimate the unknown compositional input from its output response through an unknown system after estimating the inverse of the original system with a training set. The proposed methods using artificial neural networks (ANNs) can compete with the optimal bounds for linear systems, where convex optimization theory applies, and demonstrate promising results for nonlinear system inversions. We performed extensive experiments by designing numerous different types of nonlinear systems. [105] arXiv:2001.09046 (cross-list from cs.LG) [pdf, other] Title: PDE-based Group Equivariant Convolutional Neural Networks Subjects: Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Differential Geometry (math.DG); Machine Learning (stat.ML) We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where the equation's geometrically meaningful coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation equivariance instead of being restricted to just translation equivariance as in traditional CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. Roto-translation equivariance for image analysis applications is the example we will be using throughout the paper. Our default PDE is solved by a combination of linear group convolutions and non-linear morphological group convolutions. Just like for linear convolution a morphological convolution is specified by a kernel and this kernel is what is being optimized during the training process. We demonstrate how the common CNN operations of max/min-pooling and ReLUs arise naturally from solving a PDE and how they are subsumed by morphological convolutions. We present a proof-of-concept experiment to demonstrate the potential of this framework in increasing the performance of deep learning based imaging applications. [106] arXiv:2001.09066 (cross-list from cs.MA) [pdf, other] Title: Policy Synthesis for Factored MDPs with Graph Temporal Logic Specifications Subjects: Multiagent Systems (cs.MA); Logic in Computer Science (cs.LO); Optimization and Control (math.OC) We study the synthesis of policies for multi-agent systems to implement spatial-temporal tasks. We formalize the problem as a factored Markov decision process subject to so-called graph temporal logic specifications. The transition function and the spatial-temporal task of each agent depend on the agent itself and its neighboring agents. The structure in the model and the specifications enable to develop a distributed algorithm that, given a factored Markov decision process and a graph temporal logic formula, decomposes the synthesis problem into a set of smaller synthesis problems, one for each agent. We prove that the algorithm runs in time linear in the total number of agents. The size of the synthesis problem for each agent is exponential only in the number of neighboring agents, which is typically much smaller than the number of agents. We demonstrate the algorithm in case studies on disease control and urban security. The numerical examples show that the algorithm can scale to hundreds of agents. [107] arXiv:2001.09081 (cross-list from cs.CG) [pdf, other] Title: Approximating Surfaces in $R^3$ by Meshes with Guaranteed Regularity Comments: 33 pages, 15 figures Subjects: Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Geometric Topology (math.GT) We study the problem of approximating a surface $F$ in $R^3$ by a high quality mesh, a piecewise-flat triangulated surface whose triangles are as close as possible to equilateral. The MidNormal algorithm generates a triangular mesh that is guaranteed to have angles in the interval $[49.1^o, 81.8^o]$. As the mesh size $e\rightarrow 0$, the mesh converges pointwise to $F$ through surfaces that are isotopic to $F$. The GradNormal algorithm gives a piecewise-$C^1$ approximation of $F$, with angles in the interval $[35.2^o, 101.5^o]$ as $e\rightarrow 0$. Previously achieved angle bounds were in the interval $[30^o, 120^o]$. [108] arXiv:2001.09093 (cross-list from eess.SP) [pdf, other] Title: Joint Long-Term Cache Updating and Short-Term Content Delivery in Cloud-Based Small Cell Networks Comments: Accepted by IEEE Trans. Commun Explosive growth of mobile data demand may impose a heavy traffic burden on fronthaul links of cloud-based small cell networks (C-SCNs), which deteriorates users' quality of service (QoS) and requires substantial power consumption. This paper proposes an efficient maximum distance separable (MDS) coded caching framework for a cache-enabled C-SCNs, aiming at reducing long-term power consumption while satisfying users' QoS requirements in short-term transmissions. To achieve this goal, the cache resource in small-cell base stations (SBSs) needs to be reasonably updated by taking into account users' content preferences, SBS collaboration, and characteristics of wireless links. Specifically, without assuming any prior knowledge of content popularity, we formulate a mixed timescale problem to jointly optimize cache updating, multicast beamformers in fronthaul and edge links, and SBS clustering. Nevertheless, this problem is anti-causal because an optimal cache updating policy depends on future content requests and channel state information. To handle it, by properly leveraging historical observations, we propose a two-stage updating scheme by using Frobenius-Norm penalty and inexact block coordinate descent method. Furthermore, we derive a learning-based design, which can obtain effective tradeoff between accuracy and computational complexity. Simulation results demonstrate the effectiveness of the proposed two-stage framework. [109] arXiv:2001.09094 (cross-list from cs.DM) [pdf, ps, other] Title: Results of nested canalizing functions Comments: 11 pages Subjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO) Boolean nested canalizing functions (NCF) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study the certificate complexity of NCF. We obtain the formula for $b$ - certificate complexity, $C_0(f)$ and $C_1(f)$. Consequently, we get a direct proof of the certificate complexity formula of NCF. Symmetry is another interesting property of Boolean functions. We significantly simplify the proofs of some recent theorems about partial symmetry of NCF. We also describe the algebraic normal form of the $s$-symmetric nested canalizing functions. We obtain the general formula of the cardinality of the set of all $n$-variable $s$-symmetric Boolean NCF for $s=1,\cdots,n$. Particularly, we obtained the cardinality formula for the set of all the strongly asymmetric Boolean NCFs. [110] arXiv:2001.09122 (cross-list from cs.LG) [pdf, ps, other] Title: Reasoning About Generalization via Conditional Mutual Information Subjects: Machine Learning (cs.LG); Cryptography and Security (cs.CR); Data Structures and Algorithms (cs.DS); Information Theory (cs.IT); Machine Learning (stat.ML) We provide an information-theoretic framework for studying the generalization properties of machine learning algorithms. Our framework ties together existing approaches, including uniform convergence bounds and recent methods for adaptive data analysis. Specifically, we use Conditional Mutual Information (CMI) to quantify how well the input (i.e., the training data) can be recognized given the output (i.e., the trained model) of the learning algorithm. We show that bounds on CMI can be obtained from VC dimension, compression schemes, differential privacy, and other methods. We then show that bounded CMI implies various forms of generalization. [111] arXiv:2001.09144 (cross-list from cs.SC) [pdf, other] Title: Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials Subjects: Symbolic Computation (cs.SC); Classical Analysis and ODEs (math.CA); Rings and Algebras (math.RA); Representation Theory (math.RT) Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function. Replacements for Mon, 27 Jan 20 [112] arXiv:0907.4469 (replaced) [pdf, ps, other] Title: Grassmannians and conformal structure on absolutes Comments: 8 pages. Dedicated to the memory of Waldyr Rodrigues Jr Journal-ref: Adv. Appl. Clifford Algebras 29, 5 (2019) Subjects: Differential Geometry (math.DG) [113] arXiv:1101.4711 (replaced) [pdf, other] Title: Von Neumann Normalisation of a Quantum Random Number Generator Comments: 27 pages, 2 figures. Updated to published version Journal-ref: Computability 1, 59 (2012) [114] arXiv:1601.05454 (replaced) [pdf, ps, other] Title: Bounding 2D Functions by Products of 1D Functions Comments: 13 pages Subjects: Logic (math.LO) [115] arXiv:1603.00175 (replaced) [pdf, ps, other] Title: Structure of the polynomials in preconditioned BiCG algorithms and the switching direction of preconditioned systems Subjects: Numerical Analysis (math.NA) [116] arXiv:1607.00196 (replaced) [pdf, other] Title: Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects Comments: This replacement is part I of the final version of the paper, which has been split into two parts. The second part is available from the arXiv under the title "Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation" arXiv:2001.08722 [117] arXiv:1609.09688 (replaced) [pdf, ps, other] Title: Mapping cones in the bounded derived category of a gentle algebra Comments: 23 pages, revised version with simplified proofs, there are two oversights in the statements involving band complexes, these are corrected in arXiv:2001.06435, footnotes in the text reference affected statements Journal-ref: Journal of Algebra 530 (2019), 163--194 [118] arXiv:1706.01108 (replaced) [pdf, ps, other] Title: Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory Comments: Accepted to SIAM Journal on Matrix Analysis and Applications. This arXiv version has an additional section (Section 6.2), listing several extensions done since the paper was first written. Statistics: 39 pages, 4 reformulations, 3 algorithms [119] arXiv:1707.06934 (replaced) [pdf, ps, other] Title: On extensions for gentle algebras Comments: 34 pages, updated statements on middle terms of extensions involving band modules Subjects: Representation Theory (math.RT) [120] arXiv:1709.02742 (replaced) [pdf, ps, other] Title: Pin Groups in General Relativity Authors: Bas Janssens Comments: 9 pages, added section 7 on the role of diffeomorphism invariance [121] arXiv:1709.07867 (replaced) [pdf, ps, other] Title: The center of the categorified ring of differential operators Authors: Dario Beraldo Subjects: Algebraic Geometry (math.AG) [122] arXiv:1710.11149 (replaced) [pdf, other] Title: Analysis, Identification, and Validation of Discrete-Time Epidemic Processes [123] arXiv:1802.00197 (replaced) [pdf, ps, other] Title: On commuting $p$-version projection-based interpolation on tetrahedra Journal-ref: Math. Comp.89 (2019), pp. 45-87 Subjects: Numerical Analysis (math.NA) [124] arXiv:1802.04736 (replaced) [pdf, ps, other] Title: Amenable uniformly recurrent subgroups and lattice embeddings Authors: Adrien Le Boudec Comments: v1: 44 pages, preliminary version. v2: slightly modified version. v3: modified terminology, added paragraph 6.5.4. v4: Part of Section 6 has been extracted to arXiv:2001.08689 Subjects: Group Theory (math.GR) [125] arXiv:1804.06537 (replaced) [pdf, other] Title: Understanding Convolutional Neural Networks with Information Theory: An Initial Exploration Comments: Paper accepted by IEEE Transactions on Neural Networks and Learning Systems (TNNLS). Code for 1) estimating information quantities, 2) plotting the information plane, and 3) selecting convolutional filters, is available from (MATLAB) this https URL or (Python) this https URL [126] arXiv:1804.10956 (replaced) [pdf, ps, other] Title: The Regularity Problem for Lie Groups with Asymptotic Estimate Lie Algebras Comments: 27 pages. Version as published at Indagationes Mathematicae (title refined; presentation improved; proof of Lemma 9 revised) Journal-ref: Indag. Math. (2020), Vol. 31, Issue 1, Pages 152-176 [127] arXiv:1805.07962 (replaced) [pdf, other] Title: A Nonconvex Projection Method for Robust PCA Comments: In the proceedings of Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19) Journal-ref: In the proceedings of Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19), 33(01), pp. 1468-1476, 2019 [128] arXiv:1805.12072 (replaced) [pdf, other] Title: Virtual Rational Tangles Comments: 11 pages; version 2 makes minor changes, and includes a proof that Kauffman's bracket polynomial is invariant under classical and virtual flypes Subjects: Geometric Topology (math.GT) [129] arXiv:1805.12095 (replaced) [pdf, ps, other] Title: A note on the Grothendieck group of an abelian variety Authors: Shahram Biglari Comments: 18 pages, no figure (corrected the abstract and added acknowledgment) [130] arXiv:1806.05286 (replaced) [pdf, other] Title: Bounds on the localization number Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM) [131] arXiv:1806.06267 (replaced) [pdf, ps, other] Title: Cooperative colorings of trees and of bipartite graphs Comments: 8 pages, 2 figures, accepted to the Electronic Journal of Combinatorics, corrections suggested by the referees have been incorporated Subjects: Combinatorics (math.CO) [132] arXiv:1807.05537 (replaced) [pdf, ps, other] Title: Equality in Suita's conjecture Authors: Robert Xin Dong Comments: 12 pages, after substantial revision [133] arXiv:1807.06406 (replaced) [pdf, other] Title: Eigenfunctions and the Integrated Density of States on Archimedean Tilings Comments: 22 pages, 15 figures [134] arXiv:1808.05430 (replaced) [pdf, ps, other] Title: Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences Comments: 14 pages, 1 table, Lemma 2.1 added, some additions and minor corrections made Journal-ref: Adv. in Appl. Math. (2020) Subjects: Combinatorics (math.CO) [135] arXiv:1809.05794 (replaced) [pdf, other] Title: When Lift-and-Project Cuts are Different Comments: INFORMS Journal on Computing (to appear) Subjects: Optimization and Control (math.OC); Mathematical Software (cs.MS) [136] arXiv:1809.07520 (replaced) [pdf, ps, other] Title: Variable Martingale Hardy Spaces and Their Applications in Fourier Analysis Subjects: Probability (math.PR) [137] arXiv:1810.04799 (replaced) [pdf, other] Title: Approximate controllability for Navier--Stokes equations in {\rm 3D} Cylinders under Lions boundary conditions by an explicit saturating set Authors: Duy Phan Comments: 33 pages, 4 figures. arXiv admin note: text overlap with arXiv:1712.04900 [138] arXiv:1810.06708 (replaced) [pdf, ps, other] Title: Attractors associated to a family of hyperbolic $p$-adic plane automorphisms Authors: Clayton Petsche [139] arXiv:1811.01055 (replaced) [pdf, ps, other] Title: On the sets of $n$ points forming $n+1$ directions Authors: Cédric Pilatte Comments: 7 pages, 5 figures Journal-ref: Electronic Journal of Combinatorics, Vol. 27, Issue 1 (2020); P1.24 Subjects: Combinatorics (math.CO) [140] arXiv:1811.12910 (replaced) [pdf, ps, other] Title: Improved Finite Difference Results for the Caputo Time-Fractional Diffusion Equation Comments: 22 pages, 3 tables, 1 appendix Subjects: Numerical Analysis (math.NA) [141] arXiv:1812.04094 (replaced) [pdf, other] Title: Indeterminacy loci of iterate maps in moduli space Comments: 49 pages, 3 figures. revised version. added Theorem B, Figure 1 and details in Section 4 Subjects: Dynamical Systems (math.DS) [142] arXiv:1812.07268 (replaced) [pdf, other] Title: Magnetic Skyrmions at Critical Coupling Comments: 23 pages, 1 figures; version published in Comm. Math. Phys. with note added on alternative definition of energy in this model. Commun. Math. Phys. (2020) [143] arXiv:1812.09457 (replaced) [pdf, other] Title: Prescribing Morse scalar curvatures: blow-up analysis Comments: 52 pages Subjects: Analysis of PDEs (math.AP) [144] arXiv:1901.04183 (replaced) [pdf, ps, other] Title: A Unified Approach for Solving Sequential Selection Problems Subjects: Probability (math.PR); Statistics Theory (math.ST) [145] arXiv:1901.07069 (replaced) [pdf, ps, other] Title: Minimum Age of Information in the Internet of Things with Non-uniform Status Packet Sizes Authors: Bo Zhou, Walid Saad Comments: 33 pages, 8 figures. Accepted by IEEE Transactions on Wireless Communications. Corrected the typos in Fig.1 and Fig. 2 [146] arXiv:1901.07496 (replaced) [pdf, ps, other] Title: On the isometrisability of group actions on p-spaces Comments: 8 pages, no figures [147] arXiv:1901.10375 (replaced) [pdf, other] Title: A low-rank technique for computing the quasi-stationary distribution of subcritical Galton-Watson processes Subjects: Numerical Analysis (math.NA) [148] arXiv:1902.00767 (replaced) [pdf, ps, other] Title: Properties of high rank subvarieties of affine spaces Comments: Added effective Stillman conjecture over algebraically closed fields. Some small changes + added formulation in terms of singular locus Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO) [149] arXiv:1902.01584 (replaced) [pdf, ps, other] Title: Bilipschitz equivalence of polynomials Authors: Arnaud Bodin Comments: 16 pages. v2: corrections after referees' comments [150] arXiv:1902.09623 (replaced) [pdf, other] Title: Microlocal analysis of a Compton tomography problem Comments: 31 pages, 18 figures [151] arXiv:1903.00960 (replaced) [pdf, other] Title: Supercritical Regime for the Kissing Polynomials Comments: 40 pages, 14 figures Subjects: Classical Analysis and ODEs (math.CA); Complex Variables (math.CV) [152] arXiv:1903.06143 (replaced) [pdf, other] Title: Variation of stable birational types in positive characteristic Comments: 14 pages; final version, published in EPIGA Journal-ref: \'Epijournal de G\'eom\'etrie Alg\'ebrique, Volume 3 (2019), Article no. 20 [153] arXiv:1904.02808 (replaced) [pdf, other] Title: Overlap matrix concentration in optimal Bayesian inference Authors: Jean Barbier Subjects: Information Theory (cs.IT); Disordered Systems and Neural Networks (cond-mat.dis-nn); Probability (math.PR) [154] arXiv:1904.05996 (replaced) [pdf, ps, other] Title: Deformation theory of the trivial mod $p$ Galois representation for $\mathrm{GL}_n$ Authors: Ashwin Iyengar Comments: To appear in International Mathematics Research Notices. 32 pages Subjects: Number Theory (math.NT) [155] arXiv:1904.07784 (replaced) [pdf, other] Title: The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem Subjects: Probability (math.PR); Numerical Analysis (math.NA) [156] arXiv:1904.09186 (replaced) [pdf, other] Title: Super-resolution of near-colliding point sources Subjects: Numerical Analysis (math.NA) [157] arXiv:1904.10102 (replaced) [pdf, other] Title: Sublinear-Time Non-Adaptive Group Testing with $O(k \log n)$ Tests via Bit-Mixing Coding Comments: (v2) Expanded related work section Subjects: Information Theory (cs.IT); Signal Processing (eess.SP); Probability (math.PR) [158] arXiv:1904.11744 (replaced) [pdf, other] Title: Arnold maps with noise: Differentiability and non-monotonicity of the rotation number Comments: Electronic copy of final peer-reviewed manuscript accepted for publication in the Journal of Statistical Physics Subjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD); Geophysics (physics.geo-ph) [159] arXiv:1905.00488 (replaced) [pdf, other] Title: Conformal Mechanics of Planar Curves Comments: 24 pages, 11 figures, extensive rewrite, new section added Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Soft Condensed Matter (cond-mat.soft); Mathematical Physics (math-ph); Classical Physics (physics.class-ph) [160] arXiv:1905.04945 (replaced) [pdf, ps, other] Title: Asymptotic dynamics of Young differential equations: a unified approach Comments: 32 Subjects: Probability (math.PR) [161] arXiv:1905.09641 (replaced) [pdf, other] Title: Greedy energy minimization can count in binary: point charges and the van der Corput sequence Comments: 18 pages, 7 figures, discrepancy bound added [162] arXiv:1905.12912 (replaced) [pdf, ps, other] Title: Analysis on Riemannian foliations of bounded geometry Comments: 46 pages [163] arXiv:1906.09710 (replaced) [pdf, other] Title: Uniqueness of unitary structure for unitarizable fusion categories Authors: David Reutter Comments: 14 pages; v2: generalized main theorem to semisimple C-tensor categories with possibly infinitely many simple objects [164] arXiv:1906.09839 (replaced) [pdf, ps, other] Title: Higher order regularity of nonlinear Fokker-Planck PDEs with respect to the measure component Authors: Alvin Tse [165] arXiv:1907.00284 (replaced) [pdf, ps, other] Title: The large cardinal strength of Weak Vopěnka's Principle Authors: Trevor M. Wilson Subjects: Logic (math.LO) [166] arXiv:1907.04694 (replaced) [pdf, ps, other] Title: Data-Driven Screening of Network Constraints for Unit Commitment Subjects: Optimization and Control (math.OC) [167] arXiv:1907.05727 (replaced) [pdf, ps, other] Title: Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes Authors: Wadim Gerner Comments: 20 pages, proof of proposition 4.4 was corrected Subjects: Mathematical Physics (math-ph) [168] arXiv:1907.12788 (replaced) [pdf, ps, other] Title: Properties of moduli of smoothness in $L_p(\mathbb{R}^d)$ Subjects: Classical Analysis and ODEs (math.CA) [169] arXiv:1907.13357 (replaced) [pdf, other] Title: Hybrid Spatio-Spectral Total Variation: A Regularization Technique for Hyperspectral Image Denoising and Compressed Sensing Comments: 11 pages, 3 tables, 8 figures, submitted to IEEE Trans. Geosci. Remote Sens Subjects: Signal Processing (eess.SP); Optimization and Control (math.OC) [170] arXiv:1908.01319 (replaced) [pdf, ps, other] Title: Construction of projective special Kähler manifolds Authors: Mauro Mantegazza Comments: 44 pages, 4 tables; Prop.7.9 and Def.7.10 replaced with an analysis of PSK structures differing by a U(1)-valued function (Sec.8); Cor.7.7 (and proofs relying on it) fixed by requiring H^2(M)=0 (Cor.7.9); proof of Theo.10.2 simplified and fixed by adding abelian case; proofs in Sec.3 replaced with references; Remk7.5 added; references added and fixed; typos fixed; presentation improved Subjects: Differential Geometry (math.DG) [171] arXiv:1908.03953 (replaced) [pdf, ps, other] Title: Counting pattern-avoiding integer partitions Comments: 28 Pages, 1 table Subjects: Combinatorics (math.CO); Number Theory (math.NT) [172] arXiv:1908.04433 (replaced) [pdf, other] Title: Sharp Guarantees for Solving Random Equations with One-Bit Information [173] arXiv:1908.06219 (replaced) [pdf, other] Authors: Yao Li Comments: v2: proof is simplified Subjects: Mathematical Physics (math-ph) [174] arXiv:1908.08698 (replaced) [pdf, ps, other] Title: Convergence Rate of Multiscale Finite Element Method for Various Boundary Problems Subjects: Numerical Analysis (math.NA) [175] arXiv:1908.09337 (replaced) [pdf, ps, other] Title: A stochastic MPC scheme for distributed systems with multiplicative uncertainty Comments: 10 pages, 2 figures Subjects: Optimization and Control (math.OC) [176] arXiv:1908.11320 (replaced) [pdf, other] Title: A Complete Realization of the orbits of generalized derivatives of Quasiregular Mappings Subjects: Complex Variables (math.CV) [177] arXiv:1909.00424 (replaced) [pdf, ps, other] Title: Invariant measures for stochastic damped 2D Euler equations Comments: 22 pages. This is the version accepted for publication in Commun. Math. Phys Subjects: Probability (math.PR) [178] arXiv:1909.01291 (replaced) [pdf, ps, other] Title: Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2 Comments: Accepted to Linear Algebra and Its Applications, pages 12 Subjects: Spectral Theory (math.SP); Numerical Analysis (math.NA); Probability (math.PR) [179] arXiv:1909.02974 (replaced) [pdf, other] Title: Sharp asymptotics of the first eigenvalue on some degenerating surfaces Comments: v2: 35 pages, 1 figure; many more details and much fewer typos; to appear in Trans. Amer. Math. Soc [180] arXiv:1909.07195 (replaced) [pdf, ps, other] Title: On Hausdorff Metric Spaces Comments: 11 pages Subjects: General Topology (math.GN) [181] arXiv:1909.08338 (replaced) [pdf, ps, other] Title: Singular optimal control of stochastic Volterra integral equations [182] arXiv:1909.09426 (replaced) [pdf, ps, other] Title: Biseparable extensions are not necessarily Frobenius Subjects: Rings and Algebras (math.RA) [183] arXiv:1909.09464 (replaced) [pdf, ps, other] Title: BGK and Fokker Planck Models for thermally perfect gases Authors: J. Mathiaud (CEA-CESTA), Luc Mieussens (IMB) Comments: arXiv admin note: substantial text overlap with arXiv:1904.02403 Subjects: Mathematical Physics (math-ph) [184] arXiv:1909.10436 (replaced) [pdf, ps, other] Title: Inversion of adjunction for $F$-signature Authors: Gregory Taylor Comments: 20 pages, v2: fixed typos, improved exposition [185] arXiv:1909.12813 (replaced) [pdf, ps, other] Title: Spectral-free methods in the theory of hereditarily indecomposable Banach spaces Authors: Noé de Rancourt Comments: 12 pages. Formerly entitled "New proofs of some properties of hereditarily indecomposable Banach spaces". The first version has been considerably expanded Subjects: Functional Analysis (math.FA) [186] arXiv:1909.13067 (replaced) [pdf, other] Title: Structural localization in the Classical and Quantum Fermi-Pasta-Ulam Model [187] arXiv:1910.00147 (replaced) [pdf, other] Title: Grassmann angles between real or complex subspaces Subjects: Metric Geometry (math.MG) [188] arXiv:1910.00359 (replaced) [pdf, other] Title: Truth or Backpropaganda? An Empirical Investigation of Deep Learning Theory Comments: 16 pages, 5 figures. First two authors contributed equally. Accepted as a conference paper at ICLR 2020 [189] arXiv:1910.01169 (replaced) [pdf, other] Title: Weil-Petersson translation length and manifolds with many fibered fillings Comments: v2. Added references. v1. 49 pages, 9 figures [190] arXiv:1910.01672 (replaced) [pdf, other] Title: Genus one cobordisms between torus knots Comments: 20 pages, 12 figures. V3: Minor corrections and implementation of referee's recommendations. This version has been accepted for publication by IMRN Subjects: Geometric Topology (math.GT) [191] arXiv:1910.02832 (replaced) [pdf, ps, other] Title: The distribution of divisors of polynomials Comments: v2. minor edits and corrections Subjects: Number Theory (math.NT) [192] arXiv:1910.03480 (replaced) [pdf, other] Title: The legacy of Jozef Marcinkiewicz: four hallmarks of genius Comments: 9 pages, 2 figures Subjects: History and Overview (math.HO) [193] arXiv:1910.05854 (replaced) [pdf, ps, other] Title: On the Long-Range Dependence of Mixed Fractional Poisson Process Subjects: Probability (math.PR) [194] arXiv:1910.10332 (replaced) [pdf] Title: Characterization of blood pressure and heart rate oscillations of POTS patients via uniform phase empirical mode decomposition Subjects: Quantitative Methods (q-bio.QM); Signal Processing (eess.SP); Spectral Theory (math.SP) [195] arXiv:1910.11639 (replaced) [pdf, other] Title: Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces Authors: Roland Prohaska Comments: 12 pages; expanded section 3, adding a more uniform version of the result in this section Subjects: Dynamical Systems (math.DS) [196] arXiv:1911.00974 (replaced) [pdf, ps, other] Title: Asymptotic Criticality of the Navier-Stokes Regularity Problem Comments: 42 pp; a more detailed description of the local-in-time dynamics of the `chains of derivatives' is provided (additional 10pp) [197] arXiv:1911.01096 (replaced) [pdf, ps, other] Title: Ax's theorem with an additive character Authors: Ehud Hrushovski Comments: Local corrections in section 2 Subjects: Logic (math.LO) [198] arXiv:1911.01129 (replaced) [pdf, ps, other] Title: Definability patterns and their symmetries Authors: Ehud Hrushovski Comments: Various local corrections Subjects: Logic (math.LO) [199] arXiv:1911.01204 (replaced) [src] Title: Graus Dinâmicos e Cohomológicos em Variedades Abelianas Authors: Armand Azonnahin Comments: This article has been withdrawn by arXiv administrators as it is an unauthorized translation of arXiv:1901.02618 without acknowledging the original authorship Subjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG) [200] arXiv:1911.01380 (replaced) [pdf, other] Title: A Decentralized Time- and Energy-Optimal Control Framework for Connected Automated Vehicles: From Simulation to Field Test Subjects: Optimization and Control (math.OC); Signal Processing (eess.SP) [201] arXiv:1911.01515 (replaced) [pdf, other] Title: Can the Elliptic Billiard Still Surprise Us? Comments: 19 pages, 16 figures Subjects: Dynamical Systems (math.DS); Computational Geometry (cs.CG); Robotics (cs.RO); Algebraic Geometry (math.AG) [202] arXiv:1911.03475 (replaced) [pdf, other] Title: Concurrent Optimization of Vehicle Dynamics and Powertrain Operation Using Connectivity and Automation Comments: Updating and replacing the old version of arXiv:1911.03475 with the finalized manuscript [203] arXiv:1911.08001 (replaced) [pdf, other] Title: Universality for Langevin-like spin glass dynamics Comments: 19 pages, 2 figures [204] arXiv:1911.11604 (replaced) [pdf, ps, other] Title: Unirational Differential Curves and Differential Rational Parametrizations Authors: Lei Fu, Wei Li Comments: 21 pages Subjects: Algebraic Geometry (math.AG) [205] arXiv:1911.11792 (replaced) [pdf, ps, other] Title: Quantum-classical duality for Gaudin magnets with boundary Comments: 19 pages, references added Journal-ref: Nuclear Physics B 952 (2020) 114931 [206] arXiv:1911.12701 (replaced) [pdf, ps, other] Title: Moduli theory, stability of fibrations and optimal symplectic connections Comments: 44 pages, v2: improved presentation [207] arXiv:1912.00652 (replaced) [pdf, ps, other] Title: Generalised conformal higher-spin fields in curved backgrounds Comments: 20 pages, comments and references added [208] arXiv:1912.01019 (replaced) [pdf, other] Title: Canonical analysis of $n$-dimensional Palatini action without second-class constraints Comments: Paper's title was changed, expanded analysis, notation was changed a bit, added reference, corrected typos Journal-ref: Phys. Rev. D 101, 024042 (2020) [209] arXiv:1912.03963 (replaced) [pdf, other] Title: Data Collection versus Data Estimation: A Fundamental Trade-off in Dynamic Networks Subjects: Optimization and Control (math.OC) [210] arXiv:1912.03973 (replaced) [pdf, other] Title: Deep Teams: Decentralized Decision Making with Finite and Infinite Number of Agents Comments: To appear in IEEE Transaction on Automatic Control, 16 pages Subjects: Optimization and Control (math.OC) [211] arXiv:1912.04752 (replaced) [pdf, ps, other] Title: Closed conformal Killing-Yano initial data Comments: 32 pages; v2: minor corrections [212] arXiv:1912.05263 (replaced) [pdf, ps, other] Title: Semicontinuity of Singularity Invariants in Families of Formal Power Series Comments: 35 pages. Extended version: more cases where semicontinuity of the completed fibre dimension holds, a comparison of the completed fibre with the usual fibre and for families of finite type over the base ring a version of Zariski's main theorem for modules [213] arXiv:1912.08752 (replaced) [pdf, ps, other] Title: Blow-up criteria for linearly damped nonlinear Schrödinger equations Authors: Van Duong Dinh Comments: 14 pages, the manuscript has been rewritten Subjects: Analysis of PDEs (math.AP) [214] arXiv:1912.09347 (replaced) [pdf, ps, other] Title: Euclidean structures and operator theory in Banach spaces Comments: 148 pages. Added a section on dilations. Minor typo's corrected. Submitted Subjects: Functional Analysis (math.FA) [215] arXiv:2001.00218 (replaced) [pdf, other] Title: Lossless Compression of Deep Neural Networks Comments: Under review [216] arXiv:2001.00843 (replaced) [pdf, ps, other] Title: Monte Carlo Cubature Construction Authors: Satoshi Hayakawa Comments: 10 pages Subjects: Numerical Analysis (math.NA); Probability (math.PR) [217] arXiv:2001.01662 (replaced) [src] Title: Sharp bounds on the Nusselt number in Rayleigh-Bénard convection Comments: The paper is withdrawn due to a problem in the proof of Proposition 3.1 and with a scaling argument Subjects: Analysis of PDEs (math.AP) [218] arXiv:2001.01778 (replaced) [pdf, ps, other] Title: Locally recoverable codes from automorphism groups of function fields of genus $g \geq 1$ Subjects: Algebraic Geometry (math.AG) [219] arXiv:2001.01981 (replaced) [pdf, ps, other] Title: On the real and complex zeros of the quadrilateral zeta function Authors: Takashi Nakamura Comments: 12 pages, 10 figures. The second main term in Proposition 1.5 and its proof are corrected. Some typos are corrected Subjects: Number Theory (math.NT) [220] arXiv:2001.03597 (replaced) [pdf, ps, other] Title: Torsors under Néron blowups Authors: Timo Richarz Comments: Reference added; minor corrections Subjects: Algebraic Geometry (math.AG) [221] arXiv:2001.06128 (replaced) [pdf, other] Title: Coupling constant dependence for the Schrödinger equation with an inverse-square potential Authors: A.G. Smirnov Comments: 48 pages, 6 figures, a reference added [222] arXiv:2001.06500 (replaced) [pdf, ps, other] Title: Exceptional Collections for Mirrors of Invertible Polynomials Comments: 14 pages, Counterexample retracted thanks to K. Ueda. References added Subjects: Algebraic Geometry (math.AG) [223] arXiv:2001.06726 (replaced) [pdf, ps, other] Title: Removable sets in elliptic equations with Musielak-Orlicz growth Comments: the manuscript extends arXiv:1901.03412 to the general growth Subjects: Analysis of PDEs (math.AP) [224] arXiv:2001.06843 (replaced) [pdf, ps, other] Title: Zero-divisors and idempotents in quandle rings Comments: 25 pages Subjects: Rings and Algebras (math.RA); Geometric Topology (math.GT) [225] arXiv:2001.08083 (replaced) [pdf, ps, other] Title: The Convergence of Finite-Averaging of AIMD for Distributed Heterogeneous Resource Allocations [226] arXiv:2001.08368 (replaced) [pdf, other] Title: Characterizations of annihilator $(b,c)$-inverses in arbitrary rings Subjects: Rings and Algebras (math.RA) [227] arXiv:2001.08685 (replaced) [pdf, ps, other] Title: Linear sets from projection of Desarguesian spreads Subjects: Combinatorics (math.CO) [ total of 227 entries: 1-227 ] Disable MathJax (What is MathJax?) 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